Springer Tracts in Modern Physics 259 A Novel Approach to Charge Carrier Transport in Semiconductors The Thermoballistic Transport Model Reinhard Lipperheide Uwe Wille Springer Tracts in Modern Physics Volume 259 Honorary editor G. Höhler, Karlsruhe, Germany Series editors Atsushi Fujimori, Tokyo, Japan Johann H. Kühn, Karlsruhe, Germany Thomas Müller, Karlsruhe, Germany Frank Steiner, Ulm, Germany William C. Stwalley, Storrs, CT, USA Joachim E. Trümper, Garching, Germany Peter Wölfle, Karlsruhe, Germany Ulrike Woggon, Berlin, Germany For further volumes: http://www.springer.com/series/426 http://www.springer.com/series/426 Springer Tracts in Modern Physics Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in physics. The following fields are emphasized: Elementary Particle Physics, Condensed Matter Physics, Light Matter Interaction, Atomic and Molecular Physics, Complex Systems, Fundamental Astrophysics. Suitable reviews of other fields can also be accepted. The Editors encourage prospective authors to correspond with them in advance of submitting a manuscript. For reviews of topics belonging to the above mentioned fields, they should address the responsible Editor as listed below. Special offer: For all clients with a print standing order we offer free access to the electronic volumes of the Series published in the current year. Elementary Particle Physics, Editors Johann H. Kühn Institut für Theoretische Teilchenphysik Karlsruhe Institut für Technologie KIT Postfach 69 80 76049 Karlsruhe, Germany Phone: +49 (7 21) 6 08 33 72 Fax: +49 (7 21) 37 07 26 Email:
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[email protected] www-phys.uconn.edu/faculty/stwalley.html http://www-ttp.physik.uni-karlsruhe.de/~jk http://www-ekp.physik.uni-karlsruhe.de http://www.mpe-garching.mpg.de/index.html http://www.ioap.tu-berlin.de http://www.wyvern.phys.s.u-tokyo.ac.jp/welcome_en.html http://www.wyvern.phys.s.u-tokyo.ac.jp/welcome_en.html http://www-tkm.physik.uni-karlsruhe.de http://www.physik.uni-ulm.de/theo/qc/group.html http://www-phys.uconn.edu/faculty/stwalley.html Reinhard Lipperheide • Uwe Wille The Thermoballistic Transport Model A Novel Approach to Charge Carrier Transport in Semiconductors 123 Reinhard Lipperheide Helmholtz-Zentrum Berlin für Materialien und Energie Berlin Germany e-mail:
[email protected] Uwe Wille Helmholtz-Zentrum Berlin für Materialien und Energie Berlin Germany e-mail:
[email protected] ISSN 0081-3869 ISSN 1615-0430 (electronic) ISBN 978-3-319-05923-5 ISBN 978-3-319-05924-2 (eBook) DOI 10.1007/978-3-319-05924-2 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014936423 � Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface In this book, we present a comprehensive survey of the thermoballistic approach to charge carrier transport in semiconductors. This semiclassical approach, which we have developed over the past decade, bridges the gap between the opposing drift- diffusion and ballistic (‘‘thermionic’’) models of carrier transport, whose validity is limited to values of the carrier mean free path that are very small and very large, respectively, compared with typical length scales in a semiconducting sample. The physical concept underlying the thermoballistic approach, while incorpo- rating basic features of the drift-diffusion and ballistic descriptions, constitutes a novel, unifying scheme. It is based on the introduction of ‘‘ballistic configura- tions’’ arising from a random partitioning of the length of a semiconducting sample into ballistic transport intervals across which the carriers move without collisions (for simplicity, we consider one-dimensional transport). These intervals are linked to one another at points of local thermodynamic equilibrium, where the carriers are, at the same time, both emitted and absorbed. [The joint appearance of thermal and ballistic attributes has led us to call our approach ‘‘thermoballistic’’.] During their transmission across the ballistic intervals, the carriers encounter, in general, potential energy barriers generated by a combination of internal and external electrostatic potentials in the sample, and concurrently undergo spin relaxation. The lengths of the ballistic intervals are stochastic variables, which are contingent upon the probabilities for carriers to traverse an interval without collisions with the scattering centers randomly distributed over the sample. These probabilities are controlled by the carrier mean free path, whose magnitude is arbitrary. By aver- aging the ballistic total and spin-polarized carrier currents as well as the associated densities over all ballistic configurations, position-dependent thermoballistic cur- rents and densities are derived, which are expressed in terms of two dynamical functions, viz:, an average chemical potential and a spin accumulation function. Algorithms to determine these functions are set up by appropriately connecting the thermoballistic total and spin-polarized currents with the total physical current and the spin polarizations in the contacts. The thermoballistic currents form the point of departure for the calculation of all relevant transport quantities. In our original publications, we developed the thermoballistic concept gradu- ally, starting from what we called the ‘‘generalized Drude model’’. In this model, the averaging over ballistic configurations was introduced, and the current-voltage characteristic was obtained by assuming current conservation from one ballistic v interval to the next. Relaxing this assumption, we were led to the construction of a position-dependent total thermoballistic current, which is expressed in terms of a local chemical potential. Spin-dependent thermoballistic currents and densities were introduced in conjunction with applications to spin-polarized transport in ferromagnet/semiconductor heterostructures and structures involving diluted magnetic semiconductors. While in these publications the essence of the thermoballistic concept has been outlined, none of them affords a complete, systematic elaboration of this concept and its implementation. We, therefore, believe that a comprehensive survey of the thermoballistic approach is called for, with emphasis on a self-contained and coherent presentation based on a consistent set of physical assumptions. This includes a recollection of early attempts to describe carrier transport in semicon- ductors and a transparent, step-by-step derivation of the formalism. Presenting such a kind of survey is the aim of this book. The book starts with an account of Drude’s model of carrier transport, which is at the root of all classical and semiclassical transport models. This is followed by a survey of the standard drift-diffusion and ballistic descriptions, forming the cor- nerstones of the thermoballistic approach. Thereupon, to pave the way for the fully developed form of the thermoballistic concept, we present a ‘‘prototype thermo- ballistic model’’. This model, while based on the assumptions underlying our ‘‘generalized Drude model’’, revises and refines that model. In the central part of the book, a coherent exposition of the thermoballistic approach proper is presented within a general formulation that takes into account arbitrarily shaped, spin-split potential energy profiles as well as spin relaxation during the carrier motion across ballistic intervals. The physical conditions underlying the implementation of the thermoballistic concept are established, and explicit equations governing the average chemical potential and the spin accumulation function are derived. The relevance of the work presented in this book lies in providing a unified understanding of semiclassical carrier transport, which, we believe, constitutes a valuable contribution to present-day semiconductor physics and spintronics. Moreover, by presenting explicit equations for the calculation of transport quan- tities and working out various examples, we supply tools for future, systematic analyses of experimental data. Being originally non-specialists in the field of solid-state physics, we were motivated to embark on theoretical studies of carrier transport in semiconductors by our experimental colleagues in the (then) Hahn-Meitner-Institut Berlin, who had asked for theoretical support in interpreting their data on carrier transport in poly- and microcrystalline photovoltaic materials. Having entered deeper into the field, we perceived the challenge to fill, by unifying drift-diffusion and ballistic transport within a physically consistent approach, a long-standing gap in the theory of carrier transport in semiconductors. Moreover, we realized that recent progress in device physics, spintronics, and photovoltaics called for an extension of the theoretical framework hitherto available for the semiclassical analysis of experi- mental data in these fields. vi Preface We gratefully acknowledge the invaluable contributions made by Thomas Weis in the early stages of our project. We are indebted to the Board of Directors and the staff of the Helmholtz-Zentrum Berlin für Materialien und Energie (formerly Hahn-Meitner-Institut Berlin) for generously granting access to its premises and facilities beyond the date of our retirement, thereby supporting our efforts to continue and complete our research work on charge carrier transport in semi- conductors and, ultimately, to write this book. Berlin, February 2014 Reinhard Lipperheide Uwe Wille Preface vii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Drift-Diffusion and Ballistic Transport . . . . . . . . . . . . . . . . . . . . . 5 2.1 Drift-Diffusion Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Drude’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Drift-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Current-Voltage Characteristic and Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.4 Range of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Ballistic (Thermionic) Transport . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Nondegenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Electron Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2.3 Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.4 Interface Resistances and Chemical Potential . . . . . . . . . 22 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Prototype Thermoballistic Model . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Probabilistic Approach to Carrier Transport . . . . . . . . . . . . . . . 25 3.1.1 Net Electron Current . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.2 Mean Free Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Prototype Thermoballistic Model: Concept and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.1 Ballistic Configurations . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Averaging Over Ballistic Configurations . . . . . . . . . . . . 31 3.2.3 Electron Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.4 Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.5 Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Prototype Thermoballistic Model: Examples . . . . . . . . . . . . . . . 39 3.3.1 Single Potential Energy Barrier. . . . . . . . . . . . . . . . . . . 39 3.3.2 Double Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.3 Arbitrary Number of Barriers . . . . . . . . . . . . . . . . . . . . 43 3.3.4 Chemical Potential for Field-Driven Transport . . . . . . . . 47 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ix http://dx.doi.org/10.1007/978-3-319-05924-2_1 http://dx.doi.org/10.1007/978-3-319-05924-2_1 http://dx.doi.org/10.1007/978-3-319-05924-2_1#Bib1 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2#Sec1 http://dx.doi.org/10.1007/978-3-319-05924-2_2#Sec1 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and Spin Accumulation Function . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Ballistic Spin-Polarized Transport . . . . . . . . . . . . . . . . . . . . . . 55 4.2.1 Ballistic Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2.2 Ballistic Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.3 Spin Balance Equation. . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.4 Net Ballistic Currents and Joint Ballistic Densities . . . . . 62 4.3 Thermoballistic Spin-Polarized Transport . . . . . . . . . . . . . . . . . 64 4.3.1 Thermoballistic Currents and Densities . . . . . . . . . . . . . 64 4.3.2 Drift-Diffusion Regime . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3.3 Ballistic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3.4 Thermoballistic Energy Dissipation . . . . . . . . . . . . . . . . 75 4.4 Synopsis of the Thermoballistic Concept . . . . . . . . . . . . . . . . . 78 4.4.1 Ingredients and Physical Content . . . . . . . . . . . . . . . . . 78 4.4.2 Merits and Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . 80 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Thermoballistic Approach: Implementation . . . . . . . . . . . . . . . . . 83 5.1 Physical Conditions Determining the Dynamical Functions . . . . 83 5.1.1 Determination of the Mean Spin Function . . . . . . . . . . . 83 5.1.2 Determination of the Spin Accumulation Function . . . . . 85 5.2 Average Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 The Resistance Functions. . . . . . . . . . . . . . . . . . . . . . . 86 5.2.2 Resistance Functions for Constant Spin Splitting . . . . . . 90 5.2.3 Thermoballistic Average Chemical Potential . . . . . . . . . 92 5.2.4 Sharvin Interface Resistance. . . . . . . . . . . . . . . . . . . . . 95 5.2.5 Current-Voltage Characteristic and Magnetoresistance. . . 96 5.3 Spin Accumulation Function . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3.1 Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.3.2 Differential Equation for Field-Driven Transport . . . . . . 99 5.4 Current and Density Spin Polarizations . . . . . . . . . . . . . . . . . . 104 5.4.1 Spin Polarizations in the Semiconductor . . . . . . . . . . . . 104 5.4.2 Current Spin Polarization in the Contacts. . . . . . . . . . . . 105 5.4.3 Matching the Current Spin Polarization at the Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 Homogeneous Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.1 Low-Field Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1.2 Field-Driven Transport: Numerical Results. . . . . . . . . . . 116 x 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. 136 7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Appendix A: Useful Formulae for Numerical Calculations . . . . . . . . 143 A.1 Resistance Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.2 Spin Accumulation Function . . . . . . . . . . . . . . . . . . . . . . . . . 144 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Contents xi http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec10 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec10 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec11 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec11 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec14 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec14 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec14 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec19 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec19 http://dx.doi.org/10.1007/978-3-319-05924-2_6#Sec19 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The idea that electric currents flowing inside solidmaterials are effected by the transport of “small” charged particles (“charge carriers”) was first conceived by Weber [1, 2]. Following the discovery of the electron by Thomson [3],Weber’s idea quickly found its concrete expression in attempts to understand electric (as well as thermal) conduction in solids as a manifestation of electron transport. The basic concept for a theoretical treatment of conduction in terms of the motion of individual carriers was outlined by Riecke [4, 5]. Relying to some extent on this concept, Drude [6, 7] formulated his celebrated transport model, which subsequently was refined by Lorentz [8]. In Drude’s model, the atomistic picture of matter and the kinetic theory of gases are combined to describe conduction in terms of a homoge- neous gas of non-interacting, mobile charge carriers in thermodynamic equilibrium, which are assumed to move against a background of spatially fixed, heavy atoms. When an external electric field is applied, the interplay of field-induced acceleration and subsequent thermalizing collisions with the heavy atoms gives rise to a “drift current” of the carriers. The magnitude of this current is determined by the “mean free path”, i.e., the average distance the carriers travel between two collisions. For the drift current to be a valid concept, the mean free path must be very small as compared with typical length scales in a sample. While originally devised to describe carrier transport in metals, Drude’s model later on has been frequently used in qualitative, or semi-quantitative, analyses of transport properties of semiconductors as well. To overcome the shortcomings of Drude’s model in the quantitative description of carrier transport in semiconductors, particularly in inhomogeneously doped systems, the drift current was supplemented with a diffusion current [9, 10], whereby Drude’s model has been extended to the “drift-diffusionmodel” of transport. The lattermodel, while representing a substantial improvement over Drude’s model and serving as a benchmark of semiclassical transport theories even in modern times, is again valid in the range of very small carrier mean free paths only. In the opposite case of very large mean free paths, carrier transport in semiconductors can be described in terms of the R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 1 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_1, © Springer International Publishing Switzerland 2014 2 1 Introduction ballistic (“thermionic”) model [11–15], in which carriers thermally emitted at the ends of a sample are assumed to traverse it without collisions with the background atoms. Until recently, no systematic attempts had been made to bridge the gap between the limiting cases of the drift-diffusion and ballistic descriptions within a unifying approach. In an effort to remedy this situation, we have set out to develop the “ther- moballistic approach”1 to carrier transport in semiconductors. The physical concept underlying the thermoballistic approach is the following. The length of a semiconducting sample is randomly partitioned into “ballistic transport intervals”. Here, “randomly” implies (i) an arbitrary number of intervals and (ii) arbitrary positions of the end-points, and thus arbitrary lengths, of the inter- vals. [The drift-diffusion model may be viewed as implying a partitioning of the sample length into infinitesimally short ballistic intervals; in the ballistic transport model, on the other hand, the sample length constitutes a single ballistic interval.] An individual partition defines what we call a “ballistic configuration”. The points linking adjacent intervals in a ballistic configuration are assumed to be points of local thermodynamic equilibrium characterized by a local chemical potential. Any such point acts both as a source of, and a sink for, carriers. That is, on the one hand, carriers are thermally emitted there, with a velocity distribution determined by the local temperature; subsequent to their emission, the carriers are ballistically trans- mitted across either interval to the left and right, facing, in general, potential energy barriers arising from a combination of internal and external electrostatic potentials inside the sample. On the other hand, carriers emitted at the two equilibrium points neighboring the point under consideration on either side and transmitted to it are “absorbed” there, i.e., they return to thermodynamic equilibrium instantaneously. This equilibration is assumed to result from collisions of the carriers with spatially fixed scattering centers (“impurities”) randomly distributed over the sample, a view adopted from Drude’s transport model. In the manner by which the ballistic intervals are introduced, the lengths of these intervals are stochastic variables, with associated probabilities given by the probabili- ties for carriers to traverse an interval without impurity scattering. These probabilities are governed by the carrier mean free path, which is allowed to have arbitrary magni- tude. By averaging the (in general, spin-dependent) carrier currents in the individual ballistic intervals over all ballistic configurations, a position-dependent total (i.e., spin-summed) thermoballistic current as well as a thermoballistic spin-polarized current are derived. These currents, in conjunction with the associated ballistic den- sities, represent the key element of the thermoballistic approach. They form the point of departure for the calculation of the spin-resolved equilibrium chemical potentials. With the latter functions known, the relevant transport quantities, like the current- voltage characteristic, the magnetoresistance, and (position-dependent) current and density spin polarizations, are obtained in terms of the potential energy profile, the 1 After having introduced the term “thermoballistic” into the theory of carrier transport in semicon- ductors, we became aware of Ref. [16], in which this term is used in quite a different context. 1 Introduction 3 mean free path, the ballistic spin relaxation length, and other parameters character- izing the sample. The present book is devoted to a comprehensive exposition of the physical concept underlying the thermoballistic approach, and its detailed implementation. First, we develop a prototype model of thermoballistic transport (the “prototype model”, for short), which introduces averaging over ballistic configurations as the constitutive element of the thermoballistic concept. It is based on the simplifying assumption that the carrier current is conserved across the points of local thermodynamic equilibrium linking the ballistic transport intervals. This assumption allows the derivation of an explicit, transparent expression for the current-voltage characteristic, as well as the construction of a local chemical potential in a heuristic form. After describing the prototype model, we present a general formulation of the thermoballistic concept proper, which makes use of a local chemical potential as the essential dynamical quantity, and takes into account arbitrarily shaped, spin-split potential energy pro- files (thereby allowing for arbitrary inhomogeneities in the sample) as well as spin relaxation during the ballistic carrier motion. Throughout this book, the formulation deviates in many respects from that in our original publications. Apart from correcting flaws and inconsistencies, we introduce substantial modifications and extensions of the formalism that make it more transpar- ent and more comprehensible. Aiming at a self-contained, unifying presentation, we will develop the elements of the thermoballistic concept in considerable detail. This includes a recollection of early attempts to describe charge carrier transport in semi- conductors and, at one place or another, the coherent recapitulation of background material which otherwise can be found only scattered over different textbooks. In implementing the thermoballistic approach, we aim at carrying the development up to the pointwherewe obtain explicit equations for the calculation of all relevant trans- port quantities. Realizing that our approach as such is quite intricate, we go through the mathematical derivations step-by-step in a, hopefully, easy-to-follow way. We do not enter here into applications requiring numerical calculations. Pertinent results presented in our previous publications will be briefly mentioned at various places in this book. While the thermoballistic approach, in its most general form, would allow a con- sistent treatment of three-dimensional, bipolar carrier transport in semiconducting systems, we confine ourselves here to the case of one-dimensional, unipolar trans- port throughout. To be specific, we consider electron transport in the conduction band of n-doped systems, with the understanding that all results for hole transport in the valence band of p-doped systems can be obtained by transcribing the results for electron transport in an obvious way. The detailed organization of this book is as follows. In the next chapter, we present a preparatory account of Drude’s transport model and the standard drift-diffusion and ballistic models. In Chap. 3, we begin with an outline of the probabilistic approach to carrier transport, which is prerequisite to the formulation of the thermoballistic model. This is followed by the description of the prototype model. In Chap. 4, the thermoballistic concept proper is developed at length, where particular emphasis is placed on the inclusion of spin degrees of freedom. The spin-resolved ballistic http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_4 4 1 Introduction currents and associated densities are introduced in terms of an “average chemical potential” and a “spin accumulation function” connectedwith the splitting of the spin- resolved chemical potentials. Total and spin-polarized thermoballistic currents and densities are constructed by averaging the corresponding ballistic quantities over the ballistic configurations, and are evaluated immediately in the drift-diffusion regime and in the ballistic limit. Energy dissipation (“heat production”) is analyzedwithin the thermoballistic approach. In Sect. 4.4, in particular, a synopsis of the thermoballistic concept is presented, with emphasis on its physical content as well as its merits and weaknesses. The procedures devised for the implementation of the thermoballistic concept, i.e., for the explicit calculation of the average chemical potential and of the spin accumulation function as functions of intrinsic and external physical parameters, are described in Chap. 5. Expressions are derived for the current and density spin polarizations and the magnetoresistance in terms of the values of the spin accumu- lation function at the boundaries of a semiconducting sample. As specific examples of current interest in semiconductor and spintronics research, we treat in Chap. 6 spin-polarized transport in heterostructures formed of a nonmagnetic semiconduc- tor and two ferromagnetic contacts, and spin-polarized transport in heterostructures involving diluted magnetic semiconductors in their paramagnetic phase. Finally, in Chap. 7, we summarize the contents of this book and give an outlook towards future developments. References 1. W. Weber, Ann. Phys. (Leipzig) 232, 1 (1875) 2. W. Weber, in Wilhelm Weber’s Werke, vol. 4: Galvanismus und Elektrodynamik, Part 2, ed. by H. Weber (J. Springer, Berlin, 1894) 3. J.J. Thomson, Phil. Mag. 44, 293 (1897) 4. E. Riecke, Ann. Phys. (Leipzig) 302, 353 (1898) 5. E. Riecke, Ann. Phys. (Leipzig) 302, 545 (1898) 6. P. Drude, Ann. Phys. (Leipzig) 306, 566 (1900) [reprinted in Ostwalds Klassiker der Exak- ten Wissenschaften, vol. 298, ed. by H.T. Grahn and D. Hoffmann (Verlag Harri Deutsch, Frankfurt/M., 2006)] 7. P. Drude, Ann. Phys. (Leipzig) 308, 369 (1900) 8. H.A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (Columbia University Press, New York, 1909) 9. C. Wagner, Z. Phys. Chem. B 21, 25 (1933) 10. J. Frenkel, Phys. Z. Sowjetunion 8, 185 (1935) 11. O.W. Richardson, Phil. Trans. R. Soc. (Lond.) 201, 497 (1903) 12. O.W. Richardson, Thermionic Phenomena and the Laws which Govern Them, Nobel Lecture, 12 Dec 1929 (The Nobel Foundation, Stockholm, 1929) 13. A. Sommerfeld, Z. Phys. 47, 1 (1928) 14. A. Sommerfeld, H. Bethe, Elektronentheorie der Metalle, in Handbuch der Physik, vol. 24/2, 2nd edn., ed. by H. Geiger, K. Scheel (Springer, Berlin, 1933) [reprinted in Heidelberger Taschenbücher, vol. 19 (Springer, Berlin, 1967)] 15. H.A. Bethe, MIT Radiat. Lab. Report 43–12 (1942) [reprinted in Semiconductor Devices: Pioneering Papers, ed. by S. M. Sze (World Scientific, Singapore, 1991), p. 387] 16. S.H. Hasinger, Thermoballistic Generator (United States Patent 3577022, May 1971) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_6 http://dx.doi.org/10.1007/978-3-319-05924-2_7 Chapter 2 Drift-Diffusion and Ballistic Transport As the thermoballistic approach is devised to unify the drift-diffusion and ballistic (thermionic) descriptions of carrier transport, we present in this chapter a detailed exposition of the standard formulations of these two limiting cases. We consider one-dimensional transport in three-dimensional, “plane-parallel” semiconducting samples and disregard spin degrees of freedom. In the ballistic case, effects of carrier tunneling and degeneracy are taken into account. The ranges of validity of the drift- diffusion and ballistic models are indicated. With a view towards the thermoballistic description of transport, we comment on the role of interface resistances and on the chemical potential in the ballistic description. 2.1 Drift-Diffusion Transport The drift-diffusion model is an extension of the transport model of Drude, so that we begin here with an account of the latter. Drude’s model is far from being able to describe transport properties of semiconductors quantitatively; nevertheless, its exposition provides uswith the opportunity to introduce and discuss the basic notions on which classical and semiclassical transport theories rely, and which will appear ubiquitously throughout this book. 2.1.1 Drude’s Model In the transport model of Drude [1–4], one considers a (three-dimensional) homo- geneous classical gas of non-interacting electrons in thermodynamic equilibrium at temperature T , which is subjected to an externally applied, constant electric field E. The electrons collide with randomly distributed, spatially fixed scattering centers (“impurities”). The collisions are assumed to cause the electrons to return instanta- neously to complete thermodynamic equilibrium. The average vectorial velocity of R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 5 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_2, © Springer International Publishing Switzerland 2014 6 2 Drift-Diffusion and Ballistic Transport the electrons emerging from this equilibration is equal to zero. Their mean speed u, i.e., the thermal average of the magnitude of the electron velocity as derived from the three-dimensional Maxwell-Boltzmann velocity distribution function, is given by u = ( 8 πm∗β )1/2 , (2.1) where m∗ is the effective electron mass, and β = 1/kBT , with kB the Boltzmann constant. (Note that in Drude’s original papers [1, 2], the mean electron speed was derived from Boltzmann’s equipartition theorem; the Maxwell-Boltzmann velocity distribution was introduced into the description of electron transport in metals by Lorentz [5].) 2.1.1.1 Drift Current Defining the mean free path (or momentum relaxation length) l as the average dis- tance (measured along the transport direction) traveled by the electrons between two collisions, one finds for the collision time (or relaxation time) τ , defined as the average time-of-flight between successive collisions, τ = l u . (2.2) Themean free path l characterizes a set of collision-free ballistic intervals of average length l, across which electrons move ballistically under the influence of the external field E. On average, they acquire a velocity, the drift velocity vdr , given by the acceleration−eE/m∗ times their average time-of-flight across the ballistic intervals, i.e., times the collision time τ , vdr = − eE m∗ τ ≡ −νE, (2.3) where ν ≡ e m∗ τ = e m∗ l u = e ( πβ 8m∗ )1/2 l (2.4) is the electron mobility. In the picture of Drude, the charge current density j (current, for short) is a drift current, jdr , driven by the external electric field E, j ≡ jdr = −envdr = enνE, (2.5) 2.1 Drift-Diffusion Transport 7 where n is the electron density. Since, in Drude’s model, n as well as the field E are independent of position, j is constant. In terms of the electron conductivity σ = eνn, (2.6) the current j is expressed as j = σE, (2.7) i.e., in the form of Ohm’s law. The transport mechanism in Drude’s model can be elucidated [4] by noting that Eq. (2.3), when written as − eE − m ∗ τ vdr = 0, (2.8) states that the effect of the external force −eE is balanced, on average, by that of a friction force −γvdr , with the friction coefficient γ given by γ = m ∗ τ = e ν . (2.9) The friction force reflects the reset to zero, on average, of their individual velocities when, subsequent to their acceleration through the external field, the electrons are thermally equilibrated due to impurity scattering. 2.1.1.2 Range of Validity Limits on the range of validity of Drude’s model are set by the requirement that the perturbation of the electron gas due to the external field is sufficiently small such that the gas stays close to thermodynamic equilibrium. This condition can be met by requiring the magnitude of the drift velocity to be very small as compared with the mean speed of the electrons, |vdr | ↑ u. (2.10) Using Eqs. (2.1), (2.3) and (2.4), condition (2.10) can be re-expressed in a form that restricts, for given mean free path l, the magnitude of the electric-field vector E, |E| ↑ 8 πβel . (2.11) For a homogeneous sample of length S subjected to an externally applied voltage V , so that |E| = |V |/S, one has |V | ↑ 8S πβel . (2.12) 8 2 Drift-Diffusion and Ballistic Transport Reversely, for given V , the ratio of mean free path to sample length is restricted by the condition l S ↑ 8 πβe|V | . (2.13) For room temperature, i.e., 1/β ≈ 0.025 eV, and values of e|V | of a few tens of meV, which are typical for semiconducting devices, the right-hand side of condition (2.13) is of order unity. Hence, one may use the condition l S ↑ 1 (2.14) as a rough criterion for delimiting the range of validity of Drude’s model. The value of l remains small if the density of impurities is sufficiently high. 2.1.2 Drift-Diffusion Model Electron transport in inhomogeneous semiconductors is outside of the scope of Drude’s model. The non-uniformity of the donor density in inhomogeneous sam- ples entails a position dependence of the electron density, n = n(x), and gives rise to an x-dependent internal (“built-in”) electrostatic potential [3, 6, 7]. [Throughout this book, we consider one-dimensional transport in three-dimensional, “plane-parallel” semiconducting samples, i.e., samples whose parameters do not vary in the direc- tions perpendicular to the transport direction, which is taken as the x-direction (for a discussion of this assumption, see Sect. 4.4.2). Further, the temperature T is assumed to be constant across the sample.] 2.1.2.1 Local Thermodynamic Equilibrium Transport in inhomogeneous systems near thermodynamic equilibrium can be described in terms of a local thermodynamic equilibrium [3] characterized by a local chemical potential μ(x)1 Disregarding spin degrees of freedom and adopting the effective-mass approximation [3, 6], we have for the equilibrium electron density n(x) in a nondegenerate system n(x) = 4π ( m∗ h )3 ∞∫ −∞ dvx ∞∫ 0 dwwf MB(E(v; x)− μ(x)). (2.15) 1 Throughout this book, we use the term “chemical potential” as defined in Ref. [3]. http://dx.doi.org/10.1007/978-3-319-05924-2_4 2.1 Drift-Diffusion Transport 9 Here, h is Planck’s constant, and we have introduced cylindrical coordinates in three- dimensional velocity space (with Cartesian coordinates vx, vy, vz) such that v = (v2x + w2)1/2, (2.16) with w2 = v2y + v2z . The function f MB(E) = e−βE (2.17) is the Maxwell-Boltzmann energy distribution function, and E(v; x) = λ(v)+ Ec(x) (2.18) is the total electronic energy at the equilibrium point x, with the kinetic energy λ(v) = m ∗ 2 v2. (2.19) The potential energy profile Ec(x), here assumed to be a prescribed function, com- prises the conduction band edge potential (which includes the position-dependent internal potential) and the external electrostatic potential. In general, Ec(x) exhibits a “multiple-barrier” shape associated with local maxima of the profile. Evaluation of the integrals in Eq. (2.15) results in the standard form for the density, n(x) = Nce−β[Ec(x)−μ(x)], (2.20) where Nc = 2 ( 2πm∗ βh2 )3/2 (2.21) is the effective density of states at the conduction band edge [3]. 2.1.2.2 Drift-Diffusion Current The effects of the spatial variation of the electron density and of the associated occur- rence of an internal potential in inhomogeneous semiconductors can be described within an extension of Drude’s model, the drift-diffusion model. In this model, a gen- eralized, x-dependent drift current jdr(x) is supplemented [3, 8–10] by a diffusion current jdi(x) proportional to the density gradient along the x-axis, so that the total (conserved) current j is given by j = jdr(x)+ jdi(x). (2.22) 10 2 Drift-Diffusion and Ballistic Transport The generalized drift current is obtained from expression (2.7) by replacing (i) the conductivity σ with the local conductivity σ(x) given by Eq. (2.6), with n(x) in lieu of n, and (ii) the constant external electric field E with the field [dEc(x)/dx]/e associated with the potential energy profile Ec(x), so that one obtains jdr(x) = 1 e σ(x) dEc(x) dx . (2.23) The diffusion current is written as jdi(x) = eD dn(x) dx , (2.24) where D is the diffusion coefficient, which is related to the electron mobility ν via the Einstein relation [3, 4], D = ν βe = τ m∗β , (2.25) and ν is now assumed to have the form corresponding to one-dimensional transport, ν = e ( 2β πm∗ )1/2 l (2.26) [see Eq. (2.4) for the three-dimensional form of ν]. Then, generalizing Eq. (2.6), one has σ(x) ≡ eνn(x) = βe2Dn(x). (2.27) For the total current j, one now finds from Eqs. (2.22)–(2.24), using Eq. (2.27), j = 1 e σ(x) [ dEc(x) dx + 1 βn(x) dn(x) dx ] , (2.28) which is the standard drift-diffusion expression (“drift-diffusion equation”) for the total charge current. For the derivation of this expression fromBoltzmann’s transport equation and for its application in device simulation, see, e.g., Ref. [11]. In Ref. [12], the expression is derived within a time-dependent, tutorial treatment of diffusion in the presence of an external electric field (“biased-random-walk model”). 2.1.3 Current-Voltage Characteristic and Chemical Potential Substituting expression (2.20) for the equilibrium density n(x) in Eq. (2.28), we obtain the current j in the form 2.1 Drift-Diffusion Transport 11 j = 1 e σ(x) dμ(x) dx , (2.29) which shows that, in the drift-diffusionmodel, it is the local chemical potential which provides the driving force for electron transport. Using Eqs. (2.20) and (2.27), we can rewrite Eq. (2.29) as βj νNc eβEc(x) = d dx eβμ(x). (2.30) Considering a sample extending from x1 to x2, so that S = x2 − x1 (2.31) is the sample length, and integrating Eq. (2.30) over the interval [x1, x], we can express the local chemical potential μ(x) in the form eβμ(x) = eβμ(x1) + βj νNc x∫ x1 dx′eβEc(x′). (2.32) An equivalent representation of μ(x) is obtained by integrating Eq. (2.30) over the interval [x, x2]. 2.1.3.1 Current-Voltage Characteristic Now, setting x = x2 in Eq. (2.32), identifying the chemical potentials at the end- points x1,2 with the potentials μ1,2 in the contacts connected to the semiconducting sample, μ(x1,2) = μ1,2, (2.33) and solving the resulting equation for the current j, we then obtain, using Eq. (2.20), the current-voltage characteristic of the drift-diffusion model in the form j = −n(x1)e−βElb(x1,x2) ν βS̃ (1− e−βeV ). (2.34) Here, V = μ1 − μ2 e (2.35) is the voltage applied between the points x1 and x2, and Elb(x1, x2) ≡ Emc (x1, x2)− Ec(x1) ≥ 0 (2.36) 12 2 Drift-Diffusion and Ballistic Transport is the maximum barrier height of the potential energy profile relative to its value Ec(x1) at the left end of the sample, where Emc (x1, x2) is the overall maximum of Ec(x) in the interval [x1, x2]. Finally, the quantity S̃ ≡ x2∫ x1 dxe−β[Emc (x1,x2)−Ec(x)] (2.37) is the “effective sample length”. It has the appealing property of becoming equal to the sample length S when the profile is flat, i.e., when Ec(x) = const., and otherwise satisfies S̃ < S. Writing the current-voltage characteristic in the particular form (2.34) facilitates comparison with analogous expressions given below. Expression (2.34) shows that the current-voltage characteristic of the drift- diffusion model is controlled (i) by the “barrier factor” e−βElb(x1,x2), which involves the overall maximum of the profile Ec(x), and (ii) by the ratio ν/S̃ or, owing to Eq. (2.26), by the ratio l/S̃, in which the effective sample length reflects, in an integral way, the shape of Ec(x). [In the ballistic description of transport, the barrier factor e−βElb(x1,x2) re-appears as the thermally averaged probability for elec- tron transmission from x1 to x2; see Eq. (2.68) below.] 2.1.3.2 Chemical Potential Now, inserting expression (2.34) in Eq. (2.32), using Eqs. (2.20) and (2.33), and defining Ic(x ′, x′′) = x′′∫ x′ dzeβEc(z), (2.38) we can express the chemical potential μ(x) in the explicit form eβ[μ(x)−μ1] = 1− Ic(x1, x) Ic(x1, x2) (1− e−βeV ), (2.39) i.e., solely in terms of the potential energy profile Ec(x) and the voltage V . Equiva- lently, we can write eβ[μ(x)−μ2] = 1+ Ic(x, x2) Ic(x1, x2) (1− e−βeV ) (2.40) by using for μ(x) the representation obtained by integrating Eq. (2.30) over the interval [x, x2]. It is easily seen that the two expressions (2.39) and (2.40) indeed represent a unique chemical potential μ(x). 2.1 Drift-Diffusion Transport 13 2.1.3.3 Field-Driven Transport As a special case, we consider electron transport in a homogeneous semiconductor (i.e., in a homogeneously doped, monocrystalline semiconducting sample), driven by a static external electric field of arbitrary magnitude (“field-driven transport”). When the effect of Schottky barriers is disregarded, the field is spatially constant, withmagnitude E = V/S. Assuming the field to be directed antiparallel to the x-axis, we write the corresponding potential energy profile in the form Ec(x) = Ec(x1)− λ β (x − x1), (2.41) where the field parameter ε is given by ε = βeE . (2.42) For the effective sample length S̃, we then have S̃ = 1 ε (1− e−εS), (2.43) so that, using the relation εS = βeV, (2.44) we obtain for the current j from Eq. (2.34) j = −n(x1) ν β ε = −σ V S , (2.45) where the conductivity σ is given by Eq. (2.6). For the chemical potential μ(x), we find from Eqs. (2.39) and (2.40), using relation (2.44), μ(x) = Ec(x)+ [μ1 − Ec(x1)] = Ec(x)+ [μ2 − Ec(x2)] ≡ 1 2 (μ1 + μ2)− eV x − (x1 + x2)/2 S . (2.46) The current j given by Eq. (2.45) is a pure drift current, i.e., in a homogeneous system, there is no field-induced carrier diffusion in the drift-diffusion regime. Hence, the equilibrium density n(x) given by Eq. (2.20) must be constant across the sample. As the chemical potential μ(x) given by Eq. (2.46) runs parallel to potential energy profile Ec(x) given by Eq. (2.41), this requirement is indeed fulfilled here. 14 2 Drift-Diffusion and Ballistic Transport 2.1.4 Range of Validity The drift-diffusion model is based on the assumption of a continuously varying equilibrium chemical potential μ(x), which implies that the sample length is covered by points of local thermodynamic equilibrium that lie arbitrarily dense. Then, strictly speaking, the mean free path l must be confined to arbitrarily small values. On the other hand, l (or the mobility ν) must be finite and large enough to give rise to a non- vanishing conductivity of a magnitude in the range of typical experimental values, which calls for a relaxation of the former condition. To obtain a practical criterion for the range of validity of the drift-diffusionmodel, one may require l to be so small that the effective number of points of local ther- modynamic equilibrium along the length of the sample is so large that the spatial variations in the potential energy profile Ec(x), and hence in the electron density n(x), are “resolved” with sufficient accuracy. In terms of a local, x-dependent mean free path l(x), this requirement may be expressed as l(x) ↑ Δx, (2.47) whereΔx is the length of an interval, centered at the point x, over which the relative variation of Ec(x) is very small compared to unity, i.e., 1 Ec(x) ∣∣∣∣dEc(x)dx ∣∣∣∣Δx ↑ 1. (2.48) For constant mean free path l, one may fulfill the above requirement in an overall way by adopting the condition l S̃ ↑ 1. (2.49) The effective sample length S̃ defined by the integral (2.37) tends to decrease expo- nentially when rapid variations in Ec(x) are “switched on”. Condition (2.49) tightens condition (2.14) so as to permit the “resolution” of the details of the profile Ec(x). 2.2 Ballistic (Thermionic) Transport In contrast to the drift-diffusion model, in which the points of local thermodynamic equilibrium are assumed to lie arbitrarily dense, the ballistic (thermionic) transport model presupposes the complete absence of such points inside the sample. Then, the electrons in the sample perform a collision-free ballistic motion in the field associated with the potential energy profile Ec(x). Without thermal equilibration, it is meaningless to speak of a local chemical potential. Only at the sample ends at x1,2, the electrons are forced into equilibrium, with densities 2.2 Ballistic (Thermionic) Transport 15 n(x1,2) = Nce−β[Ec(x1,2)−μ1,2] (2.50) determined by the boundary values of the potential energy profile, Ec(x1,2), and by the chemical potentials μ1,2 in the contacts. Here, the mean free path l, which has been of central importance in Drude’s model and in the drift-diffusion model, is effectively of infinite length and does not appear in the formalism of the ballistic model. Hence, roughly speaking, one may use the condition l S 1 (2.51) to delimit the range of validity of the ballistic model. 2.2.1 Nondegenerate Case In the ballistic model, the end-points x1,2 of the sample are fixed points of local thermodynamic equilibrium with chemical potentials μ(x1,2) = μ1,2, out of which thermal electron currents are symmetrically emitted towards the left and right, so that only one half of each of these currents are emitted towards the inner region of the sample. 2.2.1.1 Emitted Currents For a nondegenerate system, the classical electron current Jl(x1) emitted at the left end-point x1 towards the right, say, is expressed, in extension of Eq. (2.15) for the electron density n(x), in the form Jl(x1) = 4π ( m∗ h )3 ∞∫ 0 dvxvx ∞∫ 0 dwwf MB(E(v; x1)− μ1) = 4πm ∗2 βh3 ∞∫ 0 dvxvxf MB(E(vx; x1)− μ1) (2.52) [here and in the following, we use the symbol J to denote any electron current density (current, for short), as opposed to j used for the charge current density]. Using Eq. (2.50), we then find Jl(x1) = veNce−β[Ec(x1)−μ1] = ven(x1), (2.53) 16 2 Drift-Diffusion and Ballistic Transport where ve = ( m∗β 2π )1/2 ∞∫ 0 dvxvxf MB(m∗v2x/2) = ( 1 2πm∗β )1/2 (2.54) is the emission velocity, which is actually equal to one half of the one-dimensional mean electron speed u = ( 2 πm∗β )1/2 , (2.55) the three-dimensional analogue of which is given by Eq. (2.1). 2.2.1.2 Transmitted Currents The electrons emitted at x1 with velocity component v (1) x move along ballistic trajectories, thereby conserving their total energy, E(vx; x) = E(v(1)x ; x1), (2.56) where x is any point inside the interval [x1, x2]. Therefore, although x is not a point of local thermodynamic equilibrium, we can associate with it an equilibrium energy distribution which is equal to that at x1. Now, since only electrons with total energy larger than the overall maximum Emc (x1, x2) of Ec(x) [see Eq. (2.36)] are classically able to reach the right end-point of [x1, x2] at x2, part of the current Jl(x1) emitted at x1 will be reflected, and the electrons forming it are absorbed when they return to their origin at x1. The other, transmitted part Jl(x1, x2; x), called the (left) “ballistic current”, is absorbed into the contact connected to the sample at x2. Modifying expression (2.52), we have for this part Jl(x1, x2; x) = 4πm ∗2 βh3 ∞∫ 0 dvxvxf MB(E(vx; x)− μ1) ×Θ(E(vx; x)− Emc (x1, x2)). (2.57) In the integration, the potential energy profileEc(x) contained in the functionE(vx; x) drops out, and we obtain for the (left) ballistic current the expression Jl(x1, x2) = veNce−β[Emc (x1,x2)−μ1], (2.58) which is independent of x, i.e., the ballistic current is conserved, as expected. The (right) ballistic current Jr(x1, x2) transmitted from the right end-point x2 of the interval [x1, x2] is given by 2.2 Ballistic (Thermionic) Transport 17 Jr(x1, x2) = −veNce−β[Emc (x1,x2)−μ2], (2.59) in analogy to Eq. (2.58). The ballistic currents (2.58) and (2.59) bear a close analogy to the “thermionic emission current” associated with the evaporation of electrons from a heated metal (“Richardson effect”) [13–16]. In semiconductor physics, “thermionic emission”was introduced as a mechanism of carrier transport by Bethe [17, 18] in his treatment of electron transport across a Schottky barrier. 2.2.1.3 Ballistic Densities Associated with the ballistic currents Jl,r(x1, x2) are the “ballistic densities” nl,r(x1, x2; x) of the electrons making up the currents inside the ballistic interval [x1, x2]. These densities will turn out to be instrumental in establishing the spin- dependent thermoballistic scheme (see Sect. 4.2). The density nl(x1, x2; x) associated with the current Jl(x1, x2; x) of Eq. (2.57) is given by nl(x1, x2; x) = 4πm ∗2 βh3 ∞∫ 0 dvxf MB(E(vx; x)− μ1) ×Θ(E(vx; x)− Emc (x1, x2)), (2.60) which is evaluated to yield nl(x1, x2; x) = Nc 2 Cm(x1, x2; x)e−β[Emc (x1,x2)−μ1]. (2.61) Here, Cm(x1, x2; x) = eβ[Emc (x1,x2)−Ec(x)]erfc({β[Emc (x1, x2)− Ec(x)]}1/2), (2.62) where the function erfc(x) = 2√ π ∞∫ x dze−z2 (2.63) is the complementary error function [19]. The ballistic density is position-dependent via the x-dependence of the functionCm(x1, x2; x), i.e., of the potential energy profile Ec(x). The ballistic density nr(x1, x2; x) associated with the current Jr(x1, x2) is obtained by replacing μ1 with μ2 in expression (2.61). The function Cm(x1, x2; x) determines the “ballistic velocities” http://dx.doi.org/10.1007/978-3-319-05924-2_4 18 2 Drift-Diffusion and Ballistic Transport vl,r(x1, x2; x) ≡ J l,r(x1, x2) nl,r(x1, x2; x) = ± 2ve Cm(x1, x2; x) , (2.64) which have the same magnitude for the currents transmitted from the left and right. For constant potential energy profile, one has Cm(x1, x2; x) = 1, and the electrons move with speed 2ve, i.e., with the mean electron speed u given by Eq. (2.55). For position-dependent profiles, when Cm(x1, x2; x) < 1, the magnitude of the ballistic velocities is larger than u. 2.2.1.4 Net Currents and Joint Densities The net ballistic current J(x1, x2) in the interval [x1, x2], J(x1, x2) = Jl(x1, x2)+ Jr(x1, x2) ≡ J (2.65) equals the (conserved) total current J , which we can express, using Eqs. (2.58) and (2.59), as J = veNce−βEmc (x1,x2)(eβμ1 − eβμ2). (2.66) This can be rewritten, using Eqs. (2.35) and (2.50), in the form J = ven(x1)T̄ l(x1, x2)(1− e−βeV ). (2.67) Here, T̄ l(x1, x2) ≡ β ∞∫ 0 dλe−βλT(x1, x2; λ+ Ec(x1)) = e−βElb(x1,x2), (2.68) with Elb(x1, x2) given by Eq. (2.36), is the thermal average of the classical transmis- sion probability T(x1, x2;E) = Θ(E − Emc (x1, x2)) (2.69) for electrons emitted at x1 with total energy E = λ+ Ec(x1) to be transmitted to the point x2. If, in particular, the potential energy profile is constant across the interval [x1, x2], or if its maximum lies at the emission point x1 itself, then Emc (x1, x2) = Ec(x1) in Eq. (2.36), and hence T̄ l(x1, x2) = 1. Relation (2.67) is the current-voltage characteristic of the (classical) ballistic trans- port model. In contrast to the characteristic (2.34) of the drift-diffusion model, which involves the mean free path l and the potential energy profile Ec(x) [via the effective sample length S̃], the characteristic (2.67) is controlled by one “material parameter” only, viz., the thermally averaged transmission probability T̄ l(x1, x2). 2.2 Ballistic (Thermionic) Transport 19 For the joint ballistic density n(x1, x2; x), n(x1, x2; x) = nl(x1, x2; x)+ nr(x1, x2; x), (2.70) we have n(x1, x2; x) = Nc 2 Cm(x1, x2; x)e−βEmc (x1,x2)(eβμ1 + eβμ2), (2.71) in analogy to expression (2.66) for the net ballistic current. 2.2.2 Electron Tunneling The ballistic transport model is straightforwardly extended so as to include elec- tron tunneling by replacing the classical transmission probability T(x1, x2;E) of Eq. (2.69) with the corresponding quantal probability T (x1, x2;E). The thermally averaged quantal transmission probability is then, in generalization of expression (2.68), given by T̄ (x1, x2) = β ∞∫ 0 dλe−βλT (x1, x2; λ+ E>c (x1, x2)), (2.72) where E>c (x1, x2) ≡ max{Ec(x1),Ec(x2)}. (2.73) The probability T (x1, x2;E) is obtained by solving the stationary Schrödinger equa- tion with the potential energy function Ec(x). [Owing to time reversal invariance, the probability for transmission from the left equals that for transmission from the right.] The integration in Eq. (2.72) starts at the total energy E>c (x1, x2), so that scattering boundary conditions can be imposed on the wave function both in the ranges x ≤ x1 and x ≥ x2. In WKB approximation [20, 21], the transmission probability T (x1, x2;E) to be used in Eq. (2.72) is composed of the classical (“over-barrier”) part T(x1, x2;E) given by Eq. (2.69) and the remaining quantal (“sub-barrier”) part Tsb(x1, x2;E), T (x1, x2;E) = T(x1, x2;E)+ Tsb(x1, x2;E). (2.74) The sub-barrier contribution has the form Tsb(x1, x2;E) = Θ(Emc (x1, x2)− E)Pc(x1, x2;E), (2.75) 20 2 Drift-Diffusion and Ballistic Transport where Pc(x1, x2;E) = exp ⎛ ⎝−2 x2∫ x1 dxκc(x) ⎞ ⎠ , (2.76) with κc(x) = 1 � {2m∗[Ec(x)− E]}1/2Θ(Ec(x)− E), (2.77) is the barrier penetration factor. In writing Tsb(x1, x2;E) in the form (2.75), we disregard resonance effects that may occur when Ec(x) exhibits two or more local maxima in the interval [x1, x2], with a corresponding number of one or more minima in between. Then, when the energy E is located below the second-highest maximum and above the lowest min- imum, there is at least one “valley” in Ec(x), across which the electron motion is classically allowed, so that resonance formation due to quantum coherence becomes possible. In semiconductor physics, a concrete realization of this situation occurs in resonant tunneling in multiple-barrier quantum-well structures (see, e.g., Ref. [22]). For this case, the fullWKB tunneling probability for double-barrier and triple-barrier structures, respectively, has been presented in Ref. [23]. From Eq. (2.74), we now find for the WKB form of the thermally averaged trans- mission probability T̄ (x1, x2) of Eq. (2.72) T̄ (x1, x2) = T̄(x1, x2)+ T̄sb(x1, x2). (2.78) Here, we have T̄(x1, x2) = e−βEb(x1,x2), (2.79) with Eb(x1, x2) = Emc (x1, x2)− E>c (x1, x2), (2.80) for the over-barrier contribution, and T̄sb(x1, x2) = β ∞∫ 0 dλe−βλPc(x1, x2; λ+ E>c (x1, x2))Θ(Eb(x1, x2)− λ) (2.81) for the sub-barrier contribution. The thermally averaged quantal probability T̄ l(x1, x2) for transmission from the left end-point at x1 can be expressed in terms of T̄ (x1, x2), using Eqs. (2.36) and (2.80), as 2.2 Ballistic (Thermionic) Transport 21 T̄ l(x1, x2) = T̄ (x1, x2)e−β[E>c (x1,x2)−Ec(x1)]. (2.82) In analogy to Eq. (2.67) for the classical case, we have J = ven(x1)T̄ l(x1, x2)(1− e−βeV ) (2.83) for the current-voltage characteristic of tunneling-enhanced ballistic transport. 2.2.3 Degenerate Case In the degenerate case, when the electron system obeys Fermi-Dirac statistics, we write the ballistic current Jl(x1, x2) in the form [see Eqs. (2.52) and (2.57) for the nondegenerate case] Jl(x1, x2; x) = 4π ( m∗ h )3 ∞∫ 0 dvxvx ∞∫ 0 dww × f FD(E(v; x)− μ1)Θ(E(v; x)− Emc (x1, x2)), (2.84) where f FD(E) is the Fermi-Dirac energy distribution function, f FD(E) = 1 1+ eβE . (2.85) Expression (2.84) for the current Jl(x1, x2; x) formally agrees with the expression for the current of evaporated electrons encountered in the degenerate treatment of the Richardson effect [15, 16]. Following the procedure of Ref. [15], we can reduce the threefold integration in Eq. (2.84) to a single integration over the kinetic energy λ = m∗w2/2, obtaining Jl(x1, x2) = 4πm ∗ βh3 ∞∫ 0 dλ ln(1+ e−β(λ−μ1))Θ(λ− Emc (x1, x2)). (2.86) The ballistic current Jr(x1, x2) transmitted from the right end-point of the sample at x2 is the negative of expression (2.86), with μ2 substituted for μ1. The total current J [see Eq. (2.65)] is thus obtained as J = veNcβ ∞∫ 0 dλ[ln(1+ e−β(λ−μ1))− ln(1+ e−β(λ−μ2))]Θ(λ−Emc (x1, x2)). (2.87) 22 2 Drift-Diffusion and Ballistic Transport Here, we have expressed the factor preceding the integral in Eq. (2.86) in terms of the emission velocity, ve, and the effective density of states, Nc, which are nonde- generate quantities given by Eqs. (2.21) and (2.54), respectively. Expression (2.87) for the total current J is the degenerate analogue to the nondegenerate current-voltage characteristic (2.66). For low external electric fields (“low-field transport”) characterized by the condition εS = βeV ↑ 1 (2.88) [see Eq. (2.44)], we find, expanding the right-hand side of Eq. (2.87) to first order in βeV , J = veNcβeV ⎡ ⎣ ∂ ∂μ ∞∫ 0 dλ ln(1+ e−β(λ−μ))Θ(λ− Emc (x1, x2)) ⎤ ⎦ μ=μ1 = veNcβeV ln(1+ e−β[Emc (x1,x2)−μ1]), (2.89) where we have used Eq. (2.35). For the conductance per unit area in the ballistic transport model, we now have g ≡ eJ V ∣∣∣∣ V→0 = βe2veNc ln(1+ e−β[Emc (x1,x2)−μ1]). (2.90) In highly doped, degenerate semiconductors, we may have Emc (x1, x2)− μ1 < 0 (2.91) (see, e.g., Ref. [24], where grain-boundary-limited transport in polycrystalline mate- rials is considered). Then, if β[μ1 − Emc (x1, x2)] 1, (2.92) we find from Eq. (2.90) g = β2e2veNc [ μ1 − Emc (x1, x2) ] , (2.93) which equals the conductance of a ballistic point contact [25, 26]. 2.2.4 Interface Resistances and Chemical Potential In closing this section, we note that the ballistic transport model does not provide information onwhere the resistance causing the voltage drop is located along the sample. Evidently, it cannot be inside the collision-free sample. In the quantal 2.2 Ballistic (Thermionic) Transport 23 description of ballistic electron transport in mesoscopic systems as formulated by Landauer [27–31], the resistance is made up solely of the interface resistances aris- ing from the abrupt change in the density of states (“transverse modes”) that the electrons encounter when they move across the interfaces separating the contacts (with infinitely many modes) from the sample (with a few modes only). The voltage drop is located, therefore, in the immediate vicinity of the interfaces, so that, when a chemical potential is introduced ad hoc, this must be constant inside the sample and discontinuous at the interfaces. Prior to the work of Landauer, the importance of interface resistances in ballistic transport had been emphasized by Sharvin [25]. Anticipating the later development, we remark at this juncture that in the ther- moballistic approach, i.e., for arbitrary, finite magnitude of the mean free path, the (local) chemical potential is a constitutive element of the transport mechanism; it is defined, and can be explicitly calculated, all along a semiconducting sample. This potential has discontinuities at the contact-sample interfaces, whose magnitude increases from near-zero in the small-l, drift-diffusion regime to the Sharvin value in the large-l, ballistic regime. For more details, see Sect. 5.2.4. References 1. P. Drude, Ann. Phys. (Leipzig) 306, 566 (1900) [reprinted in Ostwalds Klassiker der Exak- ten Wissenschaften, vol. 298, ed. by H.T. Grahn and D. Hoffmann (Verlag Harri Deutsch, Frankfurt/M., 2006)] 2. P. Drude, Ann. Phys. (Leipzig) 308, 369 (1900) 3. N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt Brace College Publishers, Fort Worth, 1976) 4. B. Sapoval, C. Hermann, Physics of Semiconductors (Springer, Berlin, 1995) 5. H.A. Lorentz, The Theory of Electrons and its Applications to the Phenomena of Light and Radiant Heat (Columbia University Press, New York, 1909) 6. K.W. Böer, Survey of Semiconductor Physics: Electrons and Other Particles in Bulk Semicon- ductors (Van Nostrand Reinhold, New York, 1990) 7. K.W. Böer, Introduction to Space Charge Effects in Semiconductors. Springer Series in Solid- State Sciences, vol. 160 (Springer, Berlin, 2010) 8. C. Wagner, Z. Phys. Chem. B 21, 25 (1933) 9. J. Frenkel, Phys. Z. Sowjetunion 8, 185 (1935) 10. J. Bardeen, in Handbook of Physics, 2nd edn., ed. by E.U. Condon, H. Odishaw (McGraw-Hill, New York, 1967), Part 8, Chap. 4 11. C. Jacoboni, Theory of Electron Transport in Semiconductors. Springer Series in Solid-State Sciences, vol. 165 (Springer, Berlin, 2010) 12. J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, I. Žutić, Acta Phys. Slovaca 57, 565 (2007) 13. O.W. Richardson, Phil. Trans. R. Soc. (Lond.) 201, 497 (1903) 14. O.W. Richardson, Thermionic Phenomena and the Laws which Govern Them, Nobel Lecture, 12 Dec 1929 (The Nobel Foundation, Stockholm, 1929) 15. A. Sommerfeld, Z. Phys. 47, 1 (1928) 16. A. Sommerfeld, H. Bethe, Elektronentheorie der Metalle, in Handbuch der Physik, vol. 24/2, 2nd edn., ed. by H. Geiger, K. Scheel (Springer, Berlin, 1933) [reprinted in Heidelberger Taschenbücher, vol. 19 (Springer, Berlin, 1967)] 17. H.A. Bethe, MIT Radiat. Lab. Report 43–12 (1942) [reprinted in Semiconductor Devices: Pioneering Papers, ed. by S. M. Sze (World Scientific, Singapore, 1991), p. 387] http://dx.doi.org/10.1007/978-3-319-05924-2_5 24 2 Drift-Diffusion and Ballistic Transport 18. S.M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981) 19. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1965), Chap. 7 20. L.I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), Chap. 8.34 21. C.B. Duke, Tunneling in Solids, in Solid State Physics Supplements, ed. by F. Seitz, D. Turnbull, H. Ehrenreich, vol. 10 (Academic Press, New York, 1969) 22. L.L. Chang, E.E. Méndez, C. Tejedor (eds.), Resonant Tunnneling in Semiconductors: Physics and Applications. NATO ASI Series: Series B, vol. 277 (Plenum Press, New York, 1991) 23. J.M. Xu, V.V. Malov, L.V. Iogansen, Phys. Rev. B 47, 7253 (1993) 24. M.W.J. Prins, K.-O. Grosse-Holz, J.F.M. Cillessen, L.F. Feiner, J. Appl. Phys. 83, 888 (1998) 25. Yu.V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965) [Sov. Phys. JETP 21, 655 (1965)] 26. C.W.J. Beenakker, H. van Houten, in Solid State Physics, vol. 44, ed. by H. Ehrenreich, D. Turnbull (Academic Press, Boston, 1991), p. 1 27. R. Landauer, Z. Phys. B: Condens. Matter 68, 217 (1987) 28. R. Landauer, J. Phys.: Condens. Matter 1, 8099 (1989) 29. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995) 30. Y. Imry, R. Landauer, Rev. Mod. Phys. 71, S306 (1999) 31. Yu.V. Nazarov, Ya.M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University Press, Cambridge, 2009) Chapter 3 Prototype Thermoballistic Model In the drift-diffusion and ballistic transport models, the parameter of central impor- tance is the electron mean free path l or, equivalently, the collision time π originally introduced in Drude’s model. The collision time represents the average time-of-flight that elapses between successive electron collisions with the randomly distributed scattering centers in the sample. The drift-diffusion and ballistic transport mechanisms are limiting cases associated with very small and very large (effectively, infinite) collision times, respectively. In the present chapter, we proceed to the “pro- totype thermoballistic model” [1], which represents an attempt to unify the drift- diffusion and ballistic models within a stationary (time-independent) description in which the mean free path is not limited in magnitude and enters as a parameter that controls collision probabilities. [In our original paper [1], we have called this model the “generalized Drude model”; the new name appears to give a better description of its features.] We begin this chapter by introducing the probabilistic definition of l (or π ). 3.1 Probabilistic Approach to Carrier Transport In the time-dependent probabilistic approach, one introduces the probability 1/π that an electron undergoes a collision, i.e., is equilibrized with its surroundings, within unit time. That is, dt/π is the probability for the collision to occur within an infinitesimally short time interval dt. Then, assuming the collision time π to be constant, the probability p(t) that an electron moves without collision over a finite time interval t is given [2, 3] by p(t) = e−t/π , (3.1) and the conditional probability P(t)dt that an electron undergoes a collision within the time interval dt after a collision-free flight over a time interval t, by R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 25 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_3, © Springer International Publishing Switzerland 2014 26 3 Prototype Thermoballistic Model P(t)dt = p(t)dt π . (3.2) The mean time-of-flight between successive collisions now becomes t̄ ∗ ≡∫ 0 dttP(t) = π . (3.3) Identifying t̄ with the collision time π of the Drude model, one sees that the inverse of this quantity is just equal to the collision probability per unit time, 1/π , of the probabilistic approach. We recall that the term “collision” used here is that proffered by Drude, meaning a collision leading to instantaneous, complete equilibration of the momentum of an electron with its surroundings. In that picture, π is the average time of collision-free flight between two points of equilibration, hence the appellation. However, 1/π is, more generally, the probability for a complete equilibration to occur in unit time; this may happen as a consequence of a “complete collision” (in the Drude sense) after the average time π , or by “incomplete collisions” (leading to incomplete equilibration) in a sequence of shorter collision times, which add up, on average, to the time π . 3.1.1 Net Electron Current Within the time-dependent probabilistic picture, one can set up [2] an expression for the net electron current J(x, t) at position x and time t, which embodies features that are relevant for devising the thermoballistic concept (see Sect. 4.3.1). Assuming a sample with position-dependent electron density n(x), one writes J(x, t) = Jl(x, t)+ Jr(x, t). (3.4) Considering for the moment electrons with arbitrary, constant velocity vx , the currents Jl,r(x, t) are expressed, using Eqs. (3.1) and (3.2), as Jl,r(x, t) = ±vx t∫ −≡ dt↑ π e−(t−t↑)/πn(x ≈ vx(t − t↑)). (3.5) The contributions to these currents for fixed time t arise from electrons that are transmitted, subsequent to their emission at the points x≈ ∗ x≈vx(t−t↑) [lying to the left (right) of the point x], without collision from x− (x+) to the point x. The intervals [x−, x] and [x, x+] define “ballistic intervals”; the currents transmitted across these intervals contribute to the total current with weight e−(t−t↑)/π . Note that, in contrast to the situation considered above, where ballistic motion occurs before equilibration http://dx.doi.org/10.1007/978-3-319-05924-2_4 3.1 Probabilistic Approach to Carrier Transport 27 in the time interval dt (with probability dt/π ), here the ballistic motion occurs after the electrons have been equilibrated in the time interval dt↑ (with probability dt↑/π ). When the collision time π is sufficiently small, such that vxπ is much smaller than the average length over which the density n(x) changes appreciably, one has, by expanding in expressions (3.5) n(x) to first order, J(x, t) = −2v2x dn(x) dx t∫ −≡ dt↑ π e−(t−t↑)/π (t − t↑) = −2v2x π dn(x) dx ∗ J(x). (3.6) Now, taking into account that only electrons with vx > 0 (vx < 0) contribute to the current arriving at the point x from the left (right), one obtains the thermal average of v2x as ∞v2x 〉 = 1 2m≥β , (3.7) and hence the local expression J(x) = − π m≥β dn(x) dx (3.8) for the thermally averaged electron current. The charge current−eJ(x) is then found, in view of expression (2.25) for the diffusion coefficient D, to agree with relation (2.24) for the diffusion (charge) current jdi(x) of the drift-diffusionmodel. This result may thus be regarded as a proof of the Einstein relation (2.25). When π → ≡, only electrons emitted at the left and right end, respectively, of the sample contribute to the currents Jl,r(x, t). Then, the (thermally averaged) net electron current J(x, t) essentially reduces to the ballistic current given by expres- sion (2.66). In the intermediate regime, i.e., for nonzero, finite values of π , J(x, t) combines elements of both drift-diffusion and ballistic transport. 3.1.2 Mean Free Path In analogy to the time-dependent probability p(t) given by Eq. (3.1), one can introduce the probability p(x) for an electron to travel without collision over a finite distance x, p(x) = e−x/l, (3.9) with a constant electron mean free path l. The conditional probability for an electron to undergo a collision in the infinitesimal collision interval dx after a collision-free flight over the distance x reads http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 28 3 Prototype Thermoballistic Model P(x)dx = p(x)dx l . (3.10) Then one finds x̄ ∗ ≡∫ 0 dxxP(x) = l (3.11) for the mean distance x̄ between successive electron collisions. In the following, the mean free path l will be assumed to take on any finite value, while, as before, the collisions are assumed to lead instantaneously to a complete equilibration of the electron momenta, and are to be interpreted as a simulation of the real situationwhere the distances between successive collisions are short (of the order of magnitude of the average distance between neighboring scattering centers), while the equilibration during a collision is (usually) far less than complete. The quantity l, in parallel to the collision time π , is simply to be regarded as a parameter which determines the probability (3.9) for an electron to traverse the distance x without suffering a complete collision (or a sequence of incomplete collisions effectively “adding up” to a complete collision). It is noted here that when treating inhomogeneous systems, one should, strictly speaking, allow for a position dependence of the collision time, π = π (x), and of the mean free path, l = l(x). In that case, expressions (3.1) and (3.9) for the probabilities p(t) and p(x), respectively, must be replaced with the more general forms p(t) = exp − ≡∫ 0 dt↑ π (x(t↑)) (3.12) and p(x) = exp − ≡∫ 0 dx↑ l(x↑) , (3.13) respectively. Then, of course, the relation l = uπ [see Eq. (2.2)] ceases to be valid. 3.2 Prototype Thermoballistic Model: Concept and Implementation The unification of the drift-diffusion and ballistic transport models within the pro- totype thermoballistic model [1] is based on the introduction of configurations of ballistic transport intervals, which cover the length of the sample. The indi- vidual ballistic intervals in a configuration are linked by points of local ther- modynamic equilibrium, and their lengths are stochastic variables occurring with http://dx.doi.org/10.1007/978-3-319-05924-2_2 3.2 Prototype Thermoballistic Model: Concept and Implementation 29 0 X Sx x x x*ii -1 N N E (x) E (i) m C C Fig. 3.1 Schematic diagram of a potential energy profile Ec(x) exhibiting a single barrier with maximum at x = X . The partitioning of the sample length S into N ballistic intervals [xNi−1, xNi ] (i = 1, . . . ,N) is indicated, where Emc (i) is the maximum of Ec(x) in interval i. For the definition of the mirror point x≥ associated with a point x, see Eq. (3.74) below probabilities determined by the probabilities for collision-free electronmotion across the intervals, where the electron mean free path is allowed to have arbitrary magni- tude. The electron current is, of course, conserved in each ballistic interval. In addi- tion, it is assumed that the current is conserved also from one interval to the next. Averaging over the ballistic configurations then leads to a transparent and intu- itively appealing expression for the current voltage-characteristic in terms of an effective transport length, in which the detailed shape of the potential energy profile is taken into account. Moreover, a local chemical potential can be constructed in a heuristic way. 3.2.1 Ballistic Configurations Here, as well as in Sect. 3.2.2, we confine ourselves to (one-dimensional) classical transport in nondegenerate systems. The effects of tunneling and degeneracy will be considered in Sects. 3.2.3 and 3.2.4, respectively. For given N (N = 1, . . . ,≡), the length S of the sample is randomly partitioned into N “ballistic intervals” labeled i (i = 1, . . . ,N), in which the electrons move ballistically across the potential energy profile Ec(x) (see Fig. 3.1). These intervals are linked by points of local thermodynamic equilibrium xNi−1 and x N i (x N i−1 < x N i ) which, in addition, include the two (true) equilibrium points at the interfaces with the left and right contacts at xN0 = 0 and xNN = S, respectively. [In this section, in order to avoid confusion with the labeling of the equilibrium points inside, we denote the end-point coordinates of the sample by 0 and S, instead of x1 and x2, as used elsewhere.] At xNi−1 and x N i , electrons are thermally emitted towards the right and left, respectively, into the interval [xNi−1, xNi ]. Any set 30 3 Prototype Thermoballistic Model {xNj } ∗ {xN0 , xN1 , . . . , xNj , . . . , xNN } (3.14) of N + 1 equilibrium points linking N ballistic intervals characterizes a ballistic configuration. 3.2.1.1 Net Ballistic Current Introducing an equilibrium chemical potential μ(x) all across the sample, in terms of which the values μ(xNi−1) and μ(x N i ) are defined, the net ballistic current J N (i) ∗ JN (xNi−1, x N i ) transmitted across the interval i in a partition with N intervals is given, in analogy to expression (2.66) for the current in the ballistic transport model, by JN (i) = veNce−βEmc (i)[eβμ(xNi−1) − eβμ(xNi )], (3.15) where Emc (i) ∗ Emc (xNi−1, xNi ) (3.16) denotes the absolute maximum of the potential energy profile Ec(x) in the interval i. In terms of the (classical) thermally averaged transmission probability T̄(i) = e−β[Emc (i)−E>c (i)] (3.17) defined in analogy to expression (2.79), we can write JN (i) in the form JN (i) = veNcT̄(i)e−βE>c (i)[eβμ(xNi−1) − eβμ(xNi )], (3.18) where E>c (i) ∗ max{Ec(xNi−1),Ec(xNi )}. (3.19) The current JN (i) is, of course, conserved across each interval i. In its general form, expression (3.18) does not lend itself to the calculation of transport properties in terms of external physical quantities. Therefore, we make the simplifying assumption that the electron current is conserved also across the points of local thermodynamic equilibrium linkingoneballistic interval to its adjacent intervals (and hence is constant all along the length of the sample), so that in Eq. (3.18) JN (i) = J = const. (3.20) for all i and N , where J is the physical current. We can then iterate Eq. (3.18) with respect to i (for fixed N), with the result eβμ N (0) − eβμN (S) = J veNc eβE m c (0,S)ZN . (3.21) http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 3.2 Prototype Thermoballistic Model: Concept and Implementation 31 Here, Emc (0, S) denotes the overall maximum of the profile Ec(x) in the interval [0, S], and the function ZN , defined as ZN ∗ ZN ({xNj }) = N∑ i=1 1 T̄(i) e−β[Emc (0,S)−E>c (i)] ∗ N∑ i=1 e−β[Emc (0,S)−Emc (i)], (3.22) can be formally viewed as the partition function corresponding to the “energy spec- trum” {Emc (0, S)−Emc (i)}. It depends on the potential energy profile at the equilibrium points xNj in the ballistic configuration {xNj } via Eq. (3.16). 3.2.2 Averaging Over Ballistic Configurations In order to arrive at results in terms of physical quantities, we average Eq. (3.21) over the complete set of ballistic configurations {xNj } (j = 0, 1, . . . ,N; N = 1, . . . ,≡). In doing so, we identify the average of the left-hand side of Eq. (3.21) with the term eβμ0 − eβμS , where μ0 and μS are the values of the chemical potential at the contact side of the left and right contact-semiconductor interface, respectively, and obtain eβμ0 − eβμS = J veNc eβE m c (0,S)R. (3.23) Here, R = ⎛ ZN ({xNj }) ⎝ (3.24) denotes the average of the partition functions ZN over all ballistic configurations {xNj }. From Eq. (3.23), the current-voltage characteristic of the prototype thermobal- listic model is now obtained in the general form J = veNce−β[Emc (0,S)−μ0] 1R (1− e −βeV ), (3.25) with the voltage V given by V = μ0 − μS e . (3.26) Aside from the barrier factor involving the absolute maximum, Emc (0, S), of the potential energy profile, the current J is controlled here by the parameter R, which alone comprises the detailed dependence of the current-voltage characteristic on the shape of the profile and on the material parameters characterizing the sample. As it appears in Eq. (3.25),R can be viewed as a (dimensionless) reduced resistance. Since 32 3 Prototype Thermoballistic Model the potential energy profile includes the external electrostatic potential, R depends on the voltage V , so that the characteristic (3.25) departs from the pure Shockley form. 3.2.2.1 Implementing the Averaging Process In the averaging process implied in Eq. (3.24), the contribution of each ballistic interval i in a partition with N intervals is to be weighted with the conditional prob- ability PNi dx N i for an electron to make a collision in the infinitesimal interval dx N i after having traveled freely across the interval i of length xNi − xNi−1, into which it was emitted after a collision in the infinitesimal interval dxNi−1. The collision prob- ability is connected with the two ends of each ballistic interval, and the collision interval dxNi is the equilibration interval for the ballistic interval i, but it is simulta- neously the emission interval for the ballistic interval i+1. This “sharing” of collision intervals dxNi between the ballistic intervals i and i + 1 implies that effectively the density of points of equilibration is only one-half of that given originally. Therefore, the conditional probability PNi dx N i is, in conformance with Eqs. (3.9) and (3.10), expressed as PNi dx N i = e−(x N i −xNi−1)/Δ dx N i Δ , (3.27) with the effective mean free path Δ given by Δ = 2l. (3.28) Here, the meaning of the mean free path l is the original one introduced within Drude’s model. The conditional probability dPN for an electron emitted at xN0 = 0 to undergo N − 1 collisions at the equilibrium points xNi (i = 1, . . . N − 1), and finally be absorbed with unit probability at xNN = S, is then found as dPN = ⎞ N−1⎠ i=1 dxNi Δ e−(x N i −xNi−1)/ΔΘ(xNi − xNi−1) ⎡ e−(S−x N N−1)/Δ (3.29) for N √ 2, while dP1 = e−S/Δ. (3.30) In the product in Eq. (3.29), the exponentials cancel out except for the factor e−S/Δ, so that dPN = e−S/Δ N−1⎠ i=1 dxNi Δ Θ(xNi − xNi−1) (3.31) 3.2 Prototype Thermoballistic Model: Concept and Implementation 33 and ≡∑ N=1 S∫ 0 dPN = 1 (3.32) [the symbol ⎣ S 0 is meant to imply (N − 1)-fold integration over xN1 , . . . , xNN−1 from 0 to S], i.e., the total probability is unity. 3.2.2.2 Reduced Resistance The reduced resistance R defined by Eq. (3.24) can now be expressed as R = ≡∑ N=1 S∫ 0 dPNZN ∗ e−S/Δ ≡∑ N=1 ONZN ({xNj }), (3.33) where the operators ON acting on the partition functions ZN are given by O1 = 1 and ON = S∫ 0 dxN1 Δ S∫ xN1 dxN2 Δ . . . S∫ xNN−2 dxNN−1 Δ (3.34) for N √ 2. It can be shown that, owing to the separable form of ZN [see Eqs. (3.22) and (3.41)], the multi-dimensional integrals in Eq. (3.33) can always be reduced to one-dimensional and two-dimensional integrals. The remaining infinite series can be summed up in terms of exponentials. For the classical partition function ZN of Eq. (3.22), the reduced resistanceR is thus obtained in the form R = e−S/ΔR̄(0, S)+ S∫ 0 dx↑ Δ [e−x↑/ΔR̄(0, x↑)+ e−(S−x↑)/ΔR̄(x↑, S)] + S∫ 0 dx↑ Δ S∫ x↑ dx↑↑ Δ e−(x↑↑−x↑)/ΔR̄(x↑, x↑↑), (3.35) where the (dimensionless) function R̄(x↑, x↑↑) is given by R̄(x↑, x↑↑) ∗ R̄(x↑↑, x↑) = 1 T̄(x↑, x↑↑) e−β[Emc (0,S)−E>c (x↑,x↑↑)] = e−β[Emc (0,S)−Emc (x↑,x↑↑)]. (3.36) 34 3 Prototype Thermoballistic Model Here, the thermally averaged classical transmission probability T̄(x↑, x↑↑) is defined in analogy to Eqs. (2.79) and (3.17), Emc (x ↑, x↑↑) ∗ Emc (x↑↑, x↑) is the absolute maximum of the potential energy profile Ec(x) in the interval [x↑, x↑↑], and E>c (x ↑, x↑↑) ∗ max{Ec(x↑),Ec(x↑↑)}. (3.37) In general, we have R̄(x↑, x↑↑) ≤ 1 (3.38) and, in particular, R̄(0, S) = 1. For a flat potential energy profile, Ec(x) ∗ const., when R̄(x↑, x↑↑) ∗ 1, expression (3.35) can be immediately evaluated with the result R ∗ R0 = 1+ S Δ , (3.39) so that in the general case, owing to Eq. (3.38), R ≤ R0, (3.40) i.e., R decreases when a position-dependent potential energy profile is introduced. In the current-voltage characteristic (3.25), this decrease ofR is overcompensated by the effect of the barrier factor e−β[Emc (0,S)−μ0], so that the current J decreases as well. By reducing expression (3.33) for R to the form (3.35), we have transcribed the prototype thermoballistic model, which, according to its original concept, relies on a discrete partitioning of the sample length into ballistic intervals, into a pure continuum model. Expression (3.35) is composed of three contributions. The first corresponds to ballistic transport all across the sample length S, with associated probability e−S/Δ. The second reflects the integrated effect of ballistic transport across intervals [0, x↑] and [x↑, S], respectively, with probabilities e−x↑/Δ and e−(S−x↑)/Δ. The third, finally, corresponds to the integrated effect of ballistic transport across intervals [x↑, x↑↑], with probabilities e−(x↑↑−x↑)/Δ. The central task in implementing the prototype thermoballisticmodel is to evaluate the reduced resistance R from Eq. (3.35) [or from the analogous equation derived from the quantal partition function (3.41)] for given potential energy profile Ec(x). In general, the computation of Ec(x) is accomplished (see, e.g., Ref. [4]) by solving, for given distribution of the space-fixed charges in the sample, a nonlinear Poisson equation. 3.2.3 Electron Tunneling In the ballistic transport model (see Sect. 2.2.2), we have taken into account the effect of electron tunneling by introducing quantal probabilities to describe electron transmission across the whole sample length. In the prototype model, we can include http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 3.2 Prototype Thermoballistic Model: Concept and Implementation 35 tunneling effects by using quantal transmission probabilities in expression (3.18) for the current JN (i) in the individual ballistic intervals [xNi−1, xNi ]. Replacing in Eq. (3.18) the probability T̄(i) with the thermally averaged quantal transmission probability T̄ l(i) ∗ T̄ l(xNi−1, xNi ) defined in analogy to expression (2.72), with x1,2 replaced with xNi−1,i, the partition functionZN in Eq. (3.21) acquires the form ZNq = N∑ i=1 1 T̄ (i) e−β[Emc (0,S)−E>c (i)] ∗ N∑ i=1 T̄(i) T̄ (i) e−β[Emc (0,S)−Emc (i)], (3.41) where Eq. (3.17) for T̄(i) has been used. In generalization of expression (3.35), the reduced resistance Rq including tun- neling effects is obtained from the quantal partition function ZNq as Rq = e−S/ΔR̄(0, S)+ S∫ 0 dx↑ Δ [e−x↑/ΔR̄(0, x↑)+ e−(S−x↑)/ΔR̄(x↑, S)] + S∫ 0 dx↑ Δ S∫ x↑ dx↑↑ Δ e−(x↑↑−x↑)/ΔR̄(x↑, x↑↑), (3.42) where R̄(x↑, x↑↑) = 1 T̄ (x↑, x↑↑) e−β[Emc (0,S)−E>c (x↑,x↑↑)] ∗ T̄(x ↑, x↑↑) T̄ (x↑, x↑↑) R̄(x↑, x↑↑). (3.43) Here, the thermally averaged quantal transmission probability T̄ (x↑, x↑↑) is defined in analogy to expression (2.72), and R̄(x↑, x↑↑) is given by Eq. (3.36). In view of Eq. (3.43), the inequality (3.40) providing an upper limit on the values of the classical function R holds also for the quantal function Rq. We now adopt the WKB approximation, for which we have, using Eq. (2.78), R̄(x↑, x↑↑) = R̄(x↑, x↑↑)+ R̄sb(x↑, x↑↑), (3.44) with R̄sb(x↑, x↑↑) = − T̄sb(x ↑, x↑↑) T̄(x↑, x↑↑)+ T̄sb(x↑, x↑↑) R̄(x↑, x↑↑). (3.45) Inserting expression (3.44) in Eq. (3.42), we find that Rq separates in the form Rq = Rcl +Rsb, (3.46) where the classical contribution Rcl is given by Eq. (3.35), and the tunneling (sub-barrier) contributionRsb byEq. (3.42),with R̄(x↑, x↑↑) replacedwith R̄sb(x↑, x↑↑). Evidently, Rsb < 0 and |Rsb| < Rcl. http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 36 3 Prototype Thermoballistic Model According to its definition in analogy to Eq. (2.81), the thermally averaged sub- barrier transmission probability T̄sb(x↑, x↑↑) is nonzero only if the potential energy profile Ec(x) exhibits at least one local maximum in the range x↑ < x < x↑↑ such that Eb(x↑, x↑↑) > 0. In the evaluation of the sub-barrier function Rsb by Eq. (3.42), ballistic intervals for which Ec(x) does not meet this requirement can, therefore, be excluded from the outset. 3.2.4 Degenerate Case Recalling the procedure developed in the nondegenerate case (see Sects. 3.2.1 and 3.2.2), in which expression (3.15) for the net ballistic current JN (i) forms the starting point for the derivation of the current-voltage characteristic (3.25), we find that an analogous procedure cannot be set up, in general, in the degenerate case. The reason is that for the argument leading toEq. (3.25), JN (i)must factorize into a termdepending on the potential energy profile Ec(x), and terms depending on the chemical potential μ(x). In the degenerate case, the current analogous to JN (i) is the current J(x1, x2) given by Eqs. (2.65) and (2.87), with x1 and x2, respectively, replaced with xNi−1 and xNi . In general, this current does not factorize. In the low-field regime, however, the chemical potentials at the sample ends, μ(0) and μ(S), differ only infinitesimally, and the increment of μ(x) across a ballistic interval i, τi = μ(xNi−1)− μ(xNi ), (3.47) is infinitesimally small as well. Then, we find in analogy to Eq. (2.89) JN (i) = veNcβτi ⎤ ⎦ ν νμ ≡∫ 0 dσ ln(1+ e−β(σ−μ))Θ(σ− Emc (i)) μ=μ(xNi−1) = veNcβτi ln(1+ e−β[Emc (i)−μ(0)]), (3.48) where we have replaced, in the second equation, μ(xNi−1) with μ(0). Expression (3.48) has the factorization property alluded to above, so that, follow- ing the procedure leading to Eq. (3.21) in the nondegenerate case, we here find β[μ(0)− μ(S)] = J veNc ZNd ln(1+ e−β[Emc (0,S)−μ(0)]) , (3.49) where the function ZNd = N∑ i=1 ln(1+ e−β[Emc (0,S)−μ(0)]) ln(1+ e−β[Emc (i)−μ(0)]) (3.50) http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 3.2 Prototype Thermoballistic Model: Concept and Implementation 37 generalizes the partition function ZN given by Eq. (3.22) to the degenerate case. Averaging Eq. (3.49) over the ballistic configurations {xNj } yields, in analogy to Eq. (3.23), β(μ0 − μS) = J veNc Rd ln(1+ e−β[Emc (0,S)−μ0]) , (3.51) where Rd = ⎛ ZNd ({xNj }) ⎝ (3.52) is the reduced resistance for the degenerate case. The conductance per unit area then reads g = βe2veNc 1Rd ln(1+ e −β[Emc (0,S)−μ0]), (3.53) in generalization of Eq. (2.90). 3.2.5 Chemical Potential Within the prototype model, we can construct, in a heuristic way, a unique chemical potential μ(x) all along the sample length by modifying relation (3.23) that connects the values of the chemical potential at the contact-semiconductor interfaces, μ0 and μS . We proceed as follows. On the one hand, by replacing the fixed position S with a variable position x inside the sample, we turn μS into a chemical-potential function μ1(x) and, simultaneously, reduce the range of averaging over the ballistic configurations in Eq. (3.24) for the reduced resistance R from [0, S] to [0, x]. We then obtain eβμ1(x) = eβμ0 − J veNc eβE m c (0,x)R1(x) (3.54) (0 < x < S), where we have introduced the “resistance function” R1(x) as R1(x) ∗ ⎛ ZN ({xNj }) ⎝ [0,x] . (3.55) On the other hand, by replacing in Eq. (3.23) the fixed position 0 with the variable x, so that μ0 turns into a chemical-potential function μ2(x) and the range of averaging is reduced to [x, S], we have eβμ2(x) = eβμS + J veNc eβE m c (x,S)R2(x) (3.56) http://dx.doi.org/10.1007/978-3-319-05924-2_2 38 3 Prototype Thermoballistic Model (0 < x < S), with the resistance function R2(x) defined by R2(x) ∗ ⎛ ZN ({xNj }) ⎝ [x,S] . (3.57) From the definitions (3.24), (3.55), and (3.57), it follows that R1(S) = R2(0) = R. (3.58) The resistance functions R1(x) and R2(x) are obtained in explicit form from expression (3.35) [or from expression (3.42), which includes tunneling effects] by transcribing it so as to correspond to samples with end-point coordinates 0, x and x, S, respectively. From the potentials μ1,2(x), we now construct a unique chemical potential μ(x) by setting eβμ(x) = 1 2 [eβμ1(x) + eβμ2(x)] (3.59) for 0 < x < S, while μ(0) = μ0, μ(S) = μS. (3.60) Using Eqs. (3.54) and (3.56), along with Eq. (3.23), in Eq. (3.59), we obtain eβμ(x) = 1 2 (eβμ0 + eβμS )− R−(x) 2R (e βμ0 − eβμS ), (3.61) where the function R−(x) is defined as R−(x) = e−βEmc (0,S)[eβEmc (0,x)R1(x)− eβEmc (x,S)R2(x)]. (3.62) With the resistance functionsR1,2(x) and the reduced resistanceR calculated from Eq. (3.35) [or Eq.(3.42)], the chemical potential in the prototype model can now be explicitly evaluated in terms of the potential energy profile Ec(x) and the mean free path l. A noteworthy feature of the chemical potential μ(x) is the occurrence of discon- tinuities at x = 0 and x = S. Using Eqs. (3.23), (3.35), (3.36), and (3.58), we find from Eq. (3.61) eβ[μ(0+)−μ0] − 1 = − J 2veNc eβ[Ec(0)−μ0] (3.63) for the interface at x = 0, and eβ[μ(S−)−μS] − 1 = J 2veNc eβ[Ec(S)−μS] (3.64) for the interface at x = S. 3.2 Prototype Thermoballistic Model: Concept and Implementation 39 In the fully thermoballistic approach (see Sect. 5.2.3.1 below), we will derive an expression for the chemical potential which closely resembles expression (3.61) and which exhibits discontinuities analogous to those expressed by Eqs. (3.63) and (3.64), but relies on different assumptions. 3.3 Prototype Thermoballistic Model: Examples To illustrate the formalism of the prototype thermoballistic model, we work out in this section a number of specific examples. With a view towards inhomogeneous semiconductors and heterostructures, in which barriers in the potential energy profile arise, e.g., at heterojunctions, grain boundaries, and metal-semiconductor contacts (“Schottky barriers”), we deal with the evaluation of the reduced resistance R for profiles Ec(x) exhibiting an arbitrary number of barriers and interjacent valleys. As a prelude to the general case, we separately treat the cases of a single barrier and of two barriers enclosing a valley. Another example considered here is that of field- driven transport in homogeneous semiconductors, for which we evaluate the position dependence of the chemical potential. 3.3.1 Single Potential Energy Barrier To begin with, we consider the case of a single barrier in the profile Ec(x), with its maximum located at some position X → [0, S] (see Fig. 3.1). Then, we have Emc (0, S) = Ec(X). If, in particular, X = 0 or X = S, the profile is monotonic. In the calculation of the corresponding reduced resistance R from expression (3.35), we must distinguish three cases. If the ballistic interval [x↑, x↑↑] contains X, we have Emc (x ↑, x↑↑) = Ec(X); if it lies to the left or right of X, we have Emc (x↑, x↑↑) = Ec(x↑↑) or Emc (x↑, x↑↑) = Ec(x↑), respectively. We then find R = 1+ S̃ Δ , (3.65) where S̃ is the effective sample length, S̃ = S∫ 0 dxe−β[Ec(X)−Ec(x)]. (3.66) This expression is formally equal to the effective sample length (2.37) introduced in the context of the drift-diffusionmodel, but it differs in physical origin.While expres- sion (3.66) results from averaging over ballistic configurations involving intervals http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_2 40 3 Prototype Thermoballistic Model of arbitrary, finite length, expression (2.37) arises from the assumption of arbitrarily short ballistic intervals, on which the drift-diffusion model is implicitly based. 3.3.1.1 Current-Voltage Characteristic In expression (3.65) for R, the unit term corresponds to ballistic transmission all across the sample, governed by the barrier maximum at X, while the effective sample length S̃ represents the effect of the potential energy profile in an integral way. The (classical) current-voltage characteristic (3.25) for the case of a single potential energy barrier now reads J = veNce−β[Ec(X)−μ0] Δ Δ+ S̃ (1− e −βeV ). (3.67) If the effectivemean free path ismuch longer than the effective sample length, Δ S̃, the characteristic (3.67) reduces to that of the ballistic transport model, Eq. (2.67). In the opposite case, Δ ∼ S̃, using the relation γ = 2βevel = βeveΔ (3.68) for the electron mobility γ [see Eqs. (2.26) and (2.54)], the characteristic of the drift-diffusion model, Eq. (2.34), is retrieved. Expression (3.67) exemplifies the uni- fication of the ballistic and drift-diffusion transport mechanisms in the prototype thermoballistic model. We note that a heuristic attempt to unify drift-diffusion and ballistic transport has been made, for the special case of transport across a Schottky barrier, by Crowell and Sze [5] (see also Ref. [6]), who assumed ballistic transport to prevail in the vicinity of the barrier maximum, and drift-diffusion transport elsewhere. They obtained an expression for the current-voltage characteristic equivalent to (3.67), J = Nce−β[Ec(X)−μ0] vevdi ve + vdi (1− e −βeV ), (3.69) where vdi = γ βeS̃ = veΔ S̃ (3.70) is an effective diffusion velocity, with the electron mobility γ given by Eq. (3.68). A picture similar to that of Ref. [5] has been developed and applied by Evans and Nelson [7] in a study of transport across a single grain boundary barrier. Other studies shedding light upon the transition region between drift-diffusion and ballistic trans- port were presented byWexler [8], de Jong [9], Bauer et al. [10], and Prins et al. [11]. http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 3.3 Prototype Thermoballistic Model: Examples 41 3.3.1.2 Effect of Tunneling Considering now, for the single barrier, the effect of electron tunneling, we obtain, following Sect. 3.2.3, Rsb = e−S/ΔR̄sb(0, S)+ S∫ X dx↑↑ Δ e−x↑↑/ΔR̄sb(0, x↑↑) + X∫ 0 dx↑ Δ e−(S−x↑)/ΔR̄sb(x↑, S) + X∫ 0 dx↑ Δ S∫ X dx↑↑ Δ e−(x↑↑−x↑)/ΔR̄sb(x↑, x↑↑) (3.71) for the sub-barrier partRsb of the reduced resistanceRq, while the classical partRcl is given by Eq. (3.65). As |R̄sb(x↑, x↑↑)| < 1, we have from Eq. (3.71) |Rsb| < 1. (3.72) Since Emc (x ↑, x↑↑) ∗ Ec(X), the transmission probabilities T̄(x↑, x↑↑) and T̄sb(x↑, x↑↑), and consequently the function R̄sb(x↑, x↑↑), depend on x↑ and x↑↑ only via the func- tion E>c (x ↑, x↑↑) [see Eqs. (2.79) and (2.81)]. Then, R̄sb(x↑, x↑↑) is a function of one coordinate only, determined by the potential energy profile Ec(x), R̄sb(x↑, x↑↑) ∗ { R̄sb(x↑) if E>c (x↑, x↑↑) = Ec(x↑), R̄sb(x↑↑) if E>c (x↑, x↑↑) = Ec(x↑↑). (3.73) As x↑↑ increases from X to S, for fixed x↑ → [0,X], R̄sb(x↑↑) changes to R̄sb(x↑) when x↑↑ = x↑≥ or, equivalently, x↑ = x↑↑≥; here, the “mirror point” x≥ is defined as that position to the right (left) of a maximum or minimum of the potential energy profile where it has the same value as at the point x to the left (right), i.e., Ec(x ≥) = Ec(x) (3.74) (see Fig. 3.1). The double integral in expression (3.71) then reduces to two single integrals which, together with the single integrals preceding it, combine to just one single integral, so that Rsb attains the simple form Rsb ∗ Rsb(Δ) = e−0≥/ΔR̄sb(0)+ 0≥∫ 0 dx Δ e−|x−x≥|/ΔR̄sb(x). (3.75) This expression bears a close formal similarity to the shape term appearing in the classical treatment of the double-barrier case [see Eq. (3.77) below]. http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 42 3 Prototype Thermoballistic Model 3.3.2 Double Barrier Next, we consider a potential energy profile of the type shown in Fig. 3.2, exhibiting two barriers with maxima at x = X0 and x = X1, respectively (without loss of generality, the maximum at X0 is assumed to be the higher one), and a valley with minimum at x = Y1 in between. In the evaluation of the reduced resistance R, a number of cases are to be dis- tinguished when calculating the function Emc (x ↑, x↑↑) in dependence on the location of the ballistic interval [x↑, x↑↑] with respect to the different ranges delimited by the points x = 0, X0, X≥1 , Y1, X1, and S. From expression (3.35), we obtain R = 1+ S̃ + Λ̃ Δ , (3.76) where the effective sample length S̃ is given by Eq. (3.66) with X = X0. The “shape term” Λ̃ has the form Λ̃ ∗ Λ̃(Δ) = X1∫ X≥1 dxe−|x−x≥|/Δe−βEc(X0)[eβEc(X1) − eβEc(x)]. (3.77) In writing Λ̃ in this compact form, we have used the identity 1− e−w1/Δ ∗ X1∫ X≥1 dx Δ e−|x−x≥|/Δ, (3.78) where w1 = X1 − X≥1 (3.79) is the width of the valley (see Fig. 3.2). The position of the minimum of the valley, Y1, enters inR implicitly via the shape of Ec(x) in the interval [X≥1 ,X1]. Comparing in Eq. (3.77) the magnitude of the integral involving the term eβEc(X1) in the brackets to that of the integral involving eβEc(x), it is seen that for b0,1 ∼ l ∼ w1, where b0,1 are the widths of the two barriers centered about X0,1, the second integral can be neglected. Hence, R = 1+ S̃ Δ + e−β[Ec(X0)−Ec(X1)], (3.80) i.e., the two barriers contribute independently to the reduced resistance (first and third term). 3.3 Prototype Thermoballistic Model: Examples 43 w1E (x) C x x*Y X S0 0 1X 1 1X* Fig. 3.2 Schematic diagram of a potential energy profileEc(x) exhibiting two barriers withmaxima at x = X0 and x = X1, respectively, which enclose a valley with minimum at x = Y1 and width w1. For the definition of the mirror point x≥ associated with a point x, see Eq. (3.74) When Δ S (ballistic limit), we have Λ̃ < w1e −β[Ec(X0)−Ec(X1)] < S, (3.81) so that both the terms S̃/Δ and Λ̃/Δ in Eq. (3.76) can be neglected. Then,R reduces to the unit term, which reflects ballistic transmission across the higher barrier maximum at X0; this maximum “eclipses” the lower maximum at X1. 3.3.3 Arbitrary Number of Barriers In the foregoing cases of a single barrier and of a valley in-between two barriers, we have obtained explicit expressions for the reduced resistance R in terms of the potential energy profileEc(x) and the effectivemean free path Δ. If the profile contains an arbitrary combination of barriers and valleys, R must be evaluated, in general, for each case anew, starting from the basic formula (3.35). However, an explicit expression for R can still be obtained in the special case where the height of the barriers monotonically decreases or increases along the sample length. 3.3.3.1 General Shape Term Assuming, without loss of generality, the barrier height to decrease monotonically when x increases from 0 to S (including the case of equal height of all barriers), we consider M + 1 barriers, with maxima at x = Xv (v = 0, 1, . . . ,M), enclosing M valleys, with minima at x = Yv (v = 1, . . . ,M) and widths wv (see Fig. 3.3). Then, using Eq. (3.35) and proceeding as in the case of a single valley, we obtainR in the general form given by Eq. (3.76), with S̃ again given by Eq. (3.66) with X = X0, where X0 is now the position of the maximum of the highest (left-most) barrier of the M + 1 barriers considered. The shape term Λ̃ entering expression (3.76), 44 3 Prototype Thermoballistic Model E (x)C wV 0 X Y X SX*0 V V V Fig. 3.3 Schematic diagram of a potential energy profile Ec(x) exhibiting M + 1 barriers with maxima at x = Xv (v = 0, 1, . . . ,M), and heights decreasing monotonically with increasing x. The valleys enclosed by two adjacent barriers have minima located at x = Yv (v = 1, . . . ,M) and widths wv Λ̃ ∗ Λ̃(Δ) = M∑ v=1 Xv∫ X≥v dxe−|x−x≥|/Δe−βEc(X0)[eβEc(Xv) − eβEc(x)], (3.82) generalizes expression (3.77) to the case of M valleys; it appears as a sum over separate contributions from the different valleys. The contribution of valley v, with the maximum of the adjoining lower barrier located at Xv , consists of an integral which extends over the width of the valley from X≥v to Xv . When the barrier heights in the potential energy profile do not behave monoton- ically, inspection of formula (3.35) shows that the reduced resistance R can always be expressed as a sum of a term formally identical to expression (3.76) [with Ec(X0) in S̃ and Λ̃ replaced with Ec(Xm), where Xm is the position of the maximum of the highest barrier] plus additional terms arising from the combined effect of barriers and valleys on the electron transport in the ballistic intervals. 3.3.3.2 Current-Voltage Characteristic Introducing now the effective transport length L as L ∗ RΔ, (3.83) we have from Eqs. (3.39), (3.40), and (3.76) L = Δ+ S̃ + Λ̃(Δ) ≤ Δ+ S. (3.84) From Eqs. (3.25) and (3.83), we then obtain the current-voltage characteristic for a potential energy profile of the type depicted in Fig. 3.3 in the form 3.3 Prototype Thermoballistic Model: Examples 45 J = veNce−β[Ec(X0)−μ0] ΔL (1− e −βeV ). (3.85) This expression is the principal result of the prototype thermoballistic model. It has been derived here in the framework of classical transport in nondegenerate systems, where its interpretation is most transparent. The properties of the characteristic (3.85) are determined by the barrier factor e−β[Ec(X0)−μ0] and by the ratio Δ/L. In the effective transport length L, the effective mean free path Δ represents the ballistic contribution to the current, which is associ- ated with the maximum Ec(X0) of the highest barrier in the potential energy profile. The remaining terms give a quantitative measure of the influence of that part of the electron motion which is not purely ballistic. Their contribution amounts to at most the sample length S. The effective sample length S̃ given by Eq. (3.66) with X = X0 represents a contribution that characterizes the potential energy profile Ec(x) in an integral way; it does not manifestly depend on Δ, only implicitly so via the profile (an indirect relationship between profile and mean free path arises from their common dependence on the donor density). The shape term Λ̃ given by Eq. (3.82), on the other hand, depends on the detailed structure of the profile as well as explicitly on the mean free path, and thus represents the interplay of ballistic and drift-diffusion transport. This term is a distinctive feature of expression (3.85), and therefore of the prototype thermoballistic model. 3.3.3.3 Effects of Tunneling and Degeneracy Tunneling effects in the current-voltage characteristic (3.85) canbe taken into account by generalizing the effective transport lengthL of Eq. (3.83) so as to include the sub- barrier contribution Rsb to the reduced resistance. From Eq. (3.46), we have Lq ∗ RqΔ = (Rcl +Rsb)Δ = (1− |Rsb(Δ)|)Δ+ S̃ + Λ̃(Δ). (3.86) Here, as exemplified by expression (3.75) for the single-barrier case, Rsb depends explicitly on the effective mean free path Δ. From expression (3.86), the inclusion of tunneling is seen to lower the ballistic contribution to the effective transport length [see Eq. (3.83)]. This behavior is in line with the role of Rq as a resistance, which ought to decrease when the barriers become increasingly “transparent” to ballistic electron transport. The (classical) reduced resistance in the degenerate case,Rd [see Eqs. (3.50) and (3.52)], is expressed, in analogy to Eqs. (3.83) and (3.84) for the nondegenerate case, in terms of a (classical) effective transport length Ld given by Ld ∗ RdΔ = Δ+ S̃d + Λ̃d(Δ), (3.87) where, in generalization of Eq. (3.66), 46 3 Prototype Thermoballistic Model S̃d = S∫ 0 dx ln(1+ e−β[Ec(X0)−μ0]) ln(1+ e−β[Ec(x)−μ0]) (3.88) is the effective sample length for the degenerate case, and Λ̃d ∗ Λ̃d(l) = M∑ v=1 Xv∫ X≥v dxe−|x−x≥|/Δ × { ln(1+ e−β[Ec(X0)−μ0]) ln(1+ e−β[Ec(Xv)−μ0]) − ln(1+ e−β[Ec(X0)−μ0]) ln(1+ e−β[Ec(x)−μ0]) } (3.89) generalizes the shape term (3.82). Using Eq. (3.87) in expression (3.53), we now have g = βe2veNc ΔLd ln(1+ e −β(Emc −μ0)) (3.90) for the conductance per unit area in the prototype thermoballistic model. The inclu- sion of tunneling effects is, in the degenerate case, a highly intricate task and will not be considered here. 3.3.3.4 Applications Previously, we have applied [1, 12–14] the prototype model in calculations of trans- port properties of poly- and microcrystalline semiconducting materials, in particular, ofmaterials relevant to photovoltaics. The occurrence of grain boundaries in this kind of materials gives rise to a multi-barrier structure of the band edge profile. Adopt- ing the trapping model [15–18] to describe the grain boundaries, the corresponding nonlinear Poisson equation has been solved [12] to obtain potential energy profiles for chains of grains. Then, taking into account tunneling corrections and using a phenomenological relation [19] to express the mean free path l in terms of the donor density, conductivities and electron mobilities have been calculated as a function of l and of the number and lengths of the grains. It turns out that neither the ballistic (thermionic) model nor the drift-diffusionmodel can provide an adequate description of electron transport in poly- and microcrystalline materials. On the other hand, the application of the prototype model in the analysis of experimental data [20–25] has led to promising results. 3.3 Prototype Thermoballistic Model: Examples 47 3.3.4 Chemical Potential for Field-Driven Transport The chemical potential μ(x) given by Eq. (3.61) can be expressed in closed form for the case of field-driven electron transport in a homogeneous semiconducting sample. The corresponding potential energy profile reads Ec(x) = Ec(0)− λ β x (3.91) see Eq. (2.41), with the field parameter λ defined by Eq. (2.42). Using this profile in the properly transcribed expression (3.35) [or, alternatively, in Eq. (3.65), considering the profile (3.91) a particular case of a barrier] and using the mean free path l [instead of the effective mean free path Δ; see Eq. (3.28)], we obtain the resistance functions R1,2(x) in the form R1(x) = 1+ 1 2λl (1− e−λx) (3.92) and R2(x) = 1+ 1 2λl [1− e−λ(S−x)] ∗ R1(S − x). (3.93) From Eq. (3.58), we then have R = R1(S) = 1+ 1 2λl (1− e−λS), (3.94) and from Eq. (3.62), R−(x) = R1(x)− e−λxR2(x) = 1− e−λx + 1 2λl (1− 2e−λx + e−λS). (3.95) Here, both R and R−(x) depend on the voltage V via their dependence on the parameter λ [see Eq. (2.44)]. Inserting expressions (3.94) and (3.95) in Eq. (3.61) gives μ(x) in closed form. In the drift-diffusion regime, l/S ∼ 1, μ(x) is readily found to agree with the chemical potential of the standard drift-diffusion model [see Eq. (2.46)]. In the low-field regime characterized by condition (2.88), we have to lowest order R = 1+ S 2l (3.96) and R−(x) = 2x − S 2l (3.97) http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 48 3 Prototype Thermoballistic Model i.e., the voltage dependence drops out. Since β|μS − μ0| ∼ 1 (3.98) and β|μ(x)− μ0| ∼ 1 (3.99) in the low-field regime, we obtain from Eq. (3.61), using Eq. (3.26), μ(x) in the form μ(x) = 1 2 (μ0 + μS)− eV x − S/2 2l + S , (3.100) which generalizes, for low fields, the drift-diffusion expression (2.46) to arbitrary values of the mean free path l. References 1. R. Lipperheide, T. Weis, U. Wille, J. Phys.: Condens. Matter 13, 3347 (2001) 2. B. Sapoval, C. Hermann, Physics of Semiconductors (Springer, Berlin, 1995) 3. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I (Addison- Wesley, Reading, Mass., 1964), Chap. 43 4. C. Jacoboni, Theory of Electron Transport in Semiconductors. Springer Series in Solid-State Sciences, vol. 165 (Springer, Berlin, 2010) 5. C.R. Crowell, S.M. Sze, Solid-State Electron. 9, 1035 (1966) 6. S.M. Sze, Physics of Semiconductor Devices (Wiley, New York, 1981) 7. P.V. Evans, S.F. Nelson, J. Appl. Phys. 69, 3605 (1991) 8. G. Wexler, Proc. Phys. Soc. (London) 89, 927 (1966) 9. M.J.M. de Jong, Phys. Rev. B 49, 7778 (1994) 10. G.E.W. Bauer, A. Brataas, K.M. Schep, P.J. Kelly, J. Appl. Phys. 75, 6704 (1994) 11. M.W.J. Prins, K.-O. Grosse-Holz, J.F.M. Cillessen, L.F. Feiner, J. Appl. Phys. 83, 888 (1998) 12. T. Weis, Ph. D. thesis, Free University Berlin, 1999 13. R. Lipperheide, T. Weis, U. Wille, Sol. Energy Mater. Sol. Cells 65, 157 (2001) 14. T. Weis, R. Lipperheide, U. Wille, S. Brehme, J. Appl. Phys. 92, 1411 (2002) 15. T.I. Kamins, J. Appl. Phys. 42, 4357 (1971) 16. J.Y.W. Seto, J. Appl. Phys. 46, 5247 (1975) 17. T. Weis, R. Lipperheide, and U. Wille, in Proceedings of the 2nd World Conference and Exhi- bition on Photovoltaic Solar Energy Conversion, ed. by J. Schmid, H. Ossenbrink, P. Helm, H. Ehmann, E. D. Dunlop (European Commission Joint Research Centre, Ispra, 1998), p. 1438 18. T. Weis, S. Brehme, P. Kanschat, W. Fuhs, R. Lipperheide, U. Wille, J. Non-Cryst. Solids 299–302, 380 (2002) 19. N.D. Arora, J.R. Hauser, D.J. Roulston, IEEE Trans. Electron. Dev. 29, 292 (1982) 20. X.Y. Chen, W.Z. Shen, Phys. Rev. B 72, 035309 (2005) 21. G. Mugnaini, G. Iannaccone, IEEE Trans. Electron. Dev. 52, 1795 (2005) 22. X.Y. Chen, W.Z. Shen, H. Chen, R. Chang, Y.L. He, Nanotechnology 17, 595 (2006) 23. A. Oprea, N. Barsan, U. Weimar, J. Phys. D: Appl. Phys. 40, 7217 (2007) 24. A. Bikowski, K. Ellmer, J. Mater. Res. 27, 2249 (2012) 25. A. Bikowski, K. Ellmer, J. Appl. Phys. 114, 063709 (2013) 26. G. Rey, C. Ternon, M. Modreanu, X. Mescot, V. Consonni, D. Bellet, J. Appl. Phys. 114, 183713 (2013) http://dx.doi.org/10.1007/978-3-319-05924-2_2 Chapter 4 Thermoballistic Approach: Concept The prototype thermoballistic model developed in the preceding chapter has been based on the random partitioning of the length of a semiconducting sample into ballistic transport intervals linked by points of local thermodynamic equilibrium, which make up a ballistic configuration. Electrons thermally emitted at either end- point of a ballistic interval, and subsequently transmitted across the potential energy profile in the sample, form the net ballistic electron current in the interval. This current is conserved across an individual ballistic interval. A distinctive assumption of the prototype model has been that the current is conserved also across the points of local thermodynamic equilibrium linking the ballistic intervals, and equals the physical current. By averaging over all ballistic configurations, the current-voltage characteristic can then be expressed, without requiring knowledge of the equilibrium chemical potential inside the sample, essentially in terms of a reduced resistance that comprises the effect of the sample parameters. The position dependence of the chemical potential has been constructed in a heuristic way only. While adhering to the idea of introducing ballistic configurations and averaging thereover, the thermoballistic concept proper refines the prototype model in that it abandons the assumption of current conservation across the points of local ther- modynamic equilibrium. [A simple example in contradiction to this assumption is provided by the case of the ballistic currents in a homogeneous semiconductor at low external electric field considered in Sect. 6.1.1.2 below.] Position-dependent total and spin-polarized thermoballistic currents as well as the associated densities are defined in terms of an average chemical potential and a spin accumulation function related to the splitting of the spin-resolved chemical potentials. By imposing appropriately cho- sen physical conditions on these dynamical functions, procedures for their explicit determination can be devised. R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 49 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_4, © Springer International Publishing Switzerland 2014 http://dx.doi.org/10.1007/978-3-319-05924-2_6 50 4 Thermoballistic Approach: Concept We have developed the thermoballistic approach to stationary semiclassical1 carrier transport in a series of papers. In Ref. [2], the concept of a thermoballistic current was introduced and implemented without regard to spin degrees of free- dom. The extension of this concept to spin-polarized electron transport across a spin-degenerate potential energy profile was presented in Ref. [3], in which spin injection out of ferromagnetic contacts into a nonmagnetic semiconducting sample was treated in detail. In Ref. [4], we have generalized the thermoballistic concept to the case of arbitrary spin splitting of the profile, thereby covering, in particular, spin- polarized transport in diluted magnetic semiconductors in their paramagnetic phase. In the present chapter, we formulate the thermoballistic concept within the frame set by Ref. [4]. Classical transport in nondegenerate systems is considered through- out. Effects of electron tunneling and degeneracy can be included, in principle, by resorting to the corresponding developments in Sects. 2.2 and 3.2. For comprehensive surveys of the fundamentals of spin physics in semiconductors and their application in the field of spintronics, see Refs. [5–8]. 4.1 Electron Densities at Local Thermodynamic Equilibrium In Sect. 3.2, we have introduced ballistic transport intervals with end-point coordinates x Ni−1, x N i characterized by discrete labels N , i . When the ballistic con- figurations made up of these intervals are averaged over, the description in terms of discrete coordinates turns into one in terms of the continuous coordinates x ∗, x ∗∗ in expressions (3.35) and (3.42) for the reduced resistanceR. Transferring this feature into the formulation of the thermoballistic concept, we work here with ballistic inter- vals [x ∗, x ∗∗] with continuous end-point coordinates x ∗ and x ∗∗ representing points of local thermodynamic equilibrium (see also Sect. 3.1). For notational convenience, we will henceforth label the end-points of a semiconducting sample by x1 and x2, respectively, so that we have x1 ≡ x ∗ < x ∗∗ ≡ x2, (4.1) and the sample length S is given by S = x2 − x1. In Fig. 4.1, a paramagnetic semiconducting sample enclosed between two metal contacts is depicted in a schematic diagram. Various physical quantities appearing in the thermoballistic description are indicated. 1 We use the term “semiclassical” essentially as defined in Chaps. 12 and 29 of Ref. [1], where the band structure of the unperturbed system is assumed to be generated from quantum-mechanical calculations, whereas the effect of a static, externally applied electric field is treated classically (the effect of a static external magnetic field is assumed here to be included in the band edge potential). We go beyond this scope when considering corrections to the ballistic currents due to electron tunneling and degeneracy. http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 4.1 Electron Densities at Local Thermodynamic Equilibrium 51 x x x x x1 paramagnetic semiconductormetal contact metal contact µ , A µ , A1 1 2 2 S E (x)↑ E (x)↓ E (x) C Δ (x) | || 2 µ(x )~ | A(x ) || J (x , x )+ | || µ (x )~ || A(x ) | , , J (x , x , x ) v - | || } Fig. 4.1 Schematic diagram showing, in a one-dimensional representation, a paramagnetic semi- conducting sample of length S enclosed between two metal contacts. The spin splitting Δ(x) of the electrostatic potential energy profile Ec(x) gives rise to the spin-dependent potential energy profiles E↑,≈(x). The coordinates x ∗ and x ∗∗ denote points of local thermodynamic equilibrium which delimit a ballistic transport interval [x ∗, x ∗∗]. Electrons thermally emitted at x ∗(x ∗∗) toward the right (left) move ballistically across the profiles E↑,≈(x) to reach the end-point x ∗∗(x ∗) of the ballistic interval, where they are absorbed (equilibrated). Two net electron currents are indicated: The (conserved) net total ballistic current J+(x ∗, x ∗∗) determined by the values μ̃(x ∗) and μ̃(x ∗∗) of the average chemical potential μ̃(x) and the values A(x ∗) and A(x ∗∗) of the spin accumulation function A(x) according to Eq. (4.67) [with the mean spin function Ã(x) expressed in terms of μ̃(x) via Eq. (4.14)], and the net relaxing ballistic spin-polarized current J̌−(x ∗, x ∗∗; x) determined by A(x ∗) and A(x ∗∗) according to Eq. (4.70). The quantities μ1,2 and A1,2 are the values of the equilibrium chemical potential and the spin accumulation function, respectively, at the contact side of the contact-semiconductor interfaces [see Eqs. (4.97) and (4.99)] 4.1.1 Spin-Resolved Densities We write the spin-dependent potential energy profiles E↑,≈(x ∗) corresponding to spin-up (↑) and spin-down (≈) conduction band electrons, respectively, in the form E↑,≈(x ∗) = Ec(x ∗)± 1 2 Δ(x ∗), (4.2) where the spin-independent part Ec(x ∗) is assumed to comprise the conduction band edge potential and the external electrostatic potential, andΔ(x ∗) is the spin splitting 52 4 Thermoballistic Approach: Concept of the conduction band [here, we assume both Ec(x ∗) and Δ(x ∗) to be prescribed functions]. Having in mind electron transport in diluted magnetic semiconductors in their paramagnetic phase, we identify this splitting with the (giant) Zeeman splitting due to an external magnetic field [9–14] [we restrict ourselves to considering a single Landau level whose energy is assumed to be included in Ec(x ∗)]. In presenting the general formalism, we assume both Ec(x ∗) and Δ(x ∗), and hence E↑,≈(x ∗), to be continuous functions of x ∗ in the interval [x1, x2]. The case of abrupt changes in one or the other of these functions, which occur at the interfaces in heterostructures, may be described, in a simplified picture, in terms of discontinuous functions (see Sect. 6.3 below). The (static) local spin polarization of the conduction band electrons, P(x ∗), is determined by the Boltzmann factors e−πE↑,≈(x ∗), P(x ∗) = B−(x ∗) B+(x ∗) , (4.3) where B±(x ∗) = e−πE↑(x ∗) ± e−πE≈(x ∗), (4.4) so that we have P(x ∗) = − tanh(πΔ(x ∗)/2). (4.5) We introduce the function Q(x ∗) defined by Q(x ∗) ∞ { 1− [P(x ∗)]2 }1/2 = 1 cosh(πΔ(x ∗)/2) , (4.6) which will be frequently used below. For the spin-resolved equilibrium electron densities n↑,≈(x ∗), we have, in gener- alization of expression (2.20), n↑,≈(x ∗) = Nc 2 e−π[E↑,≈(x ∗)−μ↑,≈(x ∗)]. (4.7) Here, μ↑,≈(x ∗) are the spin-resolved chemical potentials associated with the local thermodynamic equilibrium at x ∗, and Nc/2, with Nc given by Eq. (2.21), is the effective density of states of either spin at the conduction band edge (for simplicity, the effective electron mass m∗ entering Nc is assumed here to be independent of position and of the external magnetic field). The spin-resolved chemical potentials μ↑(x ∗) and μ≈(x ∗) are independent dynam- ical functions whose position dependence is to be determined within the thermobal- listic approach and from which, subsequently, all transport properties are to be derived. However, in implementing the thermoballistic approach, we will work not with μ↑(x ∗) and μ≈(x ∗), but with suitably defined combinations of these functions: (i) an “average chemical potential”, and (ii) a “spin accumulation function” related to the splitting of the potentials μ↑(x ∗) and μ≈(x ∗). http://dx.doi.org/10.1007/978-3-319-05924-2_6 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 4.1 Electron Densities at Local Thermodynamic Equilibrium 53 4.1.2 Average Chemical Potential and Spin Accumulation Function We define the “spin functions” A↑,≈(x ∗) as A↑,≈(x ∗) = eπμ↑,≈(x ∗), (4.8) and the “mean spin function” Ã(x) as Ã(x ∗) = 1 2 A+(x ∗), (4.9) where A+(x ∗) ∞ A↑(x ∗)+ A≈(x ∗). (4.10) We can express Ã(x ∗) in the form Ã(x ∗) = eπμ̄(x ∗) cosh(πμ−(x ∗)/2), (4.11) where μ̄(x ∗) = 1 2 [μ↑(x ∗)+ μ≈(x ∗)] (4.12) is the mean value of μ↑(x ∗) and μ≈(x ∗) [mean chemical potential], and μ−(x ∗) ∞ μ↑(x ∗)− μ≈(x ∗) (4.13) their splitting [In the following, the “±” notation introduced in Eqs. (4.10) and (4.13), respectively, will be frequently used mutatis mutandis to denote spin-summed and spin-polarized quantities.] Further, writing Ã(x ∗) = eπμ̃(x ∗), (4.14) we introduce the “average chemical potential” μ̃(x ∗). This function is not to be con- fused with the dynamical function μ̃(x ∗) introduced in Ref. [4], which is defined as the common spin-relaxed chemical potential, i.e., as the chemical potential charac- terizing the spin equilibrium. The “spin accumulation function” A−(x ∗) is defined as A−(x ∗) ∞ A↑(x ∗)− A≈(x ∗) = 2eπμ̄(x ∗) sinh(πμ−(x ∗)/2). (4.15) 54 4 Thermoballistic Approach: Concept This function agrees with the “spin transport function” A(x ∗) of Ref. [4], but differs by a factor of two from the function with the same name introduced in Ref. [3]. In the limit |πμ−(x ∗)| ≥ 1, the function A−(x) becomes proportional to the splitting μ−(x ∗), which is the dynamical quantity used in the drift-diffusion approach to spin- polarized transport, andwhich is commonly [6] referred to as the “spin accumulation” there. Using Eq. (4.11), we can rewrite A−(x ∗) in the form A−(x ∗) = 2 Ã(x ∗) tanh(πμ−(x ∗)/2). (4.16) Introducing the “reduced” spin accumulation function Ă(x ∗) as Ă(x ∗) ∞ A−(x ∗) Ã(x ∗) = 2 tanh(πμ−(x ∗)/2), (4.17) we then obtain, using Eqs. (4.14)–(4.16), the relations μ̃(x ∗) = μ̄(x ∗)+ 1 π ln(cosh(πμ−(x ∗)/2)) = μ̄(x ∗)− 1 2π ln(1− Ă2(x ∗)/4), (4.18) which express the difference between average and mean chemical potential in terms of the splitting μ−(x ∗) and of Ă(x ∗), respectively. The spin functions A↑,≈(x ∗) are proportional to the spin-resolved electron den- sities n↑,≈(x ∗) [see Eq. (4.7)], so that their use results in a formulation in terms of linear equations, instead of the nonlinear description ensuing from using the spin- resolved chemical potentials μ↑,≈(x ∗) themselves. [This aspect has been emphasized previously [15, 16] within a study, based on the standard drift-diffusion approach, of electric-field effects on spin-polarized transport in nondegenerate semiconductors.] From Eq. (4.7), the total (i.e., spin-summed) equilibrium density, n+(x ∗), and the spin-polarized equilibriumdensity, n−(x ∗), are nowobtained in terms of the functions A±(x ∗) as n±(x ∗) = Nc 4 B+(x ∗)[A±(x ∗)+ P(x ∗)A∓(x ∗)], (4.19) where Eqs. (4.3) and (4.4) have been used. It should be noted that in Ref. [4], we have required n+(x ∗) to be unaffected by spin relaxation, i.e., to equal the total density at spin equilibrium (see Eq. (13) of Ref. [4]). This density is expressed in terms of the “common spin-relaxed chemical potential”, which represents one of the dynamical functions in the formalism of Ref. [4]. Here, where we work with the average chemical potential instead, we do not impose such a kind of requirement. 4.2 Ballistic Spin-Polarized Transport 55 4.2 Ballistic Spin-Polarized Transport In this section, the spin-resolved electron currents transmitted across a ballistic transport interval as well as the associated densities (called “ballistic currents” and “ballistic densities”) are constructed by closely following the development in the bal- listic (see Sect. 2.2) and the prototype thermoballistic (see Sect. 3.2.1) transport mod- els, in which spin degrees of freedom were disregarded. Introducing spin relaxation during the ballistic electron motion, we obtain the ballistic spin-polarized currents and densities, whose dynamics are determined by a balance equation. 4.2.1 Ballistic Currents The left end-point, x ∗, as well as the right end-point, x ∗∗, of the ballistic interval [x ∗, x ∗∗] are points of local thermodynamic equilibrium but are, in general, not points of spin equilibrium. 4.2.1.1 Disregarding Spin Relaxation We first assume that the electrons thermally emitted at x ∗ towards the right are not affected by spin relaxation during their motion across the interval [x ∗, x ∗∗]. The spin- resolved equilibrium densities n↑,≈(x ∗) [see Eq. (4.7)] then give rise to conserved ballistic spin-resolved currents in that interval (i.e., currents that are independent of the position x √ [x ∗, x ∗∗]), which have the form J l↑,≈(x ∗, x ∗∗) = ven↑,≈(x ∗)T̄ l↑,≈(x ∗, x ∗∗) (4.20) (see Sect. 2.2.1), where the thermally averaged (classical) transmission probabilities T̄ l↑,≈(x ∗, x ∗∗) are given [see Eqs. (2.68) and (2.69)] by T̄ l↑,≈(x ∗, x ∗∗) = e−π[E m↑,≈(x ∗,x ∗∗)−E↑,≈(x ∗)], (4.21) with Em↑,≈(x ∗, x ∗∗) the overall maxima of the potential energy profiles E↑,≈(x) in the interval [x ∗, x ∗∗]. For later use (see Sects. 4.2.4, 5.2.1, 5.3.1, and A.1 below), we introduce here the symmetric extensions of the functions Em↑,≈(x ∗, x ∗∗), which are originally defined for x ∗ < x ∗∗ only, into the range x ∗ > x ∗∗ by defining Em↑,≈(x ∗∗, x ∗) ∞ Em↑,≈(x ∗, x ∗∗). (4.22) Corresponding extensions then hold for all functions of x ∗ and x ∗∗ that are defined solely in terms of Em↑,≈(x ∗, x ∗∗). http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 56 4 Thermoballistic Approach: Concept Using the spin functions A↑,≈(x ∗) given by Eqs. (4.8), we can now rewrite the currents J l↑,≈(x ∗, x ∗∗) in the form J l↑,≈(x ∗, x ∗∗) = ve Nc 2 e−πE m↑,≈(x ∗,x ∗∗)A↑,≈(x ∗). (4.23) We then have for the total (i.e., spin-summed) ballistic current, J l+(x ∗, x ∗∗), and the ballistic spin-polarized current, J l−(x ∗, x ∗∗), J l±(x ∗, x ∗∗) = ve Nc 4 Bm+ (x ∗, x ∗∗)[A±(x ∗)+ Pm(x ∗, x ∗∗)A∓(x ∗)], (4.24) where Bm± (x ∗, x ∗∗) = e−πE m↑ (x ∗,x ∗∗) ± e−πEm≈ (x ∗,x ∗∗) (4.25) and Pm(x ∗, x ∗∗) = B m− (x ∗, x ∗∗) Bm+ (x ∗, x ∗∗) , (4.26) which are nonlocal extensions of expressions (4.4) and (4.3) for the functions B±(x ∗) and the local polarization P(x ∗), respectively. 4.2.1.2 Introducing Spin Relaxation We now consider the effect of spin relaxation on the ballistic currents inside the interval [x ∗, x ∗∗]. In this case, the spin-resolved currents will acquire a dependence on the position x in the interval [x ∗, x ∗∗]. We take into account this dependence by extending the functions A±(x ∗) in Eq. (4.24) to functions Al±(x ∗, x ∗∗; x) that depend on x as well as on x ∗ and x ∗∗, and satisfy the conditions Al±(x ∗, x ∗∗; x = x ∗) = A±(x ∗) (4.27) [the explicit form of Al±(x ∗, x ∗∗; x) will be determined in Sect. 4.2.3 below]. Using these functions, we generalize the definitions of the currents J l±(x ∗, x ∗∗) given by Eq. (4.24) to J l±(x ∗, x ∗∗; x) = ve Nc 4 Bm+ (x ∗, x ∗∗)[Al±(x ∗, x ∗∗; x)+ Pm(x ∗, x ∗∗)Al∓(x ∗, x ∗∗; x)]. (4.28) Now, also in the presence of spin relaxation, the total ballistic current must be con- served, J l+(x ∗, x ∗∗; x) = J l+(x ∗, x ∗∗; x = x ∗) ∞ J l+(x ∗, x ∗∗), (4.29) 4.2 Ballistic Spin-Polarized Transport 57 where, from Eq. (4.24), J l+(x ∗, x ∗∗) = ve Nc 4 Bm+ (x ∗, x ∗∗)[A+(x ∗)+ Pm(x ∗, x ∗∗)A−(x ∗)]. (4.30) We note that in Ref. [4], we have required J l+(x ∗, x ∗∗) to be equal to the corresponding spin-relaxed current (see Eqs. (33), (34), and (43) of Ref. [4]). Here, proceeding similarly as for total equilibrium density n+(x) [see the remark following Eq. (4.19)], we do not impose such a kind of requirement. As a consequence, the total ballistic current becomes, in general, dependent on the spin accumulation function A(x) [see Eq. (4.67) below]. Equating expression (4.30) to expression (4.28) for J l+(x ∗, x ∗∗; x), we obtain the relation Al+(x ∗, x ∗∗; x) = A+(x ∗)+ Pm(x ∗, x ∗∗)[A−(x ∗)− Al−(x ∗, x ∗∗; x)]. (4.31) Therefore, the spin-polarized current can be written as J l−(x ∗, x ∗∗; x) = J l+(x ∗, x ∗∗)Pm(x ∗, x ∗∗)+ ve Nc 4 Bm(x ∗, x ∗∗)Al−(x ∗, x ∗∗; x), (4.32) where Bm(x ∗, x ∗∗) = Bm+ (x ∗, x ∗∗)[Qm(x ∗, x ∗∗)]2, (4.33) and Qm(x ∗, x ∗∗) ∞ { 1− [Pm(x ∗, x ∗∗)]2 }1/2 (4.34) is the nonlocal extension of expression (4.6) for the function Q(x ∗). The ballistic spin-polarized current of Eq. (4.32) has the form J l−(x ∗, x ∗∗; x) = Ĵ l−(x ∗, x ∗∗)+ J̌ l−(x ∗, x ∗∗; x), (4.35) where the first (x-independent, non-relaxing) term is the “persistent” ballistic spin- polarized current, Ĵ l−(x ∗, x ∗∗) = J l+(x ∗, x ∗∗)Pm(x ∗, x ∗∗), (4.36) while the second (x-dependent) term is the “relaxing” ballistic spin-polarized current, J̌ l−(x ∗, x ∗∗; x) = ve Nc 4 Bm(x ∗, x ∗∗)Al−(x ∗, x ∗∗; x). (4.37) The latter describes the spin dynamics in the current J l−(x ∗, x ∗∗; x) via the “spin relaxation function” Al−(x ∗, x ∗∗; x). The x-dependence of Al−(x ∗, x ∗∗; x) will be 58 4 Thermoballistic Approach: Concept determined explicitly in Sect. 4.2.3, while the procedure for calculating the spin accumulation function A−(x ∗) ∞ Al−(x ∗, x ∗∗; x = x ∗) (4.38) will be described in Sect. 5.3. Thepersistent ballistic spin-polarized current Ĵ l−(x ∗, x ∗∗)depends, via the total bal- listic current J l+(x ∗, x ∗∗), on the spin accumulation function A−(x ∗). When A−(x ∗) is calculatedwithin the thermoballistic approach, spin relaxation in all ballistic intervals (including the interval [x ∗, x ∗∗] under consideration) is taken into account. Therefore, while being not directly affected by spin relaxation inside the interval [x ∗, x ∗∗], the persistent current Ĵ l−(x ∗, x ∗∗) depends, via the dependence of the factor J l+(x ∗, x ∗∗) on A−(x ∗) [see Eqs. (4.30) and (4.36)], in an indirect way on spin relaxation. 4.2.2 Ballistic Densities With the ballistic spin-resolved densities nl↑,≈(x ∗, x ∗∗; x) defined in generalization of Eq. (2.61) for the densities nl(x1, x2; x), the total and spin-polarized ballistic densities nl±(x ∗, x ∗∗; x) associated with the respective ballistic currents J l±(x ∗, x ∗∗; x) of Eq. (4.28) are obtained as nl±(x ∗, x ∗∗; x) = Nc 8 Dm+(x ∗, x ∗∗; x)[Al±(x ∗, x ∗∗; x)+ PmC (x ∗, x ∗∗; x)Al∓(x ∗, x ∗∗; x)]. (4.39) Here, we have taken into account that the ballistic densities correspond to one half of the thermal currents emitted symmetrically at a point of local thermodynamic equilibrium [see, e.g., Eq. (2.52)]. The functions Dm±(x ∗, x ∗∗; x) and PmC (x ∗, x ∗∗; x) are defined as Dm±(x ∗, x ∗∗; x) = Cm↑ (x ∗, x ∗∗; x)e−πE m↑ (x ∗,x ∗∗) ± Cm≈ (x ∗, x ∗∗; x)e−πE m≈ (x ∗,x ∗∗), (4.40) where Cm↑,≈(x ∗, x ∗∗; x) = eπ[E m↑,≈(x ∗,x ∗∗)−E↑,≈(x)]erfc({π[Em↑,≈(x ∗, x ∗∗)− E↑,≈(x)]}1/2) (4.41) [see Eq. (2.62)], and PmC (x ∗, x ∗∗; x) = D m−(x ∗, x ∗∗; x) Dm+(x ∗, x ∗∗; x) , (4.42) in generalization of expression (4.26) for the function Pm(x ∗, x ∗∗). http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 4.2 Ballistic Spin-Polarized Transport 59 Now, inserting expression (4.31) for Al+(x ∗, x ∗∗; x) in Eq. (4.39), we find that the total ballistic density, nl+(x ∗, x ∗∗; x), and the ballistic spin-polarized density, nl−(x ∗, x ∗∗; x), have the form nl±(x ∗, x ∗∗; x) = n̂l±(x ∗, x ∗∗; x)+ ňl±(x ∗, x ∗∗; x), (4.43) Here, the first (x-dependent, but non-relaxing) term is the persistent part, n̂l±(x ∗, x ∗∗; x) = Nc 8 Dm±(x ∗, x ∗∗; x)[A+(x ∗)+ Pm(x ∗, x ∗∗)A−(x ∗)], (4.44) and the second term is the relaxing part, ňl±(x ∗, x ∗∗; x) = Nc 8 Dm±(x ∗, x ∗∗; x)Al−(x ∗, x ∗∗; x), (4.45) where Dm±(x ∗, x ∗∗; x) = Dm∓(x ∗, x ∗∗; x)− Pm(x ∗, x ∗∗)Dm±(x ∗, x ∗∗; x). (4.46) In contrast to the x-independent currents J l+(x ∗, x ∗∗) and Ĵ l−(x ∗, x ∗∗), the persistent parts of the ballistic densities, n̂l±(x ∗, x ∗∗; x), dependonposition via the x-dependence of the potential energy profiles E↑,≈(x). For zero spin splitting of the conduction band, Δ(x) ∞ 0, when Pm(x ∗, x ∗∗) = Dm−(x ∗, x ∗∗; x) = Dm+(x ∗, x ∗∗; x) = 0, (4.47) the persistent part of the spin-polarized density as well as the relaxing part of the total density vanish, n̂l−(x ∗, x ∗∗; x) = ňl+(x ∗, x ∗∗; x) = 0, (4.48) a result that conforms with the vanishing of the static spin polarization P . 4.2.3 Spin Balance Equation In order to determine the x-dependence of the spin relaxation function Al−(x ∗, x ∗∗; x) introduced in Sect. 4.2.1.2, we have to consider spin relaxation explicitly. In a time-dependent formulation, spin relaxation in a system described by spin- resolved electron densities n↑,≈(x, t) and currents J↑,≈(x, t) is governed by the local coupled spin balance equations [16] β βt n↑(x, t)+ β βx J↑(x, t) = −n↑(x, t) τ↑≈ + n≈(x, t) τ≈↑ (4.49) 60 4 Thermoballistic Approach: Concept and β βt n≈(x, t)+ β βx J≈(x, t) = −n≈(x, t) τ≈↑ + n↑(x, t) τ↑≈ , (4.50) where 1/τ↑≈ (1/τ≈↑) is the rate for spin-flip scattering from spin-up (spin-down) to spin-down (spin-up) states. In the stationary case, when β βt n↑,≈(x, t) = 0, (4.51) this leads to the balance equation d dx J̌−(x) = − 1 τs ň−(x)+ ( 1 τ≈↑ − 1 τ↑≈ ) ň+(x) (4.52) connecting the relaxing part of the spin-polarized current, J̌−(x), to the relaxing parts of the spin-polarized, ň−(x), and of the total density, ň+(x). Here, τs , defined as 1 τs = 1 τ↑≈ + 1 τ≈↑ , (4.53) is the spin relaxation time. We apply Eq. (4.52) to spin relaxation during ballistic transport. In doing this, we will disregard the term involving ň+(x) for two reasons: (i) As it is preceded by the difference of the two relaxation rates 1/τ↑≈ and 1/τ≈↑, which are estimated to be of comparable magnitude, the term may generally be considered small in comparison with the term involving ň−(x). (ii) Since both ň−(x) and ň+(x) are proportional to the spin relaxation function Al−(x ∗, x ∗∗; x) [see Eq. (4.45)], we can account for ň+(x) by combining its prefactors with those of ň−(x) in an effective spin relaxation time (still denoted by τs) depending, in general, on the potential energy profiles. [Note that wemust not assume ň+(x) = 0 from the outset by adopting the arguments leading to Eq. (2.7) of Ref. [16]. The density n↑+n≈ appearing in that equation is the deviation of the total density from its spin equilibrium value. By contrast, the relaxing total density ň+(x), when used in the thermoballistic description, gives the deviation of the total density from the persistent total density. In the latter, a spin non-equilibrium part enters via the spin accumulation function A−(x) [see Eq. (4.44)].] In the thermoballistic approach, it is assumed that the thermally emitted electrons spend only an infinitesimally short time span at the emission point, and it is only during their motion across the ballistic interval that they can undergo spin relaxation. Spin relaxation in ballistic transport is commonly described [17–19] in terms of a (ballistic) spin relaxation length ls given by ls = 2veτs . (4.54) 4.2 Ballistic Spin-Polarized Transport 61 [As in the case of the effective electron mass m∗, we assume τs , and hence ls , to be independent of position and of the external magnetic field; we consider ls here an effective quantity, in line with the interpretation of τs .] In our descrip- tion of spin-polarized electron transport, spin relaxation is thus completely sepa- rated from momentum relaxation at the points of local thermodynamic equilibrium and, in this respect, is similar to the D’yakonov-Perel’ relaxation mechanism [5–8, 20–22]. In terms of the relaxing ballistic spin-polarized current J̌ l−(x ∗, x ∗∗; x) and the corresponding density ňl−(x ∗, x ∗∗; x), the balance equation governing spin relaxation during ballistic transport now reads d dx J̌ l−(x ∗, x ∗∗; x)+ 2ve ls ňl−(x ∗, x ∗∗; x) = 0, (4.55) where the spin relaxation length ls is given by Eq. (4.54). Inserting in Eq. (4.55) the expressions for J̌ l−(x ∗, x ∗∗; x) and ňl−(x ∗, x ∗∗; x) from Eqs. (4.37) and (4.45), respectively, we obtain a first-order differential equation for the spin relaxation function Al−(x ∗, x ∗∗; x), d dx Al−(x ∗, x ∗∗; x)+ Cm(x ∗, x ∗∗; x) ls Al−(x ∗, x ∗∗; x) = 0, (4.56) where Cm(x ∗, x ∗∗; x) = D m−(x ∗, x ∗∗; x) Bm(x ∗, x ∗∗) . (4.57) The solution of Eq. (4.56) obeying the initial condition (4.27) is Al−(x ∗, x ∗∗; x) = A−(x ∗)e−C m (x ∗,x ∗∗;x ∗,x)/ ls , (4.58) where Cm(x ∗, x ∗∗; z1, z2) = ∫ z> z< dzCm(x ∗, x ∗∗; z), (4.59) with z< = min(z1, z2), z> = max(z1, z2). Now, inserting expression (4.58) in Eqs. (4.37) and (4.45), respectively, we obtain the relaxing ballistic spin-polarized current and density explicitly in terms of the spin accumulation function A−(x ∗), J̌ l−(x ∗, x ∗∗; x) = ve Nc 4 Bm(x ∗, x ∗∗)A−(x ∗)e−C m (x ∗,x ∗∗;x ∗,x)/ ls (4.60) and ňl−(x ∗, x ∗∗; x) = Nc 8 Dm−(x ∗, x ∗∗; x)A−(x ∗)e−C m (x ∗,x ∗∗;x ∗,x)/ ls . (4.61) 62 4 Thermoballistic Approach: Concept The x-dependence of the relaxing spin-polarized current and density in the ballistic interval [x ∗, x ∗∗] is hence governed by the factor e−Cm (x ∗,x ∗∗;x ∗,x)/ ls . It departs from a purely exponential behavior unless the potential energy profiles E↑,≈(x) are constant over the interval, in which case Cm(x ∗, x ∗∗; x ∗, x) = x − x ∗ (4.62) [see Eqs. (4.40), (4.41), (4.46), (4.57), and (4.59)]. 4.2.4 Net Ballistic Currents and Joint Ballistic Densities So far,we have only considered thermal emission at the left end-point, x ∗, of the ballis- tic interval [x ∗, x ∗∗], obtaining a variety of ballistic currents and densities summarily denoted here by J l(x ∗, x ∗∗; x) and nl(x ∗, x ∗∗; x), respectively. The analogous ballistic currents and densities Jr (x ∗, x ∗∗; x) and nr (x ∗, x ∗∗; x) corresponding to emission at the right end-point x ∗∗ can be expressed in terms of those emitted at x ∗ as Jr (x ∗, x ∗∗; x) = −J l(x ∗∗, x ∗; x) (4.63) and nr (x ∗, x ∗∗; x) = nl(x ∗∗, x ∗; x). (4.64) [Note that, owing to relation (4.22) for the functions Em↑,≈(x ∗, x ∗∗), the functions Bm± (x ∗, x ∗∗), Bm(x ∗, x ∗∗), Dm±(x ∗, x ∗∗; x), and Dm±(x ∗, x ∗∗; x) entering the expressions for J l(x ∗, x ∗∗; x) and nl(x ∗, x ∗∗; x) are invariant under the exchange of x ∗ and x ∗∗.] We then have J (x ∗, x ∗∗; x) ∞ J l(x ∗, x ∗∗; x)+ Jr (x ∗, x ∗∗; x) = J l(x ∗, x ∗∗; x)− J l(x ∗∗, x ∗; x) (4.65) and n(x ∗, x ∗∗; x) ∞ nl(x ∗, x ∗∗; x)+ nr (x ∗, x ∗∗; x) = nl(x ∗, x ∗∗; x)+ nl(x ∗∗, x ∗; x) (4.66) for the net ballistic currents and joint ballistic densities summarily denoted by J (x ∗, x ∗∗; x) and n(x ∗, x ∗∗; x), respectively. 4.2.4.1 Net Currents For the (conserved) net total ballistic current J+(x ∗, x ∗∗) inside the ballistic interval, we now find, using Eq. (4.30), 4.2 Ballistic Spin-Polarized Transport 63 J+(x ∗, x ∗∗) = ve Nc 2 Bm+ (x ∗, x ∗∗) × { [ Ã(x ∗)− Ã(x ∗∗)] + 1 2 Pm(x ∗, x ∗∗)[A(x ∗)− A(x ∗∗)] } . (4.67) Here, the function A+(x ∗) has been replaced with 2 Ã(x ∗) [see Eq. (4.9)], and the subscript attached to the spin accumulation function A−(x ∗) has been omitted, i.e., we have set A(x ∗) ∞ A−(x ∗). (4.68) The current J+(x ∗, x ∗∗) is seen to be dynamically determined, in general, both by the average chemical potential μ̃(x ∗) [via the mean spin function Ã(x ∗)] and the spin accumulation function A(x ∗). The same then holds for the net persistent ballistic spin-polarized current Ĵ−(x ∗, x ∗∗), for which we have, using Eq. (4.36), Ĵ−(x ∗, x ∗∗) = J+(x ∗, x ∗∗)Pm(x ∗, x ∗∗). (4.69) For zero spin splitting, when Δ(x) = 0 and hence Pm(x ∗, x ∗∗) = 0, the dependence of J+(x ∗, x ∗∗) on A(x) drops out and, further, Ĵ−(x ∗, x ∗∗) = 0. For the net relaxing ballistic spin-polarized current, we have, using Eq. (4.60), J̌−(x ∗, x ∗∗; x) = ve Nc 4 Bm(x ∗, x ∗∗) [ A(x ∗)e−Cm (x ∗,x ∗∗;x ∗,x)/ ls −A(x ∗∗)e−Cm (x ∗,x ∗∗;x,x ∗∗)/ ls ], (4.70) which is dynamically determined by the spin accumulation function A(x) alone. 4.2.4.2 Joint Densities For the joint total ballistic density, n+(x ∗, x ∗∗; x), and the joint ballistic spin-polarized density, n−(x ∗, x ∗∗; x), respectively, we have from Eq. (4.43) n±(x ∗, x ∗∗; x) = n̂±(x ∗, x ∗∗; x)+ ň±(x ∗, x ∗∗; x), (4.71) with persistent parts n̂±(x ∗, x ∗∗; x) and relaxing parts ň±(x ∗, x ∗∗; x) given by n̂±(x ∗, x ∗∗; x) = Nc 4 Dm±(x ∗, x ∗∗; x) { [ Ã(x ∗)+ Ã(x ∗∗)] + 1 2 Pm(x ∗, x ∗∗)[A(x ∗)+ A(x ∗∗)] } (4.72) 64 4 Thermoballistic Approach: Concept [see Eq. (4.44)] and ň±(x ∗, x ∗∗; x) = Nc 8 Dm±(x ∗, x ∗∗; x) [ A(x ∗)e−Cm (x ∗,x ∗∗;x ∗,x)/ ls +A(x ∗∗)e−Cm (x ∗,x ∗∗;x,x ∗∗)/ ls ] (4.73) [see Eqs. (4.45) and (4.61)], in analogy to Eqs. (4.67), (4.69), and (4.70) for the net currents. From expressions (4.70) and (4.73), we now derive, using Eqs. (4.57) and (4.59), the balance equation d dx J̌−(x ∗, x ∗∗; x)+ 2ve ls ň−(x ∗, x ∗∗; x) = 0 (4.74) connecting the net relaxing ballistic spin-polarized current J̌−(x ∗, x ∗∗; x) and the associated joint density ň−(x ∗, x ∗∗; x). This equation can be obtained, of course, simply by adding the balance equation Eq. (4.55) and the corresponding equation for emission at x ∗∗. 4.3 Thermoballistic Spin-Polarized Transport In a significant advance over its prototype presented in Sect. 3, where overall current conservation was introduced via the condition (3.20), the thermoballistic concept proper rests on the introduction of a “reference coordinate” x that characterizes an arbitrary point inside the semiconducting sample extending from x1 to x2, as shown in Fig. 4.1. Singling out this coordinate, we consider the net ballistic currents and joint ballistic densities within the ensemble of all ballistic intervals [x ∗, x ∗∗] enclosing x . These currents and densities form the building blocks for establishing, at x , the corresponding thermoballistic quantities. The point labeled by the reference coordinate x is not a point of local thermodynamic equilibrium (we may call it a “ballistic point”). However, the “equilibrium points” x ∗, x ∗∗ may come infinitesimally close to x . 4.3.1 Thermoballistic Currents and Densities The thermoballistic currents and densities at the point x are constructed by per- forming weighted summations of the corresponding net ballistic currents and joint ballistic densities over all ballistic intervals [x ∗, x ∗∗] subjected to the condition x1 ≡ x ∗ < x < x ∗∗ ≡ x2. (4.75) http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 4.3 Thermoballistic Spin-Polarized Transport 65 Just as in the prototype thermoballistic model (see Sect. 3.2.2), we adopt here the probabilistic picture outlined in Sect. 3.1: the contributions from the interval [x ∗, x ∗∗] are weighted with the probability e−(x ∗∗−x ∗)/ l (one-dimensional transport is assumed here; see the remark at the beginning of Sect. 2.1.2) that the electrons traverse the interval without collisions, multiplied by the probability dx ∗/ l (dx ∗∗/ l) that they are absorbed or emitted in an interval dx ∗ (dx ∗∗) around the end-point x ∗ (x ∗∗). At the ends of the semiconducting sample at x1,2, absorption and emission occur with unit probability. Like the effective electron mass m∗ and the spin relaxation length ls , the momentum relaxation length l is assumed here to be independent of position and of the external magnetic field. [Here and in the following, we refer to the quantity l as the “momentum relaxation length”, as opposed to the term “mean free path” used in the preceding chapters.] 4.3.1.1 General Expression Representing the net ballistic currents and joint ballistic densities of Eqs. (4.67)– (4.73) summarily by a function F(x ∗, x ∗∗; x), and the corresponding thermoballistic currents and densities by F(x), we write F(x) in the form F(x) ∞ F(x1, x2; x; l) = e−(x2−x1)/ l F(x1, x2; x)+ ∫ x− x1 dx ∗ l e−(x2−x ∗)/ l F(x ∗, x2; x) + ∫ x2 x+ dx ∗∗ l e−(x ∗∗−x1)/ l F(x1, x ∗∗; x) + ∫ x− x1 dx ∗ l ∫ x2 x+ dx ∗∗ l e−(x ∗∗−x ∗)/ l F(x ∗, x ∗∗; x), (4.76) where x± = x ± ν, and the infinitesimal ν > 0 has been introduced in accordance with condition (4.75). Further, we set F(x1,2) = F(x1,2 ± σ) ∣∣ σ≤0+ . (4.77) In parallel to Eq. (3.32) for the total probability in the prototype model, we have F(x) = 1 for F(x ∗, x ∗∗; x) = 1. Specifically, the total thermoballistic current J+(x) is given by expression (4.76), with J+(x ∗, x ∗∗) of Eq. (4.67) substituted for F(x ∗, x ∗∗; x). The persistent ther- moballistic spin-polarized current Ĵ−(x) follows by identifying F(x ∗, x ∗∗; x) with Ĵ−(x ∗, x ∗∗) of Eq. (4.69). The relaxing thermoballistic spin-polarized current J̌−(x) is obtained by replacing F(x ∗, x ∗∗; x) with J̌−(x ∗, x ∗∗; x) of Eq. (4.70). Further, the thermoballistic densities n+(x), n̂−(x), and ň−(x) corresponding to the currents J+(x), Ĵ−(x), and J̌−(x) follow by substituting the respective joint ballistic densi- ties [see Eqs. (4.71)–(4.73)] for F(x ∗, x ∗∗; x) in Eq. (4.76). http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_3 66 4 Thermoballistic Approach: Concept II I III IV dx dx x x x x x1 2 | || | || Fig. 4.2 Schematic diagram illustrating the four types of ballistic current contributing to the total thermoballistic current J+(x) according to Eq. (4.76) In expression (4.76) when specialized to the total thermoballistic current J+(x), the first termon the right-hand side represents the net electron current passing through the point x while being ballistically transmitted between x1 and x2 [which occurswith probability e−(x2−x1)/ l ], the second term refers to ballistic electron motion between any point x ∗ (x1 ≡ x ∗ ≡ x−) and x2 [probability e−(x2−x ∗)/ ldx ∗/ l], the third term to ballistic electron motion between x1 and any point x ∗∗ (x+ ≡ x ∗∗ ≡ x2) [probability e−(x ∗∗−x1)/ ldx ∗∗/ l], etc. The four types of ballistic current appearing in expression (4.76) are illustrated by the double arrows labeled I to IV in Fig. 4.2. 4.3.1.2 Derivatives The derivatives with respect to x of the various thermoballistic quantities can be written in the general form d dx F(x) = d dx F(x) ∣∣∣∣ F +D(x), (4.78) where the first term on the right-hand side comprises the contributions arising from differentiating the functions F(x ∗, x ∗∗; x) in the integrands of expression (4.76), and the second, those from differentiating the limits of integration: 4.3 Thermoballistic Spin-Polarized Transport 67 D(x) ∞ D(x1, x2; x; l) ∞ ( β βx+ + β βx− ) F(x1, x2; x; l) = −1 l {F1(x; [F])− F2(x; [F])}. (4.79) Here, the functionals F1(x; [F]) and F2(x; [F]), defined by F1(x; [F]) = e−(x−x1)/ l F(x1, x+; x)+ ∫ x− x1 dx ∗ l e−(x−x ∗)/ l F(x ∗, x+; x) (4.80) and F2(x; [F]) = e−(x2−x)/ l F(x−, x2; x)+ ∫ x2 x+ dx ∗∗ l e−(x ∗∗−x)/ l F(x−, x ∗∗; x), (4.81) respectively, represent the contributions of the function F(x ∗, x ∗∗; x) arising from the ranges to the left and right of the point x . For the total thermoballistic current J+(x) constructed from the x-independent ballistic current J+(x ∗, x ∗∗) given by Eq. (4.67), the first term on the right-hand side of Eq. (4.78) vanishes, so that we have d dx J+(x) = D+(x), (4.82) withD+(x) given by expression (4.79) for F(x ∗, x ∗∗; x) ∞ J+(x ∗, x ∗∗). The function D+(x) is not, in general, equal to zero, so that the total thermoballistic current J+(x) is not conserved. The quantityD+(x)dx is the increment of J+(x) across the infinitesimal interval dx at position x of the sample. The four types of ballistic current appearing in expression (4.79) which contribute to this increment, are depicted by the double arrows labeled I to IV in Fig. 4.3, with the arrows I and II representing the term F1(x; [F]), and III and IV the term F2(x; [F]). If we set F(x ∗, x ∗∗; x) ∞ J̌−(x ∗, x ∗∗; x) in expressions (4.76) and (4.79), we find from Eq. (4.78) for the relaxing thermoballistic spin-polarized current d dx J̌−(x) = −2ve ls ň−(x)+ Ď−(x). (4.83) In the first term on the right-hand side of this equation, we have introduced the relaxing thermoballistic spin-polarized density ň−(x) by using the balance equa- tion (4.74) to replace the derivatives β J̌−(x ∗, x ∗∗; x)/βx which we encounter when differentiating the integrals in expression (4.76). The thermoballistic currents and densities, Eq. (4.76), are expressed, via the corre- sponding ballistic currents and densities, Eqs. (4.67)–(4.73), in terms of two dynam- ical functions, vi z., the mean spin function Ã(x) and the spin accumulation function A(x), or, equivalently, the average chemical potential μ̃(x) and the splitting of the 68 4 Thermoballistic Approach: Concept x1 x | x x || x2 II I III IV dx dx | || dx Fig. 4.3 Schematic diagram illustrating the four types of ballistic current contributing to the infini- tesimal incrementD+(x)dx of the total thermoballistic current J+(x) according to Eqs. (4.79) and (4.82) spin-resolved chemical potentials, μ−(x) [see Eqs. (4.14), (4.16), and (4.68)]. To implement the thermoballistic concept, we must establish algorithms for calculating Ã(x) and A(x) in terms of the intrinsic parameters of the semiconducting system, like momentum and spin relaxation lengths, as well as of the external parameters, like applied voltage and spin polarization in the external leads. Before establishing these algorithms within the thermoballistic approach, i.e., for arbitrary values of the momentum relaxation length l, we directly evaluate the ther- moballistic currents and densities in the drift-diffusion regime (small l) and for the ballistic case (l ≤ →). The results, when specialized to the case of a spin-degenerate conduction band, turn out to be equivalent to the respective standard physical expres- sions summarized in Chap. 2. This shows that the thermoballistic description indeed bridges the gap between the drift-diffusion and ballistic descriptions of carrier trans- port. 4.3.2 Drift-Diffusion Regime In the drift-diffusion regime, when l/S ≥ 1 and l/ ls ≥ 1, nonzero contributions to the thermoballistic currents and densities defined by Eq. (4.76) arise only from very short ballistic intervals [x ∗, x ∗∗] enclosing the point x , so that x − x ∗ and x ∗∗ − x are infinitesimals and only the double integral over x ∗ and x ∗∗ contributes. 4.3 Thermoballistic Spin-Polarized Transport 69 4.3.2.1 Thermoballistic Densities Then, from Eq. (4.72), the persistent part of the total thermoballistic density, n̂+(x), is seen to reduce to the total equilibrium density n+(x) [see Eq. (4.19)], n̂+(x) = n+(x), (4.84) with n+(x) expressed in terms of the functions Ã(x) and A(x) as n+(x) = Nc 2 B+(x) [ Ã(x)+ 1 2 P(x)A(x) ] . (4.85) Further, since Dm+(x ∗, x ∗∗; x) = 0 in the drift-diffusion regime, the relaxing part of the joint total ballistic density, ň+(x ∗, x ∗∗; x) [see Eq. (4.73)], vanishes. Consequently, we have ň+(x) = 0 (4.86) for the relaxing part of the total thermoballistic density, ň+(x), and hence n+(x) ∞ n̂+(x)+ ň+(x) = n+(x) (4.87) for the total thermoballistic density n+(x) in the drift-diffusion limit. In particular, for field-driven electron transport in a homogeneous semiconductor, n+(x) is constant (see Sect. 2.1.2), and hence also n+(x). The persistent part of the thermoballistic spin-polarized density, n̂−(x), is obtained from Eq. (4.72) as n̂−(x) = Nc 2 B−(x) [ Ã(x)+ 1 2 P(x)A(x) ] , (4.88) and the relaxing part of the thermoballistic spin-polarized density, ň−(x), from Eq. (4.73) as ň−(x) = Nc 4 [B+(x)− P(x)B−(x)]A(x) = Nc 4 B+(x)Q2(x)A(x), (4.89) with Q(x) given by Eq. (4.6). Then, re-expressing the spin-polarized equilibrium density n−(x) [see Eq. (4.19)] in the form n−(x) = Nc 2 [ B−(x) Ã(x)+ 1 2 B+(x)A(x) ] , (4.90) http://dx.doi.org/10.1007/978-3-319-05924-2_2 70 4 Thermoballistic Approach: Concept we have n−(x) ∞ n̂−(x)+ ň−(x) = n−(x) (4.91) for the thermoballistic spin-polarized density n−(x) in the drift-diffusion limit. 4.3.2.2 Total Thermoballistic Current The net total ballistic current J+(x ∗, x ∗∗) is found from Eq. (4.67), by expanding to first order in x ∗ − x and x ∗∗ − x , in the form J+(x ∗, x ∗∗) = ve Nc 2 B+(x) Â(x)(x ∗ − x ∗∗), (4.92) where Â(x) = d Ã(x) dx + 1 2 P(x) d A(x) dx . (4.93) Now, evaluating the double integral over x ∗ and x ∗∗ in Eq. (4.76) for F(x ∗, x ∗∗; x) = x ∗ − x ∗∗ and taking the limit ν ≤ 0 at fixed l > 0, we are left with a factor −2l, so that the drift-diffusion limit of the total thermoballistic current J+(x) is obtained as J+(x) = −ve Ncl B+(x) Â(x) = −γNc 2πe B+(x) Â(x) ∞ J. (4.94) Here, we have used relation (3.68) to introduce the electron mobility γ, and we have identified the constant, total thermoballistic current in the drift-diffusion regime with the (conserved) total physical current J . Then, using Eqs. (4.85) and (4.90), we can express J in terms of the equilibrium densities n±(x) [see Eqs. (4.87) and (4.91)] in the form J = −γ e [ n+(x) d Ec(x) dx + 1 π dn+(x) dx + 1 2 n−(x) dΔ(x) dx ] . (4.95) For zero spin splitting, Δ(x) = 0, this expression becomes equivalent to expression (2.28) for the total current in the standard drift-diffusion model. However, by con- trast with the latter model, we have obtained Eq. (4.95) without invoking the Einstein relation (2.25). This feature can be traced back to the probabilistic description under- lying the thermoballistic approach (see Sect. 3.1), which allows the diffusion current to be directly expressed in terms of the collision time (and hence of the momentum relaxation length l) [see Eq. (3.8)]. http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 4.3 Thermoballistic Spin-Polarized Transport 71 4.3.2.3 Average Chemical Potential To obtain an explicit expression for the mean spin function Ã(x), and hence for the average chemical potential μ̃(x), we observe that in the drift-diffusion regime the total thermoballistic density n+(x) is equal to the total equilibrium density n+(x) [see Eq. (4.87)], which, in turn, is related to Ã(x) via Eq. (4.85). Using Eq. (4.93) to solve Eq. (4.94) for d Ã(x)/dx , integrating over the interval [x1, x], and using Eqs. (4.4)–(4.6) to simplify the integrals, we can express eπμ̃(x) in the form eπμ̃(x) = eπμ̃(x1) − πeJ γNc ∫ x x1 dx ∗eπEc(x ∗)Q(x ∗)− 1 2 [P(x)A(x)− P(x1)A(x1)] −π 4 ∫ x x1 dx ∗ dΔ(x ∗) dx ∗ Q2(x ∗)A(x ∗), (4.96) which generalizes expression (2.32) so as to allow for a spin splitting of the con- duction band. Integration over the interval [x, x2] leads to another expression for μ̃(x), which is different in form, but numerically equal to expression (4.96). The drift-diffusion form of the spin accumulation function A(x) in expression (4.96) is determined by a differential equation [see Eq. (4.106) below]. We now set x = x2 in Eq. (4.96) and identify the boundary values of μ̃(x) at the interface positions x1,2 with the values, μ1,2, of the equilibrium chemical potential at the contact side of the contact-semiconductor interfaces, μ̃(x1,2) = μ1,2 (4.97) [see Eq. (2.33)], so that from Eq. (4.14) Ã(x1,2) = eπμ1,2 ∞ σ1,2. (4.98) Similarly, we identify the boundary values of A(x) with external values A1,2, A(x1,2) = A1,2. (4.99) We then obtain the drift-diffusion form of the current-voltage characteristic, which, for simplicity, is written down here for the case of constant spin splitting, when dΔ(x)/dx = 0, P(x) = P , and Q(x) = Q, J = γNc πeQS̃ e−π[Emc (x1,x2)−μ1] [ 1− e−πeV + 1 2 e−πμ1 P(A1 − A2) ] . (4.100) Here, we have used Eqs. (2.35) and (2.37), respectively, to introduce the voltage V and the effective sample length S̃. Expression (4.100) generalizes expression (2.34) so as to allow for a spin-split conduction band. http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 72 4 Thermoballistic Approach: Concept For zero spin splitting, when P(x) = 0 and Q(x) = 1, expressions (4.96) and (4.100) become equivalent to the expressions (2.32) and (2.34), respectively, in the standard drift-diffusion model. In particular, for field-driven transport in a homoge- neous system, we thus recover Eqs. (2.45) and (2.46). 4.3.2.4 Spin-Polarized Thermoballistic Currents Thedrift-diffusion limit of the persistent thermoballistic spin-polarized current Ĵ−(x) immediately follows from Eqs. (4.69) and (4.94) as Ĵ−(x) = −γNc 2πe B−(x) Â(x) = J P(x), (4.101) so that Ĵ−(x) = 0 for zero spin splitting. Proceeding in analogy to the derivation of J+(x), we find the drift-diffusion limit of the relaxing thermoballistic spin-polarized current J̌−(x) from Eqs. (4.70) and (4.76) in the form J̌−(x) = −γNc 4πe B+(x)Q2(x) d A(x) dx . (4.102) Using Eq. (4.89), we can express J̌−(x) in terms of the analogous thermoballistic density ň−(x), J̌−(x) = −γ e [ B̂+(x)ň−(x)+ 1 π dň−(x) dx ] , (4.103) where B̂+(x) = − 1 π dln(B+(x)Q2(x)) dx , (4.104) so that B̂+(x) = d Ec(x)/dx (4.105) for zero spin splitting. Inserting expressions (4.102) and (4.89) in the thermoballistic spin balance equa- tion (5.9) connecting J̌−(x) and ň−(x) [for the argument leading to the latter equation, see Sect. 5.1.2 below], we obtain a homogeneous second-order differential equation for the spin accumulation function A(x), d2A(x) dx2 − π B̂+(x)d A(x) dx − 1 L2s A(x) = 0, (4.106) where Ls = √ lls (4.107) http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 4.3 Thermoballistic Spin-Polarized Transport 73 is the spin diffusion length. In conjunction with the conditions (4.99), Eq. (4.106) defines a boundary value problem for A(x), whose solution determines ň−(x) and J̌−(x) via Eqs. (4.89) and (4.102), respectively. 4.3.2.5 Generalized Spin Drift-Diffusion Equation Equation (4.106) for the spin accumulation function A(x) can be converted, with the help of Eq. (4.89), into an analogous equation for the relaxing thermoballistic spin-polarized density ň−(x), d2ň−(x) dx2 + π d dx [B̂+(x)ň−(x)] − 1 L2s ň−(x) = 0. (4.108) For zero spin splitting, this equation generalizes, by including arbitrarily shaped potential energy profiles (i.e., total electric fieldswith arbitrary position dependence), the drift-diffusion equation commonly used to describe the effect of a constant electric field on spin-polarized transport in homogeneous semiconductors [see Eq. (2.8) of Ref. [16]; in that equation, the spin diffusion length appears as √ Dτs , with D an effective diffusion coefficient and τs the spin relaxation time]. In Ref. [23], a generalized drift-diffusion equation equivalent to the spin- degenerate limit of Eq. (4.108) has been derived for the case of spin-polarized trans- port in inhomogeneously doped semiconductors, in which built-in electric fields come into play, in addition to the external field. The influence of field inhomo- geneities on the spin diffusion length has been analyzed, and verified by numerical calculations in which nonlinear, coupled Boltzmann-Poisson equations were solved self-consistently. The results indicate that even weak inhomogeneities can have a significant effect on the spin diffusion length, and hence on spin-polarized transport in inhomogeneous semiconductors as a whole. 4.3.3 Ballistic Limit In the strict ballistic limit, when l ≤ →, there are no points of local thermodynamic equilibrium in the interval [x1, x2], and the average chemical potential μ̃(x) is not defined inside this interval. Expression (4.76) reduces to F(x) = F(x1, x2; x), (4.109) i.e., the thermoballistic currents and densities are given by the corresponding expres- sions for the net ballistic currents and joint ballistic densities (see Sect. 4.2.4), eval- uated at x ∗ = x1 and x ∗∗ = x2. In expressions (4.67), (4.70), (4.72), and (4.73), the boundary values of the mean spin function, Ã(x1,2), and those of the spin 74 4 Thermoballistic Approach: Concept accumulation function, A(x1,2), are to be identified with the corresponding values in the contacts [see Eqs. (4.98) and (4.99), respectively]. FromEq. (4.67), we then obtain the total thermoballistic current J+(x) in the form J+(x) ∞ J+ = J+(x1, x2) = ve Nc 2 Bm+ (x1, x2) [ σ−12 + 1 2 Pm(x1, x2)A − 12 ] ∞ J, (4.110) where σ±12 = σ1 ± σ2, (4.111) with σ1,2 defined by Eq. (4.98), and A±12 = A1 ± A2. (4.112) For zero spin splitting, expression (4.110) becomes equivalent to expression (2.66) for the total current in the ballistic transportmodel. The ballistic limit of the persistent thermoballistic spin-polarized current Ĵ−(x) is obtained from J+(x) by replacing in expression (4.110) the quantity Bm+ (x1, x2) with Bm− (x1, x2) [see Eqs. (4.26) and (4.69)]. For the total thermoballistic density n+(x), we have, defining A±12(x1, x2; x) ∞ A1e−C m (x1,x2;x1,x)/ ls ± A2e−Cm (x1,x2;x,x2)/ ls (4.113) and using Eqs. (4.71)–(4.73), n+(x) = n+(x1, x2; x) = n̂+(x1, x2; x)+ ň+(x1, x2; x) = Nc 4 { Dm+(x1, x2; x) [ σ+12 + 1 2 Pm(x1, x2)A + 12 ] + 1 2 Dm+(x1, x2; x)A+12(x1, x2; x) } . (4.114) For zero spin splitting, this becomes equivalent to expression (2.71). Further, using Eqs. (4.70) and (4.73), we express the relaxing thermoballistic spin- polarized current J̌−(x) and the corresponding density ň−(x) as J̌−(x) = J̌−(x1, x2; x) = ve Nc 4 Bm(x1, x2)A−12(x1, x2; x) (4.115) and ň−(x) = ň−(x1, x2; x) = Nc 8 Dm−(x1, x2; x)A+12(x1, x2; x), (4.116) respectively. In viewof Eq. (4.74), the current J̌−(x) and density ň−(x) in the ballistic limit are trivially connected by the general balance equation (5.9). http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 4.3 Thermoballistic Spin-Polarized Transport 75 4.3.4 Thermoballistic Energy Dissipation The stochastic equilibration of the electrons that occurs during their motion across the sample is associated with the dissipation of energy, i.e., the net transfer of energy out of the ensemble of conduction band electrons into a reservoir (“heat bath”) of electrons in thermodynamic equilibrium. 4.3.4.1 Ballistic Energy Currents To describe this transfer within the thermoballistic concept, we introduce the (con- served) ballistic energy currents El,r (x ∗, x ∗∗) generated by thermal electron emission at the end-points x ∗ and x ∗∗, respectively, of the ballistic interval [x ∗, x ∗∗]. Neglecting spin degrees of freedom, we have for the current El(x ∗, x ∗∗), by an obvious modifi- cation of expression (2.57) for the electron current J l(x1, x2), writing λ = m∗v2x/2, E l(x ∗, x ∗∗) = 4εm ∗ πh3 ∫ → 0 dλλe−π[λ+Ec(x ∗)−μ̃(x ∗)]Θ(λ− Elb(x ∗, x ∗∗)), (4.117) where, in analogy to Eq. (2.36), Elb(x ∗, x ∗∗) = Emc (x ∗, x ∗∗)− Ec(x ∗) (4.118) is the maximum barrier height of the potential energy profile Ec(x) relative to its value at x ∗. Evaluating the integral in expression (4.117), we obtain, in extension of Eq. (2.58), E l(x ∗, x ∗∗) = ve Nc [ 1 π + Elb(x ∗, x ∗∗) ] e−π[Emc (x ∗,x ∗∗)−μ̃(x ∗)]. (4.119) For the ballistic energy current Er (x ∗, x ∗∗), we have E r (x ∗, x ∗∗) = −El(x ∗∗, x ∗), (4.120) in line with Eq. (4.63) for the corresponding ballistic electron current, and for the net ballistic energy current in the interval [x ∗, x ∗∗], E(x ∗, x ∗∗) = El(x ∗, x ∗∗)+ Er (x ∗, x ∗∗), (4.121) in line with Eq. (4.65) for the net ballistic electron current. Now, we again introduce a “reference coordinate” x inside the sample, whose meaning, however, differs from that of the coordinate x in the definition (4.76) of the thermoballistic currents and densities.While in that definition, x is a coordinate inside ballistic intervals, we here consider x a coordinate characterizing the position of a http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 76 4 Thermoballistic Approach: Concept point of local thermodynamic equilibrium. More precisely, we consider a collection of such points, with density dx/ l, in an interval dx centered about x , in which ballistic energy currents are absorbed from, and emitted towards, either side. The equilibrium point x , where incoming electron currents are completely equilibrated and the outgoing currents are solely determined by the parameters of the reservoir (i.e., by the chemical potential), separates dynamically the two sample partitions to the left and right of x . 4.3.4.2 Locally Dissipated Energy We denote byWin(x) the (kinetic) energy transferred per unit volume and unit time into the reservoir by absorption of electrons at the point x , and correspondingly by Wout (x) the change in energy of the reservoir due to thermal electron emission out of it. The net energy W(x), i.e., the energy dissipated locally at the point x , is then given by W(x) = Win(x)+Wout (x). (4.122) The energiesWin(x) andWout (x) are each composed of two parts (see also Fig. 4.3). One part of Win(x) is given by the weighted sum of the energy currents El(x ∗, x ∗∗) over all ballistic intervals [x ∗, x ∗∗] lying to the left of x and having their right end at x ∗∗ = x , where the currents are absorbed. The other part is expressed analogously in terms of the currents Er (x, x ∗∗) in ballistic intervals to the right of the absorption point x . For the energy Win(x)dx dissipated in an interval dx centered around x , we then have Win(x)dx = dx l {F1(x; [El ])+ F2(x; [Er ])}. (4.123) Here, the functionals F1(x; [F]) and F2(x; [F]) are given, for an arbitrary ballis- tic current F(x ∗, x ∗∗; x), by Eqs. (4.80) and (4.81), respectively. The energy current Wout (x)dx is obtained from Eq. (4.123) by interchanging the role of El(x ∗, x ∗∗) and E r (x ∗, x ∗∗), Wout (x)dx = dx l {F1(x; [Er ])+ F2(x; [El ])}, (4.124) which comprises all ballistic energy currents emitted at the point x towards either side. For the dissipated energy W(x), we then have W(x) = 1 l {F1(x; [E])+ F2(x; [E])}, (4.125) where the net ballistic energy current E(x ∗, x ∗∗) is given by Eq. (4.121). We note that in obtaining expression (4.125) for W(x) we have not drawn on the thermoballistic energy current E(x) that results from identifying in Eq. (4.76) the function F(x ∗, x ∗∗; x) with net ballistic energy current E(x ∗, x ∗∗). Naively, one might expect that the dissipated energy W(x) can be represented by the derivative 4.3 Thermoballistic Spin-Polarized Transport 77 (β/βx++β/βx−)E(x) given byEq. (4.79).However, comparing expressions (4.125) and (4.79), one observes that the contributions of the ballistic energy currents from the two sides of x add up in the former, and are subtracted from one another in the latter. The result (4.125) can be obtained from the thermoballistic energy current E(x) in the form W(x) = − ( β βx+ − β βx− ) E(x), (4.126) at variance with expression (4.79). 4.3.4.3 Ballistic Limit and Drift-Diffusion Regime We do not write down here the general expression for W(x) in terms of the average chemical potential μ̃(x) obtained by using expressions (4.119)–(4.121) in Eq. (4.125), and confine ourselves to considering the ballistic limit and the drift- diffusion regime. Owing to the overall factor 1/ l in the right-hand side of Eq. (4.125), W(x) vanishes in the ballistic limit l ≤ →, which reflects the complete absence of equilibration in this limit. In the drift-diffusion regime, on the other hand, when l ≥ S, only the integral terms contribute. We evaluate these terms for the special case of a homogeneous sample subjected to a constant external electric field, so that the potential energy profile has the form (2.41). For this profile, the average chem- ical potential μ̃(x) is found to run parallel to the profile [see Eq. (2.46)], and the total equilibrium electron density is constant, n(x) = n(x1). The net ballistic energy current E(x ∗, x ∗∗) can now be expressed as E(x ∗, x ∗∗) = ve π n(x1){1− e−κ(x ∗∗−x ∗)] [ 1+ κ(x ∗∗ − x ∗)]} ∼ ve 2π n(x1)κ 2(x ∗∗ − x ∗)2, (4.127) where the second, approximate equation holds in the low-field regime, κS ≥ 1. Inserting the approximate representation of E(x ∗, x ∗∗) in the integral terms of expres- sion (4.125), we find for the locally dissipated energy W(x) in the drift-diffusion regime W(x) ∞ W = 2ve π ln(x1)κ 2 = e 2 J 2 ∂ , (4.128) where we have used Eq. (2.45) to introduce the current J = − j/e. For the total energy W = WAS (4.129) http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 78 4 Thermoballistic Approach: Concept dissipated per unit time in a sample with cross-sectional area A and resistance R = S/∂A, we then recover the usual expression W = e2 J 2A2R (4.130) (“Joule heating”; see Ref. [1]). 4.4 Synopsis of the Thermoballistic Concept We conclude this chapter with a synopsis of the concept underlying the thermobal- listic description of charge carrier transport in semiconductors, in which we briefly comment on its basic ingredients and elucidate the physical content of its formal structure. This will be followed by an assessment of its merits as well as its weak- nesses. 4.4.1 Ingredients and Physical Content The basic ingredients of the (semiclassical) thermoballistic concept are the ballistic carrier currents and densities which are constructed within the following framework. (i) Thermal emission of carriers occurs at points of local thermodynamic equilibrium randomly distributed over the sample. (ii) The equilibrium points link “ballistic transport intervals” across which the emitted carriers move ballistically under the influence of potential energy profiles arising from a combination of internal and external electrostatic potentials. During their ballistic motion, the carriers undergo spin relaxation controlled by a ballistic spin relaxation length ls . (iii)At the end-points of the ballistic intervals, instantaneous “point-like” equilibration (“absorption”) of the carriers takes place. Here, we invoke the picture of “reflectionless contacts” [24–27], according to which the carriers that enter a contact (representing a “bath” with an effectively infinite number of transverse modes) are completely absorbed there. In the thermoballistic concept, the emission and absorption of carriers are treated in thisway at all points of local thermodynamic equilibrium inside the sample. At these points, the absorption of carriers into the bath is complete (“reflectionless”), but at the same instant carriers are emitted out of the bath into the ballistic intervals on either side. The opposing collision-free currents emitted at either end into the ballistic interval combine to form a net ballistic current inside the interval, with an associated joint ballistic density. The random distribution of points of local thermodynamic equilibrium ismirrored in a random partitioning of the length of the sample into ballistic intervals, where each partition defines a “ballistic configuration”, a central notion of the concept. The ballistic carrier currents and densities in these configurations are assembled to form the corresponding thermoballistic currents and densities. These are constructed, at 4.4 Synopsis of the Thermoballistic Concept 79 a reference position x located arbitrarily inside the sample, by performing weighted summations, with weights controlled by a momentum relaxation length l, over the net ballistic currents and joint ballistic densities in all ballistic intervals containing the point x . The thermoballistic currents and densities constitute the key element of the thermoballistic concept. The physical content of the thermoballistic transport mechanism can be exhib- ited by analyzing the underlying formalism with regard to the intertwined effects of thermal electron emission and ballistic motion. Let us consider expressions (4.67), (4.69), and (4.70) for the net ballistic currents and expressions (4.71)–(4.73) for the joint ballistic densities, which are essentially composed of two factors each. On the one hand, they contain the nonlocal barrier factors Bm± (x ∗, x ∗∗) and Bm(x ∗, x ∗∗) in the expressions for the total and persistent spin-polarized currents and the relaxing spin-polarized current, respectively, and the factors Dm±(x ∗, x ∗∗; x) and Dm(x ∗, x ∗∗; x) in the analogous expressions for the densities. These factors describe the collision- free motion of the electrons across the ballistic interval [x ∗, x ∗∗], which is essentially determined by the potential energy profiles E↑,≈(x) inside the interval. They rep- resent the ballistic attribute of thermoballistic transport. The factors in braces in Eqs. (4.67) and (4.72) and in brackets in Eqs. (4.70) and (4.73), on the other hand, contain terms depending on the average chemical potentials μ̃(x ∗), μ̃(x ∗∗) and the spin accumulation functions A(x ∗), A(x ∗∗), which are directly related to the spin-resolved chemical potentials μ↑,≈(x ∗), μ↑,≈(x ∗∗) at the end-points x ∗, x ∗∗ of the ballistic inter- val. These factors, which describe the thermal emission (“thermal activation”) of the ballistic currents at the points of local thermodynamic equilibrium at x ∗ and x ∗∗, respectively, represent the thermal attribute of thermoballistic transport. The terms in braces and brackets, respectively, are the “activation terms”. [The joint appear- ance of ballistic and thermal attributes shows that the term “thermoballistic” indeed provides an appropriate characterization of our approach.] The contributions of the ballistic currents and densities (4.67) and (4.69)–(4.73), summarily denoted by F(x ∗, x ∗∗; x), to the corresponding thermoballistic currents and densities F(x) are to be read from Eq. (4.76): they are given by the current (or density) F(x ∗, x ∗∗; x) in the interval [x ∗, x ∗∗], multiplied by the probability e−(x ∗∗−x ∗)/ l that the electrons traverse this interval ballistically. The momentum relaxation length l controls the magnitude of the contribution of the ballistic configurations to the entire transport process. At the same time, it determines the average number of collisions, S/ l, in a sample of length S. In the ballistic limit, when l ≤ → and there is no point of local thermodynamic equilibrium inside the sample, the transport is purely ballistic between the end-points x1 and x2, and only the first term on the right-hand side of Eq. (4.76) contributes. In the opposite limit, when l ≤ 0, the points of local thermodynamic equilibrium at which the electrons are equilibrated, lie infinitesimally close to one another. Then only the double integral in Eq. (4.76) survives, and we arrive at J+(x) = −ve Ncl B+(x) d dx eπμ̃(x) (4.131) 80 4 Thermoballistic Approach: Concept for the total thermoballistic current J+(x) = J [see Eq. (4.94)]. This current is, except for the factor l, solely given in terms of equilibrium quantities, a property that characterizes the drift-diffusion regime. Expression (4.131) has the form of the current in the standard drift-diffusion approach [see Eqs. (2.27) and (2.29)] J+(x) = −γ e n+(x) dμ̃(x) dx , (4.132) where γ is the electron mobility given by Eq. (3.68), and n+(x) the equilibrium electron density given by Eq. (4.85). In that approach, the momentum relaxation length l is nonzero (so that γ remains nonzero), but small compared with the length scales over which the other parameters vary appreciably. The activation term reduces to a derivative, and the relation between the total thermoballistic current and the average chemical potential becomes a local one. 4.4.2 Merits and Weaknesses The principal merit of the thermoballistic concept is that it allows to establish a consistent and transparent formalism for bridging, within the semiclassical approx- imation, the gap between the standard drift-diffusion and ballistic descriptions of charge carrier transport in semiconductors. While incorporating basic features of these descriptions, the thermoballistic concept consistently unifies and generalizes them by introducing random partitionings of the sample length into ballistic config- urations. The concept is transparent in a twofoldway. First, as shown above, a lucid interpre- tation of its physical content can be given in terms of ballistic and thermal attributes. Second, owing to its semiclassical character, the concept allows the effects of the different parameters describing a semiconducting system to be clearly distinguished. In the implementation of this concept, explicit equations for various transport quan- tities can be derived, and simple solutions can be obtained in important special cases. The merit of transparency of the thermoballistic concept carries with it some simpli- fications and weak points which, however, in many cases can be remedied, albeit at the cost of increased complexity: they are not detrimental to the concept as a whole. While the formulation of the full thermoballistic concept given here describes semiclassical transport in nondegenerate semiconducting systems, we have demon- strated within the prototype model how effects of electron tunneling and degeneracy can be taken into account. Quantum interference effects in the electron motion are not treated explicitly, but they may be assumed to be implicitly incorporated via an extended interpretation of the momentum relaxation length (or mean free path) l, which from the outset has been taken as a phenomenological parameter. Being for- mally treated as the average distance that the carriers travel without collision between points of complete thermodynamic equilibrium, as in the relaxation time approxima- tion [1], it simulates various effects: the incomplete equilibration due to neutral and http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_3 4.4 Synopsis of the Thermoballistic Concept 81 ionized impurity scattering as well as phonon and carrier-carrier scattering [1, 28], the dimensionality of the system, and, in the extreme, quantal phase correlations [29]. Indeed, in the present work the momentum relaxation length l is the determining parameter in which a great diversity of detail is subsumed. It is introduced as a constant, so it must include in an average way the effect of spatial variations in the internal and external parameters characterizing the semiconducting system; in particular, this constant is chosen to be independent of the potential energy profile, to which, strictly speaking, it should be related in a self-consistent way. We may work with position-dependent momentum relaxation lengths, but this would increase the complexity of the formalism and obscure the general line of argument. Moreover, the choice of the momentum relaxation length l, rather than the relaxation time τ , is also merely one of convenience for the stationary treatment in this work. The spin relaxation mechanism in terms of the ballistic spin relaxation length ls is again a phenomenological one, having certain similarities with the D’yakonov- Perel’ mechanism [20]. We assume spin relaxation to occur only during the ballistic electron motion; however, simultaneous spin and momentum relaxation could be taken into account by introducing additional terms in the spin balance equation. In principle, the thermoballistic concept allows a fully three-dimensional treat- ment of bipolar carrier transport to be implemented. However, in the present chapter, in order to keep the formalism manageable, we work within a narrowed framework. First, we confine ourselves to unipolar transport, dealing specifically with electron transport in a spin-split conduction band. Second, we consider three-dimensional, “plane-parallel” samples whose parameters (in particular, the average density of the scattering centers associated with impurities) do not vary in the directions perpen- dicular to the transport direction (the x-direction). Nevertheless, electron transport in this kind of sample depends on the number of dimensions, n, via “no-scattering probabilities” pn(x) (see Sect. II of Ref. [2]). Here, in order to be able to write down formulae which are easy to interpret physically, we use one-dimensional no- scattering probabilities of the form p1(x) = e−x/ l (see Sect. 3.1). In concluding the synopsis of the thermoballistic concept, we come back to our earlier remarks on the transparency of this concept and its implementation. It is theo- retically appealing andmay serve as a tool for a first-attempt analysis of experimental data in an overall, readily understandable fashion.Our concept is not intended to com- pete with advanced numerical methods for the semiclasscial treatment of transport processes in semiconductors, like the simulation of the carrier dynamics within the Monte Carlo method or the direct numerical solution of the Boltzmann equation [28, 30, 31]. The latter methods clearly provide immediate access to device simu- lation and the quantitative description of any kind of experimental situation. On the whole, however, they appear to lack the transparency inherent in the thermoballistic concept. http://dx.doi.org/10.1007/978-3-319-05924-2_3 82 4 Thermoballistic Approach: Concept References 1. N.W. Ashcroft, N.D. Mermin, Solid State Physics (Harcourt Brace College Publishers, Fort Worth, 1976) 2. R. Lipperheide, U. Wille, Phys. Rev. B 68, 115315 (2003) 3. R. Lipperheide, U. Wille, Phys. Rev. B 72, 165322 (2005) 4. R. Lipperheide, U. Wille, Ann. Phys. (Berlin) 521, 127 (2009) 5. D.D. Awschalom, D. Loss, N. Samarth (eds.), Semiconductor Spintronics and Quantum Com- putation (Springer, Berlin, 2002) 6. I. Žutić, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004) 7. J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, I. Žutić, Acta Phys. Slovaca 57, 565 (2007) 8. M.I. Dyakonov (ed.), Spin Physics in Semiconductors. Springer Series in Solid-State Sciences, vol. 157 (Springer, Berlin, 2008) 9. J.K. Furdyna, J. Appl. Phys. 64, R29 (1988) 10. T.S. Moss (ed.), Handbook on Semiconductors (North Holland, Amsterdam, 1994) 11. T. Dietl, in Handbook on Semiconductors, vol. 3b, ed. by T.S. Moss (North Holland, Amster- dam, 1994), Chap. 17 12. J. Cibert, D. Scalbert, in Spin Physics in Semiconductors, ed. by M.I. Dyakonov (Springer, Berlin, 2008), Chap. 13 13. P. Bhattacharya, R. Fornari, H. Kamimura (eds.), Comprehensive Semiconductor Science and Technology (Elsevier, Amsterdam, 2011) 14. J. A. Gaj, in Comprehensive Semiconductor Science and Technology, vol. 2, ed. by P. Bhat- tacharya, R. Fornari, H. Kamimura (Elsevier, Amsterdam, 2011), Chap. 2.04 15. Z.G. Yu, M.E. Flatté, Phys. Rev. B 66, 201202(R) (2002) 16. Z.G. Yu, M.E. Flatté, Phys. Rev. B 66, 235302 (2002) 17. Y. Yafet, Phys. Rev. 85, 478 (1952) 18. R.J. Elliott, Phys. Rev. 96, 266 (1954) 19. Y. Yafet, in Solid State Physics, vol. 14, ed. by F. Seitz, D. Turnbull (Academic Press, New York, 1963), p. 2 20. M.I. D’yakonov, V.I. Perel’, Fiz. Tverd. Tela (Leningrad) 13, 3581 (1971) [Sov. Phys. Solid State 13, 3023 (1971)] 21. M.I. Dyakonov, in Spin Physics in Semiconductors, ed. by M.I. Dyakonov (Springer, Berlin, 2008), Chap. 1 22. T. Korn, Phys. Rep. 494, 415 (2010) 23. D. Csontos, S.E. Ulloa, Phys. Rev. B 74, 155207 (2006) 24. R. Landauer, Z. Phys. B: Condens. Matter 68, 217 (1987) 25. R. Landauer, J. Phys.: Condens. Matter 1, 8099 (1989) 26. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cam- bridge, 1995) 27. Y. Imry, R. Landauer, Rev. Mod. Phys. 71, S306 (1999) 28. C. Jacoboni, Theory of Electron Transport in Semiconductors. Springer Series in Solid-State Sciences, vol. 165 (Springer, Berlin, 2010) 29. Yu.V. Nazarov, Ya.M. Blanter, Quantum Transport: Introduction to Nanoscience (Cambridge University Press, Cambridge, 2009) 30. C. Jacoboni, L. Reggiani, Rev. Mod. Phys. 55, 645 (1983) 31. C. Jacoboni, P. Lugli,The Monte Carlo Method for Semiconductor Device Simulation (Springer, Wien, 1989) Chapter 5 Thermoballistic Approach: Implementation Havingpresented, in thepreceding chapter, the concept underlying the thermoballistic approach, we now turn to the implementation of this concept. We begin by estab- lishing the physical conditions from which the algorithms for calculating the two dynamical functions, viz., the average chemical potential (via the mean spin func- tion) and the spin accumulation function, are developed. Thereafter, these algorithms are described in detail. The mean spin function is determined in terms of two “resis- tance functions”, which are obtained as solutions of Volterra-type integral equations. For the spin accumulation function, a Fredholm-type integral equation is derived. 5.1 Physical Conditions Determining the Dynamical Functions We call physical conditions (i) a relation introduced to connect the total thermoballis- tic current J+(x) with the cognate total physical current J , which allows us to calcu- late the mean spin function Ã(x), and (ii) an assumption concerning the detailed spin relaxation mechanism, which leads to the determination of the spin accumulation function A(x). 5.1.1 Determination of the Mean Spin Function The current J+(x) ∗ J+(x1, x2; x; l), owing to its construction in terms of ballistic currents averaged over random ballistic configurations (with weights controlled by the momentum relaxation length l), is to be interpreted as an “ensemble average” of the electron current at the point x. This average is spatially varying, and, therefore, it is the spatial average of the ensemble average over the length of the sample which is to be identified with the (constant) total physical current J+ inside the sample. By current conservation at x1 and x2, this current is equal to the total current J in the left and right leads, so that we have R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 83 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_5, © Springer International Publishing Switzerland 2014 84 5 Thermoballistic Approach: Implementation 1 x2 − x1 x2∫ x1 dxJ+(x1, x2; x; l) = J+ ∗ J. (5.1) In this condition, the current J+(x1, x2; x; l) is defined at a ballistic point x located inside an ensemble of ballistic intervals [x≡, x≡≡], where x≡ and x≡≡, unlike x, are points of local thermodynamic equilibrium [see Eqs. (4.75) and (4.76)]. In Eq. (5.1), the integration over the ballistic coordinate x starts at the fixed equilibrium point x1 and ends at the fixed equilibrium point x2. Now, just as we have earlier introduced the ballistic reference coordinate x, we turn here to consider an equilibrium reference coordinate, i.e., a point of local thermodynamic equilibrium anywhere inside the sample, which, again, is denoted by x (x1 < x < x2) (see also Sect. 4.3.4.1). Such a point acts in the same way as the fixed equilibrium points x1 and x2. The ballistic current entering x, say, from the left, is completely absorbed, whereupon a thermal current is instantaneously emitted to either side of x. The same happens to the current entering the equilibrium point x from the right. However, in contrast to the “true”, externally controlled equilibrium points x1,2, which are located at the contact side of the contact-semiconductor interfaces, no Sharvin-type interface resistance (see Sect. 5.2.4 below) appears at x. Within this scheme, we introduce the thermoballistic current J(1)+ (x1, x; ξ; l), with x1 < ξ < x ↑ x2, where x is an equilibrium point, and ξ a ballistic point. In conformance with Eq. (5.1), the spatial average of this current over the range [x1, x] is again, by current conservation at x1, equal to the physical current in the left lead, so that 1 x − x1 x∫ x1 dξJ(1)+ (x1, x; ξ; l) = J. (5.2) Analogously, using similar arguments as above, we have for the thermoballistic current J(2)+ (x, x2; ξ; l) referred to the range [x, x2] 1 x2 − x x2∫ x dξJ(2)+ (x, x2; ξ; l) = J, (5.3) where x1 ↑ x < ξ < x2. Equations (5.2) and (5.3), when expressed in terms of the mean spin function Ã(x) via Eqs. (4.67) and (4.76), lead to two different Volterra-type integral equations [1] with solutions Ã1(x) and Ã2(x), respectively. Trivially, by sat- isfying Eqs. (5.2) and (5.3), these solutions also satisfy condition (5.1). They define, for given spin accumulation function A(x), two different total equilibrium densities n(1)+ (x) and n (2) + (x), respectively, via Eq. (4.85). The total equilibrium density at x is obtained as the mean value of these, n+(x) = 1 2 [ n(1)+ (x)+ n(2)+ (x) ] , (5.4) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.1 Physical Conditions Determining the Dynamical Functions 85 so that, using Eq. (4.85) again, we have for the unique (thermoballistic) mean spin function Ã(x) ∗ eβμ̃(x) = 1 2 [Ã1(x)+ Ã2(x)]. (5.5) The physical conditions (5.2) and (5.3), together with Eq. (5.4), determine the pro- cedure for calculating the average chemical potential μ̃(x). Using the mean value (5.4) as the point of departure for constructing μ̃(x) reflects the fact that this function expresses an intrinsic property of the semiconducting sample, with no preference for one or the other of the sample ends at x1 and x2. With Ã(x) given by Eq. (5.5), it follows from Eqs. (4.67) and (4.76) that the unique total thermoballistic current J+(x1, x2; x; l) is given by J+(x1, x2; x; l) = 1 2 [ J (1) + (x1, x2; x; l)+ J(2)+ (x1, x2; x; l) ] , (5.6) a symmetric combination as in Eq. (5.5). This completes the construction of a uniquemean spin function (or average chem- ical potential) and of a unique total thermoballistic current. We note that in Ref. [2] we have applied a different procedure. There, the combination of the functions Ã1(x) and Ã2(x) was not chosen to be symmetric as in Eq. (5.5), but was determined by the requirement that the values of the unique total thermoballistic current at the two sample ends be equal, J+(x1, x2; x+1 ; l) = J+(x1, x2; x−2 ; l). (5.7) This condition derives from postulating that the position dependence of the total ther- moballistic current is compensated by that of a “background current” [2, 3], such that these two currents add up to the (conserved) total physical current. The back- ground current is assumed to be fed by sources and sinks whose effect averages out to zero when the current is integrated over the length of the sample. The introduction of a background current represents an ad hoc hypothesis which is abandoned in the present work. However, the numerical results obtained in Ref. [2] do not lose their significance, and are in qualitative agreement with those obtained by the present procedure. 5.1.2 Determination of the Spin Accumulation Function In the thermoballistic transport mechanism, we assume that spin relaxation takes place only inside the ballistic intervals as described in Sect. 4.2.3, and that an infin- itesimal shift of the end-points of the ballistic intervals does not affect the current J̌−(x). Accordingly, we set the term arising from differentiating the limits of inte- gration in expression (4.76) for J̌−(x) equal to zero, Ď−(x) = 0. (5.8) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 86 5 Thermoballistic Approach: Implementation From Eq. (4.83), the relaxing thermoballistic spin-polarized current and density are then seen to be connected by the balance equation d dx J̌−(x)+ 2ve ls ň−(x) = 0, (5.9) which is of the same form as the balance equation (4.74) connecting the relaxing spin-polarized current and density in the individual ballistic intervals. The spin accumulation functionA(x), and hence the relaxing thermoballistic spin- polarized current J̌−(x), are determined by condition (5.8). When written in terms of A(x) by using expression (4.70) in Eq. (4.79), this condition turns into a linear, inhomogeneous Fredholm-type integral equation of the second kind [1] for A(x). The explicit form of this equation and its conversion into a differential equation in a specific case of particular importance, viz., field-driven transport in homogeneous semiconductors, will be the subject of Sect. 5.3. 5.2 Average Chemical Potential The average chemical potential μ̃(x) is determined, via expression (5.5) for the mean spin function Ã(x), by the solutions Ã1,2(x) of the integral equations (5.2) and (5.3), respectively, which are conveniently solved in terms of “resistance functions”. 5.2.1 The Resistance Functions We begin by considering Eq. (5.2). To solve it for the function Ã1(x), we define functions Ã1(x) and A(x) via Ã1(x) = veNc J Bm+(x1, x2) 2 Ã1(x) (5.10) and A(x) = veNc J Bm+(x1, x2) 2 A(x), (5.11) where the spin accumulation function A(x) is assumed to be given. Furthermore, we introduce the “resistance function” R̃1(x) (the choice of this name will be substan- tiated in Sect. 5.2.5 below) as R̃1(x) = Ã1(x1)− Ã1(x), (5.12) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.2 Average Chemical Potential 87 along with the function R1(x) = 1 2 [A(x1)− A(x)] (5.13) [for given A(x), R1(x) is a given function as well]. 5.2.1.1 Integral Equation for the Function R̃1(x) Expressing the net total ballistic current J+(x≡, x≡≡) given by Eq. (4.67) in terms of the functions R̃1(x) and R1(x), we obtain from Eq. (4.76) the total thermoballistic current J(1)+ (x1, x; ξ; l) to be inserted in Eq. (5.2) in the form J (1) + (x1, x; ξ; l) = J̃(1)+ (x1, x; ξ; l)+ J(1)+ (x1, x; ξ; l). (5.14) Here, J̃ (1) + (x1, x; ξ; l) = J { w+(x1, x; l)R̃1(x) + ξ−∫ x1 dx≡ l w+(x≡, x; l)[R̃1(x)− R̃1(x≡)] + x∫ ξ+ dx≡≡ l w+(x1, x≡≡; l)[R̃1(x≡≡)− R̃1(x1)] + ξ−∫ x1 dx≡ l x∫ ξ+ dx≡≡ l w+(x≡, x≡≡; l)[R̃1(x≡≡)− R̃1(x≡)] } , (5.15) while the term J(1)+ (x1, x; ξ; l) is obtained from expression (5.15) by replacing R̃1(x) with R1(x) throughout, and w+(x≡, x≡≡; l) with w−(x≡, x≡≡; l), where w±(x≡, x≡≡; l) = e−|x≡−x≡≡|/l B m±(x≡, x≡≡) Bm+(x1, x2) ∗ w±(x≡≡, x≡; l) (5.16) [see Eq. (4.22)]. To evaluate condition (5.2), we use the relation (as applied to expres- sion (5.15), the integration limits ξ± may be replaced with ξ) x∫ x1 dξ ξ∫ x1 dx≡ x∫ ξ dx≡≡F(x≡, x≡≡) = x∫ x1 dx≡ x∫ x≡ dx≡≡(x≡≡ − x≡)F(x≡, x≡≡) (5.17) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 88 5 Thermoballistic Approach: Implementation to carry out the integration over ξ, so that the condition can be expressed in the explicit form K+(x1, x; x; l)R̃1(x)+ x∫ x1 dx≡ l K+(x1, x; x≡; l)R̃1(x≡) = −Ω(1)− (x1, x; l). (5.18) Here, the inhomogeneity Ω(1)− (x1, x; l) is given by Ω (1) − (x1, x; l) = x − x1 l + K−(x1, x; x; l)R1(x) + x∫ x1 dx≡ l K−(x1, x; x≡; l)R1(x≡). (5.19) In terms of arbitrary coordinates ζ ≡, ζ ≡≡ such that ζ ≡ ↑ x≡ ↑ ζ ≡≡, the integral kernels K±(ζ ≡, ζ ≡≡; x≡; l) are written in the form K±(ζ ≡, ζ ≡≡; x≡; l) = u±(x≡, ζ ≡≡; l)− u±(ζ ≡, x≡; l)+ ζ ≡≡∫ ζ ≡ dx≡≡ l u±(x≡, x≡≡; l), (5.20) with u±(x≡, x≡≡; l) = x ≡≡ − x≡ l w±(x≡, x≡≡; l) ∗ −u±(x≡≡, x≡; l) (5.21) [see Eq. (4.22)]. Equation (5.18) is a linear, inhomogeneous, Volterra-type integral equation of the second kind [1] for the resistance function R̃1(x), defined in the range x1 < x ↑ x2. The kernelsK±(x1, x; x≡; l) depend on the potential energy profiles E≈,∞(x) solely via the quantities Bm±(x≡, x≡≡), and are independent of the total physical current J . Nonetheless, for nonzero spin splitting, a J-dependence of the resistance function R̃1(x) can arise, via the function R1(x), from a nonlinear J-dependence of the spin accumulation function A(x) (see Sects. 5.3 and 5.4 below). In the low-field regime, when A(x) is proportional to J , the functionR1(x), and hence R̃1(x), become inde- pendent of J . For zero spin splitting, when Bm−(x≡, x≡≡) = 0, we have Ω (1) − (x1, x; l) = x − x1 l , (5.22) and R̃1(x) does not depend on A(x). The resistance function R̃1(x) is discontinuous at x = x1: we have R̃1(x1) = 0 (5.23) http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.2 Average Chemical Potential 89 from Eq. (5.12), whereas we obtain R̃1(x + 1 ) = Bm+(x1, x2) B+(x1) − P(x1)R1(x+1 ) (5.24) by expandingEq. (5.18) to first order in x−x1 (we assume the potential energy profiles E≈,∞(x) to be continuous in the interval [x1, x2]). For arbitrary functions Ec(x) and Δ(x), the calculation of the resistance function R̃1(x) for a chosen parameter set consists of three, consecutive steps: (i) calculation of the spin accumulation function A(x) as solution of the integral equation (5.86) [see Sect. 5.3 below], using the boundary conditions (4.99) with given values A1,2 at the contact-semiconductor interfaces; (ii) with A(x) as input in expression (5.13), calcu- lation of the inhomogeneityΩ(1)− (x1, x; l) fromEq. (5.19); (iii) usingΩ(1)− (x1, x; l) in the Volterra equation (5.18), calculation of R̃1(x) by, in general, numerical methods. 5.2.1.2 Integral Equation for the Function R̃2(x) We now turn to the determination of the function Ã2(x) solving Eq. (5.3). Proceeding as for Ã1(x), we define a function Ã2(x) by Ã2(x) = veNc J Bm+(x1, x2) 2 Ã2(x). (5.25) In analogy to Eq. (5.18) for the resistance function R̃1(x), we then introduce a resis- tance function R̃2(x) as R̃2(x) = Ã2(x)− Ã2(x2), (5.26) for which we obtain from Eq. (5.3) a Volterra-type integral equation in the range x1 ↑ x < x2, K+(x, x2; x; l)R̃2(x)+ x2∫ x dx≡ l K+(x, x2; x≡; l)R̃2(x≡) = Ω(2)− (x, x2; l). (5.27) The inhomogeneity Ω(2)− (x, x2; l) is expressed in terms of the function R2(x) = 1 2 [A(x)− A(x2)] (5.28) as Ω (2) − (x, x2; l) = x2 − x l + K−(x, x2; x; l)R2(x) + x2∫ x dx≡ l K−(x, x2; x≡; l)R2(x≡), (5.29) http://dx.doi.org/10.1007/978-3-319-05924-2_4 90 5 Thermoballistic Approach: Implementation and the kernels K±(x, x2; x≡; l) are given by Eq. (5.20). The resistance function R̃2(x) is discontinuous at x = x2, R̃2(x2) = 0, (5.30) R̃2(x − 2 ) = Bm+(x1, x2) B+(x2) − P(x2)R2(x−2 ), (5.31) in analogy to the discontinuity of the function R̃1(x) at x = x1 [see Eqs. (5.23) and (5.24)]. As to the calculation of R̃2(x) from Eq. (5.27), the foregoing discussion regarding the calculation of R̃1(x) from Eq. (5.18) applies mutatis mutandis. Alternatively, one may calculate R̃2(x) by using the relation R̃2(x) = R̃∗1(x1 + x2 − x), (5.32) where R̃∗1(x) is the solution of Eq. (5.18) corresponding to spatially reversed potential energy profiles, E∗≈,∞(x) = E≈,∞(x1 + x2 − x), (5.33) using as input a spin accumulation function A(x) calculated from Eq. (5.86) below, with the reversed profiles and with the boundary values A1 and A2 interchanged. 5.2.2 Resistance Functions for Constant Spin Splitting The calculation of the resistance functions R̃1(x) and R̃2(x) simplifies considerably if the spin splitting is constant over the sample, Δ(x) ∗ Δ [while Ec(x) still may be arbitrary]. With E≈,∞(x) = Ec(x)± Δ 2 , (5.34) we have Em≈,∞(x≡, x≡≡) = Emc (x≡, x≡≡)± Δ 2 , (5.35) and hence for the functions Bm±(x≡, x≡≡) from Eqs. (4.25) and (4.26), using Eqs. (4.3)–(4.6), Bm+(x≡, x≡≡) = 1 Q Bm0 (x ≡, x≡≡) (5.36) and Bm−(x≡, x≡≡) = PBm+(x≡, x≡≡) = P Q Bm0 (x ≡, x≡≡). (5.37) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.2 Average Chemical Potential 91 Here, we have defined Bm0 (x ≡, x≡≡) = 2e−βEmc (x≡,x≡≡) (5.38) and Q ∗ (1− P2)1/2 = 1 cosh(βΔ/2) , (5.39) and P = − tanh(βΔ/2) (5.40) is the static spin polarization. Considering, for instance, the calculation of R̃1(x), we find from Eqs. (5.20) for the kernels K±(x1, x; x≡; l), using expressions (5.36) and (5.37) in Eq. (5.16), K+(x1, x; x≡; l) = 1 Q K0(x1, x; x≡; l) (5.41) and K−(x1, x; x≡; l) = PK+(x1, x; x≡; l) = P Q K0(x1, x; x≡; l), (5.42) where K0(x1, x; x≡; l) = K+(x1, x; x≡; l) ∣∣ Δ=0 . (5.43) Now, inserting expressions (5.41) and (5.42) in Eqs. (5.18) and (5.19), respectively, we can rewrite the integral equation (5.18) in the form K0(x1, x; x; l)R̃10(x)+ x∫ x1 dx≡ l K0(x1, x; x≡; l)R̃10(x≡) = −x − x1 l (5.44) (x1 < x ↑ x2), where R̃10(x) = 1 Q [R̃1(x)+ PR1(x)] (5.45) is a “reduced resistance function”, and the function R1(x) is independent of Δ (see Sect. 5.3.1 below). As a solution of Eq. (5.44), the function R̃10(x) is a universal function that describes an intrinsic property of the semiconducting sample; it is deter- mined by the potential energy profile Ec(x) and the momentum relaxation length l, and does not depend on the spin splittingΔ and the spin accumulation function A(x). The resistance function R̃1(x) for constant spin splitting can now be expressed as R̃1(x) = QR̃10(x)− PR1(x). (5.46) Its calculation here separates, in general, into two, independent steps: (i) solution of Eq. (5.44) for R̃10(x); (ii) calculation of A(x) as solution of the integral equation 92 5 Thermoballistic Approach: Implementation (5.86) and determination of R1(x) via Eqs. (5.11) and (5.13). In particular, we have R̃1(x) = R̃10(x) (5.47) for zero spin splitting. Analogously, the resistance function R̃2(x) is, for constant spin splitting, deter- mined in terms of the reduced resistance function R̃20(x) as R̃2(x) = QR̃20(x)+ PR2(x), (5.48) where R̃20(x) obeys the integral equation K0(x, x2; x; l)R̃20(x)+ x2∫ x dx≡ l K0(x, x2; x≡; l)R̃20(x≡) = x2 − x l (5.49) (x1 ↑ x < x2), which is obtained as a special case of Eq. (5.27). For the case of field-driven transport in homogeneous semiconductors with con- stant spin splitting, we present in Appendix A.1 closed-form expressions for the kernels K0(x1, x; x; l) and K0(x, x2; x; l). 5.2.3 Thermoballistic Average Chemical Potential Introducing the mean value Ã(x) of the functions Ã1,2(x), which both satisfy condi- tion (5.1), Ã(x) = 1 2 [Ã1(x)+ Ã2(x)] ∗ veNc J Bm+(x1, x2) 2 Ã(x) (5.50) [seeEq. (5.5)],weobserve that Ã(x) satisfies condition (5.1) aswell.UsingEqs. (5.12) and (5.26), we can express Ã(x) in the form Ã(x) = 1 2 [Ã1(x1)+ Ã2(x2)] − 1 2 [R̃1(x)− R̃2(x)]. (5.51) Now, writing down this expression for x = x1 and x = x2, respectively, and using Eqs. (5.10), (5.23), (5.25), and (5.30) together with the boundary conditions (4.98), we add the resulting two expressions to obtain for the quantity η+12 defined by Eq. (4.111) η+12 = J veNc 2 Bm+(x1, x2) { Ã1(x1)+ Ã2(x2)− 1 2 [R̃1(x2)− R̃2(x1)] } . (5.52) On the other hand, subtracting the two expressions, we have http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.2 Average Chemical Potential 93 η−12 = J veNc 2R̃ Bm+(x1, x2) , (5.53) with the parameter R̃, the “reduced resistance”, defined as R̃ = 1 2 [R̃1(x2)+ R̃2(x1)]. (5.54) Then, using Eqs. (5.52) and (5.53), we can eliminate from expression (5.51) the dependence on the boundary values Ã1(x1) and Ã2(x2) of the functions Ã1,2(x). Introducing the function R̃−(x) = R̃1(x)− 1 2 R̃1(x2)− [ R̃2(x)− 1 2 R̃2(x1) ] , (5.55) so that, in view of Eqs. (5.23) and (5.30), R̃−(x1,2) = ≥R̃, (5.56) we can express the mean spin function Ã(x) as Ã(x) ∗ J veNc 2 Bm+(x1, x2) Ã(x) = η + 12 2 − R̃−(x) 2R̃ η−12 (5.57) for x1 < x < x2, and Ã(x1) = η1, Ã(x2) = η2, (5.58) with η1,2 defined by Eq. (4.98). Inserting expression (5.53) for η − 12 in Eq. (5.57), we obtain Ã(x) = η + 12 2 − J veNc R̃−(x) Bm+(x1, x2) , (5.59) where the dependence on the reduced resistance R̃ has dropped out. 5.2.3.1 General Expression for µ̃(x) Now, using Eqs. (4.14) and (4.111), we can rewrite Eq. (5.57) in the form eβμ̃(x) = 1 2 (eβμ1 + eβμ2)− R̃−(x) 2R̃ (eβμ1 − eβμ2) (5.60) for x1 < x < x2, while μ̃(x1) = μ1, μ̃(x2) = μ2. (5.61) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 94 5 Thermoballistic Approach: Implementation This is the final, general expression determining the (local) thermoballistic average chemical potential μ̃(x) [henceforth, we drop the attribute “thermoballistic” when referring to μ̃(x)]. It is seen to be uniquely determined by the values of the external parameters at the contact sides of the contact-semiconductor interfaces, viz., the values μ1,2 of the equilibrium chemical potential and the values A1,2 of the spin accumulation function [which enter via the resistance functions R̃1(x) and R̃2(x)]. Comparing expression (5.60) to expression (3.61) for the chemical potential μ(x) in the prototype thermoballistic model, we observe a formally identical structure, but differences in the explicit forms of the functions R−(x) and R−(x) and of the reduced resistances R and R. 5.2.3.2 Ballistic Regime An explicit expression for μ̃(x) can be obtained when l/S 1 (ballistic regime), in which case the probability for an electron to suffer a collision when traversing the sample of length S, 1−e−S/l √ S/l, is vanishingly small. Then, keeping terms of order S/l, we have from Eq. (5.18) R̃1(x) = B m+(x1, x2) Bm+(x1, x) − Pm(x1, x)R1(x), (5.62) and similarly from Eq. (5.27) R̃2(x) = B m+(x1, x2) Bm+(x, x2) − Pm(x, x2)R2(x). (5.63) Since from Eqs. (4.99), (5.10), (5.13) and (5.28) R1(x2) = R2(x1) = veNc 2J Bm+(x1, x2) 2 A−12, (5.64) we now find from Eq. (5.54) R̃ = 1− veNc 2J Bm+(x1, x2) 2 Pm(x1, x2)A − 12. (5.65) For the function R̃−(x), we have from Eq. (5.55), using Eq. (5.64), R̃−(x) = R̃1(x)− R̃2(x). (5.66) With expressions (5.65) and (5.66) inserted in Eq. (5.60), we obtain the average chemical potential μ̃(x) in explicit form. In the particular case of zero spin splitting, when Bm+(x≡, x≡≡) = 2e−βEmc (x≡,x≡≡) and Pm(x≡, x≡≡) = 0, we have http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.2 Average Chemical Potential 95 R̃−(x) R̃ = e−βEmc (x1,x2) [ eβE m c (x1,x) − eβEmc (x,x2) ] . (5.67) Inserting this expression in Eq. (5.60), we see that in the ballistic regime, l/S 1, the chemical potential μ̃(x) is well defined in the entire interval x1 < x < x2 and takes on a definite value in the limit l ≤ →. This result contrasts with the strict ballistic limit, l ≤ →, considered in Sect. 4.3.3, when there is absolutely no point of local thermodynamic equilibrium inside the sample. In this extreme case, no meaning can be attached to an average chemical potential, and there is nothing a calculation of such a quantity can be based upon. 5.2.4 Sharvin Interface Resistance The discontinuities in the resistance functions R̃1(x) and R̃2(x) at the points of local thermodynamic equilibrium x = x1 and x = x2, respectively, give rise to discontinuities in the function R̃−(x) both at x = x1 and x = x2. From Eq. (5.55), we find, using Eqs. (5.23) and (5.24), R̃−(x+1 )− R̃−(x1) = R̃1(x+1 ) = Bm+(x1, x2) B+(x1) − P(x1)R1(x+1 ), (5.68) and similarly, using Eqs. (5.30) and (5.31), R̃−(x2)− R̃−(x−2 ) = R̃2(x−2 ) = Bm+(x1, x2) B+(x2) − P(x2)R2(x−2 ). (5.69) Therefore, according to Eq. (5.60), the average chemical potential μ̃(x) exhibits dis- continuities at these points as well (“Sharvin effect”; see Ref. [4]), eβ[μ̃(x + 1 )−μ1] − 1 = −η − 12e −βμ1 2 R̃1(x + 1 ) R̃ = −1 2 βe2Jρ̃1, (5.70) eβ[μ̃(x − 2 )−μ2] − 1 = η − 12e −βμ2 2 R̃2(x − 2 ) R̃ = 1 2 βe2Jρ̃2, (5.71) where we have used Eqs. (5.53) and (5.54) to introduce the total physical current J . The quantities ρ̃1,2 are the Sharvin interface resistances [2, 4], ρ̃1 = 2 βe2veNcBm+(x1, x2)eβμ1 R̃1(x + 1 ), (5.72) ρ̃2 = 2 βe2veNcBm+(x1, x2)eβμ2 R̃2(x − 2 ). (5.73) http://dx.doi.org/10.1007/978-3-319-05924-2_4 96 5 Thermoballistic Approach: Implementation For zero spin splitting, when P(x1,2) = 0, so that R̃1(x + 1 ) = Bm+(x1, x2)/B+(x1) (5.74) and R̃2(x − 2 ) = Bm+(x1, x2)/B+(x2), (5.75) we can express ρ̃1,2 in the form ρ̃1,2 = 1 βe2ven (0) + (x1,2) , (5.76) where n(0)+ (x1,2) = Nc 2 B+(x1,2)eβμ1,2 = Nce−β[Ec(x1,2)−μ1,2] (5.77) are the total equilibrium electron densities at the semiconductor side of the contact- semiconductor interfaces [see Eqs. (4.19) and (4.85)]. Equations (5.70) and (5.71), respectively, are analogous to Eqs. (3.63) and (3.64) for the discontinuities of the chemical potential μ(x) in the prototype thermoballistic model. 5.2.5 Current-Voltage Characteristic and Magnetoresistance The current-voltage characteristic of the thermoballistic transport model is obtained from Eq. (5.53), using Eq. (2.35), in the form J = veNc 2 Bm+(x1, x2)eβμ1 1 R̃ (1− e−βeV ). (5.78) In the low-field regime, we then have R ∗ V eJ ∣∣∣∣ J≤0 = 2 βe2veNcBm+(x1, x2)eβμ1 R̃ = ρ̃1 R̃ R̃1(x + 1 ) (5.79) for the resistance times cross-sectional area of the sample. Comparing the current- voltage characteristic (5.78) to that of the (spinless) prototype thermoballistic model, Eq. (4.25), one sees that the (dimensionless) parameter R̃ directly corresponds to the reduced resistance R of the prototype model, Eq. (3.24). Therefore, R̃ is here also called the reduced resistance of the sample, and the functions R̃1(x) and R̃2(x) from which it is derived, the resistance functions. The reduced resistance R̃ is a central element of the thermoballistic description of electron transport in semiconductors. It comprises the effect of the detailed shape of the potential energy profiles as well as that of the momentum relaxation length. http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_3 5.2 Average Chemical Potential 97 Moreover, via its dependence on the spin accumulation function A(x), it takes into account the effect of spin relaxation. The dependence on A(x) gives rise, in general, to a dependence of R̃ on the total current J , in addition to its dependence on the voltage V arising from the external electric field (see Sect. 3.3.4), R̃ ∗ R̃(J, V ). (5.80) In the ballistic regime, R̃ is given by expression (5.65), whereas the explicit form of R̃ in the drift-diffusion regime can be obtained by rewriting Eq. (4.96) for x = x2 in the form of a current-voltage characteristic and comparing it to the characteristic (5.78). Defining now the relative magnetoresistance Rm of a semiconducting sample [5, 6] as Rm ∗ R − R0 R0 , (5.81) where R0 is the resistance at zero external magnetic field, i.e., at zero spin splitting, we have from Eq. (5.79) Rm = R̃− R̃0 R̃0 . (5.82) The reduced resistance at zero spin splitting, R̃0, is obtained by solving Eq. (5.44) for the function R̃10(x) and the analogous equation for R̃20(x) and using Eq. (5.54), R̃0 = 1 2 [R̃10(x2)+ R̃20(x1)]. (5.83) For constant splitting, we find from Eq. (5.54), using Eq. (5.46) and the analogous equation for R̃2(x) as well as Eq. (5.64), R̃ = QR̃0 − veNc 4J Bm+(x1, x2)PA−12, (5.84) and hence Rm = Q − 1− veNc 4JR̃0 Bm+(x1, x2)PA−12 (5.85) for the magnetoresistance. 5.3 Spin Accumulation Function The net relaxing ballistic spin-polarized current J̌−(x≡, x≡≡; x) [see Eq. (4.70)] and the joint relaxing ballistic spin-polarized density ň−(x≡, x≡≡; x) [see Eq. (4.73)], and thus also the corresponding thermoballistic current J̌−(x) and density ň−(x) evaluated http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 98 5 Thermoballistic Approach: Implementation fromEq. (4.76), are dynamically determined solely by the spin accumulation function A(x). As seen in Sect. 5.2, the calculation of this function is, in general, prerequisite to a complete determination of the average chemical potential μ̃(x). 5.3.1 Integral Equation Following Sect. 5.1.2, we invoke condition (5.8), which is equivalent to the spin balance equation (5.9), and insert expression (4.70) for the quantity F(x≡, x≡≡; x) in Eq. (4.79) [here, the integration limits x± in Eqs. (4.80) and (4.81) may be replaced with x] to obtain for the spin accumulation function A(x) an integral equation of the form W̌(x1, x; l, ls)A1 + W̌(x, x2; l, ls)A2 − W̌ (x; x1, x2; l)A(x) + x2∫ x1 dx≡ l W̌(x≡, x; l, ls)A(x≡) = 0, (5.86) where W̌(x≡, x≡≡; l, ls) = w̌(x≡, x≡≡; l)e−Cm(x≡,x≡≡)/ls (5.87) and W̌ (x; x1, x2; l) = w̌(x1, x; l)+ w̌(x, x2; l)+ x2∫ x1 dx≡ l w̌(x≡, x; l). (5.88) Here, the values A(x1,2) of the spin accumulation function at the interface positions x1,2 have been set equal to the external values, A1,2, at the contact side of the contact- semiconductor interfaces [see Eq. (4.99)]. The function Cm(x≡, x≡≡) is defined as Cm(x≡, x≡≡) ∗ Cm(x≡, x≡≡; x≡, x≡≡) (5.89) [see Eq. (4.59)], and w̌(x≡, x≡≡; l) = e−|x≡−x≡≡|/l B m(x≡, x≡≡) Bm+(x1, x2) ∗ w̌(x≡≡, x≡; l) (5.90) [see Eq. (4.22)] with Bm(x≡, x≡≡) given by Eq. (4.33). With the first two terms in Eq. (5.86) acting as an inhomogeneity, this equation is a linear, inhomogeneous, Fredholm-type integral equation of the secondkind [1] for the spin accumulation functionA(x). The correspondinghomogeneous equation is solved by A(x) ∗ 0 only, so that the unique solution A(x) of Eq. (5.86) for x1 < x < x2 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.3 Spin Accumulation Function 99 is linear and homogeneous in A1 and A2. Just like the average chemical potential μ̃(x), the spin accumulation function A(x) exhibits the Sharvin effect, i.e., it is not, in general, continuous at the interfaces with the contacts, ΔA1 ∗ A(x+1 )− A1 = 0, (5.91) ΔA2 ∗ A2 − A(x−2 ) = 0. (5.92) This will be demonstrated in Sect. 5.3.2.4 below by way of a specific example. The solution of Eq. (5.86) is, in general, found numerically by applying matrix methods after discretization. For constant spin splitting, Δ(x) ∗ Δ, when the functions Em≈,∞(x≡, x≡≡) are given by Eq. (5.35), the function Bm(x≡, x≡≡) defined by Eq. (4.33) reduces to Bm(x≡, x≡≡) = QBm0 (x≡, x≡≡), (5.93) with Bm0 (x ≡, x≡≡) given by Eq. (5.38), and Q by Eq. (5.39). Further, using Eqs. (5.34) and (5.35) in expression (4.41), we see that the splittingΔ cancels out in the functions Cm≈,∞ (x≡, x≡≡; x), Cm≈,∞(x≡, x≡≡; x) ∗ Cm0 (x≡, x≡≡; x). (5.94) Inserting this in Eq. (4.40), we then find, via Eqs. (4.46), (4.57) and (4.59), the func- tion Cm(x≡, x≡≡) defined by Eq. (5.89) to be independent of Δ, Cm(x≡, x≡≡) ∗ Cm0 (x≡, x≡≡). (5.95) With this result, in conjunction with Eqs. (5.90) and (5.93), used in Eqs. (5.87) and (5.88), the spin splittingΔ is seen to drop out from the integral equation (5.86), i.e., its solution A(x) does not depend on Δ. 5.3.2 Differential Equation for Field-Driven Transport For field-driven transport in homogeneous semiconducting sampleswithout Schottky barriers, the integral equation for A(x) can be converted into a differential equation. 5.3.2.1 Specializing the Integral Equation Assuming the external field to be directed antiparallel to the x-axis, the potential energy profile (4.2) for zero spin splitting,Δ(x) = 0, has the form (2.41). [According to the above discussion, the inclusion of a nonzero, constant spin splitting would not alter the results.] For this profile, the function Cm(x≡, x≡≡; x) defined by Eq. (4.57) reads http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_4 100 5 Thermoballistic Approach: Implementation Cm(x≡, x≡≡; x) ∗ Cm(ε(x − x≡)) = eε(x−x≡)erfc([ε(x − x≡)]1/2) (5.96) (ε = βeE), as can be shown by using Eqs. (4.40), (4.41) and (4.46). Using expression (5.96), via Eq. (4.59), in Eq. (5.89), the integral equation (5.86) can now be reduced to f1(x − x1)A1 + f2(x2 − x)A2 − f (x − x1)A(x) + x∫ x1 dx≡ l f1(x − x≡)A(x≡)+ x2∫ x dx≡ l f2(x ≡ − x)A(x≡) = 0, (5.97) where f1(x) = e−[ε+1/l+c(εx)/ls]x, (5.98) f2(x) = e−[1/l+c(εx)/ls]x, (5.99) and f (x) = 1 1+ εl {2+ εl[1+ e −(ε+1/l)x]}. (5.100) For the function c(ζ) defined by c(ζ) = 1 ζ ζ∫ 0 dζ ≡Cm(ζ ≡), (5.101) the relations 0 < c(ζ) ↑ 1, (5.102) c(ζ) ≤ 1 for ζ ≤ 0, (5.103) and c(ζ) ∼ 2(πζ)−1/2 for ζ ≤ → (5.104) hold. 5.3.2.2 Converting the Integral Equation We can convert the inhomogeneous integral equation (5.97) into a homogeneous integrodifferential equation for A(x) by supplementing Eq. (5.97) with the equations obtained by forming its first and second derivativewith respect to x, and subsequently eliminating from this set of three equations the boundary values A1 and A2. [In principle, a similar procedure could also be applied to the general equation (5.86), but this does not seem to lead to any advantage.] If, in the integral equation (5.97), we http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.3 Spin Accumulation Function 101 replace the function c(ζ) with a position-independent average value c̄, the functions f1(x) and f2(x) become pure exponentials, and the corresponding integrodifferential equation reduces to a homogeneous second-order differential equation of the form b0(x) d2A(x) dx2 + b1(x)dA(x) dx + b2(x)A(x) = 0. (5.105) Here, b0(x) = 2+ εl[1+ b(x)], (5.106) b1(x) = ε(2+ εl)[1− b(x)], (5.107) and b2(x) = l̃ − l ll̃2 {2+ ε(l + l̃ + εll̃)[1+ b(x)]}, (5.108) where b(x) = e−(ε+1/l)(x−x1) (5.109) and 1 l̃ = 1 l + c̄ ls , (5.110) with the ballistic spin relaxation length ls given by Eq. (4.54). [Note that expression (5.108) for the function b2(x) differs from the corresponding expression (3.48) of Ref. [2], which was derived by introducing the average value c̄ in the integrodiffer- ential equation, rather than in the original integral equation.] Now, to construct a properly normalized function A(x), we have to link two arbitrarily normalized, lin- early independent solutions of Eq. (5.105) to the boundary values A1,2. The pertinent procedure is described in Appendix A.2. Since, owing to the presence of the factor e−x/l in the functions f1(x) and f2(x), only the values of c(εx) in the range 0 ↑ x ↑ l contribute appreciably, we may choose c̄ as the average of c(εx) over an x-interval of length equal to the momentum relaxation length l, c̄ = 1 l l∫ 0 dxc(εx) ∗ 1 εl εl∫ 0 dζ ln(εl/ζ)Cm(ζ). (5.111) In the right-hand integral in this equation, the range of small ζ, when Cm(ζ) √ 1, is emphasized because of the weight factor ln(εl/ζ). For large εl (i.e., for high fields and/or in the ballistic regime), the variation of c(εx) with x becomes essential. However, for the purpose of demonstrating the principal effects of the transport mechanism, the approximation in terms of a constant average value c̄ of the function c(ζ), in conjunction with the choice (5.111) for c̄, appears to be sufficiently accurate. http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_3 102 5 Thermoballistic Approach: Implementation 5.3.2.3 Drift-Diffusion Regime In the drift-diffusion regime, when l/S � 1, l/ls � 1, and εl � 1, we have c̄ = 1 and hence l̃ = l̄, where 1 l̄ = 1 l + 1 ls . (5.112) Equation (5.105) then reduces to d2A(x) dx2 + εdA(x) dx − 1 L2s A(x) = 0, (5.113) where Ls is the spin diffusion length given by Eq. (4.107). Setting B̂+(x) = −ε/β in Eq. (4.106), we find Eq. (5.113) to agree with the former equation, which was obtained by directly evaluating the thermoballistic current J̌−(x) and density ň−(x) in the drift-diffusion limit, and which leads to the standard form (4.108) of the drift- diffusion equation for spin-dependent transport. 5.3.2.4 Low-Field Regime In the low-field regime, εS � 1, the integral equation (5.97) reduces to the form e−(x−x1)/l̄A1 + e−(x2−x)/l̄A2 − 2A(x)+ x2∫ x1 dx≡ l e−|x−x≡|/l̄A(x≡) = 0, (5.114) from which we obtain the differential equation d2A(x) dx2 − 1 L2 A(x) = 0. (5.115) Here, L = √ l̄ls = Ls√ 1+ l/ls (5.116) is the generalization of the spin diffusion length Ls given by Eq. (4.107). The length L becomes equal to the latter length, L = Ls, in the drift-diffusion regime, when l/ls � 1, and to the ballistic spin relaxation length, L = ls, in the ballistic case, when l/ls ≤ → and hence l̄ = ls. The general solution of Eq. (5.115) for x1 < x < x2 reads A(x) = C1e−(x−x1)/L + C2e−(x2−x)/L. (5.117) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 5.3 Spin Accumulation Function 103 Specializing now the procedure described in Appendix A.2, we insert this expression for A(x) in Eq. (5.114) and set x = x+1 and x = x−2 , respectively, thereby obtaining two coupled linear equations for the coefficients C1,2, with the solution C1 = 1 D [(1+ γ)eS/LA1 − (1− γ)A2], (5.118) C2 = − 1 D [(1− γ)A1 − (1+ γ)eS/LA2]. (5.119) Here, D = 2[(1+ γ2) sinh(S/L)+ 2γ cosh(S/L)], (5.120) with γ = L ls = l̄ L = √ l l + ls ↑ 1. (5.121) It then follows from Eqs. (5.117)–(5.121) that A(x) is discontinuous at x = x1 and x = x2 (Sharvin effect; see Sect. 5.3.1), ΔA1 ∗ A(x+1 )− A1 = − 1 2 (gA1 − hA2), (5.122) ΔA2 ∗ A2 − A(x−2 ) = − 1 2 (hA1 − gA2), (5.123) where g = h[cosh(S/L)+ γ sinh(S/L)] ↑ 1, (5.124) with h = 4γ D ↑ 2 1+ γ e −S/L. (5.125) In the drift-diffusion regime, when L = Ls and γ � 1, we have A(x) = A1e−(x−x1)/Ls + A2e−(x2−x)/Ls , (5.126) and in the ballistic case, when L = ls and γ = 1, A(x) = 1 2 [A1e−(x−x1)/ls + A2e−(x2−x)/ls ]. (5.127) The discontinuity of A(x) at x = x1, for example, is ΔA1 = A2 exp(−S/Ls) (5.128) 104 5 Thermoballistic Approach: Implementation in the drift-diffusion regime, i.e., exponentially vanishing when l ≤ 0, and ΔA1 = 1 2 [−A1 + A2 exp(−S/ls)] (5.129) in the ballistic case. 5.4 Current and Density Spin Polarizations The position dependence of the current and density spin polarizations as well as the magnetoresistance are the physical quantities of principal interest in the study of spin-polarized electron transport in paramagnetic semiconducting systems. 5.4.1 Spin Polarizations in the Semiconductor In the thermoballistic approach, we define the persistent current spin polarization in the semiconducting sample, P̂J(x), in terms of the persistent thermoballistic spin- polarized current Ĵ−(x) and the total physical current J as P̂J(x) = Ĵ−(x) J (5.130) (x1 ↑ x ↑ x2). Analogously, the relaxing current spin polarization P̌J(x) is defined in terms of the relaxing thermoballistic spin-polarized current J̌−(x) and the total current J as P̌J(x) = J̌−(x) J . (5.131) For the total current spin polarization PJ(x), we then have PJ(x) ∗ P̂J(x)+ P̌J(x) = Ĵ−(x)+ J̌−(x) J . (5.132) [Note that in Refs. [2, 7], we have defined the current spin polarization in terms of the total thermoballistic current J+(x), rather than in terms of the total physical current J .] The persistent density spin polarization P̂n(x) and the relaxing density spin polar- ization P̌n(x) are introduced by replacing in expressions (5.130) and (5.131) the thermoballistic currents Ĵ−(x) and J̌−(x) with the respective densities n̂−(x) and ň−(x), and the current J in the denominator of those expressions with the thermobal- 5.4 Current and Density Spin Polarizations 105 listic joint total density n+(x). Hence, we obtain Pn(x) ∗ P̂n(x)+ P̌n(x) = n̂−(x)+ ň−(x) n+(x) (5.133) for the total density spin polarization Pn(x). In expression (5.132) for the total current spin polarization inside the semicon- ductor, the current Ĵ−(x) depends linearly on the reduced spin accumulation function Ă(x), and hence on the boundary values Ă1,2, via the dependence of the total ballistic current J+(x≡, x≡≡) on Ă(x), Ĵ−(x) ∗ Ĵ−(x; Ă1, Ă2) (5.134) [see Eqs. (4.67) and (4.69); here, we have assumed J+(x≡, x≡≡) to be expressed in terms of Ă(x), rather than A(x), using Eq. (4.16)]. The relaxing thermoballistic spin- polarized current J̌−(x) is determined by the corresponding net ballistic current J̌−(x≡, x≡≡; x) given by Eq. (4.70), which is a linear-homogeneous functional of Ă(x). Hence, J̌−(x) is linear and homogeneous in the boundary values Ă1,2, so that we can write J̌−(x) ∗ J̌−(x; Ă1, Ă2) = veNc 2 B+(x)[F1(x)eβμ1 Ă1 + F2(x)eβμ2 Ă2], (5.135) with (dimensionless) “formfactors”F1(x) andF2(x) (the prefactors in this expression have been introduced for later convenience). Then, we can express the values of the total current spin polarization at the semiconductor side of the interfaces at x1,2 as P(sc)J (x1,2) = P̂J(x1,2; Ă1, Ă2)+ veNcB+(x1,2) 2J × [F1(x1,2)eβμ1 Ă1 + F2(x1,2)eβμ2 Ă2]. (5.136) The boundary values Ă1,2 of the reduced spin accumulation function, which are undetermined as yet, are to be expressed in terms of the current J and the parameters of the system by matching the current spin polarization in the semiconductor to the corresponding polarization in the contacts (see Sect. 5.4.3 below). 5.4.2 Current Spin Polarization in the Contacts In order to keep the formulation sufficiently general, we assume the semiconducting sample to be connected to (semi-infinite) ferromagnetic metal contacts, which are treated as fully degenerate Fermi systems. In the left contact located in the range x < x1, the spin-resolved chemical poten- tials μ≈,∞(x) have the form [8, 9] http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 106 5 Thermoballistic Approach: Implementation μ≈,∞(x) = e 2J σ(l)+ (x1 − x)± cl σ(l)≈,∞ e−(x1−x)/L (l) s . (5.137) Here, L(l)s is the spin diffusion length, and σ (l) ≈,∞ are the (x-independent) spin-up and spin-down bulk conductivities, respectively. Setting x = x−1 in Eq. (5.137), we obtain cl = σ (l) ≈ σ (l) ∞ σ (l) + μ−(x−1 ). (5.138) With the spin-resolved currents J≈,∞(x) given by J≈,∞(x) = − σ (l) ≈,∞ e2 dμ≈,∞(x) dx , (5.139) we then find for the current spin polarization in the left contact, using J+(x) ∗ J , PJ(x) ∗ J−(x) J+(x) = Pl − Gl2e2J μ−(x − 1 )e −(x1−x)/L(l)s . (5.140) Here, Pl ∗ σ (l) − σ (l) + (5.141) is the bulk (current or density) spin polarization, and the parameter Gl = 4σ(l)≈ σ (l) ∞ σ (l) + L (l) s = σ (l) + L(l)s ( 1− P2l ) , (5.142) which has the dimension of interface conductance, characterizes the spin-dependent transport in the ferromagnet. For the current spin polarization in the right contact located in the range x > x2, we have PJ(x) = Pr + Gr 2e2J μ−(x+2 )e −(x−x2)/L(r)s , (5.143) which follows by replacing in Eq. (5.137) the coordinate x with x1 + x2 − x, and the labels “l” attached to the parameters in Eqs. (5.137)–(5.142) with “r”. When spin-selective interface resistances are absent, the chemical-potential split- ting μ−(x) is continuous at the interfaces, μ−(x−1 ) = μ(ct)− (x1), (5.144) μ(ct)− (x2) = μ−(x+2 ), (5.145) 5.4 Current and Density Spin Polarizations 107 where μ(ct)− (x1,2) are its values at the contact side of the interfaces. The latter values are to be identified with the corresponding values μ(sc)− (x1,2) at the semiconductor side of the interface, μ (ct) − (x1,2) = μ(sc)− (x1,2) = 1 β ln ( 1+ Ă1,2/2 1− Ă1,2/2 ) , (5.146) where the right-hand equations have been obtained by solving Eq. (4.17) for μ−(x≡), setting x≡ = x1,2, and defining Ă1,2 = Ă(x1,2) (5.147) for the boundary values of the reduced spin accumulation function Ă(x≡). With Eqs. (5.144)–(5.146) used in Eqs. (5.140) and (5.143), respectively, we now have for the current spin polarizations at the contact side of the interfaces P(ct)J (x1) = Pl − Gl 2βe2J ln ( 1+ Ă1/2 1− Ă1/2 ) , (5.148) P(ct)J (x2) = Pr + Gr 2βe2J ln ( 1+ Ă2/2 1− Ă2/2 ) , (5.149) which are to be identified with the corresponding polarizations P(sc)J (x1,2) at the semiconductor side of the interfaces [see Eq. (5.152) below]. Spin-selective interface resistances ρ(1,2)≈,∞ give rise to discontinuities of the spin- resolved chemical potentials on the contact sides of the interfaces [10–15]. At x = x1, for example, the discontinuity has the form μ≈,∞(x−1 )− μ(ct)≈,∞(x1) = e2J≈,∞(x1)ρ(1)≈,∞. (5.150) The interface resistances ρ(1)≈,∞ are located between x = x−1 and x1 in the ferromagnetic contact, adjacent to the Sharvin interface resistance ρ̃1 [see Eq. (5.72)] between x = x1 and x + 1 in the semiconductor. The quantity μ−(x − 1 ) to be substituted in Eq. (5.140) is obtained from Eqs. (5.137), (5.139) and (5.142) as μ−(x−1 ) = 1 1+ Glρ(1)+ /4 { μ(ct)− (x1)+ e2J 2 [ Plρ (1) + + ρ(1)− ]} . (5.151) The same procedure applies mutatis mutandis to the interface at x = x2. The connec- tion ofμ(ct)− (x1,2)with the interface values of the reduced spin accumulation function, Ă1,2, is again given by Eqs. (5.146). http://dx.doi.org/10.1007/978-3-319-05924-2_4 108 5 Thermoballistic Approach: Implementation 5.4.3 Matching the Current Spin Polarization at the Interfaces To evaluate the current and density spin polarizations all across the contact- semiconductor system, we have to relate the boundary values of the reduced spin accumulation function, Ă1,2, to the total physical current J and the internal para- meters characterizing the system. To this end, we exploit the continuity of the total current spin polarization PJ(x) across the contact-semiconductor interfaces at x1,2, P(ct)J (x1,2) = P(sc)J (x1,2). (5.152) Using expressions (5.148) and (5.149) for P(ct)J (x1,2) and expressions (5.136) for P(sc)J (x1,2), we obtain a pair of coupled nonlinear equations for Ă1,2, which we will not write down here in their general form. In the low-field regime, when |βμ−(x)| � 1, we have |Ă1,2| � 1 [see Eq. (4.17)], so that P(ct)J (x1) and P (ct) J (x2) become linear in Ă1 and Ă2, respectively. Further, to first order, the term involving the reduced spin accumulation function Ă(x) in the persistent ballistic spin-polarized current Ĵ−(x≡, x≡≡) [see Eq. (4.69), with J+(x≡, x≡≡) expressed in terms of Ă(x) via Eqs. (4.17) and (4.68)] can be neglected, so that this current becomes independent of Ă1,2. The same then holds for the corresponding thermoballistic current, Ĵ−(x), as well as for the persistent current spin polarization P̂J(x) derived therefrom. Hence, the polarizations P (sc) J (x1,2) are linear in Ă1,2, and the coupled equations for Ă1,2 become linear. Since μ1 √ μ2 in the low-field regime, these equations can be expressed, using Eq. (4.19), in the form [ Gl 2βe2 + ven+(x1)F1(x1) ] Ă1 + ven+(x1)F2(x1)Ă2 = PlJ − Ĵ−(x1), (5.153) ven+(x2)F1(x2)Ă1 − [ Gr 2βe2 − ven+(x2)F2(x2) ] Ă2 = PrJ − Ĵ−(x2). (5.154) The current Ĵ−(x) is proportional to J , so that the quantities Ă1,2 are, as solutions of Eqs. (5.153) and (5.154), proportional to J as well. Hence, the relaxing thermobal- listic spin-polarized current J̌−(x) is also proportional to J , so that the total current spin polarization PJ(x) calculated from Eq. (5.132) is, in the low-field regime, inde- pendent of the total physical current J . Summing up, we obtain the total current spin polarization along the entire het- erostructure, PJ(x), as follows. In the metal contacts, it is given by expressions (5.140) and (5.143), respectively, with μ−(x−1 ) and μ−(x + 2 ) expressed in terms of Ă1,2 via Eqs. (5.144)–(5.146). In the semiconductor, it is given by expression (5.132) in terms of the persistent, Ĵ−(x), and relaxing, J̌−(x), thermoballistic spin-polarized currents, where the formfactors F1,2(x) entering the latter current are determined by the solution of the integral equation (5.86) for the spin accumulation function A(x). The total density spin polarization Pn(x) is obtained in an analogous way. http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 References 109 References 1. P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part I, Chap. 8 2. R. Lipperheide, U. Wille, Phys. Rev. B 72, 165322 (2005) 3. R. Lipperheide, U. Wille, Phys. Rev. B 68, 115315 (2003) 4. Yu.V. Sharvin, Zh. Eksp. Teor. Fiz. 48, 984 (1965) [Sov. Phys. JETP 21, 655 (1965)] 5. J.M. Ziman,Principles of the Theory of Solids (CambridgeUniversity Press, Cambridge, 1965), Chap. 7.13 6. L.M. Roth, in Handbook on Semiconductors, vol. 1, ed. by T.S. Moss (North Holland, Ams- terdam, 1994), Chap. 10 7. R. Lipperheide, U. Wille, Ann. Phys. (Berlin) 521, 127 (2009) 8. Z.G. Yu, M.E. Flatté, Phys. Rev. B 66, 235302 (2002) 9. I. D’Amico, Phys. Rev. 69, 165305 (2004) 10. P.C. van Son, H. van Kempen, P. Wyder, Phys. Rev. Lett. 58, 2271 (1987) 11. E.I. Rashba, Phys. Rev. B 62, R16267 (2000) 12. D.L. Smith, R.N. Silver, Phys. Rev. B 64, 045323 (2001) 13. A. Fert, H. Jaffrès, Phys. Rev. B 64, 184420 (2001) 14. E.I. Rashba, Eur. Phys. J. B 29, 513 (2002) 15. V.Ya. Kravchenko, E.I. Rashba, Phys. Rev. B 67, 121310(R) (2003) Chapter 6 Examples To illustrate the thermoballistic formalism and provide a framework for specific applications, we work out in this chapter a number of specific examples. As a partic- ularly transparent example, we consider low-field spin-polarized transport in homo- geneous semiconductors. Thereafter, also for the homogeneous case, we present a summarizing discussion of previously obtained numerical results for the general case of field-driven transport. The bulk of this chapter is devoted to a detailed treatment of spin-polarized transport in heterostructures involving nonmagnetic and magnetic semiconductors.We first deal with structures composed of a homogeneous, nonmag- netic semiconducting layer sandwiched between two ferromagnetic metal contacts. Thereafter, we consider structures in which a nonmagnetic semiconducting layer is enclosed between two layers formed of diluted magnetic semiconductors. 6.1 Homogeneous Semiconductors Aside from illustrating details of the thermoballistic formalism, the treatment of low- field transport provides uswith resultswhich are prerequisite to dealingwith transport in heterostructures involving homogeneous semiconducting layers (see Sects. 6.2 and 6.3 below). 6.1.1 Low-Field Transport In the low-field regime, the potential energy profiles E∗,≡(x) can be assumed to be independent of the externally applied voltage, so that the reduced resistance does not depend on the voltage and the current-voltage characteristic is of pure Shockley form (see Sect. 3.3.4, where the validity of this assumption is confirmed within the prototype model). For a homogeneous layer, the profiles are given by R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 111 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_6, © Springer International Publishing Switzerland 2014 http://dx.doi.org/10.1007/978-3-319-05924-2_3 112 6 Examples E∗,≡(x) ↑ Ec ± 1 2 Δ (6.1) (x1 ≈ x ≈ x2), when the effect of Schottky barriers is disregarded. 6.1.1.1 The Function R̃−(x) Specializing the development for constant spin splitting presented in Sect. 5.2.2 to the case Ec(x) ↑ Ec, we find from Eq. (5.20) the reduced (Δ = 0) kernel K0(x1, x; x∞; l) in the explicit form K0(x1, x; x∞; l) = e−(x∞−x1)/l − e−(x−x∞)/l. (6.2) From the corresponding solution R̃10(x) of Eq. (5.44) and the solution of the analo- gous equation for the function R̃20(x), we then obtain, using Eqs. (5.46) and (5.48), the resistance functions R̃1,2(x) for the homogeneous semiconductor for x > x1 and x < x2, respectively, as R̃1,2(x) = 1± x − x1,2 2l − PR1,2(x). (6.3) At x = x1 and x = x2, on the other hand, we have R̃1(x1) = R̃2(x2) = 0 (6.4) [see Eqs. (5.23) and (5.30)]. For the reduced resistance R̃, we then find from Eq. (5.54), using Eq. (5.64), R̃ = 1+ S 2l − veNc 2J P Q e−βEc A−12, (6.5) and for the function R̃−(x) from Eq. (5.55), using Eqs. (4.112), (5.11), (5.13), and (5.28), R̃−(x) = x − (x1 + x2)/2 l + veNc J P Q e−βEc [ A(x)− 1 2 A+12 ] (6.6) for x1 < x < x2, whereas from Eq. (5.56), R̃−(x1,2) = ∓R̃. (6.7) The spin accumulation function A(x) to be inserted in expression (6.6) is given by Eqs. (4.99) and (5.117). http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.1 Homogeneous Semiconductors 113 6.1.1.2 Net Total Ballistic Current To evaluate the net total ballistic current J+(x∞, x∞∞), we insert expression (5.59) for the mean spin function Ã(x) in expression (4.67), obtaining for the present case J+(x∞, x∞∞) = J 2 [R̃−(x∞∞)− R̃−(x∞)] + veNc 2 P Q e−βEc [A(x∞)− A(x∞∞)], (6.8) with R̃−(x) given by Eq. (6.6). We then have J+(x1, x2) = J ( 1+ S 2l ) , (6.9) J+(x∞, x2) = J ( 1 2 + x2 − x ∞ 2l ) (6.10) for x∞ > x1, J+(x1, x∞∞) = J ( 1 2 + x ∞∞ − x1 2l ) (6.11) for x∞∞ < x2, and J+(x∞, x∞∞) = J x ∞∞ − x∞ 2l (6.12) for x1 < x∞ < x∞∞ < x2, where the dependence on A(x) has dropped out. At this junc- ture, we note that the dependence of the ballistic current J+(x∞, x∞∞) on the end-point coordinates x∞, x∞∞ exhibited here contradicts the basic assumption of the prototype thermoballisticmodel as expressed by Eq. (3.20), viz., that the currents in the ballistic intervals are assumed to be conserved across the end-points, and are all equal to the physical current J . 6.1.1.3 Current Spin Polarization Now, using expressions (6.9)–(6.12) in Eq. (4.76), we find J+(x) ↑ J+ = J, (6.13) i.e., the total thermoballistic current is equal to the (constant) total physical current. Further, using Eq. (4.69), we have Ĵ−(x) ↑ Ĵ− = PJ (6.14) http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_3 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 114 6 Examples for the persistent thermoballistic spin-polarized current, and hence from Eq. (5.130) for the persistent current spin polarization P̂J(x) ↑ P̂J = Ĵ− J = P, (6.15) thereby retrieving the static spin polarization. Turning to the determination of the relaxing thermoballistic spin-polarized cur- rent J̌−(x) in a homogeneous semiconductor in the low-field regime, we have from Eq. (4.70) for the corresponding ballistic current J̌−(x∞, x∞∞; x) = veNc 2 Qe−βEc [A(x∞)e−(x−x∞)/ls − A(x∞∞)e−(x∞∞−x)/ls ] (6.16) (x1 ≈ x∞ < x∞∞ ≈ x2). Inserting this expression in Eq. (4.76) and subsequently using the derivative of Eq. (5.114) with respect to x, we obtain J̌−(x) in the form J̌−(x) = −veNcQe−βEc l̄ dA(x) dx (6.17) [see Eq. (4.11) of Ref. [1]; the factor 2 appearing in the right-hand side of the latter equation again reflects the fact that the normalization of A(x) used there differs from that used in the present book]. From Eqs. (5.132), (6.14), and (6.17), we now have for the total current spin polarization PJ(x) = P − veNc J Qe−βEc l̄ dA(x) dx . (6.18) [Note that owing to its normalization to the total thermoballistic current, the expres- sion for PJ(x) given by Eq. (136) of Ref. [2] differs from expression (6.18); see the remark following Eq. (5.132) above.] 6.1.1.4 Joint Total Ballistic Density For the persistent part, n̂+(x∞, x∞∞; x) ↑ n̂+(x∞, x∞∞), of the joint total ballistic density n+(x∞, x∞∞) , we have from Eq. (4.72) n̂+(x1, x2) = Nc 2 e−βEc Q ( η+12 + P 2 A+12 ) , (6.19) n̂+(x∞, x2) = n̂+(x1, x2)− J 2ve ( 1 2 + x ∞ − x1 2l ) (6.20) for x∞ > x1, http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 6.1 Homogeneous Semiconductors 115 n̂+(x1, x∞∞) = n̂+(x1, x2)+ J 2ve ( 1 2 + x2 − x ∞∞ 2l ) (6.21) for x∞∞ < x2, and n̂+(x∞, x∞∞) = n̂+(x1, x2)− J 2ve x∞ − x1 − (x2 − x∞∞) 2l (6.22) for x1 < x∞ < x∞∞ < x2. For the relaxing part of the joint total ballistic density, ň+(x∞, x∞∞; x) ↑ ň+(x∞, x∞∞), we find from Eq. (4.73) ň+(x∞, x∞∞) = 0, (6.23) so that n+(x∞, x∞∞) = n̂+(x∞, x∞∞). (6.24) Observing this relation, we now use expressions (6.19)–(6.22) in Eq. (4.76) to obtain for the thermoballistic joint total density n+(x) = n̂+(x) = n̂+(x1, x2)− J 2ve x − (x1 + x2)/2 l , (6.25) with n̂+(x1, x2) given explicitly by Eq. (6.19). 6.1.1.5 Density Spin Polarization Since n̂−(x∞, x∞∞) = Pn̂+(x∞, x∞∞) (6.26) from Eq. (4.72), we have for the persistent part of the joint ballistic spin-polarized density n̂−(x) = Pn̂+(x) = Pn+(x), (6.27) and hence for the persistent density spin polarization P̂n(x) = n̂−(x) n+(x) = P, (6.28) in agreement with the corresponding current spin polarization. The relaxing thermoballistic spin-polarized density ň−(x) for a homogeneous semiconductor is readily obtained by inserting expression (6.17) for the correspond- ing current J̌−(x) in the balance equation (5.9) and using Eq. (5.115) for the spin accumulation function A(x), with the result http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 116 6 Examples ň−(x) = Nc 2 Qe−βEc A(x), (6.29) so that we have for the total density spin polarization Pn(x) = P + Nc 2n+(x) Qe−βEc A(x), (6.30) with n+(x) given by Eq. (6.25). Now, neglecting in expression (6.25) for n+(x) the term proportional to J , so that n+(x) = n̂+(x1, x2), we can use expression (6.30) for Pn(x) in Eq. (6.18) to express the current spin polarization PJ(x) in terms of Pn(x) in the form PJ(x) = P − 2ven̂+(x1, x2) J l̄ dPn(x) dx . (6.31) Differentiating this equation and using Eqs. (5.115) and (6.30), we then obtain Pn(x) = P − J 2ven̂+(x1, x2) ls dPJ(x) dx (6.32) for the density spin polarization Pn(x) in terms of PJ(x). 6.1.1.6 Magnetoresistance Finally, the magnetoresistance of a homogeneous semiconductor in the low-field regime is found as Rm = Q − 1− veNc 2J P Q e−βEc A−12 2l 2l + S (6.33) by inserting the low-field limit of expression (6.5), R̃0 = 1+ S 2l , (6.34) for R̃0 in Eq. (5.85). 6.1.2 Field-Driven Transport: Numerical Results For field-driven transport in a homogeneous semiconductor at zero spin splitting, the potential energy profile has the form (2.41), which we repeat here, http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_2 6.1 Homogeneous Semiconductors 117 Ec(x) = Ec(x1)− ε β (x − x1). (6.35) In Ref. [1], we have presented numerical results for the average chemical potential μ̃(x), calculated with this profile for values of the ratio l/S ranging between 10−2 and 102 and various values of the parameter εS. These results were used to evaluate the total equilibrium density n(x) as well as the total thermoballistic current, J(x), and density, n(x) [the subscript “+” attached to total (spin-summed) quantities elsewhere in the present chapter is omitted]. Here, we summarize the qualitative features of the results shown in Figs. 2–4 of Ref. [1]. If calculated with the procedure adopted in the present work (see the remarks at the end of Sect. 5.1.1), the results for μ̃(x) and the quantities obtained therefrom would differ somewhat from the former results, but would agree with those in all qualitative respects. In the range l/S ≥ 1, the average chemical potential μ̃(x) (see Fig. 2 of Ref. [1]) is close to the linear form (2.46) found within the standard drift-diffusion model. With increasing l/S, the Sharvin-type discontinuities which μ̃(x) exhibits at the interfaces [see Eqs. (5.70) and (5.71)] become larger in magnitude. Concurrently, the slope of μ̃(x) becomes smaller and its behavior departs progressively from the linear form. The decrease in slope of μ̃(x) is associatedwith a departure of the equilibrium density n(x) from the constant value it has in the drift-diffusion regime (see Sect. 2.1.3.3). As a consequence of the discontinuities in μ̃(x), the density is lowered in the vicinity of the left interface, and enhanced close to the right interface, with a monotonic rise in between. The total thermoballistic current J(x) (see Fig. 3 of Ref. [1]) stays close to the total physical current J for all values of l/S and εS considered, i.e., close to the limiting value it attains in the drift-diffusion regime (see Sect. 2.1.3.3) and in the ballistic limit (see Sect. 4.3.3). Therefore, setting J(x) = J is expected to afford an adequate approximation in any kind of application of the thermoballistic approach. The total thermoballistic density n(x) (see Fig. 4 of Ref. [1]) is characterized by an overall decrease with increasing l/S. This behavior can be understood as reflecting the fact that when the ballistic contribution to the transport mechanism increases, the density is rapidly lowered owing to the rise of the ballistic velocity along the sample [see Eq. (2.64)]. This is impeded at the points of local thermodynamic equilibrium, which lie very dense when l/S ≥ 1 [slow decrease of n(x)] and are widely spread when l/S 1 [rapid decrease of n(x)] . For l/S > 1, the behavior of n(x) is largely determined by the function Cm(εx) [see Eq. (5.96)]; in the ballistic limit l/S √ ≤, we have n(x) = n(x1) 2 (1+ e−εS)Cm(ε(x − x1)). (6.36) Again, the qualitative behavior of n(x) does not change when εS is varied. http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 118 6 Examples 6.2 FM/NMS/FM Heterostructures We now consider heterostructures formed of a homogeneous, nonmagnetic semi- conducting (NMS) layer and two ferromagnetic metal (FM) contacts. The study of spin-polarized transport in this kind of structure, both experimentally and theoreti- cally, marked the beginning of semiconductor spintronics [3–14] (for recent surveys of the physics of semiconductor-based spintronic devices, see Refs. [15–18]). For low-field transport, we evaluate the position dependence of the current and density spin polarizations using the results of Sects. 5.4.2, 5.4.3, and 6.1. For the general case of field-driven transport, we obtain the spin polarizations “injected” from the contacts into the semiconductor, i.e., the polarizations generated inside the semicon- ductor in the vicinity of either FM/NMS interface regardless of the influence of the opposite interface. 6.2.1 Low-Field Spin Polarizations Inside the semiconducting layer, we have from Eq. (6.18) with P = 0 and Q = 1, using expression (5.117) for the spin accumulation function A(x), PJ(x) = veNc J e−βEcγ[C1e−(x−x1)/L − C2e−(x2−x)/L], (6.37) where γ is given by Eq. (5.121). The coefficients C1,2 can be expressed via Eqs. (5.118) and (5.119), and using Eq. (4.17), in terms of the values Ă1,2 of the reduced spin accumulation function Ă(x) on the contact sides of the interfaces. 6.2.1.1 Matching PJ(x) at the Interfaces In order to determine the quantities Ă1,2, we evaluate expression (6.37) on the semi- conductor sides of the interfaces, obtaining P(sc)J (x1) ↑ PJ(x+1 ) = ven (0) + 2J (gĂ1 − hĂ2), (6.38) P(sc)J (x2) ↑ PJ(x−2 ) = ven (0) + 2J (hĂ1 − gĂ2). (6.39) Here, we have introduced the total equilibrium density for zero spin splitting and low external field, n(0)+ = Nce−β(Ec−μ1), (6.40) http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 6.2 FM/NMS/FM Heterostructures 119 which is constant inside the semiconducting layer. The coefficients g and h in expres- sions (6.38) and (6.39) are given by Eqs. (5.124) and (5.125), respectively. Using the continuity ofPJ(x) at the interfaces, Eq. (5.152), we now equate expres- sions (5.148) and (6.38), and expressions (5.149) and (6.39). [When spin-selective interface resistances are included, expression (5.148) for P(ct)J (x1) is to be replaced with the more general expression obtained by using expression (5.151) for μ−(x−1 ) in Eq. (5.140), and analogously forP(ct)J (x2).] In the low-field regime, when |Ă1,2| ≥ 1, this results in a system of coupled linear equations for Ă1,2 [see Eqs. (5.153) and (5.154)], [g + G̃(0)l ]Ă1 − hĂ2 = 2J ven (0) + Pl, (6.41) hĂ1 − [g + G̃(0)r ]Ă2 = 2J ven (0) + Pr, (6.42) where G̃(0)l,r = Gl,r βe2ven (0) + . (6.43) The solutions of Eqs. (6.41) and (6.42) are found to be Ă1 = 2J ven (0) + Θ {[g + G̃(0)r ]Pl − hPr}, (6.44) Ă2 = 2J ven (0) + Λ {hPl − [g + G̃(0)l ]Pr}, (6.45) where Λ = [g + G̃(0)l ][g + G̃(0)r ] − h2. (6.46) Expressions (6.44) and (6.45) determine the boundary values of the reduced spin accumulation function, Ă1 and Ă2, in terms of the current J , of the polarizations Pl and Pr in the left and right ferromagnetic contact, respectively, and of material parameters, such as the conductivities σl,r and the spin diffusion lengths L (l,r) s of the contacts [via G̃(0)l,r ] and the momentum relaxation length l and the spin relaxation length ls in the semiconducting sample as well as the sample length S (via g and h) and the (constant) total equilibrium electron density n(0)+ . Since the quantities Ă1,2 are proportional to the total current J , the current spin polarization PJ(x) is independent of J , while the density spin polarization Pn(x) is proportional to J . http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 120 6 Examples 6.2.1.2 Parameter Dependence of PJ(x) The spin polarizations along the entire heterostructure are now obtained as follows. Inside the semiconductor, the current spin polarization PJ(x) is given by expression (6.37), with C1,2 calculated from the boundary values Ă1,2 given by Eqs. (6.44) and (6.45). The corresponding density spin polarization Pn(x) is then readily obtained fromEq. (6.32). The expressions forPJ(x) in the ferromagnetic contacts are provided by Eqs. (5.140) and (5.143), respectively, where the quantities μ−(x−1 ) and μ−(x + 2 ) are calculated fromEq. (5.151) and from its analogue forμ−(x+2 ), respectively.We do not write down the density spin polarizations in the ferromagnets, but only mention that they do not, in general, match the polarizations Pn(x + 1 ) and Pn(x − 2 ) on the semiconductor sides of the interfaces. In Refs. [1] and [19], we have presented results of detailed numerical calcula- tions of PJ(x) in the low-field regime for typical parameter values of FM/NMS/FM heterostructures, emphasizing the dependence of the polarization on the momentum and spin relaxation lengths, l and ls, and on the spin-selective interface resistances ρ (1,2) ∗,≡ (see Sect. 5.4.2). Themomentum relaxation length l, in particular, affectsPJ(x) in a twofold way. First, it determines the conductance of the semiconductor, which is small for small values of l, so that the polarization at the semiconductor side of the FM/NMS interfaces (“injected polarization”; see Sect. 6.2.2 below) is small. Second, it determines the generalized spin diffusion length L, which, according to Eq. (6.37), acts as the polarization decay length. Hence, for small l, PJ(x) dies out rapidly inside the semiconductor. When the value of l rises up to a length comparable to the sample length, i.e., when the ballistic component in the transport mechanism becomes prevalent, PJ(x) increases substantially all along the sample. Even when spin-selective interface resistances are disregarded, the polarization reaches values as large as 0.3 for typical values of the parameters characterizing the semiconducting sample (see Figs. 5–7 of Ref. [1]). 6.2.2 Injected Spin Polarizations for Field-Driven Transport We define the “injected spin polarization” as the polarization inside the semiconduc- tor in the vicinity of the interface, e.g., at x = x1, generated by the bulk polariza- tion Pl of the left ferromagnet regardless of the influence of the right ferromagnet. More explicitly, we define the injected current and density spin polarizations as the polarizations PJ(x + 1 ) and Pn(x + 1 ), respectively, in the limit of infinite sample length, S √ ≤. The injected spin polarizations at x = x+1 provide the initial values of the left-generated polarizations in the semiconductor, which propagate into the region x > x1 while being degraded by the effect of spin relaxation. http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.2 FM/NMS/FM Heterostructures 121 6.2.2.1 Spin Accumulation Function We now consider the injected spin polarizations for field-driven transport in a homo- geneous semiconductor at zero spin splitting, when the potential energy profile has the form (2.41), with the field parameter ε given by Eq. (2.42). In order to obtain the spin accumulation function A(x) for this case, one has to solve Eq. (5.105) [in general, numerically] under the asymptotic condition A(x) → e−x/λ for x √ ≤. (6.47) To determine the decay length λ, we solve Eq. (5.105) in the range x − x1 1/(ε+ 1/l), (6.48) where the x-dependence of the coefficient functions b0(x), b1(x), and b2(x) arising from the function b(x) can be disregarded. This yields 1 λ = ε 2 + [ ε2 4 + l − l̃ ll̃2 ( 1+ εl̃ 1+ εl 2+ εl )]1/2 . (6.49) Since l > l̃, λ is a real number for all values of ε. 6.2.2.2 Injected Current Spin Polarization The injected current spin polarization is obtained from Eq. (5.132) with Ĵ−(x) = 0. We calculate the relaxing thermoballistic spin-polarized current at the interface, J̌−(x+1 ), from the relation J̌−(x+1 ) = veNc 2 W̌(x1; x1, x2; l)[A1 − A(x+1 )], (6.50) where the function W̌(x; x1, x2; l) is given by Eq. (5.88). This relation can be shown to hold, for arbitrary sample length S and potential energy profiles E∗,≡(x), by (i) evaluating J̌−(x+1 ) fromEq.4.76)withF(x∞, x∞∞; x) ↑ J̌−(x∞, x∞∞; x) [seeEq. (4.70)] and (ii) eliminating from the resulting expression the boundary value A2 by using the integral equation (5.86) for x = x1. For the case considered here, we write down the integral equation (5.97) [with c(εx) = c̄] for x = x1 and take the limit x2 √ ≤ to obtain the relation 2A(x+1 )− Ā1 = A1, (6.51) http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 122 6 Examples where Ā1 = ≤∫ x1 dx l e−(x−x1)/l̃A(x). (6.52) Relation (6.51) fixes the normalization of the spin accumulation function A(x) in terms of the boundary value A1. Then, using this relation in Eq. (6.50) with W̌(x1; x1, x2; l) = 2e−βEc(x1), we can express the current J̌−(x+1 ) in the form J̌−(x+1 ) = ven (0) + (x1) 2 A(1)J Ă1, (6.53) where the quantity A(1)J = A(x+1 )− Ā1 A(x+1 )− Ā1/2 (6.54) is independent of the normalization of A(x), and n(0)+ (x1) = Nce−β[Ec(x1)−μ1]. (6.55) In obtaining Eq. (6.53), we have used Eq. (4.85), with P(x) = 0, as well as Eq. (4.17). 6.2.2.3 Injected Density Spin Polarization For the injected current spin polarization, we now have from Eq. (5.132) PJ(x + 1 ) = J̌−(x+1 ) J = ven (0) + (x1) 2J A(1)J Ă1, (6.56) which, by continuity, is equal to the polarization PJ(x1) at the contact side of the interface. Then, equating the right-hand sides of Eqs. (5.148) and (6.56), we obtain Pl − Gl2βe2J ln ( 1+ Ă1/2 1− Ă1/2 ) = ven (0) + (x1) 2J A(1)J Ă1. (6.57) This nonlinear equation for Ă1 is to be solved for given values of the parameters ε, Pl, Gl, n (0) + (x1), l, and ls, in order to determine the right-hand side of Eq. (6.56). We now turn to the calculation of the injected density spin polarization from Eq. (5.133), where n̂−(x) = 0 and n+(x) = n̂+(x) in the present case. Proceeding similarly as for the current J̌−(x+1 ), we obtain the relaxing thermoballistic spin- polarized density at the interface, ň−(x+1 ), in the form http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.2 FM/NMS/FM Heterostructures 123 ň−(x+1 ) = n(0)+ (x1) 4 A(1)n Ă1, (6.58) where A(1)n = A(x+1 ) A(x+1 )− Ā1/2 . (6.59) For the persistent total thermoballistic density at the interface, n̂+(x+1 ), we find by evaluating Eq. (4.76) with F(x∞, x∞∞; x) ↑ n̂+(x∞, x∞∞; x) [see Eq. (4.72)] and taking the limit S √ ≤ n̂+(x+1 ) = n(0)+ (x1) 2 (1+ Â1), (6.60) where the quantity Â1 = ≤∫ x1 dx l e−(x−x1)/leβ[μ̃(x)−μ1] (6.61) has been expressed in terms of the average chemical potential μ̃(x) using Eq. (4.14). The injected density spin polarization now follows as Pn(x + 1 ) = ň−(x+1 ) n̂+(x+1 ) = A (1) n Ă1 2(1+ Â1) , (6.62) where the boundary value Ă1 of the reduced spin accumulation function is again to be determined by solving Eq. (6.57). 6.2.2.4 Drift-Diffusion Regime In the drift-diffusion regime, when l/ls ≥ 1 and εl ≥ 1, the spin accumulation function A(x) is given by the exponentially decreasing solution of Eq. (5.113), A(x) → e−(x−x1)/Lεs (6.63) (x ≥ x1), with the field-dependent spin diffusion length Lεs given by 1 Lεs = ε 2 + ( ε2 4 + 1 L2s )1/2 , (6.64) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 124 6 Examples where Ls is the spin diffusion length defined by Eq. (4.107). From Eqs. (6.54) and (6.59), respectively, we then have A(1)J = 2l Lεs (6.65) and A(1)n = 2, (6.66) and hence, using Eqs. (6.56) and (6.62), respectively, PJ(x + 1 ) = ven (0) + (x1) J l Lεs Ă1 = 1 2εLεs Ă1 (6.67) and Pn(x + 1 ) = Ă1 1+ Â1 = Ă1 2 (6.68) for the injected current and density spin polarizations in the drift-diffusion regime. In deriving the right-hand equation (6.67), we have used Eq. (4.100), with P = 0, Q = 1, and βeV = εS, as well as Eq. (6.55), to express the total current J in the form J = 2ven(0)+ (x1)εl ↑ 1 βe2 σ (0) + ε, (6.69) where σ (0) + = 2βe2ven(0)+ (x1)l (6.70) is the (spin-summed) conductivity of the semiconductor [see Eq. (2.27)]. The right- hand equation (6.68) has been obtained by using Eq. (4.96) in Eq. (6.61). The bound- ary value Ă1 is determined by the equation Pl − Gl 2σ(0)+ ε ln ( 1+ Ă1/2 1− Ă1/2 ) = 1 2εLεs Ă1, (6.71) which follows from Eq. (6.57) by replacing its right-hand side with the right-hand side of Eq. (6.67) and inserting expression (6.69) for J . The effect of external electric fields on the current spin polarization injected at FM/NMS interfaces has been studied within the standard drift-diffusion approach by Yu and Flatté [20]. Comparing our description to that of these authors, we find that the field-dependent spin diffusion length Lεs given by Eq. (6.64) agrees with the “up-stream” spin diffusion length Lu given by Eq. (2.23b) of Ref. [20], provided the http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_4 6.2 FM/NMS/FM Heterostructures 125 “intrinsic” spin diffusion lengthL of that reference is identifiedwith the spin diffusion length Ls = ∼lls of the present work. Then, identifying in Eq. (3.5) of Ref. [20] (with the interface resistances set equal to zero) the spin-summed conductivity of the semiconductor, σs, with σ (0) + , and the spin injection efficiency at the interface, α0, with PJ(x + 1 ) = Ă1/2εLεs , and using Eq. (5.142), we observe the equivalence of the former equation with Eq. (6.71). Consequently, the injected current and density spin polarizations of either work are identical. Within the thermoballistic description, the dependence of the injected current spin polarization, PJ(x + 1 ), on the electric-field parameter ε, for l-values ranging from the drift-diffusion to the ballistic regime and for zero as well as nonzero interface resistances ρ(1)∗,≡, has been studied numerically in Ref. [1]. With rising magnitude of the electric field, PJ(x1) rises monotonically, with the slope tending to zero when the ballistic regime is approached (see Fig. 8 of Ref. [1]). 6.2.3 Injected Current Spin Polarization in the Low-Field Regime For arbitrary l, we now consider the injected current spin polarization in the low-field regime, when |Ă1| ≥ 1. Here, the spin accumulation function A(x) is given by the exponentially decreasing solution of Eq. (5.115), A(x) → e−(x−x1)/L (6.72) (x ≥ x1), where the generalized spin diffusion length L is given by Eq. (5.116). From Eq. (6.54), we then have A(1)J = 2γ 1+ γ ↑ γ ∗, (6.73) with γ given by Eq. (5.121), and hence from the low-field limit of Eq. (6.57), Ă1 = 2J ven (0) + (x1) Pl Gl/G̃1 + γ∗ , (6.74) where G̃1 ↑ 1 ρ̃1 = βe2ven(0)+ (x1) (6.75) is the Sharvin interface conductance [see Eq. (5.76)]. Inserting this in Eq. (6.56), we find http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 126 6 Examples PJ(x + 1 ) = Pl 1+ Gl/γ∗G̃1 , (6.76) i.e., in the low-field regime, the injected current spin polarization is independent of the total current J . 6.2.3.1 Drift-Diffusion Regime In the drift-diffusion regime, when l/ls ≥ 1, we have γ∗ = 2 √ l ls , (6.77) so that PJ(x + 1 ) reduces to PJ(x + 1 ) = Pl 1+ Gl/G1 , (6.78) where G1 = 2G̃1 √ l ls = σ (0) + Ls , (6.79) with σ(0)+ defined by Eq. (6.70), and Ls by Eq. (4.107). The quantity G1 is seen to be the semiconductor analogue of the “interface conductance” Gl characterizing the fer- romagnet [see Eq. (5.142)]. For typical values of the parameters of the FM/NMS/FM system (see, e.g., Ref. [1]), the values of Gl exceed those of G1 by several orders of magnitude. This indicates a “conductance mismatch” between ferromagnet and semiconductor [3, 4, 10, 14], which gives rise to very low values of the injected spin polarizations at FM/NMS interfaces. To remedy this mismatch, one may introduce spin-selective interface resistances, for example, in the form of tunneling barriers [3, 6–8, 10, 13]. 6.2.3.2 Ballistic Regime In the ballistic regime, when l/ls 1, we have γ∗ = γ = 1, (6.80) so that PJ(x + 1 ) = Pl 1+ Gl/G̃1 . (6.81) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.2 FM/NMS/FM Heterostructures 127 Here, the Sharvin interface conductance G̃1 takes the place of the quantity G1 in Eq. (6.78). As G̃1 is proportional to the density n(0)+ (x1), Eq. (6.81) yields values for PJ(x + 1 ) close to the bulk spin polarization of the ferromagnet, Pl, if n (0) + (x1) is sufficiently high. For typical parameter values [1], however, the high donor densities needed to obtain the required high electron densities imply very small values of the momentum relaxation length l, such that ballistic transport is excluded as the dominating transportmechanism.On the other hand, for densities so low that ballistic transport prevails, the injected spin polarization is confined to very small values. This result corroborates previous estimates [13] according to which spin injection is suppressed even in the ballistic regime unless spin-selective interface resistances are introduced. 6.3 DMS/NMS/DMS Heterostructures As an example of particular interest from the point of view of applications, we now consider spin-polarized electron transport in heterostructures involving diluted magnetic semiconductors (DMS) in their paramagnetic phase (see, e.g., Refs. [14, 21–34]). In structures of this kind, a nonmagnetic semiconducting (NMS) layer is sandwiched between two DMS layers which, in turn, are enclosed between non- magnetic metal contacts (see Fig. 6.1). In a complete thermoballistic description, the semiconductor part of aDMS/NMS/ DMS structure should be treated as a single sample. Then, the ballistic intervals [x∞, x∞∞] covering the sample may contain one or both of the DMS/NMS interfaces, at which the potential energy profiles must be expected to change abruptly. The same holds for the material parameters, in particular the momentum and spin relax- ation lengths, so that a description in terms of position-dependent parameters would become necessary. Here, we adopt a simplified treatment by assuming the different layers in a DMS/NMS/DMS heterostructure to be homogeneous and requiring the interfaces to act as fixed points of local thermodynamic equilibrium. This allows us to apply the thermoballistic description separately to the different layers (the potential energy profiles are then, in general, discontinuous at the interfaces). For each layer, we evaluate the spin accumulation function and the low-field current spin polariza- tion for a homogeneous semiconductor (see Sect. 6.1), and subsequently match the current spin polarization at the interfaces to obtain its full position dependence as well as the magnetoresistance. 6.3.1 Low-Field Current Spin Polarization Heterostructures of the kind depicted schematically in Fig. 6.1 are parametrized here by attaching labels j = 1, 2, 3 to quantities referring to the left DMS layer, the NMS layer, and the right DMS layer, respectively. The positions of the interfaces 128 6 Examples x x x x1 42 3 x S S S1 2 3 E E EC (1) C (2) C (3) Δ Δ 1 3↑E (x) CE (x) ↓E (x) DMS NMS DMS 1 32 Fig. 6.1 Schematic potential energy profiles E∗,≡(x) for a DMS/NMS/DMS heterostructure com- posed of three homogeneous semiconducting layers enclosed between nonmagnetic metal contacts (including the interfaces between the DMS layers and the contacts) are denoted by xk (k = 1, 2, 3, 4); this notation deviates from the one used in the preceding sections. 6.3.1.1 Position Dependence of PJ(x) Inside the left and right metal contacts, the position dependence of the current spin polarization is given by Eqs. (5.140) and (5.143), respectively [with x2 in the latter equation replaced with x4], with μ−(x1,4) = Ă1,4/β [see Eq. (4.17) for |βμ−(x∞)| ≥ 1] and Pl = Pr = 0, PJ(x) = ∓ Gl,r 2βe2J Ă1,4e −|x−x1,4|/L(l,r)s (6.82) for x < x1 and x > x4. Inside the semiconducting part of the heterostructure, PJ(x) is found in terms of the static spin polarizations Pj of the three layers, and of the boundary values of the spin accumulation function A(x) at the interface positions xk , A(xk) ↑ Ak , which are all points of local thermodynamic equilibrium. [The interface positions x2,3 are points of local thermodynamic equilibrium in the same sense as are the interface positions x1,4, except that there the interface is between metal contact and DMS, while here it is between DMS and NMS.] In each layer j, the function A(x) has the Sharvin discontinuities A(x+j ) − Aj and Aj+1 − A(x−j+1) at the interface positions xj and xj+1, respectively [see Eqs. (5.122) and (5.123)]. Using now, in the low-field regime, Eq. (6.18) together with Eq. (5.117) separately in each layer, we have http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.3 DMS/NMS/DMS Heterostructures 129 PJ(x) = Pj + veNc J Qje −βE(j)c γj[C(j)1 e−(x−xj)/Lj − C(j)2 e−(xj+1−x)/Lj ] (6.83) for xj ≈ x ≈ xj+1 (j = 1, 2, 3). Here, γj = l̄j Lj = Lj l(j)s , (6.84) with l̄j and Lj defined by Eqs. (5.112) and (5.116) in terms of the momentum relax- ation length, lj, and spin relaxation length, l (j) s , of layer j. Further, C(j)1 = 1 Dj [(1+ γj)eSj/Lj Aj − (1− γj)Aj+1] (6.85) and C(j)2 = − 1 Dj [(1− γj)Aj − (1+ γj)eSj/Lj Aj+1], (6.86) where Dj = 2[(1+ γ2j ) sinh(Sj/Lj)+ 2γj cosh(Sj/Lj)], (6.87) and Sj = xj+1 − xj (6.88) is the thickness of layer j. [Owing to a difference in normalization, expression (6.83) for PJ(x) differs from the corresponding expression given by Eq. (141) of Ref. [2]; see the remark following Eq. (6.18) above.] 6.3.1.2 Matching PJ(x) at the Interfaces In order to relate the quantities Ăk to the total physical current J and the parameters of the contact-semiconductor system, we invoke the continuity of the current spin polarization PJ(x) at all interfaces, PJ(x − k ) = PJ(x+k ) (6.89) (k = 1, . . . , 4). Setting P2 = 0 and Q2 = 1 in Eq. (6.83) and using Eq. (4.17) to replace Ak with Ăk , we then obtain from Eqs. (6.82)–(6.88) a set of four coupled linear equations for Ăk , (G̃(1)l + Q1g1)Ă1 − Q1h1Ă2 = − 2J ven (0;1) + P1, (6.90) http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 130 6 Examples −Q1h1Ă1 + (Q1g1 + eβδ12g2)Ă2 − eβδ12h2Ă3 = 2J ven (0;1) + P1, (6.91) −eβδ32h2Ă2 + (eβδ32g2 + Q3g3)Ă3 − Q3h3Ă4 = − 2J ven (0;3) + P3, (6.92) −Q3h3Ă3 + (G̃(3)r + Q3g3)Ă4 = 2J ven (0;3) + P3. (6.93) Here, G̃(1,3)l,r = Gl,r βe2ven (0;1,3) + , (6.94) where n(0;j)+ = Nce−β[E (j) c −μ1] (6.95) is the (constant) total equilibrium electron density in layer j for zero spin splitting, and δj,j∞ = E(j)c − E(j ∞) c (6.96) are the band offsets. Further, gj = hj[cosh(Sj/Lj)+ γj sinh(Sj/Lj)] ≈ 1 (6.97) [see Eq. (5.124)], with hj = 4γj Dj ≈ 2 1+ γj e −Sj/Lj (6.98) [see Eq. (5.125)]. The system of equations (6.90)–(6.93) can be solved easily, so that the complete position dependence of the low-field current spin polarization is obtained in explicit form. 6.3.1.3 Symmetric Heterostructure We now specialize to the case of a symmetric heterostructure, for which the parame- ters of the right DMS layer and metal contact are identical to those of the left DMS layer and metal contact. Assuming arbitrarily high conductivities of the contacts, http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.3 DMS/NMS/DMS Heterostructures 131 σ(l,r)+ √ ≤, (6.99) so that G̃(1,3)l,r √ ≤, (6.100) we have from Eqs. (6.82) and (6.94) Ă1,4 √ 0, (6.101) such that G̃(1,3)l,r Ă1,4 remains nonzero and finite. Then, from Eqs. (6.91) and (6.92) with P1 = P3 = P and Q1 = Q3 = Q, Ă2 = −Ă3 = 2J ven (0;D) + P QDN , (6.102) where QDN = QgD + (gN + hN )eβδDN . (6.103) Hence, from Eqs. (6.90) and (6.93), G̃(1)l Ă1 = −G̃(4)r Ă4 = − 2J ven (0;D) + P + QhDĂ2 = − 2JP ven (0;D) + ( 1− QhDQDN ) . (6.104) Here, the DMS parameters have been labeled by “D”, and the NMS parameters by “N”. Thus, in the symmetric case, the low-field current spin polarization PJ(x) is com- pletely determined by the quantity Ă2. Explicitly, we obtain, setting L(l)s = L(r)s = L(mc)s (6.105) (where the superscript “mc” refers to the metal contacts) in Eq. (6.82) and using Eqs. (6.94) and (6.104), PJ(x) = P ( 1− QhDQDN ) e−|x−x1,4|/L (mc) s , (6.106) if x < x1 and x > x4, respectively. Further, from Eq. (6.83), we have, using Eqs. (6.84)–(6.87) along with Eqs. (4.17), (6.40), and (6.102), PJ(x) = P { 1− QhDQDN [cosh(|x − x1,4|/LD)+ γD sinh(|x − x1,4|/LD)] } , (6.107) http://dx.doi.org/10.1007/978-3-319-05924-2_4 132 6 Examples if x1 ≈ x ≈ x2 and x3 ≈ x ≈ x4, respectively, and PJ(x) = 2e βδDN hN P QDN [cosh(SN/2LN )+ γN sinh(SN/2LN )] × cosh([x − (x2 + x3)/2]/LN ), (6.108) if x2 ≈ x ≈ x3. In Refs. [2] and [35], we have presented numerical results for PJ(x) for values of the momentum relaxation length l ranging from the drift-diffusion regime to the ballistic regime. The qualitative features of the results can be summarized as follows. First, the injected spin polarization, i.e., the polarization at the DMS/NMS inter- faces, perseveres on a high, weakly l-dependent level (in the range 0.6–0.7 for typi- cal parameter values of the semiconducting system; see Fig. 3 of Ref. [2]) when the momentum relaxation length l is varied by three orders of magnitude from the drift- diffusion to the ballistic regime. This behavior differs from that of the injected polar- ization calculated for FM/NMS/FM heterostructures (see Fig. 5 of Ref. [1]), which (for zero interface resistance) is very small in the drift-diffusion regime, and rises rapidlywith increasing l. The strong l-dependence reflects the conductivitymismatch at small l in FM/NMS/FM heterostructures, which is absent in DMS/NMS/DMS structures. Second, the calculated qualitative behavior of the polarization inside the NMS layer does not depend on the kind of spin injector (FM or DMS). However, caused by the larger injected polarization, the polarization inside the NMS layer is larger for the DMS injector. 6.3.2 Magnetoresistance The relative magnetoresistance Rm of a DMS/NMS/DMS heterostructure follows by inserting in Eq. (5.82) the reduced resistance R̃ and the corresponding zero-field resistance R̃0 obtained by summing up the respective contributions R̃j and R̃ (j) 0 of the (homogeneous) layers j. These contributions are given by the (appropriately labeled) expressions (6.5) and (6.34) multiplied with a factor Bm+(x1, x4)/Bm+(xj, xj+1), such that R̃ and R̃0 are normalized in conformance with the definition (5.10) for a single sample extending from x1 to x4. Specializing immediately to the case of the symmetric structure considered above, we have, defining Sj = 1+ Sj 2lj (6.109) ( j = D,N) and using Eqs. (6.101), R̃ = 2QeβE(D)c SD + eβE (N) c SN + veNce βμ1 2J P(Ă2 − Ă3) (6.110) http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 6.3 DMS/NMS/DMS Heterostructures 133 [we have omitted here the overall factor Bm+(x1, x4) which drops out when Rm is formed], from which R̃0 follows by setting P = 0 and Q = 1. Then, using Eq. (6.102), we obtain Rm = (Q − 1)SD + P 2/QDN SD + e−βδDNSN/2 . (6.111) Here, we note that in Ref. [2], in contrast to its definition in terms of the zero-field resistance R̃0 via Eq. (5.82), the relative magnetoresistance Rm has been defined with respect to the spin-equilibrium resistance obtained by setting Ă2 = Ă3 = 0 in expression (6.110). The latter definition does not seem to correspond to a genuine magnetoresistance, and so one should not attach quantitative significance to the numerical results for Rm shown in Fig. 4 of Ref. [2]. In expression (6.111) for Rm, the terms in the numerator differ in sign, (Q − 1)SD < 0, P 2 QDN > 0. (6.112) By varying the parameters in these terms, one can control their magnitude inde- pendently within sufficiently broad ranges, so that one can reach both positive and negative values for Rm. To exemplify this result, we consider in Sect. 6.3.3.2 below the behavior of Rm in the ballistic regime. In the limit of low external magnetic field, when the static spin polarization P depends linearly on the field strength, we have from Eq. (6.111), keeping terms of order P2, Rm = {1/[gD + (gN + hN )e βδDN ] − SD/2}P2 SD + e−βδDNSN/2 . (6.113) While based on assumptions that differ from, and are more general than, those under- lying the “two-band model” for the (transverse) magnetoresistance [36, 37], expres- sion (6.113) exhibits the quadratic field dependence that characterizes the lattermodel in the low-field limit. 6.3.3 Drift-Diffusion and Ballistic Regimes Considering first the thermoballistic description of spin-polarized transport in DMS/ NMS/DMSheterostructures in the drift-diffusion regime, we can compare our results to those of Ref. [30] obtained within the standard drift-diffusion approach. http://dx.doi.org/10.1007/978-3-319-05924-2_5 134 6 Examples 6.3.3.1 Drift-Diffusion Regime In the drift-diffusion regime, when lj ≥ Sj and lj ≥ l(j)s , we have Lj √ L(j)s = √ ljl (j) s (6.114) and γj √ lj L( j)s = √ l j l(j)s ≥ 1 (6.115) [see Eqs. (4.107), (5.112), (5.116), and (6.84)]. Hence, from Eqs. (6.97) and (6.98), gj = hj cosh(Sj/L( j)s ) (6.116) and hj = 2lj L(j)s sinh(Sj/L ( j) s ) , (6.117) respectively. Then, specializing to the symmetric heterostructure, we find for the quantity QDN defined by Eq. (6.103) QDN = 2 [ Q lD L(D)s coth(SD/L (D) s )+ lN L(N)s coth(SN/2L (N) s )e βδDN ] . (6.118) Introducing the (spin-summed) conductivity σ( j)+ for layer j ( j = D,N), σ (j) + = 2βe2ven(j)+ lj (6.119) [see Eq. (2.27)], with n(j)+ = n(0;j)+ Qj (6.120) (QD = Q,QN = 1) and n(0;j)+ given by Eq. (6.95), we can rewrite expression (6.118) in the form QDN = 2 Q lD L(D)s [ Q2 coth(SD/L (D) s )+ σ (N) + σ (D) + L(D)s L(N)s coth(SN/2L (N) s ) ] . (6.121) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_2 6.3 DMS/NMS/DMS Heterostructures 135 Now, inserting expression (6.121) in Eq. (6.102) and using the resulting expression for Ă2 in Eq. (6.83), we find that the values of the low-field current spin polarization PJ(x) at the positions x1,4, x2,3, and (x2+ x3)/2, respectively, agree with those given by Eqs. (16)–(18) of Ref. [30], if we identify in the latter equations the spin-flip lengths λD and λN with L (D) s and L (N) s , respectively, and the layer thicknesses d and 2x0 with SD and SN , respectively. For the magnetoresistance in the drift-diffusion regime, we obtain from Eq. (6.111), setting Sj = Sj/2lj and inserting expression (6.121) for QDN , Rm = SD[1/σ (D) + − 1/σ(0;D)+ ] + P2/QDN SD/σ (0;D) + + SN/2σ(0;N)+ , (6.122) where σ(0;j)+ is the conductivity for zero spin splitting, which is given by Eq. (6.119) with n( j)+ = n(0;j)+ , and QDN = Q2 σ (D) + L(D)s coth(SD/L (D) s )+ σ (0;N) + L(N)s coth(SN/2L (N) s ). (6.123) Then, identifying the parameters as above, we find that expression (6.122) for Rm formally agrees with expression (13) of Ref. [30]. In the latter expression, however, the magnetic-field dependence of the conductivity equivalent to σ(D)+ is not known. By contrast, the field dependence of σ(D)+ in expression (6.123) is determined, via the polarization P, by the field dependence of the Zeeman splitting Δ. 6.3.3.2 Ballistic Regime Turning now to the ballistic regime, when lj Sj and lj l(j)s , we have Lj √ l(j)s (6.124) and γj √ 1 (6.125) [see Eqs. (4.107), (5.112), (5.116), and (6.84)], so that gj = 1 (6.126) and hj = e−Sj/l (j) s . (6.127) http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 136 6 Examples For the symmetric heterostructure, we then find from Eq. (6.103) QDN = Q + [1+ e−SN/l (N) s ]eβδDN . (6.128) With this used in Eq. (6.102) for Ă2 as well as in Eq. (6.104) for G̃ (1) l Ă1, we obtain the position dependence of the low-field current spin polarization from Eqs. (6.82) and (6.83). For themagnetoresistance in the ballistic regime,wefind fromEqs. (6.111), setting Sj = 1 and inserting expression (6.128) for QDN , Rm = Q − 1+ P 2/{Q + [1+ e−SN/l(N)s ]eβδDN } 1+ e−βδDN /2 . (6.129) The magnitude of the term [1+ e−SN/l(N)s ]eβδDN ↑ χ (6.130) determines the sign of Rm: while Rm = 0 for χ = 1, we have Rm < 0 (> 0) for χ > 1 (< 1), independent of the magnitude of the polarization P, i.e., of the strength of the magnetic field. More specifically, the parameter βδDN ↑ δDN/kBT controls, for given values of SN and l (N) s , the sign of Rm: we have Rm < 0 unless (i) δDN = E(D)c − E(N)c < 0, i.e., the nonmagnetic semiconductor forms a barrier in the DMS/NMS/DMS potential energy profile, and (ii) the magnitude of |δDN |/kBT is sufficiently large. We note that a negative magnetoresistance in semiconductors has emerged from theoretical studies (see, e.g., Ref. [38]), and has been identified experimentally (see, e.g., Refs. [39–41]). References 1. R. Lipperheide, U. Wille, Phys. Rev. B 72, 165322 (2005) 2. R. Lipperheide, U. Wille, Ann. Phys. (Berlin) 521, 127 (2009) 3. E.I. Rashba, Phys. Rev. B 62, R16267 (2000) 4. G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip, B.J. van Wees, Phys. Rev. B 62, R4790 (2000) 5. A.T. Filip, B.H. Hoving, F.J. Jedema, B.J. van Wees, B. Dutta, S. Borghs, Phys. Rev. B 62, 9996 (2000) 6. D.L. Smith, R.N. Silver, Phys. Rev. B 64, 045323 (2001) 7. A. Fert, H. Jaffrès, Phys. Rev. B 64, 184420 (2001) 8. E.I. Rashba, Eur. Phys. J. B 29, 513 (2002) 9. M. Johnson, Semicond. Sci. Technol. 17, 298 (2002) 10. G. Schmidt, L.W. Molenkamp, Semicond. Sci. Technol. 17, 310 (2002) 11. J.D. Albrecht, D.L. Smith, Phys. Rev. B 66, 113303 (2002) 12. J.D. Albrecht, D.L. Smith, Phys. Rev. B 68, 035340 (2003) 13. V. Ya, Kravchenko, E.I. Rashba. Phys. Rev. B 67, 121310(R) (2003) 14. G. Schmidt, J. Phys. D: Appl. Phys. 38, R107 (2005) References 137 15. M.R. Hofmann, M. Oestreich, in Magnetic Heterostructures, ed. by H. Zabel, S.D. Bader. Springer Tracts in Modern Physics, vol. 227 (Springer, Berlin, 2008), p. 335 16. M. Tanaka, S. Ohya, in Comprehensive Semiconductor Science and Technology, vol. 6, ed. by P. Bhattacharya, R. Fornari, H. Kamimura (Elsevier, Amsterdam, 2011), Chap. 6.14 17. D. Saha, M. Holub, P. Bhattacharya, D. Basu, in Comprehensive Semiconductor Science and Technology, vol. 6, ed. by P. Bhattacharya, R. Fornari, H. Kamimura (Elsevier, Amsterdam, 2011), Chap. 6.15 18. I. Žutić, J. Fabian, C. Ertler, in Comprehensive Semiconductor Science and Technology, vol. 6, ed. by P. Bhattacharya, R. Fornari, H. Kamimura (Elsevier, Amsterdam, 2011), Chap. 6.16 19. R. Lipperheide, U. Wille, Mater. Sci. Eng. B 126, 245 (2006) 20. Z.G. Yu, M.E. Flatté, Phys. Rev. 66, 235302 (2002) 21. J.C. Egues, Phys. Rev. Lett. 80, 4578 (1998) 22. M. Oestreich, J. Hübner, D. Hägele, P.J. Klar, W. Heimbrodt, W.W. Rühle, D.E. Ashenford, B. Lunn, Appl. Phys. Lett. 74, 1251 (1999) 23. R. Fiederling, M. Keim, G. Reuscher, W. Ossau, G. Schmidt, A. Waag, L.W. Molenkamp, Nature (London) 402, 787 (1999) 24. B.T. Jonker, Y.D. Park, B.R. Bennett, H.D. Cheong, G. Kioseoglou, A. Petrou, Phys. Rev. B 62, 8180 (2000) 25. Y. Guo, H. Wang, B.L. Gu, Y. Kawazoe, J. Appl. Phys. 88, 6614 (2000) 26. G. Schmidt, G. Richter, P. Grabs, C. Gould, D. Ferrand, L.W. Molenkamp, Phys. Rev. Lett. 87, 227203 (2001) 27. J.C. Egues, C. Gould, G. Richter, L.W. Molenkamp, Phys. Rev. B 64, 195319 (2001) 28. K. Chang, F.M. Peeters, Solid State Commun. 120, 181 (2001) 29. G. Schmidt, C. Gould, P. Grabs, A.M. Lunde, G. Richter, A. Slobodskyy, L.W. Molenkamp, Phys. Rev. Lett. 92, 226602 (2004) 30. A. Khaetskii, J.C. Egues, D. Loss, C. Gould, G. Schmidt, L.W. Molenkamp, Phys. Rev. 71, 235327 (2005) 31. W. Van Roy, P. Van Dorpe, J. De Boeck, G. Borghs, Mater. Sci. Eng. B 126, 155 (2006) 32. D. Sánchez, C. Gould, G. Schmidt, L.W. Molenkamp, IEEE Trans. Electron Devices 54, 984 (2007) 33. A. Slobodskyy, C. Gould, T. Slobodskyy, G. Schmidt, L.W. Molenkamp, D. Sánchez, Appl. Phys. Lett. 90, 122109 (2007) 34. M. Ciorga, A. Einwanger, U. Wurstbauer, D. Schuh, W. Wegscheider, D. Weiss, Phys. Rev. B 79, 165321 (2009) 35. R. Lipperheide, U. Wille, AIP Conf. Proc. 893, 1279 (2007) 36. J.M. Ziman,Principles of the Theory of Solids (CambridgeUniversity Press, Cambridge, 1965), Chap. 7.13 37. L.M. Roth, in Handbook on Semiconductors, vol. 1, ed. by T.S. Moss (North Holland, Amsterdam, 1994), Chap. 10 38. R. Jullien, A. Dmitriev, M. Dyakonov, Rev. Mex. Fis. S 52, 185 (2006) 39. J.K. Furdyna, J. Appl. Phys. 64, R29 (1988) 40. S.J. May, A.J. Blattner, B.W. Wessels, Phys. Rev. B 70, 073303 (2004) 41. C. Butschkow, E. Reiger, A. Rudolph, S. Geißler, D. Neumaier, M. Soda, D. Schuh, G. Woltersdorf, W. Wegscheider, D. Weiss, Phys. Rev. B 87, 245303 (2013) Chapter 7 Summary and Outlook In this book, we have presented a comprehensive survey of the thermoballistic approach to charge carrier transport in semiconductors. The principal aim has been to develop the basic physical concept underlying this approach in detail, and to give a coherent exposition of the ensuing formalism, which unifies, and partly modifies, generalizes, and corrects, the formal developments presented in our original publi- cations. To make the presentation self-contained and easy to follow, we have proceeded step by step, starting with an account of Drude’s model as the origin of all semiclas- sical transport models. We then reviewed the standard drift-diffusion and ballistic transport models, basic features of which have been adopted to establish the ther- moballistic description of carrier transport in terms of averages over random config- urations of ballistic transport intervals. The contributions of the individual ballistic intervals to the total carrier current are governed by collision probabilities involv- ing the momentum relaxation length (or carrier mean free path) as the determining parameter of the thermoballistic concept. This concept finds a first, concrete expression in the prototype thermoballistic model, which is based on the simplifying assumption of current conservation across the points of local thermodynamic equilibrium linking ballistic intervals. The imple- mentationof the prototypemodel results in a current-voltage characteristic containing a reduced resistance, which can be expressed explicitly in terms of the parameters of the semiconducting system. In the fully developed thermoballistic concept, current conservation across the equilibrium points is abandoned, and position-dependent, total and spin-polarized thermoballistic currents and densities are introduced in terms of an average chemical- potential function and a spin accumulation function related to the spin splitting of the chemical potential. The algorithms for determining these dynamical functions follow from two physical conditions. First, the average of the total thermoballistic current over the length of the semiconducting sample (as well as over the range between either end and an arbitrary point of local thermodynamic equilibrium inside the sample) is required to equal the conserved physical current, whereby one is R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 139 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2_7, © Springer International Publishing Switzerland 2014 140 7 Summary and Outlook able to set up a scheme for obtaining the average chemical potential. The explicit calculation of this function is implemented in terms of resistance functions, for which Volterra-type integral equations are derived. Second, spin relaxation is assumed to act only inside the ballistic transport intervals. As a result, the thermoballistic spin- polarized current and density are connected by a spin balance equation, from which one obtains an inhomogeneous Fredholm-type integral equation for the spin accu- mulation function. In the general case of arbitrarily shaped potential energy profiles, considerable numerical effort is needed for solving the integral equations for the resistance functions and the spin accumulation function. In a number of important special cases, however, these equations reduce to a form which greatly facilitates their solution. Examples are the equations for the resistance functions when the spin splitting of the band edge profile is independent of position, or the equation for the spin accumulation function for a homogeneous sample in an external electric field. In the latter case, the integral equation can be converted into an easily tractable second-order differential equation. For homogeneous semiconductors subjected to low external electric fields, the solutions of the equations for the dynamical functions can be obtained in closed form throughout. For the purpose of demonstrating the potentialities of the thermoballistic approach in present-day semiconductor and spintronics research, we have summarized in this book the treatment of a number of specific examples. The prototype model is employed to describe electron transport across potential energy profiles exhibit- ing an arbitrary number of barriers, where effects of tunneling and degeneracy are included. This model proves to be of particular relevance to the description of grain boundary effects in electron transport in polycrystalline semiconductors and has already been applied with promising results in the analysis of experimental data. The full thermoballistic approach is used in the treatment of spin-polarized transport in heterostructures. For the prototype problem of semiconductor spintronics, vi z., the injection of spin polarization from ferromagnetic contacts into a semiconducting sample, the thermoballistic description extends the standard drift-diffusion descrip- tion so as to allow for arbitrary values of the momentum relaxation length. The same holds for low-field transport in heterostructures composed of layers of diluted mag- netic and nonmagnetic semiconductors, where the position dependence of the current spin polarization as well as the magnetoresistance are obtained in closed form. In our original publications, we have presented results of explicit calculations for a variety of specific cases. These calculations were mostly of exploratory character, with the aim to reveal qualitative trends in the parameter dependence of the relevant transport properties. In future work, emphasis should be placed on the application of the thermoballistic approach in quantitative studies, as required for the analysis of specific experimental results. For these studies to become successful, a self-consistent scheme for the calculation of the potential energy profiles is to be developed, and efficient numerical techniques for solving the integral equations for the dynamical functions are to be employed. While the fully three-dimensional treatment of ther- moballistic transport constitutes a highly ambitious task, the inclusion of corrections accounting for departures from strictly one-dimensional transport appears feasible. 7 Summary and Outlook 141 Thus, with these refinements considered, the thermoballistic approach should prove useful as a practical tool for analyzing experimental data in a transparent and readily interpretable overall fashion, contrasting with much more voluminous, numerical simulation schemes. This is one aspect of the thermoballistic approach. The other is its theoretical relevance as a bridge between the drift-diffusion and ballistic descriptions, in that way rounding off, in a physically and mathematically consistent manner, the semiclassical theory of charge carrier transport in semicon- ductors. Appendix A Useful Formulae for Numerical Calculations For the case of field-driven transport in homogeneous semiconductors with constant spin splitting Δ, i.e., for potential energy profiles of the form E∗,≡(x) = Ec(x)± Δ 2 = Ec(x1)− ε β (x − x1)± Δ 2 (A.1) [see Eqs. (2.41) and (5.34)], we collect in this appendix a variety of explicit formula that are useful for the numerical calculation of the resistance functions R̃1,2(x) and of the spin accumulation function A(x). A.1 Resistance Functions Theessential quantity to be evaluatedprior to the calculationof the resistance function R̃1(x) from Eqs. (5.44) and (5.46) is the integral kernel K0(x1, x; x ↑; l) given by Eq. (5.43).While the integral equation (5.44)must be solved, in general, numerically, the kernel can be expressed in closed form. For the profiles (A.1), the function Emc (x ↑, x ↑↑) is given by Emc (x ↑, x ↑↑) = Ec(x ↑) = Ec(x1)− ε β (x ↑ − x1) ≈ Emc (x ↑↑, x ↑) (A.2) for x ↑ < x ↑↑ [see Eq. (4.22)], so that we have from Eqs. (5.16), (5.36), and (5.38) w+(x ↑, x ↑↑; l) = e−|x ↑↑−x ↑|/ l eε(x ↑−x1). (A.3) Using this in Eq. (5.21) for the function u+(x ↑, x ↑↑, l), we obtain the integral kernel K0(x1, x; x ↑; l) from Eqs. (5.20) and (5.43) in the form R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 143 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2, © Springer International Publishing Switzerland 2014 http://dx.doi.org/10.1007/978-3-319-05924-2_2 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_4 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 144 Appendix A: Useful Formulae for Numerical Calculations K0(x1, x; x ↑; l) = e ε(x ↑−x1) κ2 { e−κ(x ↑−x1)/ l [ 1− (κ− 1)κ x ↑ − x1 l ] + κ2 [ 1− e−(x−x ↑)/ l ] − 1 } , (A.4) where κ = 1+ εl. (A.5) In the low-field regime, when we can set ε = 0 (see Sect. 6.1.1), we find expression (A.4) to reduce to expression (6.2). For the calculation of the resistance function R̃2(x), we invoke Eq. (5.32), by which R̃2(x) is expressed in terms of the resistance function R̃∞1(x) corresponding to the potential energy profile E∞c (x) = Ec(x1)− ε β (x2 − x) (A.6) obtained from Ec(x) of Eq. (A.1) by spatial inversion. We then have E∞mc (x ↑, x ↑↑) = E∞c (x ↑↑) = Ec(x1)− ε β (x2 − x ↑↑) ≈ E∞mc (x ↑↑, x ↑) (A.7) for x ↑ < x ↑↑ and, in analogy to Eq. (A.3), w∞+(x ↑, x ↑↑, l) = e−|x ↑↑−x ↑|/ l eε(x2−x ↑↑), (A.8) so that the kernel K∞0(x1, x; x ↑; l) to be used in Eq. (5.44) to calculate the reduced resistance function R̃∞10(x) is obtained from Eq. (5.20) in the form K∞0(x1, x; x ↑; l) = − eε(x2−x ↑) κ2 { e−κ(x−x ↑)/ l [ 1− (κ− 1)κ x − x ↑ l ] + κ2 [ 1− e−(x ↑−x1)/ l ] − 1 } , (A.9) in analogy to expression (A.4) for the kernel K0(x1, x; x ↑; l). A.2 Spin Accumulation Function For constant spin splitting, the spin accumulation function A(x) does not depend on the splitting (see Sect. 5.3.1). To calculate A(x) in the interval [x1, x2] for a potential energy profile Ec(x) of the form (A.1), the differential equation (5.105) is solved numerically for two arbitrarily normalized, linearly independent functions α1,2(x), so that A(x) is expressed as http://dx.doi.org/10.1007/978-3-319-05924-2_6 http://dx.doi.org/10.1007/978-3-319-05924-2_6 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 Appendix A: Useful Formulae for Numerical Calculations 145 A(x) = C1α1(x)+ C2α2(x). (A.10) In order fix the normalization of A(x) in terms of the boundary values A1,2, we evaluate the integral equation (5.97) [with c(ζ) = c̄] for x = x+1 and x = x−2 , respectively, obtaining a set of two coupled linear equations for the coefficients C1,2, whose solution reads C1 = 1 D {[M22 f1(0)− M12 f1(S)]A1 + [M22 f2(S)− M12 f2(0)]A2}, (A.11) C2 = 1 D {[M11 f1(S)− M21 f1(0)]A1 + [M11 f2(0)− M21 f2(S)]A2}. (A.12) Here, D = M11M22 − M12M21, (A.13) f1(0) = f2(0) = 1, (A.14) f1(S) = e−(ε+1/l̃)S, (A.15) and f2(S) = e−S/l̃ , (A.16) with l̃ defined by Eq. (5.110). The quantities Mi j are given by M11 = f (0)α1(x+1 )− x2∫ x1 dx ↑ l f2(x ↑ − x1)α1(x ↑), (A.17) M12 = f (0)α2(x+1 )− x2∫ x1 dx ↑ l f2(x ↑ − x1)α2(x ↑), (A.18) M21 = f (S)α1(x−2 )− x2∫ x1 dx ↑ l f1(x2 − x ↑)α1(x ↑), (A.19) M22 = f (S)α2(x−2 )− x2∫ x1 dx ↑ l f1(x2 − x ↑)α2(x ↑), (A.20) with f (0) = 2 (A.21) http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 146 Appendix A: Useful Formulae for Numerical Calculations and f (S) = 1 κ [2+ (κ− 1)(1+ e−κS/ l)], (A.22) where κ is given by Eq. (A.5). In the low-field regime, εS � 1, expressions (A.11) and (A.12) for the coefficients C1,2 reduce to expressions (5.118) and (5.119). http://dx.doi.org/10.1007/978-3-319-05924-2_5 http://dx.doi.org/10.1007/978-3-319-05924-2_5 Index A Activation term, 79 Applications of prototype model, 46 B Ballistic (thermionic) model, 14 electron degeneracy in, 21 electron tunneling in, 19 range of validity, 15 thermionic emission, 17 Ballistic configurations, 29, 78 averaging over, 31, 32, 37, 50 Ballistic current, 16, 17, 21 net, 18, 30, 62, 65 spin-polarized, 56, 57 persistent, 57 relaxing, 57, 61, 63 spin-resolved, 55 total, 56, 62 Ballistic density, 17 joint, 19, 62, 65 spin-polarized, 58 persistent, 59, 63 relaxing, 59, 61, 63 spin-resolved, 58 total, 58, 63 persistent, 59, 63 relaxing, 59, 63 Ballistic energy current, 75 Ballistic interval, 6, 26, 29, 50 Ballistic point, 64 Ballistic spin-polarized transport, 55 Ballistic velocity, 17, 117 Biased-random-walk model, 10 Boltzmann equation, 10, 73, 81 Built-in potential, 8, 73 C Chemical potential average, 53, 85, 86, 92 in ballistic regime, 95 in drift-diffusion regime, 71 thermoballistic, 92, 94 discontinuities of in prototype model, 38 in thermoballistic approach, 95 in ballistic model, 23 in drift-diffusion model, 13 in prototype model, 30, 37, 38, 47 local, 8 spin-resolved, 52 discontinuity of, 107 in ferromagnetic contact, 105 Collision time, 6, 25, 26 local, 28 Conductance in ballistic model, 22 in prototype model, 37, 46 Conductivity, 7 in contacts, 106 local, 10 magnetic-field dependence of, 135 Current background, 85 conserved, 30, 83 diffusion, 9, 27 drift, 6, 9, 13 thermally emitted, 15, 55, 58, 84 R. Lipperheide and U. Wille, The Thermoballistic Transport Model, 147 Springer Tracts in Modern Physics 259, DOI: 10.1007/978-3-319-05924-2, © Springer International Publishing Switzerland 2014 148 Index total, 9, 10, 18, 21, 83 Current-voltage characteristic in ballistic model, 18, 21, 22 in drift-diffusion model, 11, 71 in prototype model, 31, 40, 44 in thermoballistic approach, 96 D Diluted magnetic semiconductors, 50, 52, 111, 127 DMS/NMS/DMS heterostructures, 127 band offset in, 130, 136 low-field current spin polarization, 127 in ballistic regime, 135, 136 in drift-diffusion regime, 133, 135 matching at interfaces, 129 position dependence, 131 magnetoresistance, 132 for low magnetic field, 133 in ballistic regime, 136 in drift-diffusion regime, 135 negative, 133, 136 Drift-diffusion model, 8, 9 diffusion coefficient, 10, 27 drift-diffusion equation, 10 range of validity, 14 Drude model, 5 friction force in, 7 range of validity, 7 E Effective diffusion velocity, 40 Effective sample length, 12, 39, 46 Effective transport length, 44, 45 Einstein relation, 10, 27, 70 Electron mobility, 6, 10, 40, 70 Emission interval, 32 Emission velocity, 16 Energy distribution function Fermi-Dirac, 21 Maxwell-Boltzmann, 9 Equilibration interval, 32 Equilibrium electron density, 8, 50 spin-polarized, 54 spin-resolved, 52 total, 54, 84 F Ferromagnetic contact, 105 Field-driven transport, 13, 47, 77, 92, 99, 116, 121 FM/NMS/FM heterostructures, 118 injected spin polarizations, 120 drift-diffusion regime, 123 low-field regime, 125 low-field polarizations, 118 position dependence, 120 G Generalized drift-diffusion equation, 73 Giant Zeeman splitting, 52 Grain boundary, 39, 40, 46 H Heat bath, 75 Homogeneous semiconductor, 111 field-driven transport in, 116 total thermoballistic current for, 117 total thermoballistic density for, 117 low-field transport in, 111 current spin polarization for, 114 density spin polarization for, 116 magnetoresistance for , 116 reduced resistance for, 112 resistance functions for, 112 Homogeneous semiconductor field-driven transport in average chemical potential for, 117 I Impurity scattering, 5, 7, 81 Inhomogeneous semiconductor, 8, 73 Interface resistance, 22 Sharvin, 23, 95 spin-selective, 107 L Landauer’s transport theory, 23 Low-field transport, 22, 47, 96, 102, 108 M Magnetoresistance, 97, 116, 132 Mean electron speed, 6, 16 Mean free path, 6, 7, 15, 27, 28 effective, 32 local, 14, 28 Mean spin function, 53, 83, 85 Momentum relaxation length, 6, 65, 79–81, 83 Monte Carlo method, 81 Index 149 N Nonlinear Poisson equation, 34, 46, 73 P Partition function, 31, 35, 37 Plane-parallel sample, 8 Point contact, 22 Potential energy profile, 9 arbitrary number of barriers, 43 shape term, 43 discontinuities at interfaces, 127 double barrier, 42, 43 for constant external field, 13, 47 single barrier, 29, 39 spatially reversed, 90 spin-dependent, 51 Probabilistic approach to carrier transport, 25 Prototype thermoballistic model, 25 application to photovoltaic materials, 46 concept and implementation, 28 electron degeneracy in, 36, 45 electron tunneling in, 34, 45 examples, 39 R Reduced resistance in prototype model, 31, 33, 35, 39, 42, 45, 96 in thermoballistic approach, 93, 96 Reference coordinate, 64, 75 ballistic, 84 equilibrium, 84 Reflectionless contact, 78 Resistance function in prototype model, 37 in thermoballistic approach, 86, 89 discontinuities of, 88, 90 for constant spin splitting, 90 integral equation for, 88, 89, 91, 92, 143 reduced, 91, 92, 144 Resonant tunneling, 20 Richardson effect, 17 S Schottky barrier, 13, 17, 39 Semiclassical description, 50, 81 Sharvin effect, 95 Spin accumulation function, 53, 58, 86, 97 differential equation for, 101 in drift-diffusion regime, 72, 102 normalized solution of, 101, 144 discontinuities of, 99 in low-field regime, 102 integral equation for, 86, 98, 100 integrodifferential equation for, 100 reduced, 54 Spin balance equation, 60 ballistic, 61, 64 thermoballistic, 72, 85, 86 Spin diffusion length, 73 field-dependent, 123 generalized, 102 Spin equilibrium, 55 Spin polarization current, 104 in contacts, 106 in semiconductor, 104, 105 matching at interfaces, 108 density, 104 in semiconductor, 105 injected, 120 local, 52 nonlocal, 56 Spin relaxation, 56, 59, 60 function, 57, 61 length, 60 time, 60 Spin relaxation length, 65 Spin-flip scattering rate, 60 T Thermal activation, 79 Thermal emission, 15, 27, 29, 32, 62, 78, 79 Thermal equilibration, 6, 7, 14, 26, 27, 51, 77–79 complete, 26, 76, 80 incomplete, 26, 80 Thermoballistic approach ballistic limit, 73 concept, 49 synopsis of concept, 78 drift-diffusion regime, 68 energy dissipation in, 75 implementation, 83 physical content, 79 spin-polarized transport, 64 Thermoballistic current, 65 derivative of, 66 spin-polarized in drift-diffusion regime, 72 in strict ballistic limit, 74 persistent, 65 relaxing, 65, 86 150 Index total, 65 in drift-diffusion regime, 70 in strict ballistic limit, 74 interpretation as ensemble average, 83 spatial average of, 83 Thermoballistic density, 65 derivative of, 66 spin-polarized in drift-diffusion regime, 69 in strict ballistic limit, 74 persistent, 65 relaxing, 65 total, 65 in drift-diffusion regime, 69 in strict ballistic limit, 74 Thermoballistic energy current, 77 Thermodynamic equilibrium, 5, 7, 8, 75 local, 8, 50, 52 point of, 14–16, 28–30, 49–51, 55, 78, 79 Transmission probability classical, 18 quantal, 19 thermally averaged, 18, 19, 30, 34, 35, 55 WKB, 19, 35 Transverse modes, 23 Trapping model, 46 Preface Contents 1 Introduction References 2 Drift-Diffusion and Ballistic Transport 2.1 Drift-Diffusion Transport 2.1.1 Drude's Model 2.1.2 Drift-Diffusion Model 2.1.3 Current-Voltage Characteristic and Chemical Potential 2.1.4 Range of Validity 2.2 Ballistic (Thermionic) Transport 2.2.1 Nondegenerate Case 2.2.2 Electron Tunneling 2.2.3 Degenerate Case 2.2.4 Interface Resistances and Chemical Potential References 3 Prototype Thermoballistic Model 3.1 Probabilistic Approach to Carrier Transport 3.1.1 Net Electron Current 3.1.2 Mean Free Path 3.2 Prototype Thermoballistic Model: Concept and Implementation 3.2.1 Ballistic Configurations 3.2.2 Averaging Over Ballistic Configurations 3.2.3 Electron Tunneling 3.2.4 Degenerate Case 3.2.5 Chemical Potential 3.3 Prototype Thermoballistic Model: Examples 3.3.1 Single Potential Energy Barrier 3.3.2 Double Barrier 3.3.3 Arbitrary Number of Barriers 3.3.4 Chemical Potential for Field-Driven Transport References 4 Thermoballistic Approach: Concept 4.1 Electron Densities at Local Thermodynamic Equilibrium 4.1.1 Spin-Resolved Densities 4.1.2 Average Chemical Potential and Spin Accumulation Function 4.2 Ballistic Spin-Polarized Transport 4.2.1 Ballistic Currents 4.2.2 Ballistic Densities 4.2.3 Spin Balance Equation 4.2.4 Net Ballistic Currents and Joint Ballistic Densities 4.3 Thermoballistic Spin-Polarized Transport 4.3.1 Thermoballistic Currents and Densities 4.3.2 Drift-Diffusion Regime 4.3.3 Ballistic Limit 4.3.4 Thermoballistic Energy Dissipation 4.4 Synopsis of the Thermoballistic Concept 4.4.1 Ingredients and Physical Content 4.4.2 Merits and Weaknesses References 5 Thermoballistic Approach: Implementation 5.1 Physical Conditions Determining the Dynamical Functions 5.1.1 Determination of the Mean Spin Function 5.1.2 Determination of the Spin Accumulation Function 5.2 Average Chemical Potential 5.2.1 The Resistance Functions 5.2.2 Resistance Functions for Constant Spin Splitting 5.2.3 Thermoballistic Average Chemical Potential 5.2.4 Sharvin Interface Resistance 5.2.5 Current-Voltage Characteristic and Magnetoresistance 5.3 Spin Accumulation Function 5.3.1 Integral Equation 5.3.2 Differential Equation for Field-Driven Transport 5.4 Current and Density Spin Polarizations 5.4.1 Spin Polarizations in the Semiconductor 5.4.2 Current Spin Polarization in the Contacts 5.4.3 Matching the Current Spin Polarization at the Interfaces References 6 Examples 6.1 Homogeneous Semiconductors 6.1.1 Low-Field Transport 6.1.2 Field-Driven Transport: Numerical Results 6.2 FM/NMS/FM Heterostructures 6.2.1 Low-Field Spin Polarizations 6.2.2 Injected Spin Polarizations for Field-Driven Transport 6.2.3 Injected Current Spin Polarization in the Low-Field Regime 6.3 DMS/NMS/DMS Heterostructures 6.3.1 Low-Field Current Spin Polarization 6.3.2 Magnetoresistance 6.3.3 Drift-Diffusion and Ballistic Regimes References 7 Summary and Outlook Appendix AUseful Formulae for Numerical Calculations Index