Please cite Electron Sp ARTICLE IN PRESSG ModelELSPEC-46183; No. of Pages 4 Journal of Electron Spectroscopy and Related Phenomena xxx (2013) xxx– xxx Contents lists available at ScienceDirect Journal of Electron Spectroscopy and Related Phenomena j ourna l ho me page: www.elsev ier .com Resolvi f U self-en Tomasz D ga a Los Alamos N b Temple Unive a r t i c l Article history: Available onlin Keywords: ARPES f-Electrons Band renorma Self-energy alcul ation visua plex 1. Introdu Actinide iors. In the h interaction driven by h coexisting s ductivity or accompanied by renormalization of the electronic structure, which may lead to formation of heavy bands [2], novel quasiparticle exci- tations [3], Fermi surface renormalization [4] and/or gapping [5,6]. Angle-resolved photoemission spectroscopy (ARPES) is a method mo (k) and ene part of the the techniq I∼A(k, ω)|M ω), matrix ture of band photoemiss A(k, ω) = − where Im self-energy renormalize measureme sured Im . U ∗ Correspon E-mail add sefu band e m ergy terba le m d fro prod a 5f electron system USb2. Uranium diantimonide USb2 is a correlated f-electron antifer- romagnet with tetragonal structure (P4/nmm) with a high Néel temperature of 203 K [9,10] and anisotropic transport properties 0368-2048/$ – http://dx.doi.o this article in press as: T. Durakiewicz, et al., Resolving the multi-gap electronic structure of USb2 with interband self-energy, J. ectrosc. Relat. Phenom. (2013), http://dx.doi.org/10.1016/j.elspec.2013.10.005 st useful in obtaining a direct view on the momentum rgy (ω) resolved electronic structure of the occupied valence band, see, e.g., [7] or [8] for a review of ue. In ARPES, the measured photoemission intensity, |2F(ω), is a convolution of the spectral function A(k, element M and Fermi function F(ω). A detailed pic- renormalizations may be extracted from the measured ion intensity via the spectral function A: 1 � Im (ω − �k − Re)2 + Im2 (1) and Re are the imaginary and real parts of electron , and �k is the dispersion relation of the bare, or non- d band. The self-energy can be extracted from ARPES nts by, e.g., Kramers–Kronig transformation of the mea- sually, in high-temperature superconductors [7,8], the ding author. Tel.: +1 505 667 4819. ress:
[email protected] (T. Durakiewicz). [11]. The electronic specific heat of USb2 of 25 mJ K−2 mol−1 is small for heavy fermions, but typical for correlated uranium compounds with 5f-conduction band hybridization. Photoemission studies [12] have shown the existence of dispersive 5f bands, and the quasi-two- dimensional character of USb2. Higher resolution ARPES provided detailed measurement of the dispersion of the 5f band [13], and found the first kink structure in any f-electron system [3]. The origin of the kink was explained in the framework of the point-like Fermi surface renormalization [3,5]. Here we extend the analysis of the ARPES data using a new method of the 2D curvature [14] to resolve the multi-gap electronic structure of USb2, and we apply the sim- ple band renormalization model based on interband self-energy to calculate the measured structure starting from the simplified LDA bare bands. Below we provide a brief description of the model, its application to USb2 and a simple scalable derivation in a form useful for coding. 2. Interband self-energy model It has been shown [3,5,15,16] that the single-electron self- energy produces a coupling between the quasiparticle bands which see front matter © 2013 Elsevier B.V. All rights reserved. rg/10.1016/j.elspec.2013.10.005 ng the multi-gap electronic structure o ergy urakiewicza,∗, Peter Riseboroughb, Jian-Qiao Men ational Laboratory, MPA-CMMS Group, Mailstop K764, Los Alamos, NM 87545, USA rsity, Philadelphia, PA 19121, USA e i n f o e xxx lization a b s t r a c t The simple yet effective method of c interband self-energy is shown. Applic are given. The method allows a direct helping to resolve and picture the com ction systems exhibit a vast array of complex physical behav- eavy fermion family of f-electron systems, the complex of a coherent electron liquid and localized f-moments ybridization leads to emergence of new, and often tates at low temperatures, like: magnetism, supercon- exotic hidden order state [1]. This emergent behavior is most u single sure th self-en e.g., in a simp derive and re / locate /e lspec Sb2 with interband ating a renormalized band structure based on approximate to a multi-gap f-electron system USb2 is discussed and details lization of multiple band renormalizations in N band systems, interacting band structure of USb2. © 2013 Elsevier B.V. All rights reserved. l procedure of self-energy extraction is performed on a in order to establish the kink energy scale and/or mea- ass renormalization factors from the slope of Re . The may be calculated ab initio or estimated indirectly from, nd scattering models. In this work we show that such odel of band renormalization based on the self-energy m interband interactions may also be used to identify uce a much more complex, multi-gap band structure of Please cite p electronic structure of USb2 with interband self-energy, J. Electron Sp 3.10.005 ARTICLE IN PRESSG ModelELSPEC-46183; No. of Pages 4 2 T. Durakiewicz et al. / Journal of Electron Spectroscopy and Related Phenomena xxx (2013) xxx– xxx causes the bands to shift with respect to the Fermi energy. This was first shown for a two-band model in USb2, in which the full momentum-dependence of the interaction was taken into account [3,5]. Subsequently this effect was demonstrated in two- and four- band mode that the full tion, but th approximat We shal is applicabl bands N. Th the solution ω − �˛k + � where �˛k is �˛k(ω) is th the analysi interband s dent of k an the various �˛(iωn) = − where ωm local electr propagator (1/�)ImD( D(iωm) = − If the vario individual b one finds th by the solut iωn − �˛k + � where �ˇ is constant is g2˛,ˇ = �2˛,ˇ The above f the interact is a linear su acteristic en for the ˇth �ˇ = ∫ d� � On analytic frequencies by the solu resulting a requires tha cal. The coe of ωN−1, on �ˇ = �ˇk which goes order terms xamp found with the interband self-energy model. Vertical cuts in top panel, a, b, and c, correspond to the EDC profiles shown in bottom panel and marked e. is the direct band gap. See text for description. ths, which would be too restrictive. This procedure leads to proximate equation + � = ∑ ˇ g2 ˛,ˇ ω − �ˇk + � (9) determines the quasiparticle energies, and also redistributes ensity of ˛-character over the N quasiparticle bands. The l weight associated with the various characters can some- be inferred from photoemission experiments. For example, ctron systems, the photon energy dependence of M is some- e used to distinguish the f spectral weight from the weight ted with conduction bands. ig. 1 we show a typical application to f-electron systems, the hybridization of a heavy f-electron band with a strongly sive light conduction band leads to formation of a direct ization gap. The gap size is 2V, where V is hybridization ial, or – in the language used here – interband coupling ial. The indirect gap is not shown for simplicity. Cuts a, b, n Fig. 1 illustrate the energy distribution curves (EDC) of the article weight associated with the light conduction band, showing a cut through the direct gap. lication to USb2 measured Fermi surface of USb2 differs from the one calcu- y LDA [3]. Specifically, as shown in Fig. 2, the experimental surface comprises one hole-like cylindrical Fermi sheet cen- long the (0, 0, 1) axis, which includes and Z high symmetry this article in press as: T. Durakiewicz, et al., Resolving the multi-ga ectrosc. Relat. Phenom. (2013), http://dx.doi.org/10.1016/j.elspec.201 ls for iron-based pnictides [15,16] where it was implied momentum-dependence is not crucial for the descrip- at instead the effect can be described by using a local ion to the self-energy [17–19]. l consider an Occam’s razor model of this effect which e to a multi-band models with an arbitrary number of e quasiparticle energies for the ˛th band are found from s of − Re�˛k(ω) = 0 (2) the dispersion relation for electrons in the ˛th band and e corresponding single-particle self-energy. Following s of Ortenzi et al. [15] in the local approximation, the elf-energy for the thermal Green’s function is indepen- d can be expressed as a sum over contributions form bands via T ∑ ˇ,m �2˛,ˇD(iωm)Gˇ(iωn − iωm) (3) are the bosonic Matsubara frequencies, Gˇ(iωn) is the onic Green’s function and D(ω) is the local bosonic expressible in terms of the boson spectral density ω) via 2 � ∫ dω ωImD(ω) ω2m + ω2 (4) us contributions to the interband self-energy from the ands are evaluated in the single-pole approximation, at the quasiparticle energies for the ˛th band are given ions of = ∑ ˇ g˛,ˇ 2 iωn − �ˇ + � (5) a characteristic energy of the ˇth band and the coupling given by 2 � ∫ dω ImD(ω) ω (6) orm of the interband self-energy expresses the fact that ion causes the quasiparticles to have a character which perposition of the different band characters. The char- ergy �ˇ is given by an integral over the density of states band ˇ(�)� (7) ally continuing the self-energy from imaginary to real , one finds that the N quasi-particle energies are given tions of a set of Nth order polynomial equations. The pproximation is not self-consistent. Self-consistency t the coefficients of the various powers of ω be identi- fficients of ωN are identical. On equating the coefficients e finds the condition (8) beyond the local approximation. Equating the higher- would require equality between the non-zero coupling Fig. 1. E weight marked the sam streng the ap ω − �˛k which the int spectra times in f-ele times b associa In F where disper hybrid potent potent and c i quasip with b 3. App The lated b Fermi tered a le of a simple band renormalization and redistribution of spectral Please cite Electron Sp ARTICLE IN PRESSG ModelELSPEC-46183; No. of Pages 4 T. Durakiewicz et al. / Journal of Electron Spectroscopy and Related Phenomena xxx (2013) xxx– xxx 3 Fig. 2. Experimental (ARPES) Fermi surface of USb2, with one centrally located hole-like sheet and two electron-like sheets around the X-point. Note that the nonin- teracting (LDA) band calculation predicts two Fermi surface sheets centered around the point, b points (Z no around the Fermi surfa amount of appears in shown to b Fermi level In our a bands as pa the vicinity The simplifi sity for k va details dire the intensit and its intr sured in any momenta a trast at high visualisatio Compared band locali better visua In Fig. 3 to X direct in Stoughto the 2D cur Fig. 3. Multi-g in the Brilloui reduced with while maintai our renormali Note that the s structure clearly enhanced at large momentum values. Multiple kinks and gapping can be identified, driven by band renormaliza- tion at 0, −20 and −60 meV. The nature of the renormalization process within the central band at as well as the possible role of magnons discussed e energy scal the appare involved in the interba from LDA c three energ extending i seen in pan additional measured p 4. Conclus The prop ergy enor ested exhi orm sion. es th sions easu ned t tion listic wled s wo the a NL L DEFG dix A abov oded e no ut the central sheet is renormalized below the Fermi level. t shown here), and two electron-like sheets centered X points. All sheets are cylindrical, which makes the ce quasi-two dimensional in spite of a very small kz dependence. The missing, central hole-like sheet bare band calculation [3] but in experiments it was e removed by renormalization and shifted below the [3]. pplication of the model, we use both central hole-like rts of the starting bare band structure, but we simplify of the X point by using only one electron-like band. cation is justified by a rapid drop of measured inten- lues moving away from 0, which makes the measured ctly around the X point difficult to resolve. Specifically, y of the band centered at k = 0 in USb2 is very strong, insic width of only a few meV is the narrowest mea- 5f electron system so far [3], while the bands at higher re much weaker and appear broader. For improving con- values of k while maintaining true dispersion, we use a n technique based on the concept of 2D-curvature [14]. to plain second derivative, this method improves the zation and reduces the peak width, which allows for lisation especially of the weak bands. , panel a, we show the ARPES measurement in the ion, performed at the Synchrotron Radiation Center self-en band r have t which ple ren disper here us disper with m and tu descrip ing rea Ackno Thi under and LA award Appen model The easily c with th this article in press as: T. Durakiewicz, et al., Resolving the multi-gap ele ectrosc. Relat. Phenom. (2013), http://dx.doi.org/10.1016/j.elspec.2013.10.0 n, WI [3]. In panel b we show the result of applying vature method to reduce the ARPES data, with band ap band structure of USb2. ARPES measurement in the to X direction n zone is shown in panel a). Panel b shows the measured structure the 2D curvature algorithm (see text) to enhance the weak bands ning band dispersion. Panel c illustrates the product of application of zation model to the three bare bands obtained from LDA calculation. econd electron-like band at the X point was neglected for simplicity. E1 – lower b We start bands: G0 = ∣∣∣∣∣∣∣ g1(k, 0 0 where gn(k, ω) = ( and � → 0. W V = V0 ∣∣∣∣∣∣ 0 1 0 The couplin different va ctronic structure of USb2 with interband self-energy, J. 05 in establishing the additional gap structure are both lsewhere [3,20]. Here we note that the renormalization es appear to be momentum-independent, underscoring nt lack of momentum dependence in a bosonic mode renormalization. Panel c shows the result of applying nd self-energy model to the three bare bands obtained alculations, as described above. We also include the y scales shown in Fig. 3b into the model as flat bands, t to an effective 6-band model. All the kinks and gaps el b are reproduced and visualized in panel c, with structures showing around the X point where the hotoemission intensity is very low. ions osed simple model based on an approximate interband approach is a direct and effective tool in visualising the malization effects in correlated electron systems. We the model on the quasi-2D, 5f-electron system USb2, bits a rich, multi-gap electronic structure with multi- alization energy scales and kink-like features in band The three-band plus three energy scales model applied e bare LDA band structure to produce the renormalized and gap/kink features in good qualitative agreement rement. Interband coupling potentials were guessed o best agreement with measurement, but a quantitative is possible with theoretical effort devoted to establish- coupling. gements rk was performed at Los Alamos National Laboratory uspices of the U.S. Department of Energy, BES, DMSE, DRD Programs. P.R. was supported by the US DOE BES 02-84ER45872. . Simple derivation of the interband scattering e described model can be scaled to N bands and it can be . Here we provide a simple effective derivation, starting n-interacting band structure consisting of three bands: and, E2 – middle band, and E3 – higher band. by writing the Green’s function for three unperturbed ω) 0 0 g2(k, ω) 0 0 g3(k, ω) ∣∣∣∣∣∣∣ (A.1) ω − En + i�)−1 (A.2) e now introduce the interband coupling matrix as: 1 0 0 1 1 0 ∣∣∣∣∣∣ g matrix may be modified to, e.g., take into account lues of effective coupling between different bands. The Please cite p ele Electron Sp 3.10.0 ARTICLE IN PRESSG ModelELSPEC-46183; No. of Pages 4 4 T. Durakiewicz et al. / Journal of Electron Spectroscopy and Related Phenomena xxx (2013) xxx– xxx Dyson equation satisfied by Green’s function for the set of three interacting bands will be now: G = [I − G0 · V ]−1 · G0 (A.3) where I is the unit matrix: I = ⎡ ⎣ 1 0 00 1 0 0 0 1 ⎤ ⎦ (A.4) We can write the solution of equation (A.3) as: G = 1 1 − g2(g1 + g3)V20 ⎡ ⎢⎣ g1 − g1g2g3V20 g1g2V0 g1g2g3V20 g1g2V0 g2 g2g3V0 g1g2g3V20 g2g3V0 g3 − g1g2g3V20 ⎤ ⎥⎦ (A.5) The trace of G is given by Trace G = g1 + g2 + g3 − 2g1g2g3V 2 0 1 − g2(g1 + g3)V20 (A.6) The energies of the excitations are given by the poles of the trace. This means that the zeros of the denominator represent the ener- gies of the renormalized bands. For the three bands used here this can be written in terms of a function F: F = (ω − E1)(ω − E2)(ω − E3) − V20 (ω − E1) − V20 (ω − E3) (A.7) The real solutions of F = 0 represent the renormalized bands. References [1] J.A. Mydosh, P.M. Oppeneer, Rev. Mod. Phys. 83 (2011) 1301. [2] P. Schlottmann, Phys. Rep. 181 (1989) 1 (and references therein). [3] T. Durakiewicz, P.S. Riseborough, C.G. Olson, J.J. Joyce, P.M. Oppeneer, S. Elgaz- zar, E.D. Bauer, J.L. Sarrao, E. Guziewicz, D.P. Moore, M.T. Butterfield, K.S. Graham, Europhys. Lett. 84 (2008) 37003. [4] G.L. Dakovski, Y.S.M. Li, J.L. Sarrao, P.M. Oppeneer, P.S. Riseborough, J.A. Mydosh, T. Durakiewicz, Phys. Rev. B 84 (2011) 161103. [5] X. Yang, P.S. Riseborough, T. Durakiewicz, C.G. Olson, J.J. Joyce, E.D. Bauer, J.L. Sarrao, D.P. Moore, K.S. Graham, S. Elgazzar, P.M. Oppeneer, E. Guziewicz, M.T. Butterfield, Philos. Mag. 89 (2009) 1893. [6] X. Yang, P.S. Riseborough, T. Durakiewicz, J. Phys.: Condens. Matter 23 (2011) 94211. [7] A. Damascelli, Z. Hussain, Z. Shen, Rev. Mod. Phys. 75 (2003) 473. [8] P.D. Johnson, A.V. Fedorov, T. Valla, J. Electron Spectrosc. Relat. Phenom. 117–118 (2001) 153. [9] W. Trzebiatowski, R. Troc, Bull. Acad. Polon. Sci. Ser. Sci. Chim. 11 (1963) 661. [10] A. Henkie, Z. 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Prasankumar, Phys. Rev. Lett. 111 (2013) 57402. this article in press as: T. Durakiewicz, et al., Resolving the multi-ga ectrosc. Relat. Phenom. (2013), http://dx.doi.org/10.1016/j.elspec.201 ctronic structure of USb2 with interband self-energy, J. 05 Resolving the multi-gap electronic structure of USb2 with interband self-energy 1 Introduction 2 Interband self-energy model 3 Application to USb2 4 Conclusions Acknowledgements Appendix A Simple derivation of the interband scattering model References