On hopping conductivity in granular metals A.A. Likalter * Center for Applied Problems of Electrodynamics, Institute for High Temperatures RAS, Izhorskaya 13/19, Moscow 127412, Russian Federation Abstract Theory of hopping conductivity on spherical metal grains below the percolation threshold in a dielectric matrix has been developed. Screening of the potential beyond the percolation radius of grains reduces the energy of creating the electron–hole pairs to zero at the percolation threshold. The electric conductivity has been calculated as a function of the metal volume fraction and temperature. A few factors including partial localization of electrons and multiple ionization of grains have been found responsible for the temperature dependence usually referred to as variable range hopping. Ó 1999 Elsevier Science B.V. All rights reserved. 1. Introduction Vapor deposited metal-dielectric films consist of fine metallic grains of a few nanometers in di- ameter with gaps of an atomic scale between them. Such a structure manifests itself in gradual variation of electric conductivity over the metal– insulator transition identified with classical per- colation. Below the transition there is thermally activated conductivity with activation energies of a few meV [1,2]. Theoretical description of granular metals is still fragmentary [3–5]. Below the percolation threshold the electric conduction is caused by tunneling of electrons between grains, which re- quires an activation energy for charging the grains [3]. Thus, below the percolation threshold granular metals have an activated hopping conductivity. Though considerable conductivity is observed only near the insulator–metal transition, a general model of hopping conductivity does not allow for this proximity to the transition point. On the other hand, a scaling theory of the insulator–metal transition in composites [6] completely ignores thermal excitations. In this paper, we overcome these diculties taking into account a decrease of the activation energy by screening. Outline of the paper is as follows. In Section 2, we consider the weakly ionized granular system. In Section 3, we determine conductivity at the tran- sition point. In Section 4, we analyze the temper- ature dependence of conductivity allowing for multiple ionization of grains. We conclude in Section 5. 2. Electron–hole conductivity We write conductivity as a product of three multipliers r ˆ rcshqi; …1† where rc is the conductivity at the transition point, s � exp ÿ2d=a… † the probability of tunneling, d a critical barrier width, a ˆ …�h2=2mW †1=2 the decay Journal of Non-Crystalline Solids 250–252 (1999) 771–775 www.elsevier.com/locate/jnoncrysol * E-mail:
[email protected] 0022-3093/99/$ – see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 2 8 8 - 4 length of the electron wave function, and qh i the probability of activation. Here, �h is Planck’s con- stant, m is the e€ective electron mass, and W half- width of the forbidden band in the dielectric. For weakly ionized systems hqi � exp…ÿDc=2T †, where Dc is the energy for creating the electron–hole pair, and T is the temperature. The critical barrier width, which determines the hopping conductivity [7], is two-fold di€erence of the percolation radius of grains, Rc, at which they would form infinite percolation cluster, and their geometrical radius, R, d ˆ 2R fc=f… †1=3 h ÿ 1 i ; …2† where f ˆ 4pR3ng=3 is the volume fraction of grains, ng is the number density of grains, and fc is the critical volume fraction at the percolation threshold. The energy of creating an electron–hole pair is equal to the di€erence between the ionization en- ergy of a grain and the electron anity, which is decreased because of the dielectric polarization and screening of the Coulomb interaction. Al- lowing for the polarization and screening beyond the percolation radius, Rc, we obtain Dc ˆ D 1 " ÿ 1 � ÿ �d �m eÿjRc � f fc � �1=3# ; …3† where D ˆ e2=�dR the electrostatic energy, e the electron charge, �d the dielectric constant, �m the e€ective dielectric constant of media with the grains, and j the inverse radius of screening. As- suming strong screening, n ˆ exp…ÿjRc†=f� 1, close to the insulator–metal transition the energy gap is Dc ˆ D 1 h ÿ f=fc… †1=3 i : …4† We analyze Eq. (3) using an approximation of e€ective media, �m=�d ˆ 1‡ 3f=…1ÿ f=fc†, and the Debye approximation for screening by fluctuation charges of grains, j2 ˆ 4pe2ng=�mDc; …5† where we substituted the mean square of charge by T=Dc for the Gauss distribution of the fluctuation charges, p…z† ˆ exp…ÿz2Dc=2T †. Near the perco- lation threshold we estimate the Debye radius to be R= 3 p , and parameter n to be about 1=3. Thus, the screening is very e€ective compared to Dc, though in this case the Debye approximation is not very accurate. Having in mind additional screening by overlapping electron shells of grains, close en- ough to the transition point Eq. (4) is justifiabe. Expanding d and D on the distance from the percolation transition and substituting into Eq. (1), we obtain r ˆ rc…T †exp � ÿ 1 3 4R a � ‡ D 2T � Df fc � ; Df=fc � 1: …6† According to Eq. (6), approaching the transition point a large-scale parameter, R=a� 1, causes an exponential growth of conductivity, while the temperature dependence decreases. Since the acti- vation energy is comparable with the temperature, a temperature dependence of the pre-exponential factor is essential, but its density dependence is negligible as compared with the exponential. 3. Conductivity at the transition point Otherwise, the minimal energy of creating electron–hole pairs can be defined by a percolation condition of the classically accessible spheres of electrons in the Coulomb potentials of remainder ions. The electrostatic energy, D, plays a role of the e€ective ionization potential provided that in the ground state classically accessible radius is equal to the grain radius, R. At binding energy, Dÿ Dc, the volume fraction of the classically accessible spheres in the Coulomb potential wells of ions is equal to fc, therefore Eq. (4) determines the energy of excitation into a percolation level. Obviously, the classical percolation would be possible only without the polarization potential barriers, there- fore it is the case of virtual percolation. By definition, a minimal energy of electrons and holes just corresponds to the percolation thresh- old, therefore the mobility is described by a scaling theory. A notion of virtual percolation of electrons allows a reduction of the problem to a more fa- miliar case of overlapping atomic states. In such a 772 A.A. Likalter / Journal of Non-Crystalline Solids 250–252 (1999) 771–775 structure, valence electrons have a continuous spectrum of excitations with admixture of states of asymptotically free motion beyond the classically accessible spheres. The mobility is described by a scaling function of excitations [8] #…ep† ˆ …ep=Ds†m; ep < Ds; …7† where ep the energy of excitations with the as- ymptotic momentum p, m � 0:9 is the critical index of the correlation length of percolation system, and Ds is the width of a soft mobility gap. We define the upper edge of a soft mobility gap by a condition that at this energy the classically acces- sible radius in the Coulomb potential reaches the radius of closely packed Wigner–Seitz spheres [9] in the superlattice of grains. Correspondingly, the width of the soft gap is obtained from (4) substi- tuting the percolation threshold by the close packing degree, Ds ˆ D 1 h ÿ f=fs… †1=3 i ; …8† where fs ˆ 0:74 is the close packing degree of spheres. The soft mobility gap influences the mean mo- bility by a localization factor, #, that is the scaling function averaged over the excitation energies, # ˆ #…ep† � : …9† Eq. (9) determines the localization factor provided that the density of states and distribution function are known. The density of states, formed with an admixture of asymptotically free motion to the ground state, is greater than that of pure free states, because the overlapping states centered at di€erent grains can be distinguishable. It was shown that a renormalized Fermi energy of the excitations lowers proportionally to the localiza- tion factor squared, therefore Boltzmann’s statis- tics can be generally used [10]. Using the free- motion-like density of states and averaging scaling function Eq. (7) with the Boltzmann distribution, one obtains # ˆ 3T 2Ds ÿ 2 p p T Ds C 5 2 ; Ds T � �� ÿ C 3 2 ; Ds T � �� ; …10† where C…a; x† is the incomplete gamma-function. Here, we neglect the deviation of m index from unity. At zero temperature the localization factor goes to zero, and at high temperatures to unity. Note, that when temperature goes to zero, the re- normalized Fermi energy to the temperature goes to zero as #2=T / T , therefore Boltzmann’s sta- tistics is justifiable. We express the conductivity, caused by random walk of electrons on grains, by a modified Drude formula [10] that allows for partial localization within the soft mobility gap rc…T † ˆ e 2zengl# mvT ; …11† where ze is an e€ective valence of grains, l ˆ …4png=3†ÿ1=3 is the minimal free path length in the array of grains, vT ˆ …8T =pm†1=2 is the thermal electron velocity. An e€ective valence is the number of electrons at the Fermi level of isolated grains. Assuming within a jelly model [11] a shell electronic struc- ture similar to an atomic one, the e€ective valence can be determined by the number of r and p electrons in the outer valence shell. We still fur- ther assume that in the ground state only the r orbital is occupied, but from the inner shells ad- ditional electrons can be excited to the valence p shell. The minimal excitation energy is deter- mined by a rotational energy of p electrons equal to the centrifugal potential at the grain radius. In the case of polarized spins the e€ective valence is then ze ˆ 1‡ 3 exp…ÿ�h2=mR2T †: …12† 4. Multiple ionization We estimate the defect of energy of transitions between grains with di€erent charges neglecting the distribution of grains on radii, as well as the random potential of surroundings. Consider an electron exchange a chemical reaction between grains with charge numbers z and z0, z‡ z0 ˆ …z‡ 1† ‡ …z0 ÿ 1†; DE ˆ …zÿ z0 ‡ 1†Dc: …13† A.A. Likalter / Journal of Non-Crystalline Solids 250–252 (1999) 771–775 773 We set probability of exothermic processes to be unity, and use the principle of detail balance for the activation. Symmetrizing probability of the transition with respect to the permutation z and z0, we obtain at zÿ z0j jP 1 q z ��ÿ ÿ z0��� ˆ 1 2 1 � ‡ exp � ÿ Dc T 1 ÿ ‡ z�� ÿ z0����� …14† and at zÿ z0j j < 1 q z ��ÿ ÿ z0��� ˆ exp�ÿ Dc T � cosh z ÿ� ÿ z0�Dc T � : …15† The probability of activation, weighted with the probability with which the pair of charges occurs, is to be averaged over the Gauss charge distribu- tion. Taking into account continuous variation of charges, caused by the polarization of junctions, and integrating over charge numbers we obtain hqi ˆ 1 2 exp… h ÿ Dc=2T † ‡ erf c Dc=2T p� �i ; …16† where erf c…x† is the complementary error-func- tion. In the low-temperature limit the mean probability of activation, given by Eq. (16), is proportional to exp…ÿDc=2T †. This justifies the weighted averaging of the activation probability, since otherwise we would obtain incorrect as- ymptotic behavior. Shown in Fig. 1 is the temperature dependence of conductivity of granular Ni–SiO2 film [1]. The asymptotic activation dependence, observed at the temperatures smaller than the activation energy, crosses over to a more complex function at higher temperatures. The theoretical results agree with the experiment by an order of magnitude and in a wide range well reproduce observed temperature dependence. 5. Conclusion Granular metals consisting of mesoscopic grains in the dielectric matrix give an example of composites with the thermally activated conduc- tivity below the insulator–metal transition. The conductivity is described within a model of the percolation cluster with tunnel junctions between neighboring grains, in contrast with a variable range hopping characteristic of doped semicon- ductors. The energy gap, corresponding to the creation of electron–hole pairs, goes to zero at the insulator–metal transition because of screening by overlapping electron shells of neighboring grains. The temperature dependence of conductivity is determined by a few factors including a virtual percolation of electrons in the Coulomb potential wells. A large ratio of the grain radius to the decay length of the electron wave function causes steep dependence of the electric conductivity on the metal volume fraction. Describing random walk of electrons on the grains with partial localization of electrons, a modified Drude formula reasonably well estimates the conductivity at the insulator– metal transition point. While the model does not include some features of real systems such as the Fig. 1. Conductivity of Ni–SiO2 film with R ˆ 25 �A and metal volume fraction f ˆ 0:48 as a function of the temperature. Percolation threshold fc is 0:53. Experimental dots are by Git- tleman et al. [1], solid line is the theory. 774 A.A. Likalter / Journal of Non-Crystalline Solids 250–252 (1999) 771–775 size distribution of grains, or random potential of surroundings, it can reasonably well describe hopping conductivity in granular metals. Acknowledgements This work was partially supported by Russian foundation for Basic Research under grant 97-02- 16222. References [1] J.I. Gittleman, Y. Goldstein, S. Bozowski, Phys. Rev. B 5 (1972) 3609. [2] A. Milner, A. Gerber, B. Groisman et al., Phys. Rev. Lett. 76 (1996) 475. [3] N.F. Mott, E.A. Davis, Electron Processes in Noncrystal- line Materials, Clarendon, Oxford, 1979. [4] P. Sheng, Philos. Mag. B 65 (1992) 357. [5] C.J. Adkins, in: P.P. Edwards, C.N.R. Rao, Taylor, Francis (Eds.), Metal–Insulator Transitions Revisited, 1995. [6] A.K. Sarychev, F. Brouers, Phys. Rev. Lett. 73 (1994) 2895. [7] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, New York, 1989. [8] A.A. Likalter, Usp. Fiz. Nauk 162 (1992) 119; Sov. Phys. Usp. 35 (1992) 591. [9] J.M. Ziman, Models of Disorder, Cambridge University, Cambridge, 1979. [10] A.A. Likalter, ZhETF 107 (1995) 1996; JETP 80 (1995) 1105. [11] V.V. Kresin, W.D. Knight, in: V.Z. Kresin (Ed.), Pair Correlations in Many-Fermion Systems, Plenum, New York, 1988. A.A. Likalter / Journal of Non-Crystalline Solids 250–252 (1999) 771–775 775