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PhysRevB.90.115421.pdf

PHYSICAL REVIEW B 90, 115421 (2014) Multiterminal Anderson impurity model in nonequilibrium: Analytical perturbative treatment Nobuhiko Taniguchi* Institute of Physics, University of Tsukuba, Tennodai Tsukuba 305-8571, Japan (Received 1 May 2014; revised manuscript received 28 August 2014; published 16 September 2014) We study the nonequilibrium spectral function of the single-impurity Anderson model connecting with multiterminal leads. The full dependence on frequency and bias voltage of the nonequilibrium self-energy andspectral function is obtained analytically up to the second-order perturbation regarding the interaction strength U. High- and low-bias voltage properties are analyzed for a generic multiterminal dot, showing a crossover from theKondo resonance to the Coulomb peaks with increasing bias voltage. For a dot where the particle-hole symmetryis not present, we construct a current-preserving evaluation of the nonequilibrium spectral function for arbitrarybias voltage. It is shown that finite-bias voltage does not split the Kondo resonance in this order, and no specificstructure due to multiple leads emerges. Overall bias dependence is quite similar to finite-temperature effect fora dot with or without the particle-hole symmetry. DOI: 10.1103/PhysRevB.90.115421 PACS number(s): 73 .63.Kv,73.23.Hk,71.27.+a I. INTRODUCTION Understanding strong correlation effect away from equi- librium has been one of the most interesting yet challeng- ing problems in condensed matter physics. A prominent realization of such phenomena is embodied in quantumtransport through a nanostructure under finite-bias voltage.To understand the interplay of the correlation effect andnonequilibrium nature, the nonequilibrium version of thesingle-impurity Anderson model (SIAM) and its extensionshave been serving and continue to do so as a central theoretical model. The SIAM is indeed considered to be one of the best- studied strongly correlated models, and despite its apparentsimplicity, it exhibits rich physics already in equilibrium,such as the Coulomb blockade and the Kondo physics thathave been observed in experiments. Equilibrium propertiesof the SIAM have been well understood thanks to concertedefforts of several theoretical approaches over the years: byperturbative treatment, Fermi-liquid description, as well as exact results by the Bethe ansatz method, and numerical renormalization group (NRG) calculations (see, for instance,[1–3].) In contrast, the situation of the nonequilibrium SIAM is not so satisfactory. Each of the above approaches has met somedifficulty in treating nonequilibrium phenomena. A theoreticalapproach that can deal with the strong correlation effect innonequilibrium is still called for. Notwithstanding, a number of analytical and numerical methods have been devised to investigate nonequilibrium sta-tionary phenomena: nonequilibrium perturbation approaches[4–8] and its modifications [ 9–11], the noncrossing approxi- mation [ 12], the functional renormalization group treatment [13], quantum Monte Carlo calculations on the Keldysh contour [ 14,15], the iterative real-time path-integral method [16], and so on. Unfortunately, those approaches fail to give a consistent picture concerning the finite-bias effect on the dot spectral function, particularly regarding a possible splitting ofthe Kondo resonance. As for the equilibrium SIAM, the second-order perturbation regarding the Coulomb interaction Uon the dot [ 17–20] is known to capture essential features of Kondo physics *taniguchi.n.gf@u.tsukuba.ac.jpand agrees qualitatively well with exact results obtained by the Bethe ansatz and NRG approaches [ 1,2,20]. Such good agreement seems to persist in nonequilibrium stationary state at finite-bias voltage. For the two-terminal particle-hole (PH) symmetric SIAM, a recent study by M ¨uhlbacher et al. [8] showed that the nonequilibrium second-order perturbation calculation of the spectral function agrees with that calcu-lated by the diagrammatic quantum Monte Carlo simulation, excellently up to interaction strength U∼2γ(where γis the total relaxation rate due to leads), pretty well even for U/lessorsimilar8γ at bias voltage eV/lessorsimilar2γ. A typical magnitude of U/γ of a semiconductor quantum dot is roughly 1 ∼10 depending on the size and the configuration of the dot. Therefore, there is a good chance of describing a realistic system within the validity of nonequilibrium perturbation approach. The great advantage of semiconductor dot systems is to allow us to control several physical parameters. Those include changing gate voltage as well as configuring a more involved structure such as a multiterminal dot [ 21–26] or an interferom- eter embedding a quantum dot. Theoretical treatments oftenlimit themselves to a system with the PH symmetry wherethe dot occupation number is fixed to be one half per spin.Although assuming the PH symmetry makes sense and comesin handy in extracting the essence of the Kondo resonance,we should bear in mind that such symmetry is not intrinsicand can be broken easily in realistic systems, by gate voltage,asymmetry of the coupling with the leads, or asymmetric dropsof bias voltage [ 27,28]. The PH asymmetry commonly appears in a multiterminal dot or in an interferometer embeddinga quantum dot. It is also argued that the effect of the PHasymmetry might be responsible for the deviation observedin nonequilibrium transport experiments from the “universal” behavior of the PH symmetric SIAM [ 27,28]. To work on realistic systems, it is imperative to understand how the PHasymmetry affects nonequilibrium transport. In this paper, we examine the second-order nonequilibrium perturbation regarding the Coulomb interaction Uof the multiterminal SIAM. The PH symmetry is not assumed, andmiscellaneous types of asymmetry of couplings to the leadsand/or voltage drops are incorporated as a generic multitermi-nal configuration. Our main focus is to provide solid analytical results of the behavior of the nonequilibrium self-energy and 1098-0121/2014/90(11)/115421(12) 115421-1 ©2014 American Physical SocietyNOBUHIKO TANIGUCHI PHYSICAL REVIEW B 90, 115421 (2014) hence the dot spectral function for the full range of frequency and bias voltage, within the validity of the second-orderperturbation theory of U. The result encompasses Fermi-liquid behavior as well as incoherent non-Fermi-liquid contribution,showing analytically that increasing finite-bias voltage leads toa crossover from the Kondo resonance to the Coulomb block- ade behaviors. This work contrasts preceding perturbative studies [ 4–7,29] whose evaluations relied on either numerical means or a small-parameter expansion of bias voltage and fre-quency. The only notable exception, to the author’s knowledge,is a recent work by M ¨uhlbacher et al. [8], which succeeded in evaluating analytically the second-order self-energy forthe two-terminal PH symmetric dot. Intending to apply suchanalysis to a wider range of realistic systems and examine the effect that the two-terminal PH symmetric SIAM cannot capture, we extend their approach to a generic multiterminaldot where the PH symmetry may not necessarily be present. An embarrassing drawback of using the nonequilibrium perturbation theory is that when one has it naively apply to thePH asymmetric SIAM, it may disrespect the preservation ofthe steady current [ 4]. As a result, one then needs some current- preserving prescription, and different self-consistent schemes have been proposed and adopted [ 9,11,30]. As will be seen, the current-preserving condition involves all the frequencyranges, not only of the low-frequency region that validatesFermi-liquid description, but also of the incoherent non-Fermi-liquid part [see Eq. ( 6)]. Therefore, an approximation based on the low-energy physics, particularly the Fermi-liquid picture,should be used with care. The self-energy we will construct analytically is checked to satisfy the spectral sum rule at finite-bias voltage, so that we regard it as giving a consistentdescription for the full range of frequency in nonequilibrium.By taking its advantage, we also demonstrate a self-consistent,current-preserving calculation of the nonequilibrium spectralfunction for a system where the PH symmetry is not present. The paper is organized as follows. In Sec. II, we introduce the multiterminal SIAM in nonequilibrium. We review briefly how to obtain the exact current formula by clarifying the roleof the current conservation at finite-bias voltage. Section III presents analytical expression of the retarded self-energy fora general multiterminal dot up to the second order of theinteraction strength. Subsequently, in Sec. IV, we examine and discuss its various analytical behaviors including high- andlow-bias voltage limits. Section Vis devoted to constructing a nonequilibrium spectral function using the self-energy obtained in the previous section. We focus our attention on twoparticular situations: (1) self-consistent, current-preservingevaluation of the nonequilibrium spectral function for thetwo-terminal PH asymmetric SIAM, and (2) multiterminaleffect of the PH symmetric SIAM. Finally, we conclude inSec. VI. Mathematical details leading to our main analytical result ( 21) as well as other necessary material regarding dilogarithm are summarized in Appendices. II. MULTITERMINAL ANDERSON IMPURITY MODEL AND THE CURRENT FORMULA A. Model The model we consider is the single-impurity Anderson model connecting with multiple leads a=1,..., N whosechemical potentials are sustained by μa. The total Hamiltonian of the system consists of H=HD+HT+/summationtext aHa, where HD, HT, andHarepresent the dot Hamiltonian with the Coulomb interaction, the hopping term between the dot and the leads,and the Hamiltonian of a noninteracting lead a, respectively. They are specified by H D=/summationdisplay σ/epsilon1dnσ+Un↑n↓, (1) HT=/summationdisplay a,σ(Vdad† σcakσ+Vadc† akσdσ), (2) where nσ=d† σdσis the dot electron number operator with spinσandcakσare electron operators at the lead a.I nt h e following, we consider the spin-independent transport case,but an extension to the spin-dependence situation such asin the presence of magnetic field or ferromagnetic leads isstraightforward. When applying the wide-band limit, all theeffects of the lead aare encoded in terms of its chemical potential μ aand relaxation rate γa=π|Vda|2ρa, where ρais the density of states of the lead a. The dot level /epsilon1dcontrols the average occupation number on the dot; it correspondsroughly to 2, 1, 0 for /epsilon1 d/lessorsimilar−U,−U/lessorsimilar/epsilon1d/lessorsimilar0, and 0 /lessorsimilar/epsilon1d, respectively. The PH symmetry is realized when /epsilon1d=−U/2 and/angbracketleftnσ/angbracketright=1 2[see Eqs. ( 6) and ( 13)]. B. Multiterminal current and current conservation We here briefly summarize how the current through the dot is determined in a multiterminal setting. Special attentionis paid to the role of the current conservation because it hasbeen known that nonequilibrium perturbation calculation doesnot respect it in general [ 4]. We illustrate how to ensure the current conservation by a minimum requirement. The argu-ment following is valid regardless of a specific approximationscheme chosen, whether nonequilibrium perturbation or anyother approach. Following the standard protocol of the Keldysh formulation [31], we start with writing the current I aflowing from the lead ato the dot in terms of the dot’s lesser Green’s function G−+ σ and the retarded one GR σ: Ia=−e π/planckover2pi1/summationdisplay σ/integraldisplay dω/bracketleftbig iγaG−+ σ(ω)−2γafaImGR σ(ω)/bracketrightbig ,(3) where fa(ω)=1/(eβ(ω−μa)+1) is the Fermi distribution function at the lead a. As the present model preserves the total spin as well as charge, the net spin current flowing tothe dot should vanish in the steady state, which imposes theintegral relation between G −+ σandGR σ: /integraldisplay∞ −∞dω/bracketleftbig iγ G−+ σ(ω)+2γ¯f(ω)I mGR σ(ω)/bracketrightbig =0. (4) Here, we have introduced the total relaxation rates γ=/summationtext aγa and the effective Fermi distribution ¯fweighted by the leads ¯f(ω)=/summationdisplay aγa γfa(ω). (5) 115421-2MULTITERMINAL ANDERSON IMPURITY MODEL IN . . . PHYSICAL REVIEW B 90, 115421 (2014) When we ignore the energy dependence of the relaxation rates γa, we can recast Eq. ( 4) into a more familiar form nσ=−1 π/integraldisplay∞ −∞dω ¯f(ω)I mGR σ(ω)( 6 ) because 2 iπnσ=/integraltext dωG−+ σ(ω) is the definition of the exact dot occupation number. Note the quantity −ImGR σ(ω)/πis nothing but the exact dot spectral function out of equilibrium.We emphasize that Eq. ( 4) [or equivalently Eq. ( 6)] is the minimum, exact requirement that ensures the currentpreservation. It constrains the exact G −+andGRthat depend on the interaction as well as bias voltage in a nontrivial way.One can accordingly eliminate/integraltext dωG −+(ω)i nIa, to reach the Landauer-Buttiker–type current formula at the lead a, Ia=−e π/planckover2pi1/summationdisplay b,σγaγb γ/integraldisplay dω(fb−fa)I mGR σ(ω). (7) Or, the current conservation allows us to write it as Ia=eγa /planckover2pi1/summationdisplay σ[nσ−Nσ(μa)], (8) where Nσ(ε) is the exact number of states with spin σat finite temperature in general, defined by Nσ(μ)=−1 π/integraldisplay dεImGR σ(ε) eβ(ε−μ)+1. (9) It tells us that differential conductance ∂Ia/∂μawith fixing all other μ’s is proportional to the nonequilibrium dot spectral function, provided changing μadoes not affect the occupation number [ 21–24]. Such situation is realized, for instance, when a probe lead couples weakly to the dot. The case of a noninteracting dot always satisfies the current- preserving condition ( 4)a sG−+ σ(ω)=− 2i¯f(ω)I mGR σ(ω) holds for any ω; the distribution function of dot electrons fdot(ω)=G−+(ω)/(2iπ) is equal to −¯f(ω)I mGR σ(ω)/π. This is not the case for an interacting dot, however. As forthe interacting case, not so much can be said. We only see thespecial case with the two-terminal PH symmetric dot satisfyEq. ( 6) by choosing n σ=1 2irrespective of interaction strength. Except for this PH symmetric case, a general connectionbetween G −+andGRis not known so far. It is remarked that, based on the quasiparticle picture, a noninteracting relationG −+ σ(ω)=− 2i¯f(ω)I mGR σ(ω) is sometimes used to deduce an approximate form of G−+out of GRfor an interacting dot. Such approximation is called the Ng’s ansatz [ 32,33]. Although it might be simple and handy, its validity is far fromclear. We will not rely on such additional approximation below.It is also important to distinguish in Eq. ( 6) the electron occupa- tion number n σfrom the quasiparticle occupation number ˜nσ, as the two quantities are different at finite-bias voltage since theLuttinger relation holds only in equilibrium [ 34]. Contribution to the dot occupation number comes from all ranges offrequency, including the incoherent part. One sees fulfilling thespectral weight sum rule −/integraltext ∞ −∞dωImGR(ω)/π=1 crucial to have the dot occupation number nσnormalized correctly. In general, one needs to determine nσappropriately to satisfy Eq. ( 6) as a function of interaction and chemical potentials of the leads. The applicability of quasiparticle approaches thatignores the incoherent part is unclear. FIG. 1. The Hartree-type contribution of the self-energy Uτ 3n¯σ=±Un ¯σ. The double line refers to the exact Green’s function. III. ANALYTICAL EV ALUATION OF THE SELF-ENERGY In this section, we evaluate analytically the nonequilibrium retarded self-energy up to the second order of interactionstrength Ufor the multiterminal SIAM. We first examine the contribution at the first order and the role of currentpreservation. Then, we present the analytical result of thesecond-order self-energy in terms of dilogarithm. Following the standard treatment of the Keldysh formu- lation [ 35], the nonequilibrium Green’s function and the self-energy take a matrix structure ˆG=/parenleftbigg G −−G−+ G+−G++/parenrightbigg ;ˆ/Sigma1=/parenleftbigg /Sigma1−−/Sigma1−+ /Sigma1+−/Sigma1++/parenrightbigg ,(10) satisfying symmetry relations G−−+G++=G−++G+− and/Sigma1−−+/Sigma1++=−/Sigma1−+−/Sigma1+−. The retarded Green’s function is defined by GR=G−−+G−+; the retarded self- energy, by /Sigma1R=/Sigma1−−+/Sigma1−+. To proceed with the evaluation, it is convenient to classify self-energy diagrams into two types: the Hartree-type diagram(Fig. 1) that can be disconnected by cutting a single interaction line, and the rest which we call the correlation part and reassignthe symbol /Sigma1to. The latter starts at the second order. The resulting Green’s function (matrix) takes a form of ˆG σ(ω)=/bracketleftbigˆG−1 0σ(ω)−Uτ 3n¯σ−ˆ/Sigma1σ(ω)/bracketrightbig−1, (11) where τ3represents a Pauli matrix of the Keldysh structure, andn¯σrefers to the exact occupation number of the dot elec- tron with the opposite spin. Accordingly, the correspondingretarded Green’s function becomes G R σ(ω)=1 ω−Edσ+iγ−/Sigma1Rσ(ω), (12) where Edσ=/epsilon1d+Un ¯σis the Hartree level of the dot. A. Current preservation at the first order Before starting evaluating the correlation part /Sigma1Rthat starts contributing at the second order, it is worthwhile to examinethe current-preserving condition ( 6) at the first order. At this order, it reduces to the self-consistent Hartree-Fock equationfor the dot occupation number n 0 σ: n0 σ=1 2+1 π/summationdisplay aγa γarctan/bracketleftbiggμa−/epsilon1d−Un0 ¯σ γ/bracketrightbigg .(13) It shows how the two-terminal PH symmetric SIAM is special by choosing /epsilon1d+U/2=0,γa=γ/2, and μa=±eV/2; the 115421-3NOBUHIKO TANIGUCHI PHYSICAL REVIEW B 90, 115421 (2014) E+ , , , FIG. 2. The correlation part of the self-energy at the second-order contribution. second term of the right-hand side vanishes by having the solution n0 σ=1 2even at finite-bias voltage. It also indicates that the current preservation necessarily has the occupationnumber depend on asymmetry of the lead couplings as well asinteraction strength for the PH asymmetric SIAM. Indeed, fora small deviation from the PH symmetry and bias voltage, wesee the Hartree-Fock occupation number behave as n 0 σ−1 2≈¯μ−/epsilon1d−U/2 πγ/parenleftbigg 1−U πγ+···/parenrightbigg , (14) where ¯ μis the average chemical potential weighted by leads ¯μ=/summationdisplay aγa γμa. (15) Note ¯ μvanishes when no bias voltage applies, as we incorporate the overall net offset by leads into /epsilon1d. B. Analytical evaluation of the correlation part of the self-energy Following the standard perturbation treatment of the Keldysh formulation, we see there is only one diagramcontributing to /Sigma1 R σat the second order (Fig. 2) after eliminating the Hartree-type contribution. The contribution is written as ˆ/Sigma1(t1,t2)=−i/planckover2pi1U2/parenleftbigg G−− 12/Pi1−−21−G−+ 12/Pi1+−21 −G+− 12/Pi1−+21G++12/Pi1++21/parenrightbigg ,(16) where Gij 12=Gij(t1,t2) refer to to the unperturbed Green’s functions (including the Hartree term), whose concrete ex-pressions are found in Appendix A. The polarization matrix ˆ/Pi1 is defined by /Pi1ij 12=i/planckover2pi1Gij 12Gji21(Fig. 3).1 As was shown by the current formula in the previous section, we need only the dot spectral function to studyquantum transport, hence, /Sigma1 Rsuffices. Therefore, it is more advantageous to work on the representation in terms ofthe retarded, advanced, and Keldysh components, where thepolarization parts become /Pi1 R 12=i/planckover2pi1 2/bracketleftbig GR 12GK21+GK 12GA21/bracketrightbig , (17a) /Pi1A 12=i/planckover2pi1 2/bracketleftbig GA 12GK21+GK 12GR21/bracketrightbig , (17b) /Pi1K 12=i/planckover2pi1 2/bracketleftbig GK 12GK21+GR 12GA21+GA 12GR21/bracketrightbig , (17c) 1We define the polarization to satisfy the symmetric relation /Pi1−−+ /Pi1−−=/Pi1−++/Pi1+−.i j E+ , E, FIG. 3. The polarization part. and their Fourier transformations are given in Appendix B. Accordingly, we can express the retarded self-energy /Sigma1Ras /Sigma1R σ(ω)=−iU2 4π[I1(ω)+I2(ω)], (18) where I1(ω)=/integraldisplay+∞ −∞dEGR σ(E+ω)/Pi1K ¯σ(E), (19) I2(ω)=/integraldisplay+∞ −∞dEGK σ(E+ω)/Pi1A ¯σ(E). (20) The above second-order expression of /Sigma1Ris standard, but it has so far been mainly used for numerical evaluation, quiteoften restricted for the two-terminal PH symmetric SIAM. Weintend to evaluate Eqs. ( 19) and ( 20) analytically for the generic multiterminal SIAM, along the line employed in Ref. [ 8]. Delegating all the mathematical details to Appendices C andD, we summarize our result of the analytical evaluation of /Sigma1 Ras follows: /Sigma1R σ(ω)=iγU2 8π2(ω−Edσ+iγ)/bracketleftbigg/Xi11(ω−Edσ) ω−Edσ−iγ +/Xi12(ω−Edσ) ω−Edσ+3iγ+/Xi13 2iγ/bracketrightbigg . (21) Here, functions /Xi1i(i=1,2,3) are found to be (using ζaσ= μa−Edσ) /Xi11(ε)=2π2ε iγ+/summationdisplay a,b,β4γaγb γ2/bracketleftbigg Li2/parenleftbigg−ε+ζaσ βζb¯σ+iγ/parenrightbigg +Li2/parenleftbigg−ε−βζb¯σ −ζaσ+iγ/parenrightbigg +1 2Log2/parenleftbigg−ζaσ+iγ βζb¯σ+iγ/parenrightbigg/bracketrightbigg +/summationdisplay a,b,β4γaγb γ2/bracketleftbigg Li2/parenleftbigg−ε+βζa¯σ βζb¯σ+iγ/parenrightbigg +1 4Log2/parenleftbigg−ζa¯σ+iγ ζb¯σ+iγ/parenrightbigg/bracketrightbigg , (22a) /Xi12(ε)=6π2−/summationdisplay a,b,β4γaγb γ2/bracketleftbigg /Lambda1/parenleftbiggε−ζaσ+2iγ βζb¯σ+iγ/parenrightbigg +/Lambda1/parenleftbiggε−βζb¯σ+2iγ ζaσ+iγ/parenrightbigg +1 2Log2/parenleftbiggζaσ+iγ βζb¯σ+iγ/parenrightbigg/bracketrightbigg −/summationdisplay a,b,β4γaγb γ2/bracketleftbigg /Lambda1/parenleftbiggε+βζa¯σ+2iγ βζb¯σ+iγ/parenrightbigg +1 4Log2/parenleftbigg−ζa¯σ+iγ ζb¯σ+iγ/parenrightbigg/bracketrightbigg , (22b) /Xi13=/bracketleftbigg/summationdisplay a2γa γLog/parenleftbigg−ζa¯σ+iγ ζa¯σ+iγ/parenrightbigg/bracketrightbigg2 , (22c) 115421-4MULTITERMINAL ANDERSON IMPURITY MODEL IN . . . PHYSICAL REVIEW B 90, 115421 (2014) where the summations over β=± 1 as well as terminals a,b are understood. Function Li 2(z) is dilogarithm, whose definition as well as various useful properties are summarizedin Appendix C;/Lambda1(z) is defined by 2 /Lambda1(z)=Li2(z)+[Log(1 −z)−Log(z−1)] Log z.(23) The analytical formula /Sigma1Rgiven in Eqs. ( 21) and ( 22) constitutes the main result of this paper. Consequently, thenonequilibrium spectral function of the multiterminal SIAMis given analytically for a full range of frequency and biasvoltage, once one chooses n σto satisfy Eq. ( 6). The result also applies to a more involved structured system, such as aninterferometer embedding a quantum dot, by simply replacing/epsilon1 dandγato take account of those geometric effects. IV . V ARIOUS ANALYTICAL BEHA VIORS Having obtained an explicit analytical form of the second- order self-energy /Sigma1R(ω) at arbitrary frequency and bias voltage, we now examine its various limiting behaviors. Mostof those limiting behaviors have been known for the two-terminal PH symmetric SIAM, so it is assuring to reproducethose expressions in such a case. Simultaneously, our resultsfollowing provide multiterminal, PH asymmetric extensionsof those asymptotic results. A. Equilibrium dot with and without the PH symmetry We can reproduce the equilibrium result by setting all the chemical potentials equal, μaσ=Edσ=/epsilon1d+Un ¯σ. Then, we immediately see /Xi13=0 and /Xi11=8/bracketleftbiggπ2 4/parenleftbiggεσ iγ/parenrightbigg +3L i 2/parenleftbigg−εσ iγ/parenrightbigg/bracketrightbigg , (24) /Xi12=8/bracketleftbigg3π2 4−3/Lambda1/parenleftbiggεσ+2iγ iγ/parenrightbigg/bracketrightbigg , (25) where εσ=ω−Edσ. As a result, the correlation part of the self-energy in equilibrium becomes /Sigma1R σ(ω)=iγU2 π2(εσ+iγ)/bracketleftBiggπ2 4/parenleftbigεσ iγ/parenrightbig +3L i 2/parenleftbig−εσ iγ/parenrightbig εσ−iγ +3π2 4−3/Lambda1/parenleftbig 2+εσ iγ/parenrightbig εσ+3iγ/bracketrightBigg . (26) The PH symmetric case in particular corresponds to εσ=ω. It reproduces the perturbation results by Yamada and Yosida[17–19] up to the second order of U, when we expand the above for small ω. The PH symmetric result is indeed identical with the one obtained in Ref. [ 8] for arbitrary frequency [see also Eq. ( 27)]. 2The definition of /Lambda1(z) is equivalent to that given in Ref. [ 8], but we prefer writing it in this form because its analyticity is more transparent.B. Nonequilibrium PH symmetric dot connected with two terminals M¨uhlbacher et al. [8] have evaluated analytically the self-energy and the spectral function for the two-terminal PHsymmetric SIAM. In our notation, it corresponds to the caseγ L=γR=γ/2, and Edσ=0. When we parametrize the two chemical potentials by μa=ζaσ=aeV/ 2 with a=± 1i n Eqs. ( 22), the self-energy can be written as /Sigma1R σ(ω)=iγU2 8π2(ω+iγ)/bracketleftbigg/Xi11(ω) ω−iγ+/Xi12(ω) ω+3iγ/bracketrightbigg , (27) where /Xi11=2π2ω iγ+6/summationdisplay a,b/bracketleftbigg Li2/parenleftbigg−ω+aeV/ 2 beV/ 2+iγ/parenrightbigg +1 4Log2/parenleftbigg−aeV/ 2+iγ beV/ 2+iγ/parenrightbigg/bracketrightbigg , /Xi12=6π2−6/summationdisplay a,b/bracketleftbigg /Lambda1/parenleftbiggω−aeV/ 2+2iγ beV/ 2+iγ/parenrightbigg +1 4Log2/parenleftbiggaeV/ 2+iγ beV/ 2+iγ/parenrightbigg/bracketrightbigg . The above results are identical with what Ref. [ 8] obtained. C. Expansion of small bias and frequency We now employ the small-parameter expansion of /Sigma1R around the half-filled equilibrium system. Here, we assume parameters ζaσ=μa−Edσandεσ=ω−Edαare much smaller than the total relaxation rate γ. The expansion of /Xi11is found to contain the first- and second-order terms regarding ζaσ andεσ, while /Xi12,3do only the second-order terms. Therefore, the result of the expansion up to the second order of thesesmall parameters is presented as /Sigma1 R σ(ω)≈iU2 8π2γ/bracketleftbigg /Xi11−/Xi12 3−/Xi13 2/bracketrightbigg . (28) Functions /Xi1ican be expanded straightforwardly by using the Taylor expansion of dilogarithm in Appendix C.T h e ya r e found to behave as /Xi11(ε)≈8/bracketleftbigg(π2−12)ε+4¯μ 4iγ+3ε2+9μ2−6ε¯μ−2¯μ2 4(iγ)2/bracketrightbigg , (29) /Xi12(ε)≈− 8/bracketleftbigg3(−ε2+2ε¯μ+2¯μ2−μ2−2μ2) 4(iγ)2/bracketrightbigg ,(30) /Xi13≈16 ¯μ2 (iγ)2, (31) where ¯ μis defined in Eq. ( 15) and we have introduced μ2=/summationdisplay aγa γμ2 a=¯μ2+(δμ)2. (32) 115421-5NOBUHIKO TANIGUCHI PHYSICAL REVIEW B 90, 115421 (2014) Combining all of these, we reach the small-bias (-frequency) behavior of the self-energy /Sigma1R σas /Sigma1R σ(ω)≈U2 π2γ2/bracketleftbigg/parenleftbiggπ2 4−3/parenrightbigg (ω−Edσ)+¯μ/bracketrightbigg −iU2 2π2γ3[(ω−¯μ)2+3(δμ)2]. (33) Small-bias expansion of Im /Sigma1Rfor the two-terminal system has been discussed and determined by the argument basedon the Ward identity [ 36]. The dependence appearing in Eq. ( 33) fully conforms to it (except for the presence of the bare interaction instead of the renormalized one). Indeed,correspondence is made clear by noting the parameters ¯ μand (δμ) 2for the two-terminal case become ¯μ=γLμL+γRμR γ;(δμ)2=γLγR γ2(eV)2. (34) The presence of linear term in ωandVfor the two-terminal PH asymmetric SIAM was also emphasized recently [ 27]. D. Large-bias-voltage behavior One expects naively that the limit of large-bias voltage eV→∞ corresponds to the high-temperature limit T→∞ in equilibrium; it was shown to be so for the two-terminal PHsymmetric SIAM [ 36]. We now show that the same applies to the multiterminal SIAM where bias voltages of the leads arepairwisely large, i.e., half of them are positively large, and theothers are negatively large. In the large-bias-voltage limit, all the arguments of dilog- arithm functions appearing in Eqs. ( 22) become ±1, where the values of dilogarithm are known (see Appendix C). Accordingly, the pairwisely large-bias limit of /Xi1 iis found to be /Xi11(ε)≈2π2(ε−iγ) iγ, (35) /Xi12(ε)≈− 4π2, (36) /Xi13≈0. (37) Correspondingly, the retarded self-energy becomes /Sigma1R σ(ω)≈U2/4 ω−Edσ+3iγ. (38) It shows that the result of the multi-terminal SIAM is the same with that of the two-terminal PH symmetric SIAM except fora frequency shift. Accordingly, the retarded Green’s functionG R(ω) in this limit is given by GR σ(ω)≈1 ω−Edσ+iγ−U2/4 ω−Edσ+3iγ. (39) The form indicates that for sufficiently strong interaction U/greaterorsimilar 2γ, the dot spectral function has two peaks at Edσ±U/2= /epsilon1d+U(n¯σ±1/2) with broadening 2 γ, so the system is driven into the the Coulomb blockade regime. On the other hand,for weak interaction U< 2γ, it has only one peak with twodifferent values of broadening that reduce to γand 3γin the U→0 limit. What is the role of the current preservation condition ( 6) in this limit? It just determines the dot occupation numberexplicitly. In fact, the condition becomes n σ=−1 π/summationdisplay aγa γIm/integraldisplay(μa−Edσ)/γ −∞dx x+i−u2 x+3i(40) withu=U/(2γ), and nσis independent of the interaction strength because bias voltage sets the largest scale. One canevaluate the above integral exactly to have /integraldisplaydx x+i−u2 x+3i=/summationdisplay s=±1√ 1−u2+s 2√ 1−u2Log(x−αs),(41) where α±=− 2i±i√ 1−u2. As a result, expanding it up to the second order of uleads to nσ≈/summationdisplay aγa γθ(μa)−1 π/summationdisplay aγa μa. (42) The first term corresponds to the occupation number that one expects naturally from the effective distribution ¯f;i tc o r r e - sponds, for instance, to γL/(γL+γR) for the two-terminal dot withμR<0<μLwith|μR,L|→∞ . The second term is the deviation from it, which is proportional to the average of theinverse chemical potential weighted by the leads. V . NONEQUILIBRIUM SPECTRAL FUNCTION We now turn our attention to the behavior of the nonequi- librium dot spectral function, using our analytical expressionof the self-energy [Eqs. ( 21) and ( 22)]. Below we particularly focus our attention on the two cases: the two-terminal PHasymmetric SIAM where current preservation has been anissue, and the multiterminal PH symmetric SIAM where therole of multiple leads has been raising questions. In all of thecalculations below, we have checked numerically the validityof the spectral weight sum rule at each configuration of biasvoltages. A. Self-consistent current-preserving calculation As was emphasized in Sec. II B, when a dot system does not retain the PH symmetry, the stationary current is notautomatically conserved and one must impose the current-preservation condition ( 6) explicitly. As the right-hand side of Eq. ( 6) also depends on the dot occupation number n σ, this requires us to determine nσself-consistently by using the retarded Green’s function in a certain approximation; the second-order perturbation theory in the present case. Figure 4shows the result of nonequilibrium dot spectral function of the two-terminal PH symmetric SIAM at biasvoltage eV=0,0.5,1.5,3.0, and 5 .0γ, which is essentially the same result with Ref. [ 8] (of a different set of parameters). The occupation number is fixed to be n σ=1 2in this case, so its self-consistent determination is unnecessary. The resultswere compared favorably with those obtained by diagrammaticquantum Monte Carlo calculations [ 8]; a relatively good quantitative agreement was observed up to U∼8γ(where the Bethe ansatz Kondo temperature k BTK=0.055γ[1] while the 115421-6MULTITERMINAL ANDERSON IMPURITY MODEL IN . . . PHYSICAL REVIEW B 90, 115421 (2014) -5 0 50.0.10.20.3 0.0.10.20.3 ω/γLDOS [1/γ]U=8.0 d= -0.5 UV=0 V=0.5 V=1.5 V=3.0 V=5.0 FIG. 4. (Color online) Nonequilibrium dot spectral function of the two-terminal PH symmetric SIAM ( /epsilon1d=−U/2) at finite-bias voltage eV=0,0.5,1.5,3.0,5.0γ. The interaction strength is chosen asU=8γ. The dotted line represents the result of U=0a n dV=0. estimated half-width of the Kondo resonance kB˜TK=0.23γ) and bias voltage V/lessorsimilar2γ. Applying bias voltage gradually suppresses the Kondo resonance without splitting it, and thetwo peaks at ±U/2 are developed at larger bias voltages, which corresponds to the discussion in the previous section. Figure 5shows the result of our self-consistent calcu- lation of the nonequilibrium spectral function for the two-terminal PH asymmetric SIAM at (a) /epsilon1 d=− 0.625Uand -5 0 50.0.10.20.3 0.0.10.20.3 ω/γLDOS [1/γ] U=8.0 d = -0.75UV=0 V=0.5 V=1.5 V=3.0 V=5.0(b)-5 0 50.0.10.20.3 0.0.10.20.3 ω/γLDOS [1/γ] U=8.0 d = -0.625 UV=0 V=0.5 V=1.5 V=3.0 V=5.0(a) FIG. 5. (Color online) Nonequilibrium dot spectral function of the PH asymmetric SIAM at (a) /epsilon1d=− 0.625Uand (b) /epsilon1d= −0.75U. All the other parameters are the same with Fig. 4.A sa ne y e guide, the PH symmetric result of U=0a n d V=0 is shown as a dotted line.(b)/epsilon1d=− 0.75U. A paramagnetic-type solution is assumed in determining nσ. As in the PH symmetric SIAM, one sees increasing bias voltage not split but suppress the Kondoresonance while it develops a peak around E d−U/2. The Kondo resonance peak is suppressed more significantly at /epsilon1d= −0.625Uthan at −0.75Ubecause the Kondo temperature of the former ( kBTK≈0.067γ;kB˜TK≈0.49γ) is smaller than that of the latter ( kBTK≈0.12γ;kB˜TK≈0.58γ). An interesting feature of the PH asymmetric SIAM is that spectralweight of the Kondo resonance seems shifting graduallytoward E d+U/2 with increasing bias voltage, without ex- hibiting a three-peak structure in the PH symmetric case.This suggests a strong spectral mixing between the Kondoresonance and a Coulomb peak at finite-bias voltage. Becauseof it, the interval of the two peaks at finite bias is observedas roughly U/2 and gets widened up to Ufor larger eV. The bias dependence somehow looks similar to what wasobtained by assuming equilibrium noninteracting effectivedistribution for n σ[30] (which is hard to justify in our opinion), although we emphasize our present calculation only relies onthe current-preservation condition without using any furtherassumption. It is remarked that the effect shown by biasvoltage is quite reminiscent of finite-temperature effect thatwas observed in the PH asymmetric SIAM in equilibrium [ 37]. More insight can be gained by examining how the spectral structure depends on the interaction strength at finite-biasvoltage. Figures 6(a) and6(b) show a structural crossover from 5 0 50.0.10.20.3 0.0.10.20.3 ωγLDOS 1γ d = -U/2U=2 U=4 U=8(a) 5 0 50.0.10.20.3 0.0.10.20.3 ωγLDOS 1γd = -U/2-U=2 U=4 U=8(b) FIG. 6. (Color online) Nonequilibrium dot spectral function for different values of interaction strength at bias voltage eV=1.5γ. Results of the interaction strengths U=2γ,4γ,a n d8 γare shown, while dotted lines refer to the noninteracting case as an eye guide. (a) Spectral function of the PH symmetric SIAM at /epsilon1d=−U/2. (b) Spectral function of the PH asymmetric SIAM at /epsilon1d=−U/2−γ. 115421-7NOBUHIKO TANIGUCHI PHYSICAL REVIEW B 90, 115421 (2014) a noninteracting resonant peak (the dotted line) to correlation peaks, for (a) the PH symmetric SIAM /epsilon1d=−U/2, and (b) the PH asymmetric SIAM /epsilon1d=−U/2−γ. The PH symmetric SIAM shows introducing Uleads to developing the correlation two peaks as well as the Kondo peak that is suppressed byfinite-bias voltage. In contrast, the bias-voltage effect on thePH asymmetric SIAM is more involved because the Kondoresonance is apparently shifted and mixed with one of thecorrelation peaks, eventually showing the two-peak structureatE d±U/2 in the large-bias-voltage limit. B. Multiterminal PH symmetric SIAM To examine finite-bias affects further and see particularly how the presence of multiterminals affects the nonequilibriumspectral function, we configure a special setup of the multi-terminal SIAM that preserves the PH symmetry: the dot isconnected with Nidentical terminals, with bias levels being distributed equidistantly between −V/2 and+V/2, and each of relaxation rates is set to be γ/N . The latter ensures that the unbiased spectral function is the same, hence the Kondotemperature. Results of the nonequilibrium spectral functionare shown in Fig. 7. Again, we confirm that no splitting of the Kondo resonance is observed in this multiterminal setting. One sees further that increasing the number of terminals enhances the Kondo resonance. This can be understood by weakeningthe bias suppression effect on the Kondo resonance for a largerN. More precisely, one may estimate the suppressing effect from small-bias behavior, Eq. ( 33). Hence, δμis a control parameter. In the present multiterminal PH symmetric setting,the quantity δμis found to be δμ=V/radicalBigg N+1 12(N−1). (43) Therefore, δμdecreases with increasing N, which results in weakening the suppression and enhancing the Kondoresonance for a larger N. The preceding argument also tells us that if the spectral function bears any multiterminal signatures at all, they wouldbe more conspicuous by examining it with fixing δμrather -5 0 50.0.10.20.3 ω/γLDOS [1/γ] N=8, V=1.0 N=4, V=1.0 N=2, V=1.0N=2, V=3.0 N=4, V=3.0 N=8, V=3.0 FIG. 7. (Color online) Nonequilibrium dot spectral function for the PH symmetric multiterminal dot ( N=2,4, and 8). Other parameters are chosen as the same as in Fig. 4. The dotted line corresponds to the two-terminal noninteracting unbiased case, whilethe dashed line to the two-terminal interacting unbiased case.-5 0 50.0.10.20.3 0.0.10.20.3 ω/γLDOS [1/γ]N=2, 4, 8 2=3.0N=2, 4, 8 2=1.0(a) -2 -1 0 1 20.00.20.40.60.81.0 ω[2δμ]f_ (ω)5 10 15 20(b) 0.00.51.0Bias levels -1.0-0.5 Number of Terminals FIG. 8. (Color online) (a) Nonequilibrium spectral function for the PH symmetric multiterminal dot with fixing δμ(N=2,4,8). Other parameters are the same as in Fig. 7. (b) The effective Fermi distribution ¯f(ω) at zero temperature for the PH symmetric multiterminal dot ( N=2,4,8,16). The inset shows relative locations of bias levels with fixed δμas a function of the number of terminals. thanV. This is done in Fig. 8(a);F i g . 8(b) shows how the effective dot distribution ¯f(ω) and the relative locations of bias levels (in the inset) evolve for a fixed δμwhen N increases. No multiterminal signature in the nonequilibriumspectral function is seen in Fig. 8(a); results of different N actually collapse, not only around zero frequency but in theentire frequency range. It suggests that the suppression ofthe Kondo resonance deeply correlates the development ofCoulomb peaks, and a mixing between those spectral weightsis important. The parameter δμcontrols a crossover from the Kondo resonance to the Coulomb blockade structure. Wemay also understand the similarity between bias effect andtemperature effect by the connection through the large- Nlimit of the effective Fermi distribution ¯f(ω), as shown in Fig. 8(b). C. Finite-bias effect on the spectral function: Issues and speculation Although there is a consensus that bias voltage starts suppressing the Kondo peak, and eventually destroys it withdeveloping the two Coulomb peaks when bias voltage is muchlarger than the Kondo temperature, there is a controversy as towhether the Kondo resonance peak will be split or not in theintermediate range of bias voltage. All the results obtained by 115421-8MULTITERMINAL ANDERSON IMPURITY MODEL IN . . . PHYSICAL REVIEW B 90, 115421 (2014) the second-order perturbation consistently indicate that there is no split of the Kondo resonance; finite-bias voltage starts tosuppress the Kondo resonance, and develops the two Coulombblockade peaks by shifting the spectral weight from the Kondoresonance. We should mention that some other approximationsdraw a different conclusion. Here, we make a few remarks onapparent discrepancy seen in various theoretical approaches aswell as experiments, as well as some speculation based on ourresults. Typically, several approaches that rely on the infinite- Ulimit, notably noncrossing approximation, equation of motion method, and other approaches investigating the KondoHamiltonian, observed the splitting of the Kondo resonanceunder finite-bias voltage [ 12,25,38]. Those results, how- ever, have to be interpreted with great care, in our view.Generally speaking, the spectral function obtained by thoseapproaches does not obey the spectral weight sum rule:ignoring the doubly occupied state typically leads to the sumrule−/integraltext ∞ −∞ImGR σ(ω)/π=1/2[12], rather than the correct value. Therefore, only half of the spectral weight can beaccounted for in those methods. Simultaneously, such (false)sum rule in conjugation with the bias suppression of the Kondoresonance cannot help but lead to a two-peak structure ofthe spectrum within the range of attention. Splitting of theKondo resonance might be an artifact of the approximation.Not fulfilling the correct sum rule, those approaches maynot be able to distinguish whether finite bias will split theKondo resonance or simply suppresses it with developing theCoulomb peaks. As for the two-terminal PH symmetric SIAM,fourth-order contribution regarding the Coulomb interactionUhas been evaluated numerically [ 5–7]. The results seem unsettled, though. While Fujii and Ueda [ 5,6] suggested the fourth-order term may yield the splitting of the Kondoresonance in the intermediate-bias region k B˜TK/lessorsimilareV/lessorsimilarUfor sufficiently large interaction U/γ/greaterorsimilar4, which the second-order calculation fails to report, another numerical study indicatesthat the spectral function remains qualitatively the same withthe second-order result [ 7]. Experimental situation is not so transparent, either. While the splitting of the Kondo resonancewas reported in a three-terminal conductance measurement ina quantum ring system [ 26], a similar spitting observed in the differential conductance was attributed to being caused by aspontaneous formation of ferromagnetic contacts, not purelyto bias effect [ 39]. It is also pointed out that it has been recently recognized that the Rashba spin-orbit coupling induces spinpolarization nonmagnetically in a quantum ring system witha dot when applying finite-bias voltage [ 40–42]; hence, such spin magnetization might possibly lead to the splitting of theKondo resonance. The Kondo resonance is a manifestation of singlet forma- tion between the dot and the lead electrons. One may naivelythink that when several chemical potentials are connected withthe dot, such singlet formation would take place at each leadseparately , causing multiple Kondo resonances. The results of the multiterminal PH symmetric SIAM presented in theprevious section tempt us to speculate a different picture. Letus suppose that (almost) the same dot distribution functionf dot(ω)=G−+(ω)/(2iπ) is realized for a fixed δμwith a different terminal number N,a sF i g . 8(a) suggests. Note the assumption is fully consistent to the Ward identity for low bias,but it invalidates a quasiparticle ansatz −¯f(ω)I mGR(ω)/π that explicitly depends on N. In the large- Nlimit with a fixed δμ, the effective Fermi distribution ¯f(ω) resembles the Fermi distribution at finite temperature kBT∼δμ. Accordingly, bias voltage may well give effects similar to finite temperature.It is seen in the low- and large-bias limits for a dot withor without the PH symmetry. It implies that a dot electroncannot separately form a singlet with the lead at each chemicalpotential because it needs to implicate states at differentchemical potentials through coupling with other leads. Oursecond-order perturbation results seem to support this view. VI. CONCLUSION In summary, we have evaluated analytically the second- order self-energy and Green’s function for a generic multi-terminal single-impurity Anderson model in nonequilibrium.Various limiting behaviors have been examined analytically.Nonequilibrium spectral function that preserves the current isconstructed and is checked to satisfy the spectral weight sumrule. The multiterminal effect is examined for the PH sym-metric SIAM, particularly. Within the validity of the presentapproach, it is shown that the Kondo peak is not split due to biasvoltage. It is found that most of the finite-bias effect is similar tothat of finite temperature in low- and high-bias limits with andwithout the PH symmetry. Such nature could be understoodby help of the Ward identity and the connection through theN/greatermuch1 terminal limit. The present analysis serves as a viable tool that can cover a wide range of experimental situations.Although there is still a chance that high-order contributionsmight generate a new effect such as split Kondo resonances in alimited range of parameters, it is believed that the second-orderperturbation theory can capture the essence of the Kondophysics in most realistic situations. Moreover, having a con-crete analytical form that satisfies both the current conservationand the sum rule, this work provides a good, solid, workableresult that more sophisticated future treatment can base on. ACKNOWLEDGMENTS The author gratefully acknowledges A. Sunou for fruitful collaboration that delivered some preliminary results in thiswork. The author also appreciates R. Sakano and A. Oguri forhelpful discussion at the early stage of the work. The work waspartially supported by Grants-in-Aid for Scientific Research(C) No. 22540324 and No. 26400382 from MEXT, Japan. APPENDIX A: NONINTERACTING GREEN’S FUNCTIONS WITH FINITE BIAS We start with the nonequilibrium Green’s function G without the Coulomb interaction on the dot. Its Keldyshstructure is specified by G σ(ω)=/parenleftbigg ω−/epsilon1d+iγ(1−2¯f) +2iγ¯f −2iγ(1−¯f)−(ω−/epsilon1d)+iγ(1−2¯f)/parenrightbigg−1 , (A1) where ¯fis the effective Fermi distribution defined in Eq. ( 5). We incorporate the Hartree-type diagram into the unperturbedpart by replacing /epsilon1 dto/epsilon1d/mapsto→Edσ=/epsilon1d+Un ¯σ.N o t e n¯σis the 115421-9NOBUHIKO TANIGUCHI PHYSICAL REVIEW B 90, 115421 (2014) exact dot occupation that needs to be determined consistently later. Its retarded, advanced, and Keldysh components aregiven by G R,A σ(ω)=1 ω−Edσ±iγ, (A2) GK σ(ω)=[1−2¯f(ω)]/bracketleftbig GR σ(ω)−GA σ(ω)/bracketrightbig . (A3) T h ef u n c t i o n1 −2¯f(ω) reduces to/summationtext a(γa/γ)s g n (ω−μa)a t zero temperature. APPENDIX B: NONEQUILIBRIUM POLARIZATION PART Taking the Fourier transformation of Eqs. ( 17), using Eq. ( A3), and making further manipulations, we can rewrite /Pi1Rand/Pi1Kas /Pi1R(ε)=/summationdisplay aγa γγBaa(ε) πε(ε+2iγ)=[/Pi1A(ε)]∗, (B1) /Pi1K(ε)=2i/summationdisplay a,bγaγb γ2cothβ(ε−μab) 2Im/bracketleftbiggγBab(ε) πε(ε+2iγ)/bracketrightbigg , (B2) where μab=μa−μb,βis the inverse temperature, and Bab(ε)i sg i v e nb y Bab(ε)=/integraldisplay dε/prime[fb(ε/prime)−fa(ε/prime+ε)] [GA(ε/prime)−GR(ε/prime+ε)]. (B3) In this work, we are interested in the zero-temperature limit, for which coth( βx) becomes sgn( x). The function Babin this limit is evaluated as (with ζaσ=μa−Edσ) Bab(ε)=− log/parenleftbiggε−ζaσ+iγ −ζbσ+iγ/parenrightbigg −log/parenleftbiggε+ζbσ+iγ ζaσ+iγ/parenrightbigg . (B4) This corresponds to a multiterminal extension of the result obtained for the two-terminal PH symmetric SIAM. APPENDIX C: DILOGARITHM WITH A COMPLEX V ARIABLE To complete evaluating the remaining integral over Eof Eqs. ( 19) and ( 20), we take full advantage of various properties of dilogarithm function Li 2(z). A concrete integral formula we have utilized will be given in Appendix D. For the sake of completeness, we here collect its definition and propertiesnecessary to complete our evaluation. 1. Definition Dilogarithm Li 2(z) with a complex argument z∈Cis not so commonly found in literature. As it is a multivaluedfunction, we need to specify its branch structure properly. Oneway to define dilogarithm Li 2(z) all over the complex plane consistently is to use the integral representation Li2(z)=−/integraldisplayz 0dtLog(1 −t) t. (C1)The multivaluedness of dilogarithm Li 2originates from the logarithm in the integrand. Here, we designate the principalvalue of logarithm as Log, which is defined by Logz=ln|z|+iArgz(for−π< Argz/lessorequalslantπ).(C2) According to Eq. ( C1), Li 2(z) has a branch cut just above the real axis of x> 1. Accordingly, its values just above and below the real axis are different for x> 1: Li 2(x−iη)= Li2(x)b u tL i 2(x+iη)=Li2(x)+2iπlnx. Some special values are known analytically. We need Li 2(0)=0, Li 2(1)= π2/6, Li 2(−1)=−π2/12, and Li 2(2)=π2/4−iπln 2 for evaluation later. 2. Functional relations Dilogarithm Li 2(z) has interesting symmetric properties regarding its argument z; values at z,1−z,1/z,1/(1−z), (z−1)/z, andz/(z−1) are all connected with one another. Those points are ones generated by symmetric operations S andTdefined by Sz=1 z;Tz=1−z, (C3) and{I,S,T,ST,TS,TST }forms a group. Other operations correspond to STz=1 1−z;TSz=z−1 z, (C4) TSTz =STSz =z z−1. (C5) Applying a series of integral transformations in Eq. ( C1), one can connect the values of dilogarithm at these values withone another [ 43]. Note, those functional relations are usually presented only for real arguments. Extending them for complexvariables needs examining its branch-cut structure carefully.By following and extending the derivations in Ref. [ 43] for complex z∈C, we prove that the following functional relations are valid for any complex variable z: Li 2(Sz)=− Li2(z)−π2 6−1 2[Log(TSz )−Log(Tz)]2,(C6) Li2(Tz)=− Li2(z)+π2 6−Log(Tz)L o gz, (C7) Li2(TSTz )=− Li2(z)−1 2Log2(STz ) −[Log(Tz)+Log(STz )] Log z,(C8) Li2(TSz )=Li2(z)−π2 6 −1 2Log2(Sz)−Log(Sz)L o g (Tz),(C9) Li2/parenleftbig STz/parenrightbig =Li2(z)+π2 6 +1 2Log2(Tz)+Log(Tz)L o g (TSTz ).(C10) To our knowledge, the above form of extension of functional relations of dilogarithm has not been found in literature. 115421-10MULTITERMINAL ANDERSON IMPURITY MODEL IN . . . PHYSICAL REVIEW B 90, 115421 (2014) 3. The Taylor expansion To examine various limiting behaviors, we need the Taylor expansion of dilogarithm, which is derived straightforwardlyfrom Eq. ( C1): Li 2(z)=Li2(z0)−∞/summationdisplay k=1(z−z0)k k!dk−1 dzk−1Log(1 −z) z/vextendsingle/vextendsingle/vextendsingle/vextendsingle z=z0. (C11) The presence of Log(1 −z) reflects the branch-cut structure of Li 2(z). In particular, we utilize the following expansion in our analysis: Li2(z)≈z+z2 4+z3 9+z4 16+··· , (C12) /Lambda1(2+z)≈π2 4−z2 4+z3 6−5z4 48+··· . (C13) APPENDIX D: INTEGRAL FORMULA Here, we derive and present the central integral formula for evaluating Eqs. ( 19) and ( 20). By performing a simple integral transformation in Eq. ( C1), we have the integration /integraldisplayz −bLog/parenleftbigx+b c/parenrightbig x−adx=/integraldisplayz+b a+b 0Log/parenleftbiga+b cy/parenrightbig y−1dy (D1) =Log/parenleftbigz+b c/parenrightbig Log/parenleftbig 1−z+b a+b/parenrightbig +Li2/parenleftbigz+b a+b/parenrightbig , (D2) where all the parameters ( a,b,c )a sw e l la s zmay be taken as complex numbers. Combined with fractional decomposition,we see the following integral can be evaluated in terms ofdilogarithm: /integraldisplay z −bLog/parenleftbigx+b c/parenrightbig dx (x−a1)(x−a2)(x−a3) =3/summationdisplay i=1Log/parenleftbigz+b c/parenrightbig Log/parenleftbig 1−z+b ai+b/parenrightbig +Li2/parenleftbigz+b ai+b/parenrightbig /producttext j/negationslash=i(ai−aj).(D3)APPENDIX E: CALCULATION OF THE CORRELATED PART OF THE SELF-ENERGY The remaining task to complete calculating /Sigma1Rin the form of Eqs. ( 21) and ( 22) is to collect all the relevant formulas and organize them in a form that conforms to Eq. ( D3). To write concisely, we introduce the following notations: μab=μa−μb, (E1) ζaσ=μa−Edσ, (E2) εσ=ω−Edσ, (E3) where the Hartree level Edσis defined as before. We express the terms I1andI2defined in Eqs. ( 19) and ( 20)a s I1=−/summationdisplay a,b/summationdisplay α,β=±1αγaγb πγ ×/integraldisplay+∞ −∞dEsgn(E−βμab) (E+εσ+iγ)Log/parenleftbigE−βζa¯σ+iαγ −βζb¯σ+iαγ/parenrightbig (E+2iαγ)E, (E4) I2=−/summationdisplay a,b/summationdisplay α,β=±1αγaγb πγ ×/integraldisplay+∞ −∞dEsgn(E+εσ−ζaσ) (E+εσ+iαγ)Log/parenleftbigE+βζb¯σ−iγ βζb¯σ−iγ/parenrightbig E(E−2iγ). (E5) Here, the leads a,b inI1as well as binI2carry spin ¯ σ, while ainI2does spin σ. Singularity on energy integration is prescribed by the principal values. Equation ( D3) enables us to perform and express the above integrals in terms ofdilogarithm. The resulting expressions are still complicated,but we can simplify them further using functional relations ofdilogarithm Eqs. ( C6)–(C10). These require straightforward but rather laborious manipulations. In this way, we reach thefinal expression of /Sigma1 R σof Eq. ( 21). [1] A. M. Tsvelick and P. B. Wiegmann, Adv. Phys. 32,453 (1983 ). 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PhysRevB.20.3543.pdf

PHYSICAL REViEW B VOLUME 20,NUMBER 9 1NOVEMBER 1979 Modelsofelectronic structure ofhydrogen inmetals: Pd-H P.Jena'andF.Y.Fradin Argonne National Laboratory, Argonne, Illinois60439 D.E.Ellis Northwestern University, Evanston, Illinois60201 (Received 16August1978;revised manuscript received 12June1979) Local-density theoryisusedtostudytheelectron charge-density distribution aroundhydrogen andhostpalladium metalatoms.Self-consistent calculations usingafinite-size molecular-cluster modelbasedonthediscrete variational method arereported. Calculations arealsodoneina simple"pseudojellium" modeltostudytheelectron response tohydrogen withintheframework ofthedensity-functional formalism. Resultsofthissimpleapproach agreeverywellwiththe molecular-cluster model. Partialdensitiesofstatesobtained intheclustermodelarecompared withband-structure resultsandconclusions regarding theimportance ofthelocalenvironment ontheelectronic structure aredrawn. Calculated core-level shiftsandchargetransfer frommetl- alionstohydrogen arecompared withtheresultsofx-ray—photoelectron spectroscopy experi- mentsinmetalhydrides andarediscussed intermsofconventional anionic, covalent, andpro- tonicmodels. Theeffectofzero-point vibration ontheelectron chargeandspin-density distri- butionisstudied byrepeating theabovecalculations forseveraldisplaced configurations ofhy- drogeninsidethecl'uster. Theresultsareusedtointerpret theisotopeeffectontheelectron distribution aroundprotonanddeuteron. I.INTRODUCTION Thestudyoftheelectronic structure ofhydrogen inmetalsisatopicofgreatcurrentinterest. Apro- tonwithnocoreelectronic structure isthesimplest kindofanimpurity thatcanbeimplanted intoa solid.However, theabsenceofcoreelectrons results inaneffective electron-proton potential thatissingu- larattheprotonsite.Consequently, thescreening of suchastrongperturbing impurity cannotbehandled wellbyconventional pseudopotential perturbation 'theories' orstatistical methods. Nonlinear theories''mustbeusedtostudytheelectron response tohydrogen. Aknowledge ofthisnonlinear screening oftheprotonisusefulinunderstanding the electronic properties ofhydrogen inmetals.The motivation behindsuchamicroscopic understanding ofmetal-hydrogen systems isnotonlyacademic, but isalsoduetoitspractical importance inproblems suchasembrittlement duetodissolved hydrogen anduseofhydrogen inenergy-related technology.~ Inthispaperwehavestudiedvariouselectronic properties associated withdissolved hydrogen in transition-metal systems. Although. specific calcula- tionsareperformed forthepalladium-hydrogen sys- tem,ourdiscussions andconclusions aregeneral and should applytoanymetal-hydrogen system. Three common theoretical approaches havebeentaken:(i) Thejelliummodel—inthismodel5(meaningful only fornearly-free-electron systems) theperiodic struc-tureofthehostisneglected andthepositive charges onthehostionsaresmeared outuniformly toforma homogeneous background ofdensity np.Thescreen- ingofaprotonisthentreated instandard linear'or nonlinear screening theories.3'(ii)Theband- structure model—mostapplications basedonthe augmented-plane-wave (APW)method havebeen usedtointerpret electronic properties of stoichiometric metalhydrides. Calculations' based onthecoherent-potential approximation" aregen- erallyusedtostudymetalscontaining smallamounts ofrandomly distributed hydrogen. Thesecalculations emphasize theimportance oflatticestructure. (iii) Themolecular-cluster model'—thismodelissome- whatintermediate between theabovetwomodels. It isgenerally assumed thattheelectronic properties of theimpurity aredictated mainly byitslocalenviron- ment.Thus,onetreatstheimpurity andnearneigh- borsasforming amolecular cluster.Theeigenstates andelectron chargedensities arethencalculated self-consistently usingthelocal-density approxima- tion.Inametallic environment, thepotentials asso- ciatedwithbothhostandimpurity ionsareshort rangeduetoefficient screening oftheioniccharge. Consequently, amolecular-cluster modelmayprovide meaningful resultsfortheelectronic structure ofim- purities innon-free-electron-like systems. Although theabovemodelshavebeenextensively usedinthe past,ithasnotbeenclearwhichfeatures aremodel dependent, andwhichareintrinsic totheimpurity 20 3543 O1979TheAmerican Physical Society 3544 P.JENA,F.Y.FRADIN, ANDD.'E.ELLIS 20 system. Aconsistent comparison oftheresultsob- tainedinagivensystemfromthedifferent models will,therefore, beuseful.Inaddition, wehaveex- tendedthescopeofbothjellium andclustermodels toobtainmoredetailed information aboutthe hydrogen-metal interaction. Usingtheabovetheoretical models, weshall analyze avarietyofproblems relating totheelectron- icstructureofhydrogen inmetals. Historically, there arethreesimplemodels" thatareusedtodescribe thebehaviorofhydrogen incondensed matter.The anionic modelisbasedupontheassumption thatan electron fromthemetalionistransferred tothehy- drogen. Inthecovalent hydrogen model,itisassumed thatthehydrogen iscovalently bondedtometalions. Intheprotonic model,theelectron isassumed toleave theprotonandtoparticipate infillingtheoccupied metallic band.Itisnotclearwhether anyofthese descriptions isappropriate fortheproblemofdilute quantities ofhydrogen inmetals' wherescreening wouldcertainly playadominant role.Weshallstudy thepossible electron transfer fromthemetalionto hydrogen andtheaccompanying shiftinthebinding energyofthecorelevels.Comparison canbemade withx-ray—photoelectron spectroscopy''measure- mentsofthecore-level shiftsofthemetalioninthe hydride phasecompared tothatinthepuremetallic state. Through nuclear-magnetic-resonance experi- ments,"theprotonspin-lattice relaxation timeis usedtoprovide information onthecontact spinden- sityatthehydrogen site.Acomparison ofthiswith thedeuteron spin-relaxation rateinmetaldeuterides yieldsinformation ontheisotopeeffect.'Thepro- tonanddeuteron arebothlightimpurities, andthe effectoftheirzero-point vibration ontheelectronic structure willbediscussed. Theoutlineofthepaperdealing withthediscus- sionoftheaboveproperties isasfollows: InSec.II wediscusstheself-consistent density-functional for- malismforaninhomogeneous electron gas.We prescribe ahomogeneous-density schemefortreating thescreening ofhydrogen innon-free-electron-like metals.Thismodelcanbeviewedasapseudojellium model.InSec.III,theessentials ofthemolecular —clusterapproach areoutlined. Theresultsofelectron chargedistribution around ahydrogen atomalong different crystallographic directions obtained inthe abovetwomodelsarecompared inSec.IV.This-sec- tionalsocontains acomparison ofthepartialdensity ofstatesobtained inourmolecular-cluster model withthatoftheAPWband-structure approach. The problemofchargetransfer frommetaliontohydro- genisdiscussed inSec.Vinthelightofrecentexper- imentsusingx-ray—photoelectron spectroscopy. In Sec.VIwediscusstheeffectofzero-point vibration ontheelectron chargeandspindistribution arounda lightimpurity. Ourresultsaresummarized inSec.VII.II.HOMEGENEOUS-DENSITY APPROXIMATION TOMOLECULAR CLUSTERS: APSEUDOJELLIUM MODEL Inthissection weprescribe ascheme tostudythe screening ofaprotoninanon-free-electron-like me- tal.Intheconventional jelliumapproach, theelec- trondensityofthehomogeneous background isgiven byadensityparameter r,where ,w(r,a—p)'=I/np .4 np(f}=Xnp(rR„)— V where np(rR„)is—thefree-atom chargedensitycen- teredontheR„thlatticesiteandcanbecomputed fromaknowledge oftheone-electron orbitals,' P„p„(r), namely,(2) np(r)=X(y„((r)(' nlm =2XRJ(r) (3) „(4m where2(2I+1)isthespinandorbitaldegeneracy factorandR,~(r)istheradialwavefunctionofthe quantum statenl.Thus,thedensityparameter r,is itselfafunctionofr,i.e.,3mr,'(r)ap=1/np(r). In palladium (fcc)crystal,forexample, theprotonis knowntooccupytheoctahedral site.Theambient densityatthispointcanbeevaluated fromEq.(2). Inpractice, however, itissufficient toconsider only thenearest-neighbor hostionssincethesecondand furtheroutneighbors makeanegligible contribution totheambient electron density. Havingdetermined theambient electron densityat apointr;inspace,theresponse oftheelectrons toaTheconduction-electron density, no,isdetermined by accounting forthenumberof"free"electrons, Z (usually thevalence) peratomicvolume, Qp,i.e., np=Z/Qp.Theelectron distribution aroundthepro- tonisthenstudied byembedding thepointchargein thishomogeneous medium. Inextending this scheme tonon-free-electron-like systems, thefirst difficulty istoestimate thequantity, Z.Inkeeping withthespiritofthejelliummodel,oneshouldin- tegratethespcomponent oftheelectron densityof statesuptotheFermienergyEFtoestimate Z.This obviously requires apriorknowledge ofthepartial densityofstatesobtained intheband-structure calcu- lation.Inaddition, oneassumes thattheinteraction between theimpurity andthehostdelectrons is negligible. Inthefollowing wesuggest analternate scheme. Todetermine theambient electron chargedensity np(r)oftheperfecthostatanypointinspacetoa firstapproximation, weusethenoninteracting atom model.Inthismodel, 20 MODELS OFELECTRONIC STRUCTURE OFHYDROGEN. .. 3545 protonatthatpointiscalculated byassuming thatthe electrons respond totheprotonasiftheprotonissit- uatedinahomogeneous electron gasofdensity Ilp(f~).Thismodelwillbereferred toasthe"pseu- dojellium" modelandisobviously anapproximation toamorecomplicated molecular-cluster model(dis- cussedinSec.III)wherehydrogen andthesurround- ingmetalionsareallowed tointeract amongeach otherinestablishing theground-state distribution of theelectron density. Thejustification fortheuseof thispseudojellium modelcanonlybemadeafter comparing theresults(seeSec.IV)withthatob- tainedinthemoresophisticated molecular-cluster model. Wehaveusedthedensity-functional formalism ofHohenberg, Kohn,andSham(HKS)totreatthe screening oftheprotoninthepseudojellium model. MuchhasbeenwrittenabouttheHKStheoryandwe referthereadertotherecentpapersbyJenaetal." forfurtherdetails.Thenumerical workforthe density-functional formalism hasbeencarriedoutin amanner described earlier.'Thechargedensity n(r)andspin-density n(r)distribution around hy- drogenhavebeencalculated self-consistently toa- precision ofbetterthan2%inn(r)inthevicinityof theproton. III.SELF;.CONSISTENT MOLECULAR-CLUSTER MODEL Wealsousethelocal-density formalism described earlierincarrying outmolecular-orbital (MO)calcula- tionsonfiniteclusters representative ofthesolid. TheMOeigenstates areexpanded asalinearcombi- nationofatomicorbitals, y„(r)=Xaj(r—R~)Cq„ J(4) isapproximately solvedbyminimizing certainerror moments onasampling gridinr.Theeffective Hamiltonian forstatesofspino-isgivenby +Vcoul+ Vexch,u wherethefirsttwotermsarethekineticenergyand Coulomb potential. Theexchange. potential istaken intheusualform, V,„,„=—6n[3n(r)/4m]'~' . Thevaluea=0.7,closetothatofKohnandSham,Thevariational coefficients (Cj„lareobtained by solvingthesecularequationofthediscrete variation- almethod.'Thismethod hasbeendescribed inde- tailelsewhere.'Here,weonlynotethatthe single-particle equation, (h—a„)y„(r) =0wasusedinallcalculations. Thereexistmoreela- boratelocal-density exchange andcorrelation poten- tialswhicharefoundtoleadtosmalldifferences in self-consistent energylevelsandchargedensities for transition metals. Thesedifferences aretoosmall tobeofanyconsequence forthepresentwork. Calculations weremadefortheoctahedral Pd6and PdqHclusters withbondlengthtakenforthebulkPd metal.Theprotonwasplacedeitheratthe(0,0,0) octahedral site,ordisplaced alongthe[100)direction. Aspin-restricted (assuming n= 2n)modelwas used,withtheiteration procedure startingfromsu- perimposed atomicchargedensities, Interaction of theclusterwiththecrystalline environment wasig- nored,sinceweplantoconcentrate onproperties as- sociated withthecenterofthecluster. However, for anyreasonable treatment ofbulkmetalproperties, it isnecessary toembedtheclusterinaneffective medium. Inordertocompare theseresultswith band-structure calculations andexperiments on stoichiometric PdH,itisnecessary tostudythesensi- tivityofourcalculated electron densities around hy- drogentoitschemical environment. Wehave,there- fore,repeated ourpseudojellium calculations bycon- sidering thelatticeparameters andgeometrical ar- rangements ofPdandHinPdH.Thedecrease inthe ambient chargedensityduetothesurrounding Pd atomsinPdHasaresultoflatticeexpansion isfound tobesomewhat compensated bytheadditional con- tribution ofthehydrogen atomstotheambient chargedensity. Asaresult,thecalculated self- consistent electron densityatthehydrogen sitein PdHdoesnotdiffersignificantly fromthatofasingle octahedrally coordinated hydrogen atominpurePd. Thiscalculation wasnotrepeated fortheself- consistent molecular-. clustermodel. However, wedo notexpecttheresultstobequalitatively different. Thus,thecomparison oftheelectronic properties as- sociated withhydrogen, inSecs.IV—VI,inthepseu- dojellium andmolecular-cluster models withband- structure calculations andexperiments onPdHis meaningful. IV.COMPARISON BETWEEN PSEUDOJELLIUM, MOLECULAR-CLUSTER, ANDBAND-STRUCTURE MODELS Thissection isdivided intotwoparts.Firs&,we discusstheelectron chargedensityaroundaproton octahedrally coordinated tosixneighboring Pdatoms obtained self-consistently inboththepseudojellium andmolecular-cluster model.Second, thepartial densityofstatesobtained inthemolecular-cluster ap- proach willbecompared withAPWband-structure calculations.9 InFig.1wepresent acomparison oftheambient chargedensityobtained byasuperposition ofthe 3546 P.JENA,F.Y.FRADIN, ANDD.E.ELLIS 20 O.I5- 1o t- CI I— CA LLIC)O.IO— 4J CK cK 0.05— 4J COII III [II III III III I [»0]'' I'' I'' I [»I] [I»~) .W~.A~ I~-[IIO] ----dl~JV 0Pd-ATOM xOCTAHEDRAL INTERSTITIAL SITE0.3 O 0.2 0.1II III III I'III. III III I&II III III I [IOO] [»Ol 0IIIIIIIIIgIIIIIIIII 00,6I.2 I.800.6 l.2 I.8IIII 006II»I, 12 I8IIIIIII~ItIIIIIII IIIIIIIII 00.30.60.900.30.60.900.30.60.9 r{0,] FIG.1.Electron charge-density distribution insidethe unitcellofPd.Thesolidlinerepresents aself-consistent molecular-cluster calculation basedonasix-Pd-atom cluster; thedashedlineisobtained byalinearcombination offree- atomchargedensities centered atindividual nuclearsitesof theabovecluster. Theoctahedral site(equilibrium confi- gurationofhydrogen givenby&&)definestheoriginofthe real-space coordinate system. free-atom chargedensities (dashed curve)withthat calculated inthemolecular-cluster modelconsisting ofsixinteracting Pdatomslocatedatthefacecenters ofthecube(solidcurve). Attheoctahedral site (takenastheorigin)thechargedensityduetothein- teracting metalatomsisaboutafactorof2larger thanthatduetothesimplesuperposition model. Theanisotropy remains small,asexpected, fordis- tancesuptolao(Bohrradius)fromtheorigin. However, forfartherdistances, thechargedensity alongthe[100]direction increases muchmorerapidly thanalongthe[110]and[111]directions sincethe nearest-neighbor Pdatomliesalongthe[100]direc- tion.Thisanisotropy intheambient chargedistribu- tionisalsoapparent fromthesimplenoninteracting atommodel. Theelectron distribution around aprotonembed- dedattheoctahedral interstitial siteinPdmetalis calculated self-consistently inthepseudojellium modelandiscompared withthemolecular-cluster (Pd6H)resultinFig.2.Theelectron densities atthe protonsiteinthesetwocalculations differfromeach otherbyabout17%whilethediscrepancy getsnar- rowerasonegoesfarther awayfromtheproton. The chargedistribution remains isotropic withinasphere ofoneBohrradiusaroundtheproton. Thisresult alongwiththeagreement between pseudojellium and molecular-cluster models may,atfirst,besurprising. Ananalysisofthedifferent angular momentum com- ponentsofthechargedensitybasedonthejellium modelrevealsthattheelectrons aroundtheproton havepredominantly ssymmetry. Thisresultiscon- sistentwiththeangular momentum resolved partial densityofstatesforthePd6Hclusterinsidethehy-FIG.2.Comparison between theelectron chargedensities alongthe[100],[110],and[111]directions aroundanoc- tahedrally coordinated hydrogen atomcalculated self- consistently inthemolecular-cluster (solidcurve)andpseu- dojellium (dashed curve)models. drogensphere(seebelow)aswellaswiththe predominant s-wavescattering fromthehydrogen determined fromdeHaas—vanAlphen experiments incoppercontaining diluteamountsofhydrogen, If oneweretousetheambient densityattheoctahedral position inPdfromthemolecular-cluster calculations forthePd6complex insteadofthatobtained fromthe noninteracting atommodel,thepseudojellium model forPd-Hwouldyieldanelectron densityatthepro- tonsitethatis35%higherthanthePd6Hclustercal- culation. However, withthisapproach thepseudojel- liummodelloosesitsattractiveness, sincethere- quiredPd6clustercalculation neededtodetermine theambient density isasdifficult asthefullPd6H calculation. Itisinteresting thatthechargedensityat theprotonsiteinthepseudojelliurn modelishigher thanthatobtained inthemolecular-cluster calcula- tion.Thisresultisconsistent withone'sphysical in- tuitionthatinthemolecular-cluster model,afraction oftheelectrons around hydrogen willbepulledaway toscreenthePdatomsandtoformthePd-Hbondas well.Inaddition, thepseudojellium modeltreatsthe ambient interstitial electrons asfree-electron-like. Sincetheinterstitial densityincludes ad-statecontri- butionandthedelectrons arelesspolarizable thans electrons ofthesamedensity, thepseudojellium modelwouldtendtooverestimate theprotonscreen- ing. Tocompare theenergyeigenvalues ofelectrons between molecular-cluster andband-structure models, weusetheconceptofpartialdensityof states(PDOS). Wedecompose thechargedensity intocontributions fromdifferent sitesandobtainin- formation aboutthemetal-hydrogen bond.Inaddi- tion,itispossible tomakeacomparison withthe PDOSfoundinAPWband-structure calculations on stoichiometric PdH.TheclusterPDOSisfoundasa 20 MODELS OFELECTRONIC STRUCTURE OFHYDROGEN. .. 3547 sumofLorentzian linesofwidthycentered atthe molecular-orbital energies, D„(E)=Xf~E—Eg2+y2(8) I— COa CL Ol XQtJ Ct (b) CAl— CA CODO CLI—~CZI CL 2eVHereywaschosenas0.4eV(consistent withthe discrete levelstructure oftheclusteranduncertainty of-0.1eVinclusterlevelsduetobasis-set limita- tions),andf+weretakentobeatomicpopulations obtained fromaMulliken population analysis ofthe eigenvectors. TheclusterPDOSforPd4dandhy- drogen1sstates-are showninFig.3.ThePd-Hbond- ingbandcentered at-8eVbelowtheFermienergy hasastrongresemblance tothatfoundfortheor- deredcompound bytheAPWmethod. Thissug- geststhattheseverydifferent modelsareconverginguponacommon description. Thetotaldensityof statesforthecluster, containing sizablemetalspcon- tributions, isalsoshowninFig.3.Withthemain features alignedtoremove levelshiftsduetosmall clustersize,weseethatthedensityofstatesforPd6 andPd6Hclusters differslittle,exceptforthebond- ingPd-HpeaknotedinthePDOScurves. Wenow turntoadiscussion oftheelectron-spin densityat theprotonsiteinPdHasobtained fromband- structure andpseudojellium models. Usingthemethod inSec.II,wehavecalculated the spin-density enhancement, [nt(0)—nf(0)]/ (not—not)atthehydrogen siteinPdtobe10.7.The corresponding band-theory result9forPdHis6.8.A criticalcomparison between thepseudojellium and theband-theory resultforthespindensityisham- peredsincetheAPWband-structure9 calculation was notcarriedoutself-consistently. Itis,however, en- couraging thatourresultisins~iquantitative agree- mentwithbandcalculation. Neglectofaperiodic ar- rangement ofPdatomsinthepseudojellium model givesrisetoaspindensitythatislargerinmagnitude thantheband-theory result.Thissystematic trend, asdescribed inSec.IV,alsoexistsinthechargeden- sityattkeprotonsite. Thenuclear-spin-lattice relaxation rateatthehy-' drogensitecalculated inthepseudojellium model (withthesdensityofstatesattheFermienergytak- enfromband-theory result)isabout57%higherthan experiment.'7'8Itisworthmentioning -thatthe Knightshift(whichalsomeasures thespindensity) atthepositive muon(alightisotopeofhydrogen) sitesinparamagnetic metalscalculated5 inthejellium modelareconsistently higherthanthecorresponding experimental values.'Thus,thejelliummodelis foundtoconsistently overestimate theelectron chargeandspindensityaithehydrogen site.Theef- fectofintroducing theperiodic arrayofmetalions wouldbetoreducethemagnitude oftheseelectron densities—atrendintherightdirection forexplaining theexperimental data. LaJl— cn+ X C/yccKI— WC5 I—C)I—(c)EF EF ENERGY (eV) FIG.3.Partialdensityofstatesinarbitrary unitsfor(a) hydrogen 1s,(b)Pd4dstates,and(c)totaldensityofstates forPd6H(solidline),andPd6(dotted line)clusters.V.CHARGE TRANSFER ANDCORE-LEVEL SHIFTS DUETOHYDROGENATION Thissectiondealswithadiscussion ofmodelsof thechemical bondbetween hydrogen andmetalions andtheeffectsassociated withpossible charge transfer fromthemetalionstohydrogen. Insolving thesetofself-consistent HKSequations2 inSec.II, wehavefoundthattheeffective potential isstrong enough toformweaklyboundstateswithtwoelec- tronsashavebeenfoundearlierbyseveralwork- ers'throughout themetallic densityrange.Even thoughsingle-particle eigenvalues havenofunda- mentalmeaning inHKStheory,thewholeofband theorybasedonHKSformalism restsontheirin- 3548 P.JENA,F.Y.FRADIN, ANDD.E..E.ELLIS 20 (a) 0.50—I"T~TT'tll~ 0.25 0.20 Ica 0.15 t- O.IO 0.05 -0.02— I II I III 0 I 2 r(a) (b)' ['[ 1.0 0.8terpretation. Thespintandspin)bound-state wave functions extendoverseverallatticesacin droeni' gepicturethattheelectronic strtfh ucureohy- geninmetalsisthatofanextended H aneuallqayextended holeinthecontinuum. Sinceeionwith thecalculated lifetime broadening fth ningotesestates duetoelectron-electron intert'acionislargecomparedtotheirbinding energies theg',hephysical significance of eseboundstatesisnotwellestablish d matterosaise.Asa o~act,experiments usingp oooortheseboundstateshaveb unsuccessful.veeen oftheInordertoprovide ammorephysical understanding otheelectronic configuration ofhd environment, wecompute thedifference inthe electron densityaroundtheoceoctahedral sitebetween e6andPd6cluster,i.e., hn(r)=np,,„(r)—np,,(r) Thisdifference, indicative ofh duetoocargereadjustment uetohydrogenation isplotted inFi.4a [100]direction fortheinig.'aalongthe ionortemolecular-cluster calculation. enegative regionofelectron de beyond-2Bohrradiisuensityfordistances orradiisuggests thatthechargefrom tevicinityofthemetalionh thehydroens'nhasbeentransferred to eyrogensphere. Thus,acomparison ofthe numberofelectrons, Z(R radius8ar~,contained inasphereof raiusaroundtheprotoninametal, tRZ(R)= ~d3r5(nr)nr, (10) withthatoffree-h-hydrogen atomwouldindicate the extentofexcessscreening ofhydroen r,=2.7)isalsoapparent fromourseudl.SiZ()ht e~astobeequaltounityinall calculations toensureelectr' 1h observed chargetransfer couklb ericacargeneutralit th rcouedescribed asthe yrogenbeingslightlyanionic andthe ingslightlycationicicantemetalionbe- Adirectconsequence ofthereductionofelectrons unemetalionistoalterthe n''hecore-level ener- naleIwecompare theenergiesofthe4 s,and31corelevelsofth 11d'epaaiumatominthe e:-0.6TABLEI.ComariparisonofPdcore-level energies (eV)re- lativetoFermienergy'oft -self-consistent l1-daomandclusterinno oca-ensitymodel.d nonrelativistic 0.4 Level Atom Pd6 Pd6H 0.2 000.40.8 l.2 I.62.02.42.84p 4s 3d46.3 75.5 328.346.7 75.9 329.247.0 76.2 329.5 R(oo) FIG.4.a PdHandPdcDifference intheelectron chardcargeensityin an~clusters. Thenegative regionindicates the zonefromwhi hydrogenation.wichmetalchargehasbeentransf ddserreueto gaion.bNumberofelectrons ct'd'onaineina oraiusRaround aprotonembedded in toZratoms)(corresponding tohydrogen tetrahedrall b s(solidcurve)vsthataround aprotoninfree- hydrogen atom(dashedcurve).'HereweHerewedefinetheFermienergytobetheeienv thelastoccupied level.Bre'ieso eve. yrelating thecoreeneriesof atomandvariousclusterth'rsoteirrespective Fermien wecompensate forshiftsinbiienergies, sisinindingenergies whichare moeldependent, i.e.,depend uonclu' conitions.Thisr co't.T'procedure makesitpossible touse ground-state eigenvalues toestimate bindinene xciestateortransition state(seeRef.34) calculations neededtodetermine absoltb'd'ueiningenergies. MODELS OFELECTRONIC STRUCTURE OFHYDROGEN. .. 3549 free-atom, andPd6Pd6Hclusterconfigurations. In thesix-Pd-atom cluster,somechargefromeachatom isdonated totheconduction searesulting inanin- creaseofabout0.4eVintheioncorelevels.Thead- ditionofhydrogen accentuates thistrend.ThePd-H bonding chargeisbeingdrawnfromthevicinityof themetalioncore[seeFig.4(a)],leavingcorelevels stillmoretightlybound. Thiseffecthasbeenseeninarecentexperiment by Vealetal.'6involving x-ray—photoelectron spectros'- copy.Theseauthors havecompared thecore-level shiftsofZr4pand3dlevelsinZrH~65withthatin pureZrandfindthatthelevelsshifttohigherbind- ingenergies by0.7and1eV,respectively. This resultisconsistent withourclustercalculation inthe Pd-Hsystem. Aquantitative comparison ofthese core-level shiftsatthisstageisunwarranted sincewe expecttheseshiftstodependonthelocalenviron- ment.Inthehydride phase,forexample, thecon- centration ofhydrogen ishigh.Thus,weexpectthe magnitude ofshiftsinTableIduetohydrogenation tobesignificantly largerthanthepresentestimate. Aspointedoutearlier,themolecular clusterhasto beembedded inapotential background simulating thecrystalline environment. Wearepresently carry- ingoutthesecalculations forseveraltransition-metal hydrides. VI.ISOTOPE EFFECT ONTHEELECTRON DISTRIBUTION AROUND ~HAND2D Studiesofneutron inelastic scattering'"onmetals containing hydrogen reveallocalized modesforhy- drogenwhichinpalladium occursat56meV.As- suming thattheprotonmovesinaharmonic poten- tialwell,thislocalized modecorresponds toamean- squarehydrogen vibration amplitude of0.07A2.In thissectionwediscussbrieflytheeffectofthiszero- pointvibration ontheelectron distribution around thepointcharge. Jenaetal.'haverecently analyzed theisotopeef- fectusingasemiempirical modelbasedontheband- structure calculation andafirstprinciples calculation basedonthepseudojellium model. Theyhaveshown thatthesetwodistinctly different models yieldphysi- callysimilarresultsontheelectron-spin densityat'H and'DsitesinPdH.Theresultssuccessfully ex- plainedthehighernuclear spin=-lattice relaxation rate"ofDcompared to'Hasduetolargerzero- pointvibrational amplitude ofhydrogen. Thereader isreferred tothepaperofJenaetal.'fordetails.In thissection wemakeacomparison oftheelectron chargedistribution aroundadisplaced protonob- tainedinboththepseudojellium andmolecular- clustermodels. Thiscomparison shouldprovide someinsightintothequantitative significance ofthe resultsofthepseudojellium calculation.oo o/ 1.0—L2 /:~/ / 00.60.2 ISPLACEMENT(a l/CD LLI -0 0.8— CA CD Lalo06-9 lD 0.4— CL 0.2O 0I I I 00.40.8 l.2 l.6 r(a&) FIG.5.Self-consistent molecular-cluster resultforelec- troncharge-density distribution alongthe[100jdirection aroundahydrogen atomlocatedat(0,0,0)(curve), (0.3,0,0)(——-curve), (0.8,0,0)(——curve), and(1.2,0.0) (—-—-curve). Theinsetshowsacomparison between the electron chargedensityattheprotonsiteinamolecular- cluster(solidcurve)andpseudojel)ium (dashed curve) models.Inordertogaugethereliability ofthepseudojelli- ummodelininterpreting effectsassociated withthe zero-point vibration, wehavecarriedoutthe molecular-cluster calculation (seeSec.III)forfour different configurations ofthehydrogen atominside thePdoctahedron, i.e.,theequilibrium siteandcon- figurations ofhydrogen displaced by0.3ao,0.8ao,and 1.2aaalongthe[100)axis.Theresultsareplotted in ,Fig.5. Thefactthattheelectrons followtheprotonfaith- fullycanbeseenfromthefigure.Twootherin- teresting pointsareworthnoting.First,theelectron chargedistribution aroundtheprotonisverycloseto beingisotropic evenforaprotondisplaced byas muchas0.8aofromtheequilibrium configuration. Second, theelectron densityattheprotonsiteasa functionofdisplacement (seeinsetofFig.5)in- creases rapidlyastheprotonapproaches thenearest- neighbor Pdatom.Whiletheambient densityata .point1.2aofromtheequilibrium configuration along the[1001direction increases byafactorof2(seeFig. I)theself-consistent proton-site densityincreases by morethanafactorof3(seeFig.5).Thisenhance- mentcanbeattributed totheformation ofastronger Pd-Hbondasthenearest-neighbor Pd-Hdistance is reduced to2,4Qp.Asimilardisplacement inother directions produces asmallerenhancement. Thisan- isotropyoftheprotonenvironment isprimarily responsible forthedeviation between thepseudojelli- umandtheclusterresultsforlargeprotondisplace- ments. 3550 P.JENA,F.Y.FRADIN, ANDD.E.ELLIS 20 Tocompare theaboveresultswiththepredictions ofthepseudojellium model, wehavefollowed the sameprocedure asoutlined forthespindensity. The resultsarecompared withthemolecular-cluster modelintheinsetofFig.5.Notethatboththecal- culations areincloseagreement witheachotherfor displacements upto0.5aofromtheequilibrium config,- uration. However, forlargerdisplacements, the pseudojellium modelfailstoaccountforthesharp riseintheelectron chargedensityattheprotonsite. Theconfiguration-averaged chargedensity following theprescription ofJenaetal.'inthemolecular- clustermodelis0.35/ao3,whereas itis0.405/ao' inthe pseudojellium model.Thenatureofthisagreement between twomodels issimilartothatattheequilibri- umconfiguration discussed earlier. Thiscloseagree- mentbetween theconfiguration-averaged chargeden- sities(inspiteofthelargediscrepancy forlargerdis- placements) isnotsurprising sincetheprobability of theprotonbeingatadisplaced position becomes con- siderably smallerasthedisplacernent increases. Itis encouraging thatthepseudojellium modelgivesnot onlyqualitatively thesameresultfortheconfig- uration-averaged chargedensityasthemoresophisti- catedmolecular-cluster model,butitisalsoinsemi- quantitative agreement withthelatter.Itistobe notedthatclustercalculations willbequantitatively influenced bybothclustersizeandboundary condi- tion.A10%deviation isareasonable estimateof theseeffects. Calculations ofelectron-spin densityat theprotonsiteinthemolecular-cluster model,in- cludingeffectsduetozero-point vibration, arenot available atthepresent timetocompare withthe pseudojellium model. However, wedonotexpect anymajordifferences. VIII.CONCLUSION Inthispaperwehaveattempted togiveacompre- hensive discussion oftheelectronic structure ofhy- drogen inmetals. Although specific calculations for thePd-hydrogen systemwereperformed, thetheoret- icalmodelsandsubsequent discussions areapplicable toageneral metal-hydrogen system. Ourresultsare summarized inthefollowing: (i)Ahomogeneous densityresponse modelwithin theframework ofdensity-functional formalism wasusedtocalculate thenonlinear electron charge andspindistribution around hydrogen inpalladium. Theresultswerecompared withourself-consistent molecular-cluster model.Thechargedensity inthe vicinityoftheprotoninthesetwomodelsisfound toagreetowithin12%.We,therefore, suggest that forsemiquantitative analysis, ourpseudojelliummodelwouldserveasanefficient calculational method. Thismodelisparticularly attractive when onerealizes thatthenumerical effortisconsiderably lessthanthatinvolved inaself-consistent molecular- clustercalculation' letalonethatinaself-consistent supercell bandcalculation.'Theelectron-spin densi- tyattheprotonsiteinthePd-Hsystemwasfoundto beinfairagreement withthenon-self-consistent bandcalculation forPdH. (ii)Acomparison between ourmolecular-cluster calculation andthebandstructure indicates agree- mentinthenatureofthepalladium-hydrogen bond andinthequalitative shapeofthepartialdensityof states. (iii)Fromacomparison oftheelectron chargedis- tribution around palladium initspurestatewiththat uponhydrogenation, wefindthatthereisasignifi- cantchargetransfer fromthevincinityofthemetal iontothehydrogen sphere. Thisconsequently resultsinashiftinthecore-level binding energiesof themetalioninthehydride phasetowards higher binding ascompared toitspurestate.Thisresultis consistent withasimilareffectobserved' inZrH~65 fromx-ray—photoelectron spectroscopy measure- ments.Theresulting excesselectron densityaround hydrogen inametallic environment compared tothat infreespacegivesrisetoaphysical picturethathy- drogen inmetalsremains inaslightly"anionic" state. (iv)Theelectron-spin densityattheequilibrium protonsitewascalculated self-consistently usingthe generalized density-functional formalism.'Combined withtheenergy-band densityofstatesofselectrons attheFermienergy, thiscalculation yieldedthepro- tonspin-lattice relaxation ratethatwas57%higher than'the experimental value."Theeffectofthefin- itemassoftheprotonanddeuteron ontheelectron chargeandspindistribution ofthesurrounding elec- tronswasstudied inPdHandPdDintwodistinctly different models. Bothcalculations yieldalarger electron-spin densityatthe'Dsitethanatthe'H site—aresultinagreement withrecentexperimental data.Theeffectofzero-point vibration ontheelec- tronicstructure wasalsostudied intheself-consistent molecular-cluster modelforvarious displacements of theproton. Thetime-averaged chargedensityatthe protonsitewasfoundtobeingoodagreement with thepseudojellium model. 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PhysRevB.96.161403.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 96, 161403(R) (2017) Nonlocal Andreev entanglements and triplet correlations in graphene with spin-orbit coupling Razieh Beiranvand,1Hossein Hamzehpour,1,2and Mohammad Alidoust1 1Department of Physics, K.N. Toosi University of Technology, Tehran 15875-4416, Iran 2School of Physics, Institute for Research in Fundamental Sciences (IPM), 19395-5531 Tehran, Iran (Received 27 February 2017; published 4 October 2017) Using a wave function Dirac Bogoliubov–de Gennes method, we demonstrate that the tunable Fermi level of a graphene layer in the presence of Rashba spin-orbit coupling (RSOC) allows for producing an anomalous nonlocalAndreev reflection and equal spin superconducting triplet pairing. We consider a graphene nanojunction of aferromagnet-RSOC-superconductor-ferromagnet configuration and study scattering processes, the appearance ofspin triplet correlations, and charge conductance in this structure. We show that the anomalous crossed Andreevreflection is linked to the equal spin triplet pairing. Moreover, by calculating current cross-correlations, ourresults reveal that this phenomenon causes negative charge conductance at weak voltages and can be revealed in aspectroscopy experiment, and may provide a tool for detecting the entanglement of the equal spin superconductingpair correlations in hybrid structures. DOI: 10.1103/PhysRevB.96.161403 Introduction . Superconductivity and its hybrid structures with other phases can host a wide variety of intriguingfundamental phenomena and functional applications such asHiggs mechanism [ 1], Majorana fermions [ 2], topological quantum computation [ 3], spintronics [ 4], and quantum entan- glement [ 5–8]. The quantum entanglement describes quantum states of correlated objects with nonzero distances [ 6,8] that are expected to be employed in novel ultrafast technologiessuch as secure quantum computing [ 3,6]. From the perspective of BCS theory, s-wave singlet super- conductivity is a bosonic phase created by the coupling of twocharged particles with opposite spins and momenta (forming a so-called Cooper pair) through an attractive potential [ 9]. The two particles forming a Cooper pair can spatially havea distance equal or less than a coherence length ξ S[9]. Therefore, a Cooper pair in the BCS scenario can serve asa natural source of entanglement with entangled spin andmomentum. As a consequence, one can imagine a heterostruc-ture made of a single s-wave superconductor and multiple nonsuperconducting electrodes in which an electron and hole excitation from different electrodes are coupled by means ofa nonlocal Andreev process [ 7,10–13]. This idea has so far motivated numerous theoretical and experimental endeavoursto explore this entangled state in various geometries andmaterials [ 12,14–27]. Nonetheless, the nonlocal Andreev process is accompanied by an elastic cotunneling current that makes it practically difficult to detect unambiguously the signatures of a nonlocal entangled state [ 10,11,13–17]. This issue, however, may be eliminated by making use of agraphene-based hybrid device that allows for locally controlledFermi level [ 26]. On the other hand, the interplay of s-wave supercon- ductivity and an inhomogeneous magnetization can convertthe superconducting spin singlet correlations into equal spintriplets [ 28,29]. After the theoretical prediction of the spin triplet superconducting correlations much effort has beenmade to confirm their existence [ 4,30–43]. For example, a finite supercurrent was observed in a half-metallic junctionthat was attributed to the generation of equal spin tripletcorrelations near the superconductor–half-metal interface [ 30]. Also, it was observed that in a Josephson junction madeof a holmium–cobalt–holmium stack, the supercurrent as a function of the cobalt layer decays exponentially withoutany sign reversals due to the presence of equal spin tripletpairings [ 36,37]. One more signature of the equal spin triplet pairings generated in the hybrid structures may be detected insuperconducting critical temperature [ 43–46] and density of states [ 47–50]. Nevertheless, a direct observation of the equal spin triplet pairings in the hybrid structures is still lacking. In this Rapid Communication, we show that the existence of the equal spin superconducting triplet correlations canbe revealed through charge conductance spectroscopy of agraphene-based ferromagnet–Rashba SOC–superconductor–ferromagnet junction. We study all possible electron/holereflections and transmissions in such a configuration andshow that by tuning the Fermi level a regime is accessiblein which spin reversed cotunneling and usual crossed Andreevreflections are blocked while a conventional cotunneling andanomalous nonlocal Andreev channel is allowed. We justifyour findings by analyzing the band structure of the system.Moreover, we calculate various superconducting correlationsand show that, in this regime, the equal spin triplet correlationhas a finite amplitude while the unequal spin triplet componentvanishes. Our results show that the anomalous crossed Andreevreflection results in a negative charge conductance at lowvoltages applied across the junction and can be interpretedas evidence for the generation and entanglement of equalspin superconducting triplet correlations in hybrid structures[51–55]. Method and results . As seen in Fig. 1, we assume that the ferromagnetism, superconductivity, and spin-orbit couplingare separately induced into the graphene layer through theproximity effect as reported experimentally in Refs. [ 56–58] for isolated samples. Therefore, the low-energy behavior ofquasiparticles, quantum transport characteristics, and thermo-dynamics of such a system can be described by the DiracBogoliubov–de Gennes (DBdG) formalism [ 34,59]: /parenleftbigg H D+Hi−μi/Delta1eiφ /Delta1∗e−iφμi−T[HD−Hi]T−1/parenrightbigg/parenleftbigg u v/parenrightbigg =ε/parenleftbigg u v/parenrightbigg , (1) 2469-9950/2017/96(16)/161403(6) 161403-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS BEIRANV AND, HAMZEHPOUR, AND ALIDOUST PHYSICAL REVIEW B 96, 161403(R) (2017) in which εis the quasiparticles’ energy and Trepresents a time-reversal operator [ 34,59]. Here HD=¯hvFs0⊗(σxkx+ σyky) with vFbeing the Fermi velocity [ 59].sx,y,z andσx,y,z are 2×2 Pauli matrices, acting on the spin and pseudospindegrees of freedom, respectively. The superconductor region with a macroscopic phase φis described by a gap /Delta1in the energy spectrum. The chemical potential in a region iis shown byμiwhile the corresponding Hamiltonians read Hi=⎧ ⎪⎨ ⎪⎩HF=hl(sz⊗σ0),x /lessorequalslant0 HRSO=λ(sy⊗σx−sx⊗σy),0/lessorequalslantx/lessorequalslantLRSO HS=−U0(s0⊗σ0),L RSO/lessorequalslantx/lessorequalslantLS+LRSO HF=hr(sz⊗σ0),L S+LRSO/lessorequalslantx.(2) The magnetization /vectorhl,rin the ferromagnet segments are assumed fixed along the zdirection with a finite intensity hl,r.λis the strength of Rashba spin-orbit coupling and U0is an electrostatic potential in the superconducting region. Previous self-consistent calculations have demonstrated that sharp interfaces between the regions can be an appropriate approximation [ 34,59–62]. The length of the RSO and S regions are LRSOandLS, respectively. To determine the properties of the system, we diagonalize the DBdG Hamiltonian equation ( 1) in each region and obtain corresponding eigenvalues: ε=⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩±μ Fl±/radicalBig/parenleftbig kFlx/parenrightbig2+q2n±hl,x /lessorequalslant0 ±μRSO±/radicalBig/parenleftbig kRSOx/parenrightbig2+q2n+λ2±λ, 0/lessorequalslantx/lessorequalslantLRSO ±/radicalbigg /parenleftbig μS+U0±/radicalBig/parenleftbig kSx/parenrightbig2+q2n/parenrightbig2+|/Delta10|2,L RSO/lessorequalslantx/lessorequalslantLRSO+LS ±μFr±/radicalBig/parenleftbig kFrx/parenrightbig2+q2n±hr,L RSO+LS/lessorequalslantx.(3) The associated eigenfunctions are given in Ref. [ 63]. The wave vector of a quasiparticle in region iiski=(ki x,qn) so that its transverse component is assumed conserved upon scattering.In what follows, we consider a heavily doped superconductorU 0/greatermuchε,/Delta1 which is an experimentally relevant regime [ 59]. We also normalize energies by the superconducting gap at zerotemperature /Delta1 0and lengths by the superconducting coherent length ξS=¯hvF//Delta10. Since the magnetization in F regions is directed along the zaxis, which is the quantization axis, it allows for unam- biguously analyzing spin-dependent processes. Therefore, weconsider a situation where an electron with spin-up (described by wave function ψ F,+ e,↑) hits the RSO interface at x=0 due to a voltage bias applied. This particle can reflect back ( ψF,− e,↑(↓)) with probability amplitude r↑(↓) Nor enter the superconductor as a Cooper pair and a hole ( ψF,− h,↑(↓)) with probability amplitude FIG. 1. Schematic of the graphene-based F-RSO-S-F hybrid. The system resides in the xyplane and the junctions are located along the xaxis. The length of the RSO and S regions are denoted by LRSOand LS. The magnetization of the F regions ( /vectorhl,r) are assumed fixed along thezaxis. We assume that the ferromagnetism, spin-orbit coupling, and superconductivity is induced into the graphene layer by meansof the proximity effect.r↑(↓) Areflects back, which is the so-called Andreev reflection. Hence, the total wave function in the left F region is (seeRefs. [ 53,63]) /Psi1 Fl(x)=ψF,+ e,↑(x)+r↑ NψF,− e,↑(x)+r↓ NψF,− e,↓(x) +r↓ AψF,− h,↓(x)+r↑ AψF,− h,↑(x). (4) The total wave function in the RSO and S parts are su- perpositions of right- and left-moving spinors with differentquantum states n;ψ RSO nandψS n(see Ref. [ 63]):/Psi1RSO(x)=/summationtext8 n=1anψRSO n(x) and /Psi1S(x)=/summationtext8 n=1bnψS n(x), respectively. The incident particle eventually can transmit into the right F region as an electron or hole ( ψF,+ e,↑↓,ψF,+ h,↑↓) with probability amplitudes t↑↓ eandt↑↓ h: /Psi1Fr(x)=t↑ eψF,+ e,↑(x)+t↓ eψF,+ e,↓(x)+t↓ hψF,+ h,↓(x)+t↑ hψF,+ h,↑(x). (5) The transmitted hole is the so-called crossed Andreev reflec-tion (CAR). By matching the wave functions at F-RSO, RSO- S, and S-F interfaces we obtain the probabilities described above. Figure 2exhibits the probabilities of usual electron cotunneling |t↑ e|2, spin-flipped electron |t↓ e|2, usual crossed Andreev reflection |t↓ h|2, and anomalous crossed Andreev reflection |t↑ h|2. To have a strong anomalous CAR signal, we setLS=0.4ξSwhich is smaller than the superconducting coherence length and LRSO=0.5ξS[11]. We also choose μFl=μFr=hl=hr=0.8/Delta10,μRSO=2.6/Delta10,λ=/Delta10and later clarify physical reasons behind this choice using band-structure analyses. In terms of realistic numbers, if thesuperconductor is Nb [ 62] with a gap of the order of /Delta1 0∼ 1.03 meV and coherence length ξS∼10 nm, the chemical potentials, magnetization strengths, and the RSO intensityareμ Fl=μFr=hl=hr=0.824 meV, μRSO=2.68 meV, 161403-2RAPID COMMUNICATIONS NONLOCAL ANDREEV ENTANGLEMENTS AND TRIPLET . . . PHYSICAL REVIEW B 96, 161403(R) (2017) FIG. 2. (a) Spin-reversed cotunneling probability |t↓ e|2. (b) Anomalous crossed Andreev reflection probability |t↑ h|2. (c) Conventional cotunneling |t↑ e|2. (d) Usual CAR |t↓ h|2.T h e probabilities are plotted vs the transverse component of wave vector qnand voltage bias across the junction eV.W es e t μFl=μFr=hl= hr=0.8/Delta10,μRSO=2.6/Delta10,λ=/Delta10,L RSO=0.5ξS,L S=0.4ξS. λ=1.03 meV, respectively [ 56,57], andLS=4n m , LRSO= 5 nm. We see that the anomalous CAR has a finite amplitudeand its maximum is well isolated from the other transmissionchannels in the parameter space. Therefore, by tuning the localFermi levels the system can reside in a regime that allows for astrong signal of the anomalous CAR. According to Fig. 2this regime is accessible at low voltages eV/lessmuch/Delta1 0. The eigenvalues, Eqs. ( 3), determine the propagation critical angles of moving particles through the junction. Byconsidering the conservation of transverse component of wavevector throughout the system, we obtain the following criticalangles [ 59]: α c e,↓=arcsin/vextendsingle/vextendsingle/vextendsingleε+μFr−hr ε+μFl+hl/vextendsingle/vextendsingle/vextendsingle, (6a) αc h,↓=arcsin/vextendsingle/vextendsingle/vextendsingleε−μFr+hr ε+μFl+hl/vextendsingle/vextendsingle/vextendsingle, (6b) αc e,↑=arcsin/vextendsingle/vextendsingle/vextendsingleε+μFr+hr ε+μFl+hl/vextendsingle/vextendsingle/vextendsingle, (6c) αc h,↑=arcsin/vextendsingle/vextendsingle/vextendsingleε−μFr−hr ε+μFl+hl/vextendsingle/vextendsingle/vextendsingle. (6d) These critical angles are useful in calibrating the device properly for a regime of interest. For the spin-reversedcotunneling, the critical angle is denoted by α c e,↓, while for the conventional CAR we show this quantity by αc h,↓. Hence, to filter out these two transmission channels, we set μFr=hrand choose a representative value 0 .8/Delta10. In this regime, we see that αc e(h),↓→0 at low energies, i.e., μFr,hr,/Delta1/greatermuchε→0 and thus, the corresponding transmissions are eliminated. This is clearlyseen in Figs. 2(a) and2(d) ateV/lessmuch/Delta1 0. At the same time, the critical angles to the propagation of conventional electroncotunneling and anomalous crossed Andreev reflection reachnear their maximum values α c e(h),↑→π/2 consistent with Figs. 2(b) and 2(c). We have analyzed the reflection and transmission processes using a band-structure plot, presented-0.0400.04 0.40 0.80 -0.0400.04 0.9 1.5 2 2.5-0.400.4 0.9 1.5 2 2.5-0.400.4(a) (b) (c) (d) FIG. 3. (a)–(d) Real and imaginary parts of opposite spin f0and equal spin pairings f1within the Frregion x/greaterorequalslantLRSO+LSat weak voltages eV/lessmuch/Delta10. The parameter values are the same as those of Fig. 2except we now compare two cases where μFl=μFr=hl= 0.8/Delta10andhr=0.4/Delta10,0.8/Delta10. in Ref. [ 63], that can provide more sense on how a particle is scattered in this regime. To gain better insights into the anomalous CAR, we calculate the opposite ( f0) and equal ( f1) spin-pair correlations in the Frregion [ 31,34]: f0(x,t)=+1 2/summationdisplay βξ(t)[u↑ β,Kv↓,∗ β,K/prime+u↑ β,K/primev↓∗ β,K −u↓ β,Kv↑∗ β,K/prime−u↓ β,K/primev↑∗ β,K], (7a) f1(x,t)=−1 2/summationdisplay βξ(t)[u↑ β,Kv↑,∗ β,K/prime+u↑ β,K/primev↑∗ β,K +u↓ β,Kv↓∗ β,K/prime+u↓ β,K/primev↓∗ β,K], (7b) where KandK/primedenote different valleys and βstands forAandBsublattices [ 34,59]. Here, ξ(t)=cos(εt)− isin(εt) tanh( ε/2T),tis the relative time in the Heisenberg picture, and Tis the temperature of the system [ 31,34]. Figure 3shows the real and imaginary parts of opposite and equal spin pairings in the Frregion, extended from x= LRSO+LSto infinity, at eV/lessmuch/Delta10. For the set of parameters corresponding to Fig. 2, we see that f0pair correlation is vanishingly small, while the equal spin triplet pair correlationf 1has a finite amplitude. We also plot these correlations for a different set of parameters where μFl=μFr=hl=0.8/Delta10, whilehr=0.4/Delta10. The opposite spin triplet pairing f0is now nonzero too. Therefore, at low voltages and the parameterset of Fig. 2, the nonvanishing triplet correlation is f 1, which demonstrates the direct link of f1andt↑ h. This direct connection can be proven by looking at the total wave function in the right 161403-3RAPID COMMUNICATIONS BEIRANV AND, HAMZEHPOUR, AND ALIDOUST PHYSICAL REVIEW B 96, 161403(R) (2017) 0 1201 0G G(a) 01201 (d) (c)(b) 01201hG↓hG↑ eG↓eG↑ 0eVΔ01201 0eVΔ FIG. 4. Charge conductance (top panels) and its components (bottom panels). (a) and (c) charge conductance associated withthe probabilities presented in Figs. 2and3(h r=0.8/Delta10) and its components, respectively. (b) and (d) the same as panels (a) and (c) except we now consider hr=0.4 (see Fig. 3). The conductance is normalized by G0=G↑+G↓. F region, Eq. ( 5), transmission probabilities shown in Fig. 2, and the definition of triplet correlations, Eqs. ( 7). One can show that when t↓ eandt↓ hvanish, f0disappears and f1remains nonzero, which offers a spin triplet valve effect. We calculate the charge conductance through the BTK formalism: G=/integraldisplay dqn/summationdisplay s=↑,↓Gs/parenleftbig/vextendsingle/vextendsinglets e/vextendsingle/vextendsingle2−/vextendsingle/vextendsinglets h/vextendsingle/vextendsingle2/parenrightbig , (8) where we define G↑↓=2e2|ε+μl±hl|W/hπ in which W is the width of the junction. Figures 4(a) and4(b) exhibit the charge conductance as a function of bias voltage eVacross the junction at hr=0.8/Delta10and 0.4/Delta10, while the other parameters are set the same as those of Figs. 2and3. As seen, the charge conductance is negative at low voltages when hr=0.8/Delta10, whereas this quantity becomes positive for hr=0.4/Delta10.T o gain better insights, we separate the charge conductance into G↑↓(↑↓) e,(h), corresponding to the transmission coefficients t↑↓(↑↓) e,(h) used in Eq. ( 8). Figures 4(c)and4(d)illustrate the contribution of different transmission coefficients into the conductance.We see in Fig. 4(c) thatG↑ hdominates the other components and makes the conductance negative. As discussed earlier,this component corresponds to the anomalous CAR whichis linked to the equal spin triplet pairing, Fig. 3.T h i scomponent, however, suppresses when h r=0.4/Delta10so that the other contributions dominate, and therefore the conductanceis positive for all energies. Hence, the nonlocal anomalousAndreev reflection found in this work can be revealed ina charge conductance spectroscopy. There are also abruptchanges in the conductance curves that can be fully understoodby analyzing the band structure. We present such an analysisin Ref. [ 63]. In line with the theoretical works summarized in Ref. [ 59], we have neglected spin-dependent and -independent impuritiesand disorders as well as substrate and interface effects inour calculations [ 64–66]. Nonetheless, a recent experiment has shown that such a regime is accessible with today’sequipment [ 62]. Moreover, the same assumptions have already resulted in fundamentally important predictions such as thespecular Andreev reflection [ 59] that was recently observed in experiment [ 61]. The experimentally measured mean free path of moving particles in a monolayer graphene depositedon top of a hexagonal boron nitride substrate is around /lscript∼ 140 nm [ 67]. The coherence length of induced superconduc- tivity into a monolayer graphene using a Nb superconductorwas reported as ξ S∼10 nm [ 62]. In this situation, where /lscript/greatermuchξS, the Andreev mechanism is experimentally relevant. On the other hand, it has been demonstrated that the equal-spinpairings discussed here are long range and can survive evenin systems with numerous strong spin-independent scattering resources [ 40–42]. Therefore, as far as the Andreev mechanism is a relevant scenario in a graphene-based F-RSO-S-F devicecontaining spin-independent scattering resources, i.e., /lscript/greatermuchξ S, we expect that the negative conductance explored in this RapidCommunication is experimentally accessible. In conclusion, motivated by recent experimental achieve- ments in the induction of spin-orbit coupling into a graphenelayer [ 56,57], we have theoretically studied quantum trans- port properties of a graphene-based ferromagnet-RSOC-superconductor-ferromagnet junction. Our results reveal thatby manipulating the Fermi level in each segment, one cancreate a dominated anomalous crossed Andreev reflection. 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