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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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PhysRevB.99.155304.pdf

PHYSICAL REVIEW B 99, 155304 (2019) Strong electron-electron interactions of a Tomonaga-Luttinger liquid observed in InAs quantum wires Yosuke Sato,1,*,†Sadashige Matsuo,1,2,3,*,‡Chen-Hsuan Hsu,3Peter Stano,1,3,4Kento Ueda,1Yuusuke Takeshige,1 Hiroshi Kamata,3Joon Sue Lee,5Borzoyeh Shojaei,5,6Kaushini Wickramasinghe,7Javad Shabani,7Chris Palmstrøm,5,6,8 Yasuhiro Tokura,9Daniel Loss,3,10and Seigo Tarucha1,3,§ 1Department of Applied Physics, University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan 3Center for Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan 4Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia 5California NanoSystems Institute, University of California Santa Barbara, Santa Barbara, California 93106, USA 6Materials Engineering, University of California Santa Barbara, Santa Barbara, California 93106, USA 7Center for Quantum Phenomena, Department of Physics, New York University, New York, New York 10003, USA 8Electrical and Computer Engineering, University of California Santa Barbara, Santa Barbara, California 93106, USA 9Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan 10Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Received 29 October 2018; revised manuscript received 5 March 2019; published 16 April 2019) We report strong electron-electron interactions in quantum wires etched from an InAs quantum well, a material generally expected to have strong spin-orbit interactions. We find that the current through the wires as a functionof the bias voltage and temperature follows the universal scaling behavior of a Tomonaga-Luttinger liquid. Usinga universal scaling formula, we extract the interaction parameter and find strong electron-electron interactions,increasing as the wires become more depleted. We establish theoretically that the spin-orbit interaction causeonly minor modifications of the interaction parameter in this regime, indicating that genuinely strong electron-electron interactions are indeed achieved in the device. Our results suggest that etched InAs wires provide aplatform with both strong electron-electron interactions and the strong spin-orbit interaction. DOI: 10.1103/PhysRevB.99.155304 I. INTRODUCTION A one-dimensional electron system displays the physics of a Tomonaga-Luttinger liquid (TLL), which is strikinglydifferent to Fermi liquids in higher dimensions. A spinful TLLis described by the Hamiltonian [ 1,2] H=/summationdisplay ν=c,s/integraldisplay¯hdx 2π/braceleftbigg uνgν[∂xθν(x)]2+uν gν[∂xφν(x)]2/bracerightbigg .(1) Here,ν∈{c,s}labels the charge and spin sector, respectively, while uνare the velocities, and θνandφνthe bosonic fields, describing the two elementary excitations [ 3]. The electron- electron (e-e) interactions are parameterized by gcandgs, which range between 0 and 1 [ 4]. The spin-charge separation, meaning the independence of the charge and spin sectorsdisplayed by Eq. ( 1), appears as one of the key features of a TLL. The coupling of spin and charge degrees of freedom, in various forms of the spin-orbit interaction (SOI), plays animportant role in semiconductors and spintronics [ 5,6]. The *These authors contributed equally to this work. †yosuke.sato0530@gmail.com ‡sadashige.matsuo@riken.jp §tarucha@riken.jpresearch on SOI has been further accelerated by predictions of the emergence of Majorana fermions in an accessible setupcomprising a quantum wire with superconductivity, SOI, and amagnetic field [ 7–10]. Unfortunately, the practical realization is impeded by the incompatibility of a strong magnetic fieldand superconductivity. Recently, it has been suggested thatwires with strong e-e interactions could solve this conflictby disposing of the magnetic field [ 11,12]. More importantly, strong e-e interactions allow a realization of parafermions[11], more advanced topological particles than the Majorana fermions [ 13,14]. They rely on Cooper pair splitting into two quantum wires with high efficiency, which is achieved throughstrong e-e interactions. We note that efficient Cooper pairsplitting [ 15,16] and a transparent interface with a supercon- ductor has been recently demonstrated in self-assembled InAsnanowires [ 17,18] and quantum wells [ 19]. With this outlook, providing wires with both strong e-e interactions and strongSOI seems beneficial. Motivated by such prospects, there appeared several the- oretical works concerned with a TLL in the presence ofSOI. The SOI mixes the spin and charge sectors and a richrange of phenomena was predicted, from mild modificationsto a breakdown of the TLL phase [ 20–32]. Despite active discussions in theory, there are only few experimental studiesof TLL physics in wires with strong SOI. Concerning InAs,we note the self-assembled nanowires [ 33] and nanowire 2469-9950/2019/99(15)/155304(14) 155304-1 ©2019 American Physical SocietyYOSUKE SATO et al. PHYSICAL REVIEW B 99, 155304 (2019) quantum point contacts [ 34] experiments. In the former, a small interaction parameter was deduced, but it remainedunclear whether this was due to the SOI, intrinsically stronge-e interactions, or even some other physics. The situationcontrasts to TLLs without SOI, with a number of reports, forexample on GaAs wires [ 35–38], carbon nanotubes [ 39–44], all in which the SOI is negligible. Overall the effects of SOI inTLLs have been considered in theory but have been exploredin few experimental studies. II. SUMMARY OF THE MAIN RESULTS Here we investigate the TLL behavior of quantum wires fabricated in an InAs quantum well. Even though we donot quantify its strength in this experiment, it is generallyexpected that the SOI in InAs is strong [ 45–47]. We measure the electric current through the wires as a function of the biasvoltage at various temperatures and find that the data falls ontoa single curve upon rescaling. Such universal scaling is con-sistent with the TLL theory, allowing extraction of the valueof the interaction parameter g cin Eq. ( 1). The extracted values reach as low as 0.16–0.28 (these minimal values are for wiresclose to depletion), indicating a strong-interaction regime. In addition to transport measurements, we provide theoretical understanding of one-dimensional systems with strong e-einteractions and SOI. Overall our results demonstrate thatInAs wires offer a platform fulfilling the requirements for therealization of topological particles. III. DEVICE The data presented in this paper were measured on a single device,1shown in Fig. 1(a). It is composed of ten parallel quantum wires, which were chemically etched from an InAsquantum well. Ohmic contacts, created using Ti /Au [48], and aT i/Au top gate, deposited on top of a cross-linked PMMA serving as an insulating layer, give electrical access and con-trol. A single wire has a length of 20 μm and a nominal width (estimated from the depth of the etching) of 100 nm. The stackmaterials of the InAs quantum well are given in Fig. 1(b). Prior to measurements of the wires, the two-dimensionalelectron gas mobility of 7 .2×10 4cm/(Vs), electron density of 3.4×1011cm−2, and mean free path of 690 nm were extracted from measurements on a Hall-bar device at 560 mK.The electric current Iflowing through the parallel wires upon applying a bias voltage Vis measured by the standard four- terminal dc measurement. These measurements are performedat temperature Tin the range 2–4 K. Figure 1(c) shows Ias a function of the top gate voltage V gfor a fixed V=1m V . The device shows a pinch-off at about Vg=−0.86 V . A small current remaining below that voltage is most probably dueto a tunneling conductance through quantum dots formed inthe disorder potential of the wires. The parallel quantum wire 1We fabricated several devices with various lengths and numbers of wires in search for characteristic features of TLL. These early devices were plagued by typically large contact resistances and a wafer-dependent voltage range of the few-channel regime. The nextgeneration of devices will build on this experience. FIG. 1. (a) Microscope photographs of the parallel-wire device before (left) and after (right) depositing a top gate above a cross-linked PMMA layer. The wires are along [010]. (b) Schematic struc- ture of the wafer, grown along [001]. (c) Gate voltage dependence of the current through the wires measured at a constant source-drainvoltage as given in the figure caption. This measurement is performed at 2.9 K. structure reduces the total resistance such that the total current is still within the measurable range. Though measuring manyparallel wires precludes observing conductance plateaus, italso results in averaging out the potential fluctuations fromimpurities and other disorder. We believe that such averagingis crucial for observing the universal scaling we report. IV . UNIVERSAL SCALING OF THE CURRENT-VOLTAGE CURVE Before presenting our main results, we first review the transport properties predicted by the existing theory. It hasbeen established that, assuming the spin-charge separation , a current through a single TLL with several tunnel barriers(their number and positions are discussed below) displaysuniversal scaling [ 39,49]. Explicitly, the tunnel current is I=I 0T1+αsinh/parenleftbiggγeV 2kBT/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/Gamma1/parenleftbigg 1+α 2+iγeV 2πkBT/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 .(2) Here, I0is an unspecified overall scale dependent on a typical barrier strength, /Gamma1(z) is the Gamma function, eis the positive elementary charge, kBis the Boltzmann constant, and the parameters αandγdepend on the number and character of the barriers, or, more generally, source of resistance.2 2Below, we consider the current-voltage relation (alternatively, the conductance) under more general conditions than that under whichEq. ( 2) has been originally derived. In such general considerations, we use the name “sources of resistance” without specifying whether 155304-2STRONG ELECTRON-ELECTRON INTERACTIONS OF A … PHYSICAL REVIEW B 99, 155304 (2019) FIG. 2. Current ( I) flowing through the wires as a function of the bias voltage ( V) for the top gate voltage Vg=−0.6V . The expression γVcorresponds to a voltage drop across the tunnel barrier. Thus, in Ref. [ 49], one has γ=1. One can generalize this result to several, say N, tunnel barriers; assuming that they induce comparable resistances, a typicalvoltage drop over a single one will be V/N. For this case, 3 the inverse of γtherefore gives the number of tunnel barriers [39,50]. The parameter αdepends in an intricate way on the e-e interaction strength parameters gcandgs, the SOI strength, and the number and character of the sources of resistance.The expressions for αfor the case of zero SOI are already known. For the case of finite SOI, we provide them hereand in Ref. [ 51]. Extracting this parameter from the data and inferring from it the e-e interaction strength is the essenceof this paper. Let us first describe the former task, beforediscussing the latter one. The extraction of αis rather straightforward once the I-V curve in Eq. ( 2) is plotted on a log-log scale. This exercise reveals different slopes for γeVmuch smaller and much larger thank BT. For the (differential) conductance, G≡dI/dV,t h i s corresponds to power laws G∝TαandG∝Vα, respectively. The power law in the conductance, G∝Tα, in the regime of eV/lessmuchkBT, was observed in numerous previous experiments [38–43,50,52–57]. If the universal scaling curve is obtained for a large enough range of its natural parameter, eV/kBT, such that the crossover is seen, one can extract both γandα. V . MEASUREMENT OF I-VCURVES AND FIT TO EQ. (2) To extract both αandγ, we measure the current Ias a function of the bias voltage Vat various temperatures. A set tunnel barriers, weak impurities or other scatterers. We argue that Eq. ( 2) is still valid in this more general situation, upon proper interpretation of parameters I0,α,a n dγ. 3We stress that the connection between γand the number of barriers Nisγ=1/Nonly if all the barriers result in identical resistances. Otherwise, γcounts only the barriers which dominate the voltage drop. The typical (length) density of these dominating barriers might have no direct relation to the transport mean free pathfound for the 2DEG, as will be the case here.FIG. 3. A rescaled current I/T1+αas a function of eV/kBTfrom the data points in Fig. 2. The black-solid curve is drawn using Eq. ( 2) with the parameters ( α,γ, I0)=(1.3,0.38,3.6×−10A/K2.3), which were extracted by fitting the data in Fig. 2to Eq. ( 2). We note that the unit of I0scales with α. of such curves, for top-gate voltage of Vg=−0.6V ,i ss h o w n in Fig. 2. One can see that the current generally decreases with decreasing temperature T, and that for a fixed Tdifferent slopes for the high- Vand low- Vregimes can be observed. These features are qualitatively consistent with Eq. ( 2). For a fixed top-gate voltage Vg, we fit the whole set of I-Vcurves to Eq. ( 2) with I0,α, andγas the fitting parameters. The rescaled data, together with the fitted curve, is plotted in Fig. 3.W e observe that the rescaled data indeed collapses onto a singlecurve, confirming the universal scaling behavior of a TLL. After confirming that the universal scaling holds, and there- fore the parameters αandγare reasonably assigned by the fit, we examine their dependence on the carrier density. As thelatter is tunable through the top gate voltage, we repeat theabove measurements and fittings for various V g, and summa- rize the results in Fig. 4(Fig. 7in Appendix A3shows three more sets, for Vg=−0.2,−0.4, and −0.8 V, with both raw and scaled I-Vdata from which the fittings are performed). One can see that both parameters change with Vg, suggesting FIG. 4. Fitted values of αas a function of the top gate voltage Vg (red circle, left axis). The error bars of the fitting are smaller than the markers. Fitted values of γas a function of Vgare also plotted (blue squares, right axis). For Vg>−0.25 V, the fitted values of γare not very reliable, as reflected by the larger error bars. 155304-3YOSUKE SATO et al. PHYSICAL REVIEW B 99, 155304 (2019) that the e-e interaction strength varies with the carrier density. To convert the extracted parameters to physical parametersof the system is rather involved and will be addressed later.Before that, we consider the observation of the fitted values ofγcollapsing to 1 for voltages V g>−0.25 V. As we already stated, the fitted value of γis determined by the position of the kink in the current-bias curve (for example, in Fig. 3, the kink is at eV/kBT≈10). However, the smaller the value ofαthe smaller the variation in gradient between low- and high- Vregimes, complicating the determination of the kink position. Our fits are less sensitive to γforα/lessorsimilar0.5, where the fit returns γ=1. Though error bars become larger, the value γ=1 is consistent with the trend observed where γ is accurately extracted for higher α. Nevertheless, the most interesting part of this plot is on its left end, for large negativevalues of the top gate V g. Here, αis large which corresponds to strong e-e interactions, as we will see. Also, in the sameregion, γis around 0.5, corresponding to two tunnel barriers. We are primarily interested in extracting the strength of thee-e interactions in this regime. VI. DEDUCING THE STRENGTH OF ELECTRON-ELECTRON INTERACTIONS A. Description of the theoretical methods used We now describe our theoretical analysis, which allows us to extract the strength of the e-e interactions from theobserved α. Motivated by the expected strong SOI in InAs, we model each of the wires as a TLL subject to SOI. Tothis end, we incorporate the SOI-induced band distortion,which is parametrized by the ratio δv/v Fwithδvthe velocity difference between the two branches of the distorted energybands [ 58] (see Appendix Bfor details). This band distortion breaks the spin-charge separation of a TLL in Eq. ( 1)[20,21] and leads to a coupling between the charge and spin sectors[see Eq. ( B1) in Appendix B]. In addition, the SOI can cause the value of g sto depart from unity [ 22,30]. In deriving the current-voltage characteristics, we include both the charge-spin coupling in the Hamiltonian and a general value for theg sparameter. The theoretical analysis is complicated not only by the presence of the SOI, but also by the fact that the conductancedepends on the characters and positions of the resistancesources (strong or weak, and inside the wire or around itsboundary) and also on the value of αitself (larger or smaller than 0.5). Including these features is what sets our work apartfrom preceding studies. For the sake of brevity, we delegate the full analysis to Ref. [ 51] and state the main results from there in Appendix B. Here, in the main text, we distill that results further, and only give and comment on the formulaswhich are used to fit the experimental data. We start with that, first, we observe γroughly between 1 and 1/2, and, second, that we expect disorder to be generally present in the wires. 4Correspondingly, we begin with consid- ering the following types of resistance sources: a single tunnel 4As the wires are much longer than the bulk mean free path of 690 nm, the disorder (perhaps, in the form of weak potentialmodulation due to impurities) should play role in the wire resistance. FIG. 5. (a) A schematic illustration of scenario A: there are bulk barriers (crosses), each acting as a TLL-TLL junction, and many weak impurities (not shown). (b) A schematic illustration of scenario B: there are boundary barriers, each acting as a TLL-Fermiliquid (FL) junction, and many weak impurities. In (a) and (b), for illustration, we plot two barriers, motivated by the observed 1 /γ/lessorsimilar2. (c) Extracted values of the interaction parameter g cas a function of the top gate voltage Vgfor the two scenarios. The red-solid and green-dashed curves are the fits to Eq. ( 6), with the fitting parameter w(the wire transverse size) being 87 and 47 nm, respectively. barrier located in the wire bulk, a single tunnel barrier located near the wire boundary, and a disorder potential from manyweak impurities. We first calculate the corresponding resis-tances separately, and then discuss the total wire resistancewhen they coexist. In the presence of a single bulk or boundary barrier, we compute the tunnel current through it using the method ofRef. [ 49], which allows one to obtain the full current-bias curve. In addition, we use the renormalization-group (RG)method of Refs. [ 59,60] to obtain the current power-law scal- ing in the high-temperature and high-bias limits. We verifythat, in these limits, the two theoretical approaches give thesame exponent αand are therefore consistent. For many weak impurities (that is, disorder potential), the method of Ref. [ 49] is not applicable. Instead, we calculate the exponent of the current power law in the high-temperatureand high-bias limits using the RG method of Refs. [ 59,60]. In this case, we understand Eq. ( 2) as an interpolation formula, with the parameter αreplaced by the computed exponent of the power law and with γ=1 regardless of the number of weak impurities. With the above results, we consider the situation with coexisting tunnel barriers and weak impurities. There arefollowing possible scenarios: (A) all barriers are in the wirebulk, being TLL-TLL junctions, as illustrated in Fig. 5(a), (B) all barriers are near the wire boundaries, being junc-tions between a TLL and a Fermi-liquid lead; see Fig. 5(b), (C) There are both bulk and boundary tunnel barriers. In eachcase, in addition to the barriers, the disorder potential is alsopresent. Fortunately, as in the data we observe γclose to 1 /2 or larger, we can restrict out treatment to considering up to 155304-4STRONG ELECTRON-ELECTRON INTERACTIONS OF A … PHYSICAL REVIEW B 99, 155304 (2019) two barriers in total. This assumption reduces scenario C to a single case, with one bulk and one boundary barrier. Wecan then exclude this case, as it would give different scalingbehavior in the high-bias and high-temperature regimes [ 50], in contrast to what we observe. Left with only scenarios A and B, the single barrier re- sistance can be directly generalized to multiple barriers ofcomparable strengths by replacing V→γV. To estimate the total wire resistance, we assume that the contributions fromtunnel barriers and weak disorder are additive. To treat them inthe same way, we use the RG method to determine the powerlaw of the current-voltage dependence for both. Due to thedistinct power laws for these sources, we can identify a singleterm which dominates the resistance (in the RG sense) for agiven strength of e-e interactions. In each scenario, keepingonly the dominant term, we conclude that the current-voltagerelation is described by Eq. ( 2), and obtain the expressions for αin terms of the intrinsic interaction parameters g candgs, and the ratio δv/vF, as given below. Finally, in order to convert the observed power αto the value of gc, we need the values of the SOI-induced parameters, δv/vF, and the departure of gsfrom 1. For parameters relevant to our experiment, we estimate δv/vF/lessorsimilar0.1. As 1 −gsscales with the same quantity, δv/vF[30], we find that the modifi- cation in gsis similarly small [ 22]. Our quantitative analysis presented in Appendix B1concludes that such small values have negligible influence on the relation between αandgc, meaning that to interpret the data of our device, we can use thezero-SOI expressions for α. 5In addition, when the wires are close to being depleted, which is the strong-interaction regimeof our primary interest, δv/v Fbecomes vanishingly small, making our approximations even more accurate. We notethat, even with these approximations, the conversion from theobserved αtog cis still complicated due to various types of resistance sources. In the following, we present the derivedexpression of αand its approximated form for scenarios A and B. We use the approximate form to extract the value of g c from the value of αobserved. B. Conversion of αtogcin scenario A (bulk tunnel barriers and weak disorder) In scenario A, the tunnel current is given by Eq. ( 2) with α given by (see Appendix B) αbulk=/parenleftbigg1 g/primec+1 g/primes/parenrightbigg/parenleftbig cos2θ+g2 0sin2θ/parenrightbig −2 (3a) ≈1 gc−1, (3b) where the approximation is valid for parameters relevant here. In the above, θis a small parameter characterizing the strength of the SOI, and the explicit forms of g/prime ν,g0, andθare given in 5This conclusion also means that, even though the TLL is spin-orbit coupled, the strength of the SOI cannot easily be extracted from the quantities that we measure; for that purpose, different quantities orexperiments would have to be pursued.Appendix B. On the other hand, the interpolation formula for weak impurities is given by Eq. ( 2), with γ=1 andαbeing αimp=2−cos2θ(g/prime c+g/prime s)−g2 0sin2θ/parenleftbigg1 g/primec+1 g/primes/parenrightbigg ,(4a) ≈1−gc, (4b) where the second line again stems from the approximation valid for our parameters. Importantly, for any repulsive inter-action g c/lessorequalslant1, the approximated value is bounded αimp/lessorequalslant1, allowing us to rule out the weak impurities as the sourceof the observed value α> 1i nt h el o w - V gregime. Further, for any gc<1, one has 1 /gc−1>1−gc, such that the resistance from the bulk tunnel barriers dominates that fromweak impurities. We therefore assign the observed power lawto bulk barriers and use α A=1 gc−1, (5) to extract the gcvalues from the data in Fig. 4.I nF i g . 5(c), we plot the extracted gcas a function of Vg. The lowest value gc=0.28 corresponds to very strong e-e interactions in a wire with low electron density. To further check the consistency of our procedure, we fit the extracted gcto the formula [ 59,61] gc=/bracketleftbigg 1+e2 π2ε¯hvFln/parenleftbiggD2 dw/parenrightbigg/bracketrightbigg−1/2 . (6) In this equation, derived by estimating the compressibility of the electron gas with the Coulomb interaction screened by aconducting plane (the top gate), Dis the distance between the wire and the top gate, dis the quantum well thickness, wis the wire width, and εis the dielectric constant. For our device, we have D=300 nm, d=7 nm, and ε=15.15ε 0 [62]. Using was a fitting parameter, we get the red curve in Fig.5(c), showing a good correspondence with gcfitted from the data. Further, the fitted value w=87 nm is consistent with the nominal width of 100 nm. Given w, we estimate the wire subbands level spacing¯h2 2m∗(2π w)2≈8.64 meV corresponding toEFatVg=−0.54 V.6This estimate suggests that our device is in the single-channel regime for Vg<−0.54 V, where an approximately constant value of γ/similarequal1/2 is seen in Fig. 4.7 Given all these cross-checks, we conclude that scenario A provides a consistent interpretation of the measured data. 6From the stacking structure of the wafer, we estimate the Fermi energy EF=1.13×102×[Vg−(−0.86)]2(meV) and the Fermi velocity vF=1.31×1013×[Vg−(−0.86)]w(m/s) (see Appendix A2). Here, Vgis in units of V , wis in units of m. 7The level spacing is large enough that we can ignore higher subbands at the temperatures of our measurements. In Ref. [ 77], the subband level spacing in a 100-nm-diameter InAs nanowire with isotropic cross-section as found to be approximately 8 meV , which issimilar to our estimate here. 155304-5YOSUKE SATO et al. PHYSICAL REVIEW B 99, 155304 (2019) TABLE I. Deduced interaction parameters ( gc) of one-dimensional systems as reported in experiments, including the present work (shaded row). The description of the entries is as follows. The first column gives the material(s) used in the listed references. The second column lists the extracted αparameter (if available) from the observed quantity given in the sixth column. Based on the resistance sources attributed in the references, the corresponding parameters αbulk,αendandαimpare given (we label those with unspecified sources with an unsubscripted α). The third and fourth columns list the interaction parameter gceither quoted from the references (in black) or deduced from the αvalue (in red) using Table IIbelow. The third column includes the gcvalue deduced from αbulkor those with unknown sources. The fourth column includes those from either αendorαimp.F o rαvalue with unknown resistance sources, we deduce gcvalues for all impurity types considered here. The extracted γvalue (if available) is given in the fifth column. The notations G,T,a n d Rdenote the conductance, temperature, and resistance, respectively. The abbreviations NW, CNT, VG, and CE stand for nanowire, carbon nanotube, V groove, and cleaved edge, respectively. Extracted α gcdeducedagcdeducedbObserved Material [Ref] from experiment from αbulk fromαendorαimp γ quantity MoSe NW [ 50] αbulk=0.61–6.6; αend=0.94–5.2 0.13−0.62 0 .09−0.35(αend) 0.25cG∝Tα InAs NW α=0.35–2.5 0.28−0.74 0.16−0.65(both) 0.5–1.0 Eq. ( 2) Multiwall CNT [ 41] αend=0.36–0.95 – 0.21–0.41 ( αend) 0.05–0.24 Gd InAs NWe[33]– 0.23f–– Gmax∝T1 g−2g Single-wall CNT [ 40] αbulk=1.4 0.26 – 0.6cG∝Tα Multiwall CNT [ 42] αbulk=1.24;αend=0.6h0.29 0 .29(αend)– G∝Tα Single-wall CNT [ 39] αend=0.6h– 0.29(αend) 0.46–0.63cG∝Tα NbSe 3NW [ 53] αbulk=2.15–2.2 0.31−0.32 –1 100–1 77cR∝T−α GaAs VG [ 37] – – 0.45–0.66 ( αimp)– δG1i GaAs/AlGaAs CE [ 63] αimp=0.5– 0.50(αimp)– δG1 GaAs VG [ 38] – 0.54–0.66 – – G∝T1 gc−1 Single-wall CNT [ 44] – 0.55 – – STM imaging GaAs/AlGaAs [ 36] – 0.6 – – /Delta1Rbsj GaAs/AlGaAs [ 35] – – 0.65–0.7 ( αimp)– δG1 GaAs/AlGaAs CE [ 64] – 0.66–0.82 – – /Gamma1i∝T1 gc−1k GaAs NWl[43] α=0.02–0.23 0.81−0.98 0 .77−0.98(αimp)m– G∝Tα Multiwall CNT [ 55] α=0.02–0.05 0.91−0.96 0 .90−0.96(αimp)n– G∝Tα aHere we use αbulk=1/gc−1 for NWs and αbulk=(1/gc−1)/2 for CNTs. Note that here we intentionally use the same notation gcfor both NWs and CNTs; see Table IIfor general expressions. bHere we use αend=(1/gc−1)/2a n d αimp=1−gcfor NWs, and αend=(1/gc−1)/4a n d αimp=(1−gc)/2 for CNTs; see Table IIfor details. cIn this reference, while the universal scaling behavior was observed and thus the γvalue was obtained, the value for αwas extracted from the power-law conductance rather than from the full current-voltage curve. dOn top of the universal scaling conductance, additional phenomenological parameters are required for their fitting. eIn this reference, the device forms a quantum dot. fThis reference reported a small value 0.38 for the effective interaction parameter g=(1/2gc+1/2gs)−1, which was attributed to gs<1 due to the SOI. In contrast, our work indicates that the effects of the SOI on the interaction parameter are negligible for relevant strength of the SOI. With the assumption gs=1, the value of gcin this reference becomes 0.23. We use the latter value for the table entry here. gThe notation Gmaxdenotes the conductance value of the Coulomb peak. hIn this reference, the tunnel conductance from a FL lead into the bulk of a TLL is also measured. It leads to a different power law, whose exponent is, however, not discussed in our work. iThe notation δG1denotes the conductance correction of the first conductance plateau. jThe notation /Delta1Rbsdenotes the backscattering resistance due to Bragg reflection. kThe notation /Gamma1idenotes the full width at half maximum of a Coulomb peak. lIn this reference, a core-shell nanowire was used. mAlternatively, assuming that disorder is absent within the wire, the gcvalue deduced from αendfollows as 0.68–0.96. nAlternatively, assuming that disorder is absent within the nanotube, the gcvalue deduced from αendfollows as 0.83–0.93. C. Conversion of αtogcin scenario B (boundary tunnel barriers and weak disorder) We now consider scenario B, in which the tunnel current through the boundary barriers is given by Eq. ( 2) with (see Appendix B) αend=1 2/parenleftbigg1 g/primec+1 g/primes/parenrightbigg/parenleftbig cos2θ+g2 0sin2θ/parenrightbig −1 (7a) ≈1 2gc−1 2. (7b)The contribution of weak impurities is the same as in scenario A, characterized by γ=1 and αimp≈1−gc.I n contrast to scenario A, weak impurities now become dominantover boundary tunnel barriers for α< 0.5. The observed parameter αis therefore related to the interaction parameter g cthrough αB≈/braceleftBigg1 2gc−1 2,forα/greaterorequalslant0.5 (barriers); 1−gc, forα/lessorequalslant0.5 (impurities).(8) 155304-6STRONG ELECTRON-ELECTRON INTERACTIONS OF A … PHYSICAL REVIEW B 99, 155304 (2019) The extracted gcis shown in Fig. 5(c) along with the fit to Eq. ( 6). Here, the gcvalues are even smaller than in scenario A and reach as low as 0.16. Next, we discuss how additional features of the extracted values for αandγfit with the assumptions of scenario B. Namely, as Vgdecreases the extracted γvalue decreases from unity and approaches 0.5 around Vg=−0.4V . A t t h e s a m e voltage, αsteps across 0.5, which is the transition point between the two expressions in Eq. ( 8). Such a feature can be well captured by scenario B, where γshould be unity when weak impurities dominate and 1 /2 when the boundary tunnel barriers dominate. The fit to Eq. ( 6) gives a value w= 47 nm and the associated subband level spacing of 29.6 meV ,indicating that the wire is in the single-channel regime for V g<−0.23 V, where the extracted γdrops below 1. Thus we conclude that scenario B is also in agreement with severalaspects of the data. D. Conclusion on the considered scenarios Both scenarios A and B are reasonable and capture salient features of the experimental data such that it is difficult toexclude one or the other. Scenario A gives somewhat betteragreement with Eq. ( 6); however, we do not deem a quantita- tive discrepancy to such a simple theory as very informative.Arguably, the weak point of scenario A is the observationthat each wire contains two tunnel barriers in its interior. 8 On the contrary, in scenario B, the tunnel barriers are formednear the wire ends and having two per wire is more natural.Nevertheless, we emphasize that regardless of which scenariois realized, both support our main conclusion that strong andgate-tunable e-e interactions are present in the wires. VII. COMPARISON TO E-E INTERACTION STRENGTHS REPORTED IN LITERATURE Before concluding, we compare the e-e interaction strength found here with previous experiments. To make sensiblecomparison of numerous references, we convert—wheneverpossible—to unified parameters, being g candαin the notation of this paper. We include one-dimensional systems regardlessof materials or measurement types and arrive at Table I, with entries ordered by the lowest value of g cachieved in a given reference. In general, systems with well-defined single chan-nels (e.g., single-wall carbon nanotubes) tend to have smallervalues of g c(stronger interactions9) due to suppression of scattering and stronger spatial confinement. A smaller mass 8If they originate in random disorder, there is no reason for such uniformity. On the other hand, one could argue that disorder averaged over many parallel wires might result in a scaling curve with someeffective number of tunnel barriers, being here close to 2. However, performing such type of fitting would require the adoption of some ad hoc assumptions about the statistical distributions of the strength and position of the tunnel barriers. We, therefore, do not follow this method of analysis. 9In the TLL model that we work with here, the constants gare the only parameters defining the strength of the electron-electron inter-actions. The value of the Fermi velocity, indicating the relation of the kinetic to interaction energies, would also need to be considered toof InAs compared to GaAs is also beneficial for a smaller gc, giving a larger level spacing and a well-defined single channel. VIII. CONCLUSIONS To conclude, we investigate quantum wires etched from an InAs quantum well and find that they possess strong e-einteractions. This finding is based on observation of universalscaling of the current as a function of the bias voltage andtemperature, from which the TLL interaction parameter canbe fitted. The fitting requires a theory for the conversion ofthe observed exponent αof the power-law dependence of the conductance to the e-e interaction strength parameter g cin the TLL Hamiltonian. The relation between αandgcdepends on the character and positions of the sources of resistance. For thecase of finite SOI, we provide the main results of such theoryhere. Its most important conclusion is that for strong e-einteractions, the effects of the SOI on the relation between α andg care negligible. This reassures us that the large values of αthat we observe are due to genuinely strong e-e interactions, and not, for example, an artifact of strong SOI. All together,our work demonstrates that an etched InAs quantum wire is apromising platform offering a quasi one-dimensional channelwith strong and gate-tunable e-e interactions. ACKNOWLEDGMENTS This work was partially supported by a Grant-in-Aid for Young Scientific Research (A) (Grant No. JP15H05407),a Grant-in-Aid for Scientific Research (B) (Grant No.JP18H01813), a Grant-in-Aid for Scientific Research (A)(Grant No. JP16H02204), a Grant-in-Aid for Scientific Re-search (S) (Grant No. JP26220710), Japan Society for thePromotion of Science Research Fellowship for Young Sci-entists (Grant No. JP18J14172), Grants-in-Aid for ScientificResearch on Innovative Area “Nano Spin Conversion Science’(Grants No. JP17H05177), a Grant-in-Aid for Scientific Re-search on Innovative Area “Topological Materials Science”(Grant No. JP16H00984) from MEXT, Japan Science andTechnology Agency CREST(Grant No. JPMJCR15N2), JSTPRESTO (Grant No. JPMJPR18L8), the ImPACT Programof Council for Science, and Technology and Innovation(Cabinet Office, Government of Japan). H.K. acknowledgessupport from RIKEN Incentive Research Projects and JSPSEarly-Career Scientists (Grant No. JP18K13486). Y .T. ac-knowledges support from Japan Society for the Promotionof Science through Program for Leading Graduate Schools(MERIT). The authors would like to thank Michihisa Ya-mamoto and Ivan V . Borzenets for support of experiment andmeasurement equipment, and Russell S. Deacon for proof-reading. judge the “strength” of the interactions in a broader context. Here, we do not consider such implications and when we discuss the e-e interaction strength we are solely making statements on the value ofconstants g. 155304-7YOSUKE SATO et al. PHYSICAL REVIEW B 99, 155304 (2019) FIG. 6. SEM images of quantum wires chemically etched out from the wafer grown along [001] direction. The wires are along (a) [1 ¯10], (b) [110], and (c) [010] direction, respectively. The scale bar in each figure indicates the length of 500 nm. From the brightness of the edges compared to dots surrounding the wires, each wire’s cross-section is inferred as a trapezium, a reverse trapezium and arectangle, respectively. APPENDIX A: EXPERIMENTAL DETAILS 1. Chemical etching and crystal axis To form our Hall-bar and quantum-wire devices, we use chemical etching by diluted H 2O2and H 2SO4.I ti sw e l l known that the rates of etching speed depend on the crystalaxis of the samples, consequently so do the edge shapes ofthe devices. We tested the etching process on a trial waferand confirmed such dependency by SEM. Figure 6shows SEM images of wires formed in (a) [1 ¯10], (b) [110], and (c) [010] direction. The SEM image reflects the slope ofedges, and therefore it enables us to identify cross-sectionsof these wires as a trapezium, a reverse trapezium and a rectangle, respectively. Based on these findings, we chooseto measure on wires formed along [010] direction so that wecan determine the width of the quantum wires more precisely,being the same as the width of their top-surface. 2. Estimation of the gate dependence of electronic density in wires From the stacking structure of the quantum well and 260- nm-thick cross-linked polymethyl methacrylate (PMMA), weestimate the top gate capacitance of 2 .71×10 −16F. We take the dielectric constants of PMMA and InAlAs to be 4and 13.59, respectively [ 65,66]. With this we estimate the Fermi energy E F=1.13×102×[Vg−(−0.86)]2(meV) and the carrier density n=8.31×105×[Vg−(−0.86)] (cm−1), respectively. Owing to the high mobility of the quantum well,longer uniform quantum wires can be realized compared toself-assembled nanowires [ 33]. 3. Current-bias data for various Vg Here, we show additional plots of current as a function of bias voltage, for various Vg(Fig. 7). All the sets of raw data in Figs. 7(a)–7(c) show good universal scaling, as evidenced in Figs. 7(d)–7(f). (a) (b) (c) (d) (e) (f) FIG. 7. Current as a function of bias voltage with various Vg. The first row is the raw data and the second is fitted and scaled data. We evaluate ( α,γ, I0)=(0.42,1.0,2.9×10−9A/K1.42), (0.51,0.61,2.2×10−9A/K1.51), and (2 .5,0.38,7.7×10−13 A/K3.5)f o r Vg= −0.2,−0.4a n d−0.8 V, respectively. 155304-8STRONG ELECTRON-ELECTRON INTERACTIONS OF A … PHYSICAL REVIEW B 99, 155304 (2019) APPENDIX B: THEORETICAL ANALYSIS In this Appendix, we present the main results of our theoretical analysis. We first discuss the effects of thespin-orbit interaction (SOI). We then provide formulaswhich we use to extract the interaction parameters in variousscenarios. In addition, we summarize the expressions of theexponent αfor various Tomonaga–Luttinger liquid (TLL) systems in existing literature. 1. Effects of SOI In this section, we discuss the effects of SOI on the power- law conductance and current-voltage curve of a TLL. The mo-tivation for this calculation is to examine how the SOI affectsthe observed parameter α. Namely, whereas the observed uni- versal scaling behavior in the current-voltage characteristicsunambiguously establishes the TLL behavior of our quantumwires, it remains to be clarified whether the rather large α value (implying small g c) in the low-density regime is not an illusion owing to the expected strong SOI in InAs. First of all, we remark that it is known that in the absence of a magnetic field, the SOI can be gauged away in strictlyone-dimensional system, thereby having no influence on ob-servable quantities [ 30,67,68]. In a quasi-one-dimensional geometry such as the etched quantum wires in our experiment,however, the interplay between the SOI and the transverseconfinement potential that defines a finite width of the quan-tum wire can modify the band structure, leading to differentvelocities for different branches in the spectrum [ 58,68]. It was shown that such an effect destroys the spin-chargeseparation [ 20,21], leading to a coupling between the spin and charge sectors in Eq. ( 1) in the main text. To investigate whether such a coupling alters the ob- served αvalue, we theoretically analyze its effects on the current-voltage characteristics. In the following, we first out-line our calculation based on the TLL formalism, and thengive our results on various types of resistance sources. Tobe specific, we consider impurities which are either strong orweak (acting as tunnel barriers or potential disorder), and forthe former type we further consider whether they locate in thebulk or at the boundaries (ends) of the wires. Before continuing, let us comment on possible origins of the tunnel barriers at the boundaries of the wire. We firstclarify that these “boundary barriers” may be located close to,but not exactly at the physical boundary between the wire anda lead. As discussed in Ref. [ 49], a barrier can be considered a boundary one if its distance from the wire boundary isshorter than the scales ¯ hv F/(kBT) and ¯ hvF/(eV). Since for our experiments these length scales are typically of orderO(100 nm)– O(1μm), observing boundary tunnel barriers is plausible. To proceed, we follow Refs. [ 20,21] and add the following term to Eq. ( 1)o ft h em a i nt e x t : H so=δv/integraldisplay¯hdx 2π{[∂xφc(x)][∂xθs(x)]+[∂xφs(x)][∂xθc(x)]}. (B1) It reflects the presence of SOI as a velocity difference δv between the two branches of the energy spectrum. Since thefull Hamiltonian H+H sois still quadratic in the bosonic fields, we can diagonalize it to get H/prime≡H+Hso=/summationdisplay ν=c,s/integraldisplay¯hdx 2π/braceleftbigg u/prime νg/prime ν[∂rθ/prime ν(x)]2+u/prime ν g/primeν[∂xφ/prime ν(x)]2/bracerightbigg , (B2a) where the modified TLL parameters and velocities are given by g/prime c=gcg0 gs/bracketleftBigg/parenleftbig g2 0+g2 s/parenrightbig +/parenleftbig g2 s−g2 0/parenrightbig cos(2θ)+g0g2 sδv vFsin(2θ) /parenleftbig g2 0+g2c/parenrightbig +/parenleftbig g2 0−g2c/parenrightbig cos(2θ)+g0g2cδv vFsin(2θ)/bracketrightBigg1/2 , (B2b) g/prime s=gsg0 gc/bracketleftBigg/parenleftbig g2 0+g2 c/parenrightbig +/parenleftbig g2 c−g2 0/parenrightbig cos(2θ)−g0g2 cδv vFsin(2θ) /parenleftbig g2 0+g2s/parenrightbig +/parenleftbig g2 0−g2s/parenrightbig cos(2θ)−g0g2sδv vFsin(2θ)/bracketrightBigg1/2 , (B2c) u/prime c=vF 2g0gcgs/bracketleftbigg/parenleftbig g2 0+g2 c/parenrightbig +/parenleftbig g2 0−g2 c/parenrightbig cos(2θ)+g0g2 cδv vFsin(2θ)/bracketrightbigg1/2 ×/bracketleftbigg/parenleftbig g2 0+g2 s/parenrightbig +/parenleftbig g2 s−g2 0/parenrightbig cos(2θ)+g0g2 sδv vFsin(2θ)/bracketrightbigg1/2 , (B2d) u/prime s=vF 2g0gcgs/bracketleftbigg/parenleftbig g2 0+g2 s/parenrightbig +/parenleftbig g2 0−g2 s/parenrightbig cos(2θ)−g0g2 sδv vFsin(2θ)/bracketrightbigg1/2 ×/bracketleftbigg/parenleftbig g2 0+g2 c/parenrightbig +/parenleftbig g2 c−g2 0/parenrightbig cos(2θ)−g0g2 cδv vFsin(2θ)/bracketrightbigg1/2 , (B2e) with the parameters g0=√ 2gcgs/radicalbig g2c+g2s, (B2f) θ=1 2arctan/parenleftBigg δv vF√ 2gcgs/radicalbig g2c+g2s g2s−g2c/parenrightBigg . (B2g) 155304-9YOSUKE SATO et al. PHYSICAL REVIEW B 99, 155304 (2019) In the absence of the SOI, we have ( δv,θ)→(0,0), and therefore ( g/prime c,g/prime s,u/prime c,u/prime s)→(gc,gs,uc,us). Using the model in Ref. [ 58], we have estimated that for the parameters relevant to our experiments, the value of δv/vFis at most around 0.1 and becomes vanishingly small when the system is close to beingdepleted. We remark that Ref. [ 20] obtains a similar estimate, ofδv/v F≈0.1–0.2. With the diagonalized Hamiltonian Eq. ( B2a), we are able to compute the tunnel current and the conductance of thequantum wires. Leaving the details for a separate publication[51], here we state our results and discuss their relevance to our experiment. As mentioned in the main text, we consider several sce- narios in which different types of resistance are present. Wefirst consider a single tunnel barrier in the bulk, modeled asa TLL-TLL junction, and compute the current through it. Forrelevant strength of SOI, we obtain Eq. ( 2) in the main text, with the parameters γ=1 andαreplaced by α bulk(g/prime c,g/prime s,θ)=/parenleftbigg1 g/primec+1 g/primes/parenrightbigg/parenleftbig cos2θ+g2 0sin2θ/parenrightbig −2, (B3) where the arguments ( g/prime c,g/prime s,θ) are themselves functions of (gc,gs,δv). The exponent Eq. ( B3)i sg i v e ni nE q .( 3)i n the main text. In the presence of several bulk barriers withcomparable resistances, the tunnel current through the wire isgiven by Eq. ( 2) with the same α bulkas Eq. ( B3) and with γ equal to the inverse of the barrier number. An alternative approach based on the renormalization- group tools [ 59,60] can be employed to compute the power- law conductance in the high-temperature ( kBT/greatermucheV) and high-bias ( eV/greatermuchkBT) limits. In the presence of a single bulk barrier, the power-law conductance can be summarized as Gbulk(T,V)∝Max( kBT,eV)αbulk, (B4) which is characterized by the same parameter αbulk. Similar to the tunnel current, the above formula can be generalizedfor several bulk barriers upon replacing V→γVwith 1 /γ being the barrier number. It can be shown that G bulk(T,V)i s consistent with the current-voltage characteristics [Eq. ( 2)i n the main text] in the high-temperature and high-bias limits,demonstrating the compatibility of the two approaches. We now analyze how the SOI influences the current- voltage characteristics through the parameter α bulk. It is useful to define an effective interaction parameter gc,eff, such that all the effects of δv/vFare incorporated into a single parameter. To be specific, we define gbulk c,effby the following relation: αbulk(g/prime c,g/prime s,θ)≡αbulk/parenleftbig gbulk c,eff,1,0/parenrightbig =1 gbulk c,eff−1. (B5) This leads to the following definition for gbulk c,eff, 1 gbulk c,eff≡/parenleftbigg1 g/primec+1 g/primes/parenrightbigg/parenleftbig cos2θ+g2 0sin2θ/parenrightbig −1, (B6) which describes the relation between the apparent interaction parameter gc,eff(corresponding to the extracted gcfrom our experimental observation) and the intrinsic parameters gc, gs, and δv. We remark that the exponent of the power-law conductance does not depend on the number of barriers, so theΔ Δ Δ Δgc,effbulk Δ Δ Δ Δ gcgc,effimp FIG. 8. Effective interaction parameter gc,effas a function of the actual interaction parameter gcfor various values of the ratio δv/vF. (Top) In the case of tunnel barriers, gbulk c,effis defined in Eq. ( B6). (Bottom) In the case of weak impurities, gimp c,effis defined in Eq. ( B9). definition of gc,effis the same for single and multiple barriers in the wire. To visualize the effects of the SOI on gbulk c,eff,w ep l o ti ta s a function of gcfor several values of δv/vF, as displayed in the top panel of Fig. 8. Note that we intentionally include exaggerated values of δv/vF/greaterorequalslant0.2 in the plot; a more realistic value δv/vF/lessorsimilar0.1 leads to barely visible changes. Further, while rather strong SOI does modify the parameter gbulk c,eff,w e find two important features relevant to our experiments. First,g bulk c,effincreases with an increasing strength of SOI. Therefore, the SOI cannot make the apparent interaction constant gbulk c,eff smaller than gc. Second, the SOI-induced increase of gbulk c,effis sizable in the weak- or moderate-interaction regime (0 .5/lessorequalslant gc/lessorequalslant1), but becomes negligible for the strong-interaction regime ( gc/lessorequalslant0.5). Thus, these features allow us to neglect the SOI when extracting the value of gcin the case of bulk barriers. We emphasize that such an approximation is moreaccurate (becoming practically exact) in the low- V g(small gc) regime, which is our primary interest. We now move on to the case of a tunnel barrier located around a boundary of the wire, which acts as a TLL-Fermiliquid (FL) junction. Again, we compute the tunnel currentfor generic temperatures and bias voltages, as well as thepower-law conductance G end∝Max( kBT,eV)αendin the high- temperature and high-bias limits. In this case, the tunnelcurrent and the power-law conductance are the same as thosefor a bulk barrier, except that the exponent reads α end=/parenleftbigg1 2g/primec+1 2g/primes/parenrightbigg/parenleftbig cos2θ+g2 0sin2θ/parenrightbig −1, (B7) 155304-10STRONG ELECTRON-ELECTRON INTERACTIONS OF A … PHYSICAL REVIEW B 99, 155304 (2019) from which we can define the same parameter gc,effas in Eq. ( B6). Again, in the presence of several barriers, we have the same exponent αendandV→γV. Similar to αbulk, a real- istic value of the ratio δv/vFcause only a minor modification of the αendvalues, justifying our procedure on the extraction of the gcvalue using the zero-SOI formula [Eq. ( 7b)i nt h e main text]. We now turn to the case of weak impurities, modeled as a backscattering potential. In this case, the calculation for atunnel barrier in Ref. [ 49] is not applicable. We therefore com- pute the conductance using the method of Refs. [ 59,60]. The corresponding exponent in the high-temperature and high-biaslimits is G imp(T,V)∝Max( kBT,eV)αimp, (B8a) αimp=2−cos2θ(g/prime c+g/prime s)−g2 0sin2θ/parenleftbigg1 g/primec+1 g/primes/parenrightbigg . (B8b) Since the conductance is similar to the tunnel barrier case (upon replacing the exponent αbulk,αend→αimp), the power- law conductance can mimic the scaling behavior observed inour experiment. We therefore take α→α impin Eq. ( 2) and treat it as an interpolation formula for the current-voltagecurve of a TLL in the presence of weak impurities. In thiscase, Vis the voltage difference across the entire wire, so γ=1 regardless of the number of impurities. Equation ( B8b) allows us to define the effective interaction parameter for theweak-impurity case, g imp c,eff≡cos2θ(g/prime c+g/prime s)+g2 0sin2θ/parenleftbigg1 g/primec+1 g/primes/parenrightbigg −1.(B9) In the bottom panel of Fig. 8,w ep l o t gimp c,effvsgc. We see that the value of gimp c,effis barely changed, so neither in this case the SOI leads to substantial effects on the extracted value of theinteraction parameter. In summary, for all the types of resistance sources we consider here, the effects of SOI on the extracted value of g c are negligible. Therefore the experimental values of gccan be extracted using equations without including the spin-orbiteffects, as given in the main text. 2. Extracting the interaction parameters in various scenarios In this section, we discuss how the theoretically developed results in the previous section are applied to our experimentaldata, in order to extract the interaction parameters of our quan-tum wires. Since SOI leads to negligible changes in the param-eterα, in the following we use its zero-SOI form. We express αas a function of g cconsidering the resistance contributions arising from up to two tunnel barriers and many weak impuri-ties. The former is suggested by the observed value of 1 /γ/lessorsimilar 2, and the latter is believed to be present since our wires arerelatively long on the scale of the bulk mean free path. We examine the following scenarios: (A) all barriers are in the bulk and (B) all are around the boundaries of thewire (between the TLL and FL). For both we also add theresistance contributions from weak impurities, and the contactresistance.In scenario A, weak impurities, both the tunnel barriers in the wire and the contact resistance R 0=h/2e2contribute to the total resistance, RA(T,V)=1 Gbulk(T,V)+1 Gimp(T,V)+R0 =1/γ/summationdisplay b=1Rb/bracketleftbigg/Delta1a Max( kBT,γeV)/bracketrightbiggαbulk +Ri/bracketleftbigg/Delta1a Max( kBT,eV)/bracketrightbiggαimp +R0. (B10) Here, 1 /γis the number of bulk barriers, indexed by b, each with a bare resistance scale Rb. Further, Riis the bare resis- tance scale of the disorder potential and /Delta1ais the effective bandwidth introduced in the bosonization scheme. Assumingthat the two bare resistances are of the same order O(R b)= O(Ri), the relative magnitude of the barrier and disorder contributions is determined by the exponents αbulkandαimp. Because under experimental conditions /Delta1ais much larger thankBTandeV, the term with the larger exponent dominates (also over the contact resistance). We therefore consider thecase where the resistance due to the bulk barriers dominates(that is, when α bulk/greaterorequalslantαimp), which leads to the following condition: αbulk/greaterorsimilarαimp⇔gc/lessorsimilar1, (B11) where the approximation arises from the assumptions of O(Rb)=O(Ri),gs=1, and negligible effects from SOI. Therefore, when the tunnel barriers are in the bulk, the contribution from the barriers dominates over the one fromweak impurities for any repulsive interaction. Consequently,in scenario A, the impurity-induced resistance is negligible,and we obtain the conductance G A(T,V)=1 RA(T,V) ≈Gbulk(T,V)∝Max( kBT,γeV)αbulk,(B12) resulting in the universal scaling formula in Eq. ( 2), with α= αbulk. In the main text, we therefore use Eq. ( 5) to extract the gcvalue. We now turn to scenario B, in which there are many weak impurities coexisting with tunnel barrier(s) around the wireend(s). We get R B(T,V)=1 Gend(T,V)+1 Gimp(T,V)+R0 =1/γ/summationdisplay b=1Rb/bracketleftbigg/Delta1a Max( kBT,γeV)/bracketrightbiggαend +Ri/bracketleftbigg/Delta1a Max( kBT,eV)/bracketrightbiggαimp +R0. (B13) The condition for the dominant contribution from the barriers follows as αend/greaterorsimilarαimp⇔gc/lessorsimilar1 2. (B14) As a result, there is a transition of the dominant resistance source when varying gcthrough the top gate voltage. The 155304-11YOSUKE SATO et al. PHYSICAL REVIEW B 99, 155304 (2019) dominant source changes from the tunnel barriers in the strong-interaction regime ( gc/lessorequalslant1/2) to the weak impurities in the weak-interaction regime ( gc/greaterorequalslant1/2). We get GB(T,V)∝/braceleftbigg Max(k BT,γeV)αend,forgc/lessorequalslant1/2, Max( kBT,eV)αimp, forgc/greaterorequalslant1/2,(B15) with the exponents αendandαimpgiven in Eqs. ( B7) and ( B8b), respectively. Interestingly our calculation also suggests that the param- eterαshould be larger than 0.5 for gc/lessorequalslant1/2, and smaller than 0.5 for gc/greaterorequalslant1/2. Therefore, in scenario B, we are able to identify the transition of the dominant resistance source basedon the observed αvalues. When α/greaterorequalslant0.5, the resistance is due to the boundary tunnel barrier(s) in the regime g c/lessorequalslant1/2, and therefore γ/similarequal0.5 suggests two barriers. On the other hand, whenα/lessorequalslant0.5, the resistance arises from the impurities, and γbecomes unity. Consequently the interpolation formula for scenario B is given by Eq. ( 2), with the parameters (α,γ)=/braceleftbigg (αend,γ), forα/greaterorequalslant0.5 (barriers); (αimp,1),forα/lessorequalslant0.5 (impurities).(B16) I nt h em a i nt e x tw eu s eE q .( 8) to extract the value for gcfrom the observed αvalues. Remarkably, upon increasing Vg,w e observe that αdecreases below 0.5 around the same V gvalue at which γchanges from /similarequal0.5 toward unity. This observation is consistent with scenario B, which provides an explanationfor the change in γ. Finally we comment on a third scenario. Namely, one may consider having both types of barriers, bulk and boundary.However, in contrast to our experimental observations thisscenario would give conductance with different power lawsin the high-bias and high-temperature limits, as discussed inRef. [ 50]. We therefore conclude that this scenario is not relevant to our observations.In summary, the combined experimental and theoretical results for the considered scenarios indicate that our extractedvalue of g c=0.16–0.28 is not an artifact of the strong SOI in the InAs wires. This conclusion holds regardless of whetherscenario A or B is realized. 3. Summary of the power-law conductance in various TLLs In this section, we give a summary of the parameter αfor various TLL systems. This allows us to compare the interac-tion parameters in various one-dimensional systems listed inTable Iin the main text. Specifically, the summary includes a spinless TLL, a spinful TLL (without SOI), and a spinful TLLwith valley degrees of freedom (that is, a carbon nanotube). In Table II, we list the exponent α bulk/αend/αimpfor various TLLs subject to tunnel barriers and many weak impurities. Weremark that the exponents α bulk/αendare the same for single and multiple tunnel barriers in the wire. The first columngives the system type. The second column corresponds tothe scenario in which the tunnel barriers (isolated, strongimpurities) are located in the bulk of a wire (that is, TLL-TLLjunctions), with the references given in the third column.The fourth column corresponds to the tunnel barriers locatedaround the boundaries of the wire (that is, TLL-FL junctions),with the corresponding references in the fifth column. Thesixth column gives the exponent α impfor various TLLs subject to many weak impurities, with the references in the seventhcolumn. In contrast to α bulk/αendin the tunneling regime, the value of αimpis bounded. In the table, we give the allowed ranges, assuming that the electron-electron interactions onlyact on one sector. For example, for a spinful TLL, we assumethat the spin sector is noninteracting, g s=1. For repulsive interactions, the interaction parameter of the charge sector isin the range g c∈[0,1], leading to a bound αimp∈[0,1]. TABLE II. Exponent αof the power-law conductance in various TLLs subject to tunnel barriers and many weak impurities (treated as weak potential disorder). The first column lists the system types. The second (fourth) column corresponds to the exponent αbulk(αend)f o ra TLL-TLL (TLL-FL) junction. The sixth column lists the exponents αimpcorresponding to many weak impurities. The references corresponding to the entries are given in the third, fifth, and seventh columns. The eighth column lists the allowed ranges for αimp, assuming that only one of the sectors is interacting (with the interaction parameters of the other sectors set to unity). In the entries, the notation gdenotes the interaction parameter in a spinless TLL, while the notation gc/sdenotes the interaction parameter of the charge /spin sectors, respectively, in a spinful TLL (no SOI). For a spinful TLL with the valley degrees of freedom (for example, a carbon nanotube), the notation gνPdenotes the sectors of the charge /spin degrees of freedom (with ν∈{c,s}, respectively), and the symmetric /antisymmetric combination of the valleys (with P∈{S,A}, respectively). The quantities after the approximation symbols ( ≈) indicate the values of the exponents with gs,gcA,gsS,gsAset to unity. Additional references are given in the footnotes below the table. Bulk barrier Boundary barrier Weak impurities Allowed rangea TLL type αbulk Refs. αend Refs. αimp Refs. for αimp spinless 2 g−1−2[ 60,69] g−1−1[ 60]2 −2g [59,60,70][ 0 ,2] spinful g−1 c+g−1 s−2[ 59]1 2(g−1 c+g−1 s)−1[ 71]2 −gc−gs [59,70]b[0,1] ≈g−1 c−1 ≈1 2(g−1 c−1) ≈1−gc spinful with1 2ginv sum−2[ 73]c 1 4ginv sum−1[ 39,49,75]2 −1 2gsum [75,76]c[0,1/2] two valleysd≈1 2(g−1 cS−1) ≈1 4(g−1 cS−1) ≈1 2(1−gcS) aAssuming that g,gc,gcS∈[0,1]. bSee also the calculation in the presence of the multibands or multiple channels [ 72]. cSee also the calculation for multiwall nanotubes or ropes of single-wall nanotubes [ 74]. dFor this entry, we define gsum=gcS+gcA+gsS+gsAandginv sum=g−1 cS+g−1 cA+g−1 sS+g−1 sA. 155304-12STRONG ELECTRON-ELECTRON INTERACTIONS OF A … PHYSICAL REVIEW B 99, 155304 (2019) [1] S.-i. Tomonaga, Prog. Theor. Phys. 5,544(1950 ). [2] J. M. Luttinger, J. Math. Phys. 4,1154 (1963 ). [3] K. A. Matveev, P h y s .R e v .B 70,245319 (2004 ). [4] A. Furusaki and N. Nagaosa, Phys. 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PhysRevB.87.235402.pdf

PHYSICAL REVIEW B 87, 235402 (2013) Interplay between symmetry and spin-orbit coupling on graphene nanoribbons Hern ´an Santos,1M. C. Mu ˜noz,2M. P. L ´opez-Sancho,2and Leonor Chico2 1Departamento de F ´ısica Fundamental, Universidad Nacional de Educaci ´on a Distancia, Apartado 60141, E-28080 Madrid, Spain 2Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cient ´ıficas, Sor Juana In ´es de la Cruz 3, 28049 Madrid, Spain (Received 26 February 2013; published 3 June 2013) We study the electronic structure of chiral and achiral graphene nanoribbons with symmetric edges, including curvature and spin-orbit effects. Curved ribbons show spin-split bands, whereas flat ribbons presentspin-degenerate bands. We show that this effect is due to the breaking of spatial inversion symmetry in curvedgraphene nanoribbons, while flat ribbons with symmetric edges possess an inversion center, regardless of theirhaving chiral or achiral edges. We find an enhanced edge-edge coupling and a substantial gap in narrow chiralnanoribbons, which is not present in zigzag ribbons of similar width. We attribute these size effects to the mixingof the sublattices imposed by the edge geometry, yielding a behavior of chiral ribbons that is distinct from thosewith pure zigzag edges. DOI: 10.1103/PhysRevB.87.235402 PACS number(s): 71 .20.Tx, 73 .22.−f, 71.70.Ej I. INTRODUCTION The crucial interplay between structure and electronic properties of graphene is among the most attractive fea-tures of its derived nanomaterials. Both carbon nanotubes(CNTs) and graphene nanoribbons (GNRs) show promising characteristics for spintronic devices. Recent progress in experimental techniques has allowed for the fabrication ofgraphene nanostripes by using electron-beam lithography 1 or by unrolling CNTs.2These ribbons could be used in electronic devices, such as field-effect transistors,3opening new perspectives for nanoelectronics. The presence of localized edge states in GNRs, theoretically predicted,4,5and experimentally proved,6confers them distinct properties. GNRs have attracted a great amount of theoreticalwork but mostly focused on high-symmetry zigzag andarmchair achiral ribbons. Zigzag ribbons have edges stateswith different spin polarizations, while armchair nanoribbonsdo not have edge states. The edges of minimal 7,8chiral ribbons can be considered as a mixture of armchair and zigzag edges, thus having edge-localized states stemming from their zigzag part. Although the evolution of the nanoribbon band structureupon the change of chirality has been recently addressed, 9–11 these systems have been nonetheless much less studied and many aspects remain to be explored. The seminal work of Kane and Mele12triggered the interest on new quantum phases of matter and on the spin-orbit coupling (SOC) effect, which, although known to be small in graphene, gives rise to important physics. Inparticular, the quantum spin Hall (QSH) phase has been widelyaddressed. 13,14 Curvature is known to enhance spin-orbit interaction; its importance in SOC effects has been theoretically investigatedfor the honeycomb lattice, especially for CNTs, 15–20and experimentally confirmed.21Hybridization of πandσorbitals, decoupled in flat graphene, is enhanced by curvature and thusSOC effects are bolstered. 18The interplay between curvature and SOC in GNRs has been mostly focused in achiral ribbons,with highly symmetric zigzag and armchair edges. 22–24For zigzag GNRs, dispersionless edge bands in the flat geometrywere found to become dispersive because of SOC effects.Bothπandσedge states remain spin filtered in the curved geometry, still localized at the boundaries of the ribbon, albeitwith an in-plane spin component and a localization lengthlarger than for the flat case. 23Recent experiments2,25,26on chiral GNRs obtained by unzipping CNTs show a reminiscentcurvature. Scanning tunneling spectroscopy measurementsrevealed the presence of one-dimensional edge states, withan energy splitting dependent on the width of the ribbon. 26 By comparison with calculations employing a π-band model with a Hubbard term the width dependence of the edge stategap was interpreted as a consequence of spin-polarized edgestates. 11,26Hence, the study of curvature effects in GNRs is relevant from the experimental and theoretical viewpoint. In this paper we address the study of this ampler class of ribbons with chiral edges, focusing on the differences of SOCeffects in flat and curved nanoribbons. We summarize our mainresults as follows: (i) We find that the bands of both chiral and achiral flat ribbons with symmetric edges are at least twofold spindegenerate due to spatial inversion symmetry. Curving theribbons breaks this symmetry, thus yielding spin-split bandsexcept for the time-reversal protected special symmetry k points. (ii) We find a gap in all chiral ribbons, despite the fact that they have a zigzag edge component. Boundary conditions inchiral ribbons mix both sublattices at each edge. This enhancesedge-edge coupling, which results in a substantial gap withoutinvoking electron interactions. (iii) Curvature augments spin-orbit effects in GNRs, yield- ing a larger splitting in the spin-split bands. In fact, curvaturemay induce metallicity in ribbons which have a gap in theplanar form. (iv) The spatial distribution of edge states depends on curvature and chirality. While zigzag ribbons are known tohave spin-filtered states at the edges, in narrow ( ≈40˚A) chiral ribbons edge states can have nonzero density at both edges, dueto the edge-edge coupling. This size effect is more evident inribbons with chiral angle close to 30 o, i.e., that of the armchair edge, for which the sublattice mixing is stronger. This paper is outlined as follows. Section IIdescribes the structure and geometry of the ribbons studied. Section IIIgives 235402-1 1098-0121/2013/87(23)/235402(7) ©2013 American Physical SocietySANTOS, MU ˜NOZ, L ´OPEZ-SANCHO, AND CHICO PHYSICAL REVIEW B 87, 235402 (2013) some symmetry considerations concerning the role of spatial inversion in flat and curved general ribbons. Section IVcon- tains the description of the model Hamiltonian and calculationmethod. Section Vpresents the results, including spin-orbit interaction, for ribbons of different widths in flat and curvedgeometries. Finally, in Sec. VIwe discuss our results and final conclusions are drawn. II. GEOMETRY We focus on chiral ribbons with symmetric minimal edges,7 obtained from unrolling chiral carbon nanotubes. The ribbon isthus characterized by the edge vector T=na 1+ma2, where a1and a2are the primitive vectors of graphene, and the width vector W. The widths considered are therefore given by an integer multiple of H, defined as the smallest graphene lattice vector perpendicular to T, as depicted in Fig. 1. For a given T,His uniquely determined up to a global ±1 factor. As W=MH, we will denote the ribbons by M(n,m), where M states the width of the ribbon and ( n,m) indicates the minimal edge. All minimal edges can be decomposed in a zigzag andan armchair part, 7,8T=nZTZ+nATA, with TZ=a1and TA=a1+a2. T=T Z+2T A a1 a2 WH FIG. 1. (Color online) Geometry of the 2(3,2) GNR highlighted in dark gray on a graphene sheet, showing its translation vector T= TZ+2TAand its width vector W=2H,w h e r e H=−7a1+1a2. The unit cells spanned by Tand HorWare indicated with dotted lines.The chirality of the ribbon is specified by the chiral angle θ between the translation vector ( n,m) which defines the edges and the zigzag direction (1,0). We take into account different curvatures in the transversal direction for a given flat GNR, with no stretching allowed.Curvature is denoted by the angle ϕspanned by the ribbon from its curvature center, ranging from zero for a flat ribbon toa value of 2 π, which corresponds to a nanotube with cut bonds along its length. The degree of curvature is controlled by theangle and by the diameter of the cylindrical configuration. III. SYMMETRY CONSIDERATIONS Carbon nanotubes are classified as achiral and chiral according to their having a symmorphic or non-symmorphicsymmetry group, respectively. This means that chiral tubespossess a spiral symmetry, so that there are two enantiomersfor each chirality, while achiral tubes are equal to their mirrorimage; i.e., achiral tubes present space inversion symmetrywhile chiral tubes do not. 27,28Graphene nanoribbons, like their siblings carbon nanotubes, are customarily classified asachiral and chiral according to their edge shapes. In GNRsthis classification is related to the crystallographic orientationof the boundaries: ribbons with zigzag and armchair edges(derived from armchair and zigzag CNTs, respectively) arecalled achiral, and those obtained from chiral tubes are calledchiral GNRs. However, these so-called chiral ribbons withsymmetric edges do have an inversion center. Upon bendingthe ribbon the inversion symmetry is lost. This feature is crucialwhen considering SOC effects. Figure 2shows an example of a flat (left panel) and curved (right panel) unit cell of the (3,2) ribbon, the latter with ϕ=π. A symmetry center is indicated in the planar geometry. No such inversion center exists in thecurved nanoribbon. Notice that a flat nanoribbon with different edges lacks inversion symmetry. This situation is of experimental interest:most likely, actual ribbons will not have symmetric edges.Such asymmetry can be achieved either by adding or removingatoms to an originally edge-symmetric ribbon, or by alteringthe bond lengths in one of the edges. Dissimilar bond lengths OA A’B’ B FIG. 2. (Color online) Schematic geometry of a flat (left) and curved (right) 1(3,2) GNR. The curvature angle is ϕ=π. Two pairs of equivalent atoms under spatial inversion symmetry are highlighted, and the inversion center is marked as O. 235402-2INTERPLAY BETWEEN SYMMETRY AND SPIN-ORBIT ... PHYSICAL REVIEW B 87, 235402 (2013) may arise as a result of a different functionalization on the two edges of the nanoribbon. IV . THEORETICAL MODEL AND COMPUTATIONAL METHODS We calculate the band structure of graphene within the empirical tight-binding (ETB) approximation. Although theπ-orbital tight-binding model is known to capture the low- energy physics of graphene, since we are interested in SOCeffects we consider here an orthogonal four-orbital 2 s,2p x, 2py,2pzbasis set. This allows for the inclusion of the intrinsic SO terms within the conventional on-site approach. The matrixHamiltonian is built following the Slater-Koster formalismup to nearest-neighbor hopping. We use the parametrizationobtained by Tom ´anek-Louie for graphite. 29The expression of the one-electron ETB Hamiltonian is H0=/summationdisplay i,α,s/epsilon1α+/summationdisplay /angbracketlefti,j/angbracketright,β,stα,β ijcα+ i,scβ j,s+H.c., (1) where /epsilon1αrepresents the atomic energy of the orbital α,/angbracketlefti,j/angbracketright stands for all the atomic sites of the unit cell of the GNR,andc α+ i,sandcα i,sare the creation and annihilation operators, respectively, of one electron at site i, orbital α, and spin s. We focus on neutral graphene; thus, no doping effects areaddressed. SOC effects are included by adding an atomiclike term H SOto theH0Hamiltonian. Assuming that the most important contribution of the crystal potential to the spin-orbit couplingis close to the cores, the H SOcontribution takes the form HSO=/summationdisplay i¯h 4m2c21 ridVi driL·S=λL·S, where the spherical symmetry of the atomic potential Vihas been assumed and riis the radial coordinate with origin at theiatom. Lstands for the orbital angular momentum of the electron, and Sis the spin operator. The parameter λis a renormalized atomic SOC constant, which depends on theorbital angular momentum. Notice that the H SOterms only couple porbitals in the same atom. Considering the spin parts of the wave functions, the Hamiltonian matrix has 8 Na×8Na elements, Nabeing the number of the C atoms in the unit cell of the GNR and 8 corresponding to the four orbitals per spinof the sp 3basis set. The total Hamiltonian in the 2 ×2 block spinor structure is given by H=/parenleftbiggH0+λLzλ(Lx−iLy) λ(Lx+iLy)H0−λLz/parenrightbigg . (2) The total Hamiltonian Hincorporates both spin-conservation and spin-flip terms. The spin-conserving diagonal terms actas an effective Zeeman field producing gaps at the Kand K /primepoints of the graphene Brillouin zone (BZ), with opposite signs.18By exact diagonalization of the matrix Hwe obtain the band structure of GNRs. As explained in the previous section,the curved geometry is obtained by isotropically bending theribbon in the width direction, without changing the distancealong its length. Thus, no bond stretching is included along theribbon axis. We do not consider reconstruction or relaxationof the edges or passivation of the dangling bonds.The value of the SOC constant for C-based materials is not well established and it is still under debate. Some theoreticalestimates gave λ=1μeV for graphene, 30,31much smaller than the atomic SO coupling, 8 meV . Taking into account therole of dorbitals, this value rises to λ=25μeV . 32Accurate measurements of SOC are difficult to perform in graphenebecause external effects such as substrates, electric fields, orimpurities may mask its value. However, recent experimentsin CNT quantum dots have reported spin-orbit splittings sub-stantially higher than those theoretically predicted: Kuemmethet al. 21give a maximum splitting of 0.37 meV in a CNT of diameter 5 nm. One possible explanation for this energysplitting is that a higher value of λshould be considered; as indicated by Izumida et al. , 19those measurements are compatible with λ=14 meV . More recently, Steele et al.33 have presented evidence of large spin-orbit coupling in CNTs, up to 3.4 meV , an order of magnitude larger than previouslymeasured and the largest theoretical estimates. Furthermore,transport experiments report spin-relaxation times in graphene1000 times lower than predicted. 34This is compatible with a larger value of the SOC coupling than those given by previoustheoretical estimates. 30–32Although small, its effects in GNRs could have important consequences when considering thespin degree of freedom, as has been experimentally shownin CNTs. 21,33For the sake of clarity, we choose for the figures a spin-orbit interaction parameter λ=0.2e V . The spin-orbit contribution to the Hamiltonian, HSO,i s linear on λ. We have checked that, for small values of this parameter, such as the one employed here and those of physicalrelevance, the eigenvalues of the full Hamiltonian Hare basically a linear correction to those without the SOC term, H 0. Therefore, the spin-orbit splittings are proportional to λand the results presented in this work can be scaled accordingly. V. R E S U LT S We have calculated the electronic properties of many different chiral GNRs, varying their width and curvature.All calculations have been performed with the four-orbitalparametrization explained in Sec. IV. Therefore, they show some differences with respect to the widely used one-orbitalapproach. 11We present here the band structures for three representative chiral ribbons [ M(7,1),M(5,2), and M(3,2)] and, for comparison, some zigzag GNRs of different widths.TheM(7,1) ribbon has a chiral angle θ=6.58 o, close to the zigzag direction; the M(5,2) has θ=16.02o; and the M(3,2) GNR has θ=23.41o, closer to the armchair direction. The M(7,1) and M(3,2) GNRs have the same unit cell with 76 atoms, but with different orientation; i.e., the Hand Tvectors are interchanged. The M(5,2) ribbon, in the intermediate chirality range, has a unit cell with 52 atoms. The (5,2) edge hasthree armchair ( A) and two zigzag ( Z) units, so for the infinite system there are two possible arrangements of the armchairand zigzag units with the same edge vector. We choose theone with all zigzag units together, the ZZZAA . While the edge states in a semi-infinite graphene sheet or in very wideribbons are the same irrespective of the sequence, for narrowribbons some differences in the band structure of distinct edgearrangements may arise. 235402-3SANTOS, MU ˜NOZ, L ´OPEZ-SANCHO, AND CHICO PHYSICAL REVIEW B 87, 235402 (2013) -1.0 -0.5 0.5 1.0 Energy (eV) -0.100.1 -0.100.1 X XXX(a) (b) FIG. 3. (Color online) Electronic structure of the 1(7,1) GNR calculated with (black dots) and without (red and gray dots) SOC, considering (a) the four-orbital sp3basis set in a planar geometry and (b) the same sp3basis but for a curved geometry with ϕ=2π. Zooms of the edge bands are included close to each panel. A. SOC and inversion symmetry 1. Curvature effects In systems with time-reversal and spatial inversion sym- metry, spin-orbit interaction does not lift spin degeneracy,according to Kramers’ theorem. If spatial inversion symmetryis not present, the states are spin split except in the kpoints protected by time-reversal invariance. In symmetric-edgeGNRs, spatial inversion symmetry is broken by curving theribbon, as indicated in Fig. 2. The importance of the broken inversion symmetry is shown in Fig. 3, where the electronic structures of the 1(7,1) ribbon calculated with and withoutSOC terms are depicted for (a) the planar configuration and(b) the curved one with ϕ=2π. This angle corresponds to a maximally curved geometry without overlapping the edgesof the ribbon; it is equivalent to an open carbon nanotubewith circumference equal to the width of the GNR. SOCeffects are clear: all degeneracies, including spin, are liftedin the curved ribbon (b), while in the flat system (a) the bandsremain spin degenerated. The only noticeable difference inthe flat case is a small shift in the bands at the /Gamma1point, as can be observed in the zoom of Fig. 3. Otherwise, the effect of SOC is negligible. However, a large splitting is found inthe curved ribbons, greater for the conduction bands. Thisis due to the interaction of edge states with higher-energybands [see zoom in Fig. 3(b)], which in fact are completely hybridized due to curvature. Throughout most of the BZ, oneof the spin-split bands has an upward shift in energy, whereasthe other band undergoes a downshift. Thus, the bands withoutSOC mostly lie between the spin-orbit-split bands, as can beseen in Fig. 3(b), especially in the zoom. Notice that GNRs can be made metallic because of the curvature, as seen in Fig. 3: the gap observed in the flat 1(7,1) ribbon (panel a) is still present at /Gamma1in the curved ( ϕ=2π) geometry, but in this latter case the ribbon is metallic due tothe band bending produced by curvature-induced hybridization(panel b). FIG. 4. (Color online) Electronic structure of the 1(3,2) GNR with different curvatures: (a) ϕ=0.5π,( b )ϕ=π,a n d( c ) ϕ=1.8π.T h e bands with SOC are shown in black; bands without SOC are in red(gray). Although the spin is no longer a good quantum number, the expectation value of the spin operator shows that in theflat geometry the total spin is normal to the ribbon. Curvatureprovokes the appearance of a small component in the in-planedirection which increases with curvature. Figure 4illustrates the effect of curvature. It presents the band structure of the 1(3,2) ribbon for three bending angles,namely, (a) 0.5 π, (b) 1.2 π, and (c) 1.8 π, with (black dots) and without (red and gray dots) SOC. The curved geometriesare shown above each band panel. In a wide M(3,2) ribbon there are four edge bands at 0 eV extending from 2 3/Gamma1Xto X.8In the case depicted in Fig. 4a large gap opens between the occupied and unoccupied edge bands due to size effects,which we discuss later on. There is a general increase ofthe band splitting with growing curvature, as expected, dueto the increment of the σ-πhybridization produced in the curved ribbons, analogous to the effect predicted 16,18and experimentally measured in CNTs.21Moreover, band splitting in GNRs is anisotropic, band and kdependent, as also found in CNTs. 2. Edge modification As discussed above, planar GNRs with dissimilar edges also lack inversion symmetry. Different edges can be achievedeither by adding or removing atoms in a symmetric edgeribbon, or by altering the bond lengths at one edge of asymmetric GNR. We have explored the magnitude of thiseffect, calculating the changes in the band structure of a planar1(5,2) modified ribbon. We have considered two types ofmodifications: two atoms of one edge have been removed,and the bond length of the zigzag atoms at one edge has beenchanged 10%. With both modifications SOC breaks the spindegeneracy of the band structure; however, the splitting is twoorders of magnitude smaller than that achieved by the effect 235402-4INTERPLAY BETWEEN SYMMETRY AND SPIN-ORBIT ... PHYSICAL REVIEW B 87, 235402 (2013) FIG. 5. (Color online) Band structures for flat ribbons calculated with the sp3basis set, including the SOC term for the following GNRs: (a) 1(5,2), (b) 2(5,2), (c) 4(5,2), (d) 2(1,0), (e) 4(1,0), and (f) 8(1,0). Insets: Zooms of the bands near EF. of curvature. Thus, in what follows, we concentrate on the curvature mechanism as a means to break inversion symmetry. B. SOC effects and width of the ribbon In order to explore the interplay of SOC and the width of the ribbon we consider first flat GNRs, since in curvedgeometries they could be masked by other effects, such ashybridization. We have performed calculations for differentchiralities, verifying that there is a gap in planar chiral nanorib-bons that decreases with increasing width. Figures 5(a)–5(c) demonstrate this effect for the M(5,2) GNRs. The 1(5,2) ribbon has a substantial gap, around 0.4 eV , while the flatbands around E Ffor the 4(5,2) ribbon in panel c are clearly identified as edge bands for their dispersionless character nearthe BZ boundary X. Nonetheless, the gap can be discerned inthe inset of Fig. 5(c), as stated above. Comparison with high-symmetry zigzag ribbons of similar widths shows a striking difference. Figure 5demonstrates that zigzag ribbons have a negligible energy gap, around 0.1 meVfor the narrowest case depicted [Fig. 5(d),W=8.52˚A], while for the 1(5,2) ribbon of similar width (8.87 ˚A) the gap is around 0.4 eV . The gaps between edge bands in chiral ribbons are due to the stronger coupling between edge states localized at eachboundary. In zigzag ribbons the atoms at opposite edges belongto different sublattices, while in chiral ribbons boundaryconditions at each edge mix the two sublattices, coupling thestates located at the two edges. This results in a band gap dueto quantum size effects, without invoking electron-electroninteractions. -1.0 -0.5 0.5 1.0 Energy (eV) -0.0200.02 -0.00500.005 X XXX(a) (b) FIG. 6. (Color online) Band structures for curved ribbons calcu- lated with the sp3basis set, for the following GNRs: (a) 8(1,0) and (b) 4(5,2). The bands with SOC are shown in black; bands without SOC are in red (gray). Insets: Zooms of the bands near EF. In flat ribbons, SOC induces a tiny shift of the energy bands; in zigzag ribbons, it turns the flat edge bands into dispersiveones. As inversion symmetry is preserved, all bands remainspin degenerated. Now, we include curvature in order to enhance SOC effects and break spin degeneracy, as discussed in the previous section.For narrow chiral ribbons the gap due to quantum size effectsis rather large, as can be seen in Figs. 5(a) and5(b),s ot h e effect of SOC is the aforementioned energy shift and, mostimportantly, the spin splitting of the bands. We focus on thewidest ribbons, namely, 4(5,2) and the 8(1,0), with a smallerquantum size gap, and consider the same curvature radius forboth ribbons, R=6.274 ˚A, which yields an angle ϕ=1.8π and 1.73π, respectively. For these cases, the bands closer to E Fare strongly deformed. These happen to be edge states, so their behavior gives rise to a more interesting situation than inthe large gap ribbons, as is illustrated in Fig. 6. Figure 6(a) shows the zigzag case, with a noticeable dispersion in the edge bands. The zoom shows that the bandswith SOC are spin split, with a crossing point slightly shiftedwith respect to that of the bands without SOC. Figure 6(b) shows the chiral 4(5,2) GNR; here, besides the energy shift andspin splitting of the SOC bands, there is a slight displacementof the Fermi wave vector, which is no longer at X. Althoughsmall, SOC effects have important consequences for thetransport properties of curved GNRs: spin-filtered channelsarise due to the interplay of SOC and curvature, and for widerribbons even chiral GNRs present these spin-filtered channelsin the low-energy region. C. Chirality and spatial distribution of edge states Edge states are among the most important features of GNRs.9,10It is interesting to explore how chirality affects the behavior of these states. For the sake of simplicity, wefocus on flat geometries; generalization to curved geometries isstraightforward. In Fig. 7we present the electronic densities of edge states belonging to flat ribbons with different chiralities.The two states chosen correspond to the edge bands closerto the Fermi level, near the high-symmetry point to which 235402-5SANTOS, MU ˜NOZ, L ´OPEZ-SANCHO, AND CHICO PHYSICAL REVIEW B 87, 235402 (2013) (a) (b) (c) (d) (e) FIG. 7. (Color online) Probability densities corresponding to two states of the edge bands closer to EF, near a high-symmetry point, for several flat ribbons. The spin polarization is plotted in white for spin up and blue (gray) for spin down. (a) Zigzag 10(1,0) ribbon,k=0.95/Gamma1X. (b) Chiral 4(7,1) ribbon, k=0.10/Gamma1X. (c) Chiral 1(7,2) ribbon, k=0.95/Gamma1X. (d) Chiral 4(5,2) ribbon, k=0.10/Gamma1X. (e) Chiral 2(3,2) ribbon, k=0.95/Gamma1X. the edge bands converge for large widths. Opposite spin polarizations are indicated with distinct colors; since we aredealing with flat ribbons, the spin direction is perpendicular tothe plane of the ribbon. We choose a kvalue slightly displaced from the symmetry point (either /Gamma1or the BZ boundary X) in order to avoid degeneracy due to time-reversal invariance andribbons of similar width in order to compare size effects. Inpure zigzag ribbons, edge states are spin filtered: each edgestate has a well-defined spin orientation and it is located atone edge. This is illustrated in Fig. 7(a), which shows the square modulus of the wave functions for two edge statescorresponding to a 10(1,0) zigzag nanoribbon, of width equalto 42.61 ˚A. These two edge states of zigzag ribbons live in opposite sublattices, and their probability density is mostlyconfined to the atoms with coordination number 2, whichconstitute the geometrical edge. Figure 7(b) shows the density for two edge states of a 4(7,1) ribbon of width 42.89 ˚A, close in chirality to the zigzag case. Similarly to the zigzag ribbon,the density of each edge is mostly located in one sublattice. Itis not homogeneously distributed, being mostly in atoms withcoordination number 2, although there is some appreciableweight in inner atoms close to the edge. For the 1(7,2) GNR (of width 34.88 ˚A and θ=12.21 o) [Fig. 7(c)], which has a larger chirality angle, the wave function extends more into inneratoms, especially close to the armchair part of the boundary. For chiralities closer to the armchair, the states have a nonzero density at both edges: panel d shows the two statesclose to E Ffor a 4(5,2) ribbon of width 35.48 ˚A. These two states live at both edges simultaneously, with more inner atomswith nonzero density. This is more dramatic in panel e, whichdepicts the edge states for a 2(3,2) ribbon of width 37.40 ˚A. As its chirality is closer to the armchair case, edge states have agreater penetration into the inner part of the ribbon. In order tolocalize these states and obtain the quantum spin Hall phase,a larger value of the SOC constant ( λ≈4 eV) is needed. This behavior has been found for armchair ribbons within the Diracmodel 31and the ETB model.23The SOC strength required increases with the chiral angle and decreases with the width ofthe ribbon. In curved ribbons, edge states keep their localized character, even for the maximum curvature. Besides the spin splitting(Fig. 6), there are two main differences: the spatial localization length is larger for the curved ribbon than for the flat case, andthe spin direction changes from the direction normal to theribbon surface, acquiring a component in the ribbon plane, aspreviously reported. 23,24 VI. DISCUSSION AND CONCLUSIONS Our results show that the relation between the QSH edge states of graphene nanoribbons and both the crystallographicorientation of the edges and the curvature of the GNR allowsus to control the spin and spatial localization of the ribbonedge states. Consequently, this may allow us to achieve anefficient electrical control of spin currents and spin densitiesin GNRs. Taking into account both spin and valley degeneracy,Bloch states in graphene are fourfold degenerate. SOC splitsthem into two Kramers doublets and, as Kane and Melepredicted, 12this turns graphene into a topologically nontrivial material. In zigzag edge ribbons, it has already been shownthat at each edge spin-up and spin-down electrons movein opposite directions. 20,23Since backscattering in a given edge requires the reversal of spin it cannot be inducedby spin-independent scatterers. Accordingly, edge states inzigzag GNRs are topologically protected and hence theconductance of the edge states is quantized. However, in flatchiral-edge GNRs of finite width and for realistic values ofthe SOC strength, there is a non-zero probability of havingelectrons moving in opposite directions with the same spinpolarization at a given edge (see Fig. 7). Therefore, intra-edge backscattering may occur, which affects the quantization ofthe conductance. In chiral-edge GNRs, spin reversal canbe induced even by nonmagnetic disorder and thus edgestates do not present a robust conductance quantization. Theappearance of backscattering does depend on the chiralityangle, increasing for angles approaching the armchair limit.On the other hand, curvature breaks the inversion symmetryof the ribbons and Bloch states are spin split. Electrons withthe same spin and opposite propagating directions in a givenedge have different energies. As a result, backscattering is notallowed, and in curved chiral-edge GNRs edge states behave 235402-6INTERPLAY BETWEEN SYMMETRY AND SPIN-ORBIT ... PHYSICAL REVIEW B 87, 235402 (2013) as robust quantum channels. Therefore, these chiral ribbons present a magnetomechanical effect: upon curving the ribbon,the spin channels are split in energy, thus allowing for spatiallyseparated spin currents. Despite the weakness of SOC in these carbon systems—on the scale of a few meV—their effects in curved graphene andnanotubes are not negligible, due to the coupling of πandσ bands in curved geometries. Thus, although small, the effectsdiscussed in this work may be physically relevant, and since ingraphene the position of the Fermi level can be adjusted withexternal gate voltages the control of spin currents in GNR-derived devices could be possible. In summary, we have shown that, in the presence of spin-orbit interaction, curvature breaks spin degeneracy ingraphene ribbons. Flat nanoribbons with symmetric edges,either chiral or achiral, have spin-degenerate bands. This is dueto the existence of spatial inversion symmetry in flat ribbons,which is broken in the curved cases. Furthermore, spin-orbit splitting is enhanced in curved ribbons due to the hybridization of the bands, absent in flatsamples. Other mechanisms to break inversion symmetry, suchas edge modification, are much less efficient to remove spindegeneracy.We have also explored finite-size effects in GNRs. We find that narrow chiral ribbons present a sizable gap, despite theirhaving a zigzag edge component, whereas in pure zigzagGNRs of similar width the gap is negligible. We relate thisbehavior to the boundary conditions in chiral edges, whichmix the two sublattices at each edge. Finally, we have studiedthe chirality dependence on the spatial localization of edgestates. In narrow chiral ribbons, edge states have a nonzerodensity at both edges simultaneously, due to edge coupling.Due to the sublattice mixing produced by the chiral boundaryconditions, edge states have a larger penetration than thoseof achiral ribbons. For wider curved ribbons, they behave asspin-filtered states, being localized at one edge. ACKNOWLEDGMENTS H.S. gratefully acknowledges helpful discussions with J. E. Alvarellos. 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PhysRevB.79.235111.pdf

Galvanomagnetic properties and noise in a barely metallic film of V 2O3 Clara Grygiel *and Alain Pautrat† Laboratoire CRISMAT, UMR 6508 du CNRS, ENSICAEN et Université de Caen, 6 Bd. Maréchal Juin, 14050 Caen, France Pierre Rodière Institut Néel, CNRS-UJF, BP 166, 38042 Grenoble Cedex 9, France /H20849Received 22 December 2008; revised manuscript received 14 May 2009; published 8 June 2009 /H20850 We have measured the magnetotransport properties of a strained metallic V 2O3thin film. Most of the properties are similar to V 2O3single crystals that have been submitted to a large pressure. In addition, the resistance noise analysis indicates that conductivity fluctuations are freezing out at T/H1101510 K. Examination of a range of measurements leads to the conclusion that spins-configuration fluctuations dominate in the low-temperature regime. DOI: 10.1103/PhysRevB.79.235111 PACS number /H20849s/H20850: 73.43.Qt, 73.50.Td I. INTRODUCTION V2O3single crystals undergo a first-order metal-insulator /H20849MI/H20850transition at T/H11015160 K with antiferromagnetic ordering of vanadium spins associated to a structural transition.1In vanadium deficient samples, the metallic phase was shown tobe stabilized down to T/H1101510 K, 2where a spin-density wave condenses as indicated by a clear increase in the resistivity.3 At roughly the same temperature, the Hall resistance shows a maximum, which was first attributed to the skew scatteringin an ordered magnetic state. 4Stoichiometric V 2O3behaves differently. When submitted to a high hydrostatic pressure P/H1135026 kbar, its longitudinal resistivity decreases monotoni- cally with a decrease in temperature and any gap opening canbe observed. This indicates a fully suppressed metal to theinsulator transition down to the lowest temperature withoutany trace of magnetic ordering. 2On the other hand, the Hall resistance still shows a maximum at T/H1101510 K,5which may be explained by dominant spin fluctuations6or strong elec- tronic correlations7in the low-temperature range. It has been reported that the spin susceptibility of metallic V 2O3under pressure exhibits also an anomaly at T/H1101510 K which has been associated with either low-energy magnetic excitationsor a kind of pseudogap formation. 8There is a need for complementary experiments to clarify the origin of thisanomaly. When V 2O3is epitaxially grown over a substrate, clamp- ing of the film can prevent the structural transition.9As a consequence, a metallic state which mimics the highly pres-sured state is observed. Macroscopically, a distribution ofstrains in the thin film can lead to phase coexistence. Theresistivity exhibits thermal hysteresis due to metastable statesand is a nonergodic quantity in parts of the phase diagram,e.g., the quantitative analysis of the measured values isambiguous. 10However, using a microbridge as a local probe, it is possible to isolate a pure metallic state without anythermal hysteresis and no trace of the MI transition. In thispaper, we discuss the transport properties of a strained V 2O3 thin film in the so-called barely metallic side. We have per-formed conventional galvanomagnetic measurements andnoise measurements. We will show that most of the proper-ties are comparable to those observed in single crystalswhich were submitted to high pressure. The small size of thesystem makes the statistical averaging less effective and, thus, makes the extraction of information from noise mea-surements possible. As we will discuss below, the analysis ofconductivity fluctuations provide different insights into thelow-temperature ground state of metallic V 2O3. II. EXPERIMENTAL The V 2O3thin film was grown on a substrate of /H208490001 /H20850- oriented sapphire using the pulsed laser deposition from aV 2O5target. The details of the growth conditions and some structural and microstructural characterization have been re-ported previously. 9In particular, the evolution of unit-cell parameters was shown to be inconsistent with oxygennonstoichiometry. 9The film studied here is a 230-Å-thick sample, with a low rms roughness of 0.47 nm /H20849averaged over 3/H110033/H9262m2/H20850. It was patterned using UV photolithography and argon-ion etching to form a bridge of length /H11003width=200 /H11003100/H9262m2. Silver contact pads were connected using aluminum-silicon wires /H20849or copper wires for measurements at temperature below 2 K /H20850which were attached by the ultra- sonic bonding for four-probe and Hall-effect measurements.Measurements for T/H113492 K were performed in a He-3 cry- ostat. Transport measurements were taken at temperaturesbetween 2 and 400 K in a physical properties measurementsystem /H20849PPMS /H20850from Quantum Design. All magnetotransport measurements have been performed with the magnetic fieldalong the caxis, i.e., perpendicular to the film. For noise measurements, we used a home-made sample holder and ex-ternal electronics to acquire the resistance-time series andspectrum noise /H20849the acquisition part consists of home- assembled low noise preamplifiers, a spectrum analyzer SR-760 and a dynamic signal analyzer NI-4551 /H20850. The final res- olution is dominated by the low noise current supply /H20849current noise: 0.2 nA /Hz 1/2/H20850which leads to an equivalent noise of 100 nV /Hz1/2when the sample with R/H110150.5 k/H9024is biased at low temperature. This is higher than both the equivalentJohnson thermal noise and the preamplifier noise. The lengthdependence of the noise measured between different arms ofthe bridge allows to conclude that contact noise is not impor-tant here. 11PHYSICAL REVIEW B 79, 235111 /H208492009 /H20850 1098-0121/2009/79 /H2084923/H20850/235111 /H208495/H20850 ©2009 The American Physical Society 235111-1III. RESULTS A. Magnetotransport properties Figure 1shows the resistivity of the metallic microbridge as a function of the temperature. The resistivity is thermallyreversible and Ohmic for applied currents up to at least 1 mAand from T=400 down to 2 K. Both the functional form of the resistivity and the absolute value were close to what ismeasured in crystals under high pressure. 2In particular, the resistivity tends to curve downward at high temperature buttakes a value above the three-dimensional Ioffe-Regel limitfor metallic conductivity. This is often observed in so-calledbad metals, and a large /H9267sat/H110151m/H9024cm has been discussed as a possible consequence of interacting electrons.12For a correlated metal, a lot of different terms can play a role inelectronic scattering; but at high temperature a major contri-bution from phonons can be still expected. The electron-phonon contribution to the resistivity is usually described bythe Bloch-Grüneisen formula, /H9267ph=cT5//H92586/H20885 0/H9258/T x5/H20851/H20849ex−1/H20850/H208491−e−x/H20850/H20852−1dx, /H208491/H20850 where /H9258is the Debye temperature and cis a constant, describing the electron-phonon interaction. The high-temperature data were adjusted to the phenomenological par-allel resistor with a saturation resistivity acting as a shunt, 13 /H9267−1=/H9267sat−1+/H20849/H92670+/H9267ph/H20850−1. /H208492/H20850 A good fit is obtained for the temperature range 130 K /H11349T/H11349400 K. We found an approximative residual resistivity /H92670/H11015320/H9262/H9024cm which will be refined below, a saturation resistivity /H9267sat/H110151m/H9024cm, and a Debye temperature /H9258 =300 K. The deduced Debye temperature was close to half the value of bulk samples /H20849/H9258bulk/H11015560 K /H20850, which could be qualitatively explained by a size effect14and/or by the stress in the film. For T/H11349140–150 K, an additional contribution is needed to describe the resistivity. In particular, at the lowesttemperatures, where the electron-phonon scattering wasclearly negligible /H20849T/H1134920 K /H20850, a quadratic temperature de- pendence of the resistivity was observed in the form of /H9267T2 =AT2+/H92670, with A=0.073 /H9262/H9024cm K−2and a more precise value of the residual resistivity /H92670/H11015316/H9262/H9024cm /H20849Fig. 2/H20850. This Fermi-liquid-like dependence has been reported forsingle crystals of V 2O3submitted to a high pressure15,16and, more recently, in strained thin films.17This dependence is typical of electron-electron scattering18,19which is the gen- eral interpretation in V 2O3. Spin fluctuations can also lead to a quadratic temperature dependence at low temperature.6We note also that the functional form of /H9267/H20849T/H20850over the full tem- perature range was not very different from what was ex-pected taken in spin-fluctuations theories. 6,20The T2varia- tion in the resistivity did not extend down to the lowesttemperature, as can be seen in Fig. 2. Here, an excess of resistivity was observed when T/H1134910 K. After carefully sub- tracting /H9267T2from the measured resistivity at low temperature /H20849T/H1134920 K /H20850, a logarithmic dependence of the resistivity was observed from T/H1134910 K down to 2 K /H20849Fig. 3/H20850. This is the limiting temperature in our PPMS cryostat. Measurements inanother film of the same thickness at the lowest temperatureindicated that this logarithmic dependence extends down to400 mK. Kondo effect /H20849scattering by magnetic impurities /H20850or weak localization corrections can be made to explain that thescattering increases at low temperatures with such character-istics. The T 1/2-dependent conductivity which is characteris-FIG. 1. /H20849Color online /H20850Resistivity of the metallic V 2O3micro- bridge as function of temperature. Also shown is a fit of the high-temperature part with the formula /H208492/H20850/H20849 /H9258=300 K, /H9267sat=1 m/H9024cm/H20850. FIG. 2. /H20849Color online /H20850Resistivity of the metallic V 2O3micro- bridge as a function of temperature for T/H11349100 K and T2fit of the low-temperature part. Shown in the inset is a zoom of /H9267as function ofT2. The temperature variation in the resistivity is quadratic for 10 K/H11349T/H1134920 K. FIG. 3. Variation in the resistivity corrected from the quadratic temperature dependence and from the residual resistance. A loga-rithm dependence can be observed for 2 K /H11349T/H1134910 K. Shown in the inset are the same characteristics in a film of the same thicknessand measured down to the lowest temperature /H20849400 mK /H20850.GRYGIEL, PAUTRAT, AND RODIÈRE PHYSICAL REVIEW B 79, 235111 /H208492009 /H20850 235111-2tic of electron-electron scattering in a disordered medium has been proposed as an explanation for the low-temperature up-turn in pressurized V 2O3resistivity.2,16This, however, does not provide a satisfactory fit for our sample. In addition to the excess of resistivity, a negative longitu- dinal magnetoresistance /H20849MR /H20850appears for T/H11349130 K /H20849inset of Fig. 4/H20850. Its magnetic field dependence is quadratic from B=0 to 7 T and for all temperatures down to 2 K. The origin of the quadratic dependence of a negative magnetoresistanceis uncertain. 21One possible mechanism is the two- dimensional weak localization as far as B/H11270/H6036 //H208494eL/H92722/H20850, where L/H9272is the coherence length.22Quadratic negative magnetore- sistance is also a common feature of Kondo-like magneticmetals, where the magnetic field tends to suppress the spinfluctuations. Remarkably, the absolute value of the MRshows a notable increase for T/H1134910 K, clearly evidenced by tracing the Kohler’s plot. The idea behind Kohler’s rule isthat in conventional isotropic metals, the magnitude of MR isfixed by a single-scattering time /H9270/H20849T/H20850/H110081//H9267/H20849T/H20850and implies that/H9004/H9267/H20849B/H20850//H9267/H208490/H20850=F/H20849B/H9270/H20850, where Fis a function dependent on the details of the electronic structure. Due to its quite largegenerality, the Kohler’s rule is expected to apply in a Fermiliquid also and has been used as a probe of non-Fermi-liquidground states. In the limit of the carrier density being tem-perature independent, 23Kohler’s rule written in its simplest form is /H9004/H9267/H20849B/H20850//H9267/H208490/H20850=F/H20851B//H9267/H208490/H20850/H20852. In Fig. 4, it can be seen that /H9004/H9267/H20849B/H20850//H9267/H208490/H20850plotted as a function of /H20851B//H9267/H208490/H20850/H208522for different temperatures gives a single curve, i.e., Kohler’s rule is ful-filled when T/H1135010 K. The strong increase in the MR below 10 K was thus associated with a departure from Kohler’srule. The Hall resistance also presented a maximum in the10–20 K range /H20849Fig. 5/H20850, as previously reported both for single crystals and for thin films. 4,5,17Since it was observed in the absence of a clear magnetic transition, a dominanteffect of spin fluctuation has been proposed. 5,6Such a maxi- mum can also be expected from strong correlation effects.7 In summary of this part, we have confirmed that the magne-totransport properties of metallic V 2O3thin film present similarities with V 2O3crystals under pressure such as the T2 variation in the resistivity and the maximum of the Hall ef- fect at T/H1101510K. In addition, we have shown that the 10 K anomaly reflects also in the MR and in a Kondo-like increasein the resistivity at low temperature. Both electronic correla- tions and spin fluctuations can be used as qualitative expla-nations. B. Noise measurements as a complementary tool Benefiting from the small size of our sample, we have studied conductivity fluctuations by measuring the resistancenoise. We have measured a set of resistance-time series atvarious temperatures. The temperature was allowed to stabi-lize during sample cooling before each measurement series.The power spectral density /H20849or noise spectrum /H20850S RR/H20849f/H20850is then calculated. In ordinary metals, the electronic noise generallycomes from the motion of atomic impurities or defects andthe temperature dependence is dependent by the underlyingmechanisms, e.g., thermal activation or eventually tunnelingat low temperature. At high temperature T/H11022100 K, we have observed resistance switches insensitive to magnetic fieldduring long periods, which recall the hydrogen hoppingnoise common in several metals. 24The fractional change in resistance was between 2 /H1100310−5–10−4, which is large for individual hopping and implied a collective motion. Notethat this switching noise strongly evolved over a long period,as it was not observed 1 month after the first measurements,showing that the involved defects were mostly nonequilib-rium defects left in the film after deposition. In addition,reproducible 1 /fnoise can be observed and this noise will be discussed now. Shown in Fig. 6is typical noise spectra measured on our film. They are very close to 1 /fover the whole temperature range. The temperature dependence of this normalized noiseexhibits a nonmonotonic variation /H20849Fig. 7/H20850, with a plateau in the range 10 K /H11021T/H1102120 K. The sudden decrease in the noise at T/H1101510 K is remarkable. To give 1 order of magni- tude of the noise level, we use the phenomenological Hoogeparameter which is, for 1 /fnoise, /H9253=/H20851SRR/H20849f/H20850f/H20852/R2ncVwhere ncis the carriers density and Vis the /H20849probed /H20850sample volume.25For homogeneous and pure metals, typical values are in the range /H9253/H1101510−2–10−3. Taking ne=5/H110031022cm−3 deduced from our Hall-effect measurements, we found /H9253 /H1101510−1at room temperature, which is large, but not unusu- ally large for a metalliclike thin film and is consistent with amoderately higher concentration of defects in films than inbulk. Noise in metals can be usually explained with theFIG. 4. /H20849Color online /H20850A Kohler’s plot of the magnetoresistance /H9004/H9267/H20849B/H20850//H9267/H208490/H20850as a function of /H20851B//H9267/H208490/H20850/H208522for temperature from 100 to 5 K. The dashed line is guide for the eyes. Shown in the inset is themagnetoresistance at 7 T as a function of the temperature.FIG. 5. /H20849Color online /H20850Hall resistance as a function of the tem- perature, showing the maximum at low temperature.GALV ANOMAGNETIC PROPERTIES AND NOISE IN A … PHYSICAL REVIEW B 79, 235111 /H208492009 /H20850 235111-3Duttah-Dimon-Horn /H20849DDH /H20850analysis, which assumes that the electronic fluctuations come from smeared kinetics of defectswith a distribution of activation energies. 26When the defect relaxation is activated and typical energy is in the eV range, the noise generally increases with temperature. However,there is at least one report of a noise maximum at low tem-perature which, however, follows the DDH framework. It hasbeen interpreted as evidence of low-energy excitations in anoxide. 27In the DDH modeling, there is a direct relation be- tween the noise spectral exponent /H9251and the temperature dependence of the noise in the form /H9251/H20849T/H20850=1 −/H20849/H11509lnSRR //H11509lnT−1/H20850/ln/H208492/H9266f/H92700/H20850, where /H92700−1is the attempt frequency. We have checked this relation and found that un-realistically large /H92700values are needed to approach the very small variation in /H9251/H20849T/H20850. In parallel, the temperature of the noise maximum/plateau does not show any frequency depen-dence. Actually, pure 1 /fnoise should follow S RR/H11008Tin the DDH model, as we observed at high temperature /H20849Fig. 7/H20850, and a maximum is not expected. We conclude that the low-temperature noise maximum does not come from thermallyactivated excitations. It can neither be explained by tempera-ture fluctuations which would lead to a too weak temperaturedependence of the noise. In disordered metals at low tem-perature, universal conductance fluctuations /H20849UCFs /H20850can in principle take place. 28In the UCF model, the noise grows at low temperature due to the enhancement of the effective cou-pling between disorder configurations and resistancechanges. UCF can explain why S RRrises when the tempera- ture decreases but not the apparent freezing of conductivityfluctuations for T/H1134910 K. Since we observe a decrease in the resistance noise when a magnetic field is applied in the10–30 K range /H20849a factor of 0.6 at B=7 T and at T=15 K, see the inset of Fig. 7/H20850, a spin-coupling origin of this noisecan be suggested. The blocking of magnetic fluctuations at low temperature with non-Arrhenius slowing down makes usthink of spin glasses. 29Note, however, that some spin glasses do not show a decrease but a growing of noise at low tem-perature due to the UCF coupling. 30To probe, directly, spins freezing with resistance noise, the sensitivity of the resis-tance to spins should be independent of the temperature andthe coupling should be local. 30This appears to be the case with our sample. Note that in some intermetallic compounds,similar properties, such as a quadratic temperature depen-dence of the resistivity close to a peak in the Hall coefficient,have been interpreted as a spin-glass type of freezing at lowtemperature. 31,32A precise determination of the magnetic dis- order is required to better understand its role in the low-temperature galvanomagnetic properties of V 2O3. To conclude, we have measured a large array of transport properties in a V 2O3film in the barely metallic side. We found strong similarities with properties of single crystalssubmitted to a high pressure and some different features suchas a low-temperature noise maximum coupled to anomalousmagnetotransport properties. A central role of spin-configuration fluctuations, which freeze at low temperature,has been proposed. Future experiments directly sensitive tomagnetic order would be extremely complementary. Noiseexperiments on single crystals, if feasible, would be—also—very interesting to compare with our measurements in films. ACKNOWLEDGMENTS A.P. thanks W. Prellier for comments on the paper and C.G. thanks S. McMitchell.FIG. 7. Normalized resistance noise as a function of the loga- rithm of the temperature /H20849f=0.1 Hz /H20850. Note the sharp decrease at T/H1101510 K. The horizontal dotted line corresponds to the resolution of the measurement. Shown in the inset is the normalized noise as afunction of the magnetic field for T=15 K /H20849f=0.1 Hz /H20850.FIG. 6. /H20849Color online /H208501/f/H9251noise spectra obtained from the resistance-time series at T=5, 10, 30, 25, 20, and 15 K from the bottom to the top. Shown in the inset is the value of the exponent /H9251 as a function of the temperature.GRYGIEL, PAUTRAT, AND RODIÈRE PHYSICAL REVIEW B 79, 235111 /H208492009 /H20850 235111-4*Present address: Department of Chemistry, The University of Liv- erpool, Liverpool L69 7ZD, United Kingdom. †alain.pautrat@ensicaen.fr 1D. B. Mcwhan, T. M. Rice, and J. P. Remeika, Phys. Rev. Lett. 23, 1384 /H208491969 /H20850. 2S. A. Carter, T. F. Rosenbaum, J. M. Honig, and J. Spalek, Phys. Rev. Lett. 67, 3440 /H208491991 /H20850. 3W. Bao, C. Broholm, S. A. Carter, T. F. Rosenbaum, G. Aeppli, S. F. Trevino, P. Metcalf, J. M. Honig, and J. Spalek, Phys. Rev.Lett. 71, 766 /H208491993 /H20850. 4T. F. Rosenbaum, A. Husmann, S. A. Carter, and J. M. Honig, Phys. Rev. B 57, R13997 /H208491998 /H20850. 5S. Klimm, M. Herz, R. Horny, G. Obermeier, M. Klemm, and S. Horn, J. Magn. Magn. Mater. 226-230 , 216 /H208492001 /H20850. 6O. Narikiyo and K. Miyake, J. Phys. Soc. Jpn. 64, 2730 /H208491995 /H20850. 7J. Merino and R. H. McKenzie, Phys. Rev. B 61, 7996 /H208492000 /H20850. 8M. Takigawa, E. T. Ahrens, and Y . Ueda, Phys. Rev. Lett. 76, 283 /H208491996 /H20850. 9C. Grygiel, A. Pautrat, W. Prellier, B. Mercey, W. C. Sheets, and L. Mechin, J. Phys.: Condens. Matter 20, 472205 /H208492008 /H20850. 10C. Grygiel, A. Pautrat, W. Prellier, and B. Mercey, EPL 84, 47003 /H208492008 /H20850. 11A. C. Marley, M. B. Weissman, R. L. Jacobsen, and G. Mo- zurkewich, Phys. Rev. B 44, 8353 /H208491991 /H20850. 12O. Gunnarsson, M. Calandra, and J. E. Han, Rev. Mod. Phys. 75, 1085 /H208492003 /H20850. 13H. Wiesmann, M. Gurvitch, H. Lutz, A. Ghosh, B. Schwartz, M. Strongin, P. B. Allen, and J. W. Halley, Phys. Rev. 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PhysRevB.91.195432.pdf

PHYSICAL REVIEW B 91, 195432 (2015) Double-island Coulomb blockade in (Ga,Mn)As nanoconstrictions S. Geißler,1S. Pfaller,2,*M. Utz,1D. Bougeard,1A. Donarini,2M. Grifoni,2and D. Weiss1 1Institute for Experimental Physics, University of Regensburg, 93040 Regensburg, Germany 2Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany (Received 24 February 2015; revised manuscript received 27 April 2015; published 22 May 2015) We report on a systematic study of the Coulomb-blockade effects in nanofabricated narrow constrictions in thin (Ga,Mn)As films. Different low-temperature transport regimes have been observed for decreasing constrictionsizes: the Ohmic, the single-electron tunneling (SET), and a completely insulating regime. In the SET, complexstability diagrams with nested Coulomb diamonds and anomalous conductance suppression in the vicinity ofcharge degeneracy points have been observed. We rationalize these observations in the SET with a doubleferromagnetic island model coupled to ferromagnetic leads. Its transport characteristics are analyzed in terms ofa modified orthodox theory of Coulomb blockade which takes into account the energy dependence of the densityof states in the metallic islands. DOI: 10.1103/PhysRevB.91.195432 PACS number(s): 72 .25.−b,73.23.Hk,73.63.Kv I. INTRODUCTION (Ga,Mn)As, discovered by Ohno et al. [1] nearly two decades ago, is by now the best studied ferromagneticsemiconductor [ 2–4]. An interesting aspect of this material are large magnetoresistance effects which were discovered innanofabricated narrow constrictions in thin (Ga,Mn)As films[5–10]. While the effects were initially interpreted in terms of the tunneling magnetoresistance (TMR) [ 5] and tunneling anisotropic magnetoresistance (TAMR) [ 6], it was proven later that the interplay with Coulomb blockade is also relevant innarrow (Ga,Mn)As constrictions [ 10,11]. The origin of this Coulomb-blockade anisotropic magnetoresistance (CBAMR)effect is substantial nanoscale fluctuations in the hole density[2] forming puddles of high hole density separated by low conducting regions. (Ga,Mn)As is known to be a stronglydisordered material. Its hole density is close to the metal-insulator transition. Little variations in the hole density causedby local potential fluctuations can lead to an intrinsic structureconsisting of metallic islands separated by insulating areas.It was shown that the magnetoresistance depends, in thepresence of Coulomb blockade, not only on an applied gatevoltage but can also be tuned by changing the direction ofthe applied magnetic field [ 10,11]. The latter results from the dependence of the Fermi energy on changes in the magnetization δ/vectorMand was modeled phenomenologically by Wunderlich et al. [10]. If transport occurs through a narrow nanoconstriction, single-electron tunneling (SET) betweenislands of high carrier density becomes relevant. Thus, it isnot surprising that the bias and temperature dependence of themagnetoresistance for different magnetization directions couldbe fitted with a model for granular metals in which metallicislands are separated by insulating regions [ 11]. Because of the nanoscale size of the involved “metallic” islands, theCoulomb-charging energy Uis the dominating energy for transport across the nanoconstriction at low temperatures andsmall-bias voltages V b. Since usually more than one island is involved in transport, Coulomb-blockade diamonds, wherethe resistance is plotted as a function of both bias and gate *sebastian1.pfaller@ur.devoltage, revealed a very complex and irregular pattern. Up tonow, a detailed experimental and theoretical analysis of theCoulomb-blockade effects in (Ga,Mn)As nanoconstrictions inthe single-electron-transistor regime is still missing. The aim of this work is a systematic study of the Coulomb- blockade effects in nanofabricated narrow constrictions inthin (Ga,Mn)As films. By means of a two-step electronbeam lithography (EBL) technique, we fabricated well-definednanoconstrictions (NC) of different sizes. Depending onchannel width and length, for a specific material, differentlow-temperature transport regimes could be observed, namely,the Ohmic regime, the single-electron tunneling regime,and a completely insulating regime. In the SET regime,complex stability diagrams with nested Coulomb diamondsand anomalous conductance suppression in the vicinity ofcharge degeneracy points have been observed. In order tounderstand these observations we propose, for a specificnanoconstriction, a model consisting of two ferromagneticislands coupled to ferromagnetic leads. We study its transportcharacteristics within a modified orthodox theory of Coulombblockade which takes into account the energy dependence ofthe density of states in the metallic islands. The paper is structured as follows: Section IIexplains the fabrication process of the samples. In Sec. III, the measurement setup is presented. The next section, Sec. IV, summarizes the results of the measurements, giving a first interpretation interms of a double-island structure within a classical orthodoxmodel of Coulomb blockade [ 12–17]. In Sec. V, we present the details of the ferromagnetic double-island model, study itstransport characteristics, and make a direct comparison withthe experimental results in Sec. VI A . Conclusions are drawn in Sec. VII. II. SAMPLE FABRICATION Our NC devices were fabricated in a top-down approach starting from a (Ga,Mn)As layer with a Mn content of approx-imately 5%. The (Ga,Mn)As layer we used had a thickness of15 nm and was grown by low-temperature molecular beamepitaxy on top of a (001)-GaAs substrate. In contrast tothe experiments of Schlapps et al. [11], we used as-grown (Ga,Mn)As samples without additional annealing before the 1098-0121/2015/91(19)/195432(13) 195432-1 ©2015 American Physical SocietyS. GEIßLER et al. PHYSICAL REVIEW B 91, 195432 (2015) FIG. 1. (Color online) (a) Schematic of the PMMA mask (green/light green) defined by a two-step EBL process for etching the NC structure into a (Ga,Mn)As layer (orange) on top of a semi-insulating GaAs substrate (blue). (b) Electron micrograph ofan NC device after ion-beam-etching and resist removal. sample preparation. First of all, we defined contact pads for the source and drain contacts as well as alignment marks forthe nanopatterning. This was done using optical lithographyfollowed by thermal evaporation of 10 nm Ti and 90 nmAu in a standard liftoff technique. After that, the NC wasdefined by means of EBL and subsequent chemically assistedion-beam etching using Cl 2. A two-step EBL process, which allows a precise control of the geometry of the nanocontactand a reliable processing, was developed and is described inAppendix A. The structure of the poly-methyl-methacrylate (PMMA) mask, used for the two-step process, is sketched in Fig. 1(a).I t mainly consists of the crosslinked PMMA line (dark green) ofthe first, high dose (3 ×10 4C/cm) exposure step as well as of a narrow gap line from the second, usual exposure step, whichseparates the (Ga,Mn)As layer into two parts used as sourceand drain contacts. The two parts are connected with eachother only at the NC, where the lines of the two exposure stepscross each other. This procedure allows us to define the widthas well as the length of the NC by two single lines withinindependent exposure steps. This completely rules out theinter-proximity-effect between different exposed elements andreduces the minimum size of the NC to the smallest achievablelinewidth of the two EBL steps. Compared to a single-stepprocess, our approach is robust with respect to minor electrondose variations and thus well reproducible. Because of this, wewere able to fabricate a large number of comparable devicesand even to control the geometry of the NC with a precision of a few nanometers. Figure 1(b) shows an electron micrograph of the central part of a typical NC device taken after thechemically assisted ion-beam etching and resist removal usinga low-energy oxygen plasma. After the nanopatterning wecovered the whole sample with a 30-nm-thick Al 2O3layer grown by a low-temperature atomic layer deposition processat a temperature of 90 ◦C. The Al 2O3layer acts on the one hand as the gate dielectric and on the other hand it protects thetiny NC against oxidation. The top-gate contact was definedby optical lithography and covers not only the NC, but alsothe center part of the whole device. It consists, similarly to thesource and drain contacts, of a 10 /90-nm-thick Ti/Au stack evaporated thermally and structured using a standard liftofftechnique. An effective way to influence the transport behavior is to apply an annealing step after the nanopatterning. We used anannealing temperature of 150 ◦C and durations from 30 min to 3 h. The post patterning annealing removes probably some ofthe defects induced by chemically assisted ion-beam etching.This can change an initially insulating sample to one inwhich Coulomb effects prevail or even to a conducting one.Annealing before the nanopatterning [ 11,18], which removes defects induced during low-temperature molecular beam epi-taxy growth, is less effective than the post patterning annealing.Hence, the intrinsic structure of the NC is dominated by defectsinduced during the nanopatterning rather than by defectsstemming from the low-temperature molecular beam epitaxygrowth. III. MEASUREMENT SETUP All low-temperature measurements presented in this work were carried out at a temperature of about 25 mK using a 3He/4He-dilution fridge, equipped with a superconducting coil magnet. In combination with a rotatable sample holder,we were able to apply magnetic fields up to 19 T in anydirection parallel to the sample plane. In order to saturatethe magnetization of the device and to fix its direction, weapplied a constant in-plane magnetic field with a magnitude of1 T along one of the easy axes of the extended (Ga,Mn)Aslayer. This leads to a situation as sketched in Fig. 5(a). The electrical transport experiments were carried out in atwo-terminal setup. We performed ac and dc measurementssimultaneously by applying a dc bias voltage V dcmodulated with a small oscillating ac component Vac. The current I flowing through the device was measured using a currentamplifier which also converts the current into a correspondingvoltage signal. The dc measurement using a digital multimeterprovides the well-known I-V dccharacteristic, while the ac measurement using a lock-in amplifier offers the differentialconductance G=dI/dV acof the device. Our device could be tuned additionally by an external dc voltage ( Vg) applied to the top-gate electrode of the device. IV . EXPERIMENTAL RESULTS A. Room-temperature properties As mentioned in the Introduction, all nanoconstricted (Ga,Mn)As devices investigated in previous studies have 195432-2DOUBLE-ISLAND COULOMB BLOCKADE IN (Ga,Mn)As . . . PHYSICAL REVIEW B 91, 195432 (2015) shown a rather complex and irregular Coulomb diamond pattern [ 10,11]. This has been explained by assuming that several metallic islands are involved in transport across the NC.Hence, shrinking the size of the NC should reduce the numberof islands within the NC and bring up a more regular Coulombdiamond pattern. Looking for such samples, we investigatedmany different devices with widths and lengths of the NCranging from 10 to 100 nm. Our experiments revealed thatthe transport properties of these devices are very sensitiveto the width wof the NC while its length Lhas only a minor influence. Wider samples ( w> 25 nm) show a mainly Ohmic behavior while the most narrow ones ( w< 15 nm) are fully insulating. Only samples with intermediate widthsof 15–25 nm show the typical SET-like behavior, discussedin the following. In many cases, the room-temperature resis-tanceR NCof the nanocontact already indicates whether the constriction is insulating, in the Coulomb-blockade regime,or Ohmic: For R NC/Rsvalues (with the sheet resistance of Rs∼4k/Omega1at 4.2 K) between 10 and 15 the constriction was in most cases in the Coulomb-blockade regime for thisspecific material (see also Appendix A2). However, similar to the earlier experiments, all of our SET-like samples, even theshortest and narrowest ones, have shown, on a first glance, anirregular Coulomb diamond pattern. Following, we discuss inmore detail transport in the Coulomb-blockade regime. B. Coulomb-blockade regime In Fig. 2, we present a highly resolved stability diagram of one of our NC devices in the SET regime. The first impression FIG. 2. (Color online) Differential conductance as a function of the bias and gate voltage of the NC device in Fig. 1. The measurement was done at a temperature of T=25 mK. A partial irregular Coulomb diamond pattern with frequently occurring vertical discontinuities is observed. Three of those discontinuities are marked by white lines.Cutting the data set between two of these lines gives an undisturbed segment; stitching neighboring segments together as described in the text and shown in the upper inset allows to reconstruct the Coulombdiamond spectrum over a larger gate voltage range.is that the Coulomb diamond pattern is very irregular and exhibits frequent vertical discontinuities. Three of them arehighlighted by white lines. These abrupt shifts can be assignedto charging or discharging of local traps in close vicinity to theNC, which, with their electrostatic potential, act as local gates.Their effect can thus be described as an abrupt jump alongthe gate voltage axis. This observation suggests a method toreconstruct the stability diagrams with unperturbed Coulombdiamonds. We cut the data set in Fig. 2along the white lines and shift the segments on the V gaxis until the diamonds fit onto each other. An example of this procedure is shown in thetop inset of Fig. 2. In this way we obtain, for some parts of the V gscale, Coulomb diamonds which are essentially cleared of potential jumps due to charge fluctuations in local traps. Thedata set displayed in Fig. 3has been reconstructed from the data shown in Fig. 2and represents the starting point of our more detailed analysis. The stability diagram shown in Fig. 3presents characteristic features typical for metallic single-electron transistors [ 12–16] but also several anomalies. As expected, a series of diamonds FIG. 3. (Color online) (a) Differential conductance of the NC device of Fig. 2vs applied gate and bias voltages after reconstruction. Diamonds labeled 0 to 4 can clearly be identified. (b) Differential conductance as a function of the bias voltage corresponding to thevertical dashed line in (a). (c) Conductance at V b=0 as a function of the gate voltage corresponding to the horizontal dashed line in (a). It shows a conductance peak at the 0-1, and a blockade at the othercharge degeneracy points, including point P. 195432-3S. GEIßLER et al. PHYSICAL REVIEW B 91, 195432 (2015) of exponentially low differential conductance (black regions with fixed particle number) are surrounded by ridges ofhigh conductance. Moreover, by further increasing the bias,the differential conductance does not drop to zero [see e.g.Fig. 3(b)], allowing to exclude the single-particle energy quantization typical for quantum dots. Unexpectedly, though,(i) the size and the shape of the Coulomb diamonds is notregular, (ii) some of the diamonds are not closing at zero bias[e.g., corners between diamonds 1 and 2 or between diamonds2 and 3 as seen from the gate trace in Fig. 3(c)]. FIG. 4. (Color online) (a) Schematic of a double-island structure in a parallel configuration. Transport from source to drain is carried by two subsequent direct tunneling processes involving only one of the islands. The two islands are characterized by a different capacitive coupling to the leads ( Csi,Cdi) as well as by a different gate capacitance ( Cgi) with i=1,2. (b) Schematic to illustrate the parameter extraction from a regular Coulomb diamond in the framework of the orthodox model. (c) The two Coulomb diamonds (ABCD and EFGH) used to extract the parameters are marked bywhite dotted lines.Concerning the first anomaly, it is striking that all the diamonds exhibit an individual height as well as an individualwidth. Additionally the diamond labeled 1 and the diamondlabeled 3 are asymmetric: according to the classical orthodoxtheory [ 12], one would expect that all Coulomb diamonds associated to a single island have the same size and shape, andthat opposing edges of a Coulomb diamond were parallel. Inthe orthodox picture, the two different slopes of a Coulombdiamond are related to the capacitive coupling of the islandto the source ( C s) and drain leads ( Cd), as well as to the gate electrode ( Cg). Assuming Cg/lessmuchCs,d, the slope of the source line is given by Cg/Cdwhile the slope of the drain line is given by −Cg/Cs(see Fig. 4). In our case, only the diamond numbered 2 has parallel source and drain lines. The diamondslabeled 1 and 3, however, exhibit four different slopes, so thatwe would extract from each two different values for C sandCd or two different values for Cg, respectively. This suggests that our NC consists of at least two metallic islands producing aset of nested diamonds. In the following, we restrict ourselvesto a double-island structure. Figure 4(a) shows a simple schematic to illustrate our interpretation: the two islands are arranged in parallel, so thatan electron can tunnel from the source lead directly to eachof the two islands and from there in a subsequent tunnelingprocess directly to the drain lead. By taking into account theslopes of the diamond edges as well as the distance betweenneighboring charge degeneracy points, we can obtain twodifferent sets of parameters ( C s,Cd,Cg) from our experimental data. Each set of parameters characterizes one of the twoislands. One set can be extracted from the regularly shapeddiamond 2. For the other one, we have to reconstruct a secondregular Coulomb diamond by extending the outer edges ofdiamonds 1 and 3 until they cross each other [see Fig. 4(c)]. The extracted parameters are summarized in Table I.O u r analysis is limited to certain gate voltage ranges. We attributethis limitation to possible differences in the shape and evenin the number of participating islands associated to differentgate voltage regions. Nevertheless, the simple orthodox modelgives already a satisfactory agreement between experimentaland theoretical dI/dV bstability diagrams and suggests that transport, in this gate voltage range, occurs primarily in parallelacross two islands of different size in the reconstructed gatevoltage segments. However, the model presented so far can notaccount for the second anomaly, i.e., a pronounced transportblocking observed in the vicinity of the charge degeneracypoint between the diamonds 1-2 and 2-3 [see also Fig. 2(b)]. On TABLE I. Parameters for the small and large Coulomb diamonds (CD) extracted from Fig. 3(a) assuming a double-island structure in the framework of the orthodox theory. The charging energy U= e2/C/Sigma1,w i t h C/Sigma1=Cs+Cd+Cgbeing the total capacitance, is also given for reference. Small CD (ABCD) Large CD (EFGH) Cd 5.6×10−18F3 .0×10−18F Cs 8.4×10−18F4 .2×10−18F Cg 28×10−20F9 ×10−20F U 11.2×10−3eV 21 .9×10−3eV 195432-4DOUBLE-ISLAND COULOMB BLOCKADE IN (Ga,Mn)As . . . PHYSICAL REVIEW B 91, 195432 (2015) the other hand, the gap is not present at the charge degeneracy point 0-1 and is barely visible at 3-4 [see also Fig. 3(c)]. Hence, the gap is assigned to the island with the smaller chargingenergy. In order to account for this experimental observation,we resort below to a minimal transport model that includes theferromagnetic nature of the material and provides a possiblemechanism for the observed distinct blockade behavior. V . THEORETICAL MODELING In this section, we extend the orthodox theory of Coulomb blockade [ 12–16] in order to account for the ferromagnetic properties of the (Ga,Mn)As samples. Although transportthrough magnetic islands has been addressed in the literature[17], scarce consideration has been given, to our knowledge, to the role played by an energy-dependent density of statesin the metallic islands. The latter, instead, is crucial toexplain the anomalous current blocking observed in the presentexperiment. To this end, we assume that both leads and the metallic islands are spin polarized. Figure 5(a) shows a sketch of the magnetization directions expected in the experiments. Themagnetization of the ferromagnetic (Ga,Mn)As leads is ratherweak, and can be tuned by an external magnetic field. It formsin our experiment an angle of 45 ◦(easy direction) with the transport direction, set by the longitudinal axis of the NC[zaxis, cf. Fig. 5(a)]. In the constriction, however, the spin polarization axis is strongly influenced by strain effects and isexpected to be along the NC longitudinal axis. In order to explain the blockade effects we claim that the angleθbetween the leads and the constrictions magnetization lies in the range 1 2π<θ<3 2π. In other words, current FIG. 5. (Color online) (a) Sketch of the magnetization direction of the leads ( /vectormS/D) and of the islands ( /vectormI). The magnetization of the leads is determined by the direction of the external magnetic field. In the constriction, on the other hand, strain effects are dominating and the magnetization direction lies parallel to the constriction axis.In our experiment, the angle between the two magnetizations is approximately θ= 3 4π. (b) Sketch of the density of states of the two metallic islands, with the spins aligned along the magnetizationof the constriction.suppression originates from the fact that the majority-spin carriers in the islands and in the leads have effectively theopposite polarization. Since only one of the two superimposedCoulomb diamond structures shows a noteworthy blockadeeffect, we conclude, within our model, that the structure withthe blockade stems from transport through a fully polarizedisland, while the second island is only partially polarized. We describe the islands’ polarization with an upward shift in energy of the minority-spin band with respect to the majority-spin band [see Fig. 5(b)]. The electro-chemical potential is the external parameter which determines whether the islandis partially or fully polarized. Partial polarization is obtainedif the chemical potential μ α(α=1,2) lies above the bottom of the minority-spin band, full polarization when the chemicalpotential lies between the bottom of the majority- and of theminority-spin bands. In our model, the tunneling of a source electron of the majority-spin species (conventionally the spin up) to a fullydown polarized island is highly suppressed for low-biasvoltages since no spin-up states are available near the Fermilevel. For bias voltages which are large enough to accessalso the minority-spin band [ α SeVb>B1 +, cf. Fig. 5(b)], the suppression is lifted and an increase of the current is expected.For the partially polarized island, both spin species can beaccessed already at the Fermi energy and no suppression isobserved. A. Model Hamiltonian We describe the nanoconstriction with a system-bath model aimed at mimicking the structure of the two islands contactedto source and drain leads sketched in Fig. 4(a). The total Hamiltonian is ˆH=ˆH S+ˆHT+ˆHL, (1) where ˆHL=/summationdisplay η∈{S,D}/summationdisplay kσEηkσˆc† ηkσˆcηkσ (2) denotes the Hamiltonian of the two spin polarized leads. We assume to have a flat, but spin-dependent, density of states(σ=↑/↓) D η↑=1+pη 2Dη,D η↓=1−pη 2Dη, (3) which depends on the polarization pηof the leads ( −1/lessorequalslantpη/lessorequalslant 1). The metallic islands ( α∈{1,2}) in the nanoconstriction are modeled by ˆHS=/summationdisplay α∈{1,2}/braceleftBigg/summationdisplay iτ/epsilon1αiτˆd† αiτˆdαiτ+αgeVgˆNα +Uα 2ˆNα(ˆNα−1/parenrightbig/bracerightBigg , (4) and have in general a different spin quantization axis as the contacts. We define τ=±1f o rs p i n +/−, respectively, using the spin-quantization axis of the nanoconstriction. As alreadymentioned, we account for the ferromagnetic properties of themetallic islands by assigning spin-dependent energy levels /epsilon1 αiτ 195432-5S. GEIßLER et al. PHYSICAL REVIEW B 91, 195432 (2015) and, consequently, a relative shift of the density of states for the two spin directions /Delta1ex[Fig. 5(b)]. The long-range Coulomb interactions are included within a constant interaction model,where U αis the charging energy of the island α. The effective coupling of the gate electrode to the metallic islands is takeninto account by the term proportional to α geVg, with αg= Cg/C/Sigma1being an effective gate coupling parameter and Vgthe gate voltage. The two metallic islands and the leads are weaklycoupled by the tunneling Hamiltonian ˆH T=/summationdisplay iατ/summationdisplay ηkσ[tηασuστ(θ)ˆc† ηkσˆdαiτ+H.c.],(5) where we defined the function u↑+(θ)=u↓−(θ)=cos(θ/2), u↑−(θ)=u↓+(θ)=isin(θ/2). It results from the noncollinear spin-quantization axes of the islands and the leads. Since thetwo axes are rotated by an angle of θin the y-zplane with respect to each other, the transformation conserves the spinduring tunneling. B. Density of states of the metallic islands Some of the experimental observations can only be under- stood if the energy dependence of the density of states, inparticular the presence of different band edges for minorityand majority spins, is accounted for. Specifically, we definethe spin-dependent density of states of island αas g ατ(/epsilon1)=˜gατ/Theta1(/epsilon1+W−τ/Delta1 ex/2)/Theta1(W+τ/Delta1 ex/2−/epsilon1) ≈˜gατf−(/epsilon1+W−τ/Delta1 ex/2), (6) where Wis the spin-independent contribution to the band- width, and /Delta1exthe exchange band splitting of the ferromag- netic metallic island. The parameter ˜gατdefines the strength of the density of states. Since the Wis the largest energy scale considered in the following, the upper limit of thedensity of states can be set to infinity. In the last line ofEq. ( 6), we have approximated the left Heaviside function byf −=1−f+, with f+the Fermi function; this allows us to further proceed analytically in the calculation of thetransport properties. The density of states is also sketchedfor clarity in Fig. 5(b). For later reference we define B α τas the energy difference between the bottom of the band of thecorresponding spin species τand the chemical potential of the island α:B α τ=−W+τ/Delta1 ex/2−μα. C. Transport theory In the following, we briefly outline the main steps leading to the evaluation of the transport characteristics, emphasizingthe new ingredients entering our transport theory. For moredetails, we refer to the Appendix B. The framework is the orthodox theory of Coulomb blockade [ 12–16], extended to the case of ferromagnetic contacts [ 17], and valid also for fully spin polarized metallic islands. The explicit derivation ofthe tunneling rates should illustrate the crucial role played inour theory by the energy-dependent density of states. The theory is based on a master equation for the reduced density matrix of the islands, up to second order in thetunneling Hamiltonian. Since the two metallic islands areassumed not to interact with each other, the correspondingdensity matrices obey independent equations of motion (seeAppendix B). Moreover, the metallic islands are assumed large enough to posses a quasicontinuous single-particle spec-trum, but small enough that their charging energy dominatesthe tunneling processes that change their particle number.We further assume that, in-between two tunneling events,the islands relax to a local thermal equilibrium. Under theseassumptions, the reduced density matrix of island αcan be written as ˆρ α red(t)=/summationdisplay Nα/braceleftbigg PNαe−βˆHS,α ZNα/bracerightbigg PNα(t), (7) where HS,αis the part of the system Hamiltonian associated to the island α,PNαis the projection operator on the Nα-particle subspace, and ZNα=TrS(PNαe−βˆHS,α) is the corresponding (canonical) partition function. By projecting the master equa-t i o no nt h e N α-particle subspace and tracing over the islands degrees of freedom, we keep only the occupation probabilitiesP Nαof finding the island occupied by Nαelectrons as dynamical variables. In the stationary limit, we find (seeAppendix B) Tr S/braceleftbig PNα˙ˆρα ∞}=0 =/summationdisplay ησ/braceleftbig −/Gamma1Nα→Nα−1 ηασ PNα−/Gamma1Nα→Nα+1 ηασ PNα +/Gamma1Nα−1→Nα ηασ PNα−1+/Gamma1Nα+1→Nα ηασ PNα+1/bracerightbig . (8) Eventually, the stationary current through lead ηreads as Iη=−e/summationdisplay ασ/summationdisplay Nα/braceleftbig /Gamma1Nα→Nα+1 ηασ −/Gamma1Nα→Nα−1 ηασ/bracerightbig PNα.(9) In Eqs. ( 8) and ( 9), the rates are defined as /Gamma1Nα+1→Nα ηασ =/summationdisplay τ1+σpη 2e2Rησ ατ|uστ(θ)|2b−/parenleftbig /Delta1EG Nα−αηeVb/parenrightbig ×/braceleftbig F/parenleftbig /Delta1EG Nα+Bα τ−αηeVb/parenrightbig −F/parenleftbig Bα τ/parenrightbig/bracerightbig , /Gamma1Nα→Nα+1 ηασ =/summationdisplay τ1+σpη 2e2Rησ ατ|uστ(θ)|2b+/parenleftbig /Delta1EG Nα−αηeVb/parenrightbig ×/braceleftbig F/parenleftbig Bα τ/parenrightbig −F/parenleftbig /Delta1EG Nα+Bα τ−αηeVb/parenrightbig/bracerightbig , (10) and are expressed in terms of the normal-state resistance Rησ ατ=/planckover2pi1/(2πe2|tηασ|2˜gατDη) and the functions b±(x)= 1/(e±βx−1) and F(x)=x/(eβx−1), with β=1/(kBT)t h e inverse temperature. We account for the asymmetric bias drop with the bias coupling constants defined as αS/D=±Cd/s+Cg/2 C/Sigma1. Further, we defined the grand canonical addition energy /Delta1EG Nα=αgeVg+UαNα+μα−μ0 =/bracketleftbig ENα+1−μ0(Nα+1)/bracketrightbig −(ENα−μ0Nα) (11) which must be paid in order to increase the electron number on island αfromNα→Nα+1. We denote μ0the chemical potential of the leads at bias Vb=0. The rates given in Eq. ( 10)d i f f e rf r o mt h o s eo ft h e orthodox theory of Coulomb blockade [ 12–16] even in their spin-dependent variation [ 17] due to the energy-dependent 195432-6DOUBLE-ISLAND COULOMB BLOCKADE IN (Ga,Mn)As . . . PHYSICAL REVIEW B 91, 195432 (2015) density of states and the explicit dependence on the band edges. The latter introduce a new source of current suppressionassociated to the absence of states with a specific spin species.These rates represent the main theoretical contribution of thiswork. For the chemical potential lying far above the bottom ofthe bands, the theory recovers again the limit of the classicalorthodox theory of Coulomb blockade. Namely, in the limitB→− ∞ , lim B→−∞±b±(x){F(B)−F(x+B)}=F(±x). (12) VI. THEORETICAL RESULTS A. Comparison with the experiments The results of our simulation are reported in Fig. 6(a), with the differential conductance shown as a function of the biasand gate voltage. We see the same nested diamond structure asin the experiments. In our theory, the diamonds at the chargedegeneracy points labeled 0-1 and 3-4 close. Between the dia-monds 1-2 and 2-3 the differential conductance is suppressedfor bias voltages smaller than a certain threshold bias. Figure6(b) shows a bias trace calculated at the charge degeneracy point 1-2, for two different angles θbetween the magnetization vectors of the leads /vectorm αand the metallic islands /vectormI.I ts h o w s a suppression of the differential conductance at point (P) withrespect to point (Q). The width of the suppression region cor-responds to the one observed experimentally in Fig. 3(b)and is proportional to B 1 +, the energy difference between the bottom of the minority-spin band and the chemical potential of island 1[cf. Fig. 5(b)]. In contrast to the experiments, no full blockade can be observed at (P). A change of the orientation of themagnetization directions from θ=π(dashed red line) to θ= 3 4π(solid blue line) is shifting the curve upwards. Aside from the constant shift, the two curves are qualitatively the same. To emphasize the effect of the islands’ degree of polariza- tion on the suppression mechanism, a conductance trace atV b=0 of a full polarized island 1 is compared to the case of a partial polarized island 1 in Fig. 6(c). Partial polarization is achieved by shifting the electrochemical potential of island1 by 12 meV up in energy. The solid blue line shows thefull polarized case, where the two larger peaks correspond tothe larger Coulomb diamond (island 2). The peak observedin the experiment [Fig. 3(c)] we ascribe to transport across this partially polarized island. Although the theoreticallypredicted second peak is missing in Fig. 3(c)we note that the corresponding blockade between diamonds 3 and 4 is muchless pronounced than between, e.g., 2 and 3. This asymmetrybetween the degeneracy points 0-1 and 3-4, however, cannotbe accounted for by our model which predicts a periodicityof the Coulomb oscillation pattern. The four smaller peaksin Fig. 6(c) belong to the smaller Coulomb diamond struc- ture, corresponding to island 1, i.e., the fully polarized one[Fig. 5(b)]. Even though the conductance is not completely suppressed as in the experiment, the conductance peaks arestrongly reduced with respect to the partially polarized case.In the latter (dashed gray lines), no suppression is present andthe conductance peaks of island 1 are by a factor of 4 larger. Inthe following, we address a possible reason for the incompleteblocking within the model. Since the parameters of island 2are kept the same, both for the fully and partially polarized FIG. 6. (Color online) (a) Calculated differential conductance of two spin polarized metallic islands with spin polarized leads. Theisland with the larger charging energy is assumed to be partially polarized, while the island with the smaller charging energy is fully polarized. (b) Bias trace through a charge degeneracy point of thefully polarized island. The gate position of the line trace is marked as a dashed line in (a). (c) Gate traces at V b=0 for a full (solid blue line) and a partial polarization (dashed gray line) of island 1.Island 2 remains partially polarized. In the fully polarized case, the conductance peaks of island 1 are suppressed with respect to the partially polarized case. For island 2, both curves are identical. Theparameters used to obtain this figure are α S1=0.4,αS2=0.42,U1= 11.2m e V ,a n d U2=21.9 meV in accordance with the parameters for the capacitive couplings of Table I. Moreover, Rησ 1τ=0.57× 103h/e2,Rησ 2τ=1.4×103h/e2,B1 +=2m e V , B1 −=−18 meV , B2 += −10 meV , B2 −=−35 meV , μ1=−42 meV , μ2=−32 meV , pη= 0.8, and kBT=0.07 meV . For the full polarized island 1 in (c) B1 += −10 meV , B1 −=−30 meV . 195432-7S. GEIßLER et al. PHYSICAL REVIEW B 91, 195432 (2015) FIG. 7. (Color online) All tunneling rates for the island 1 plotted at a charge degeneracy point as a function of the bias voltage Vb.T h e angle between the two magnetization directions is θ=π. cases the corresponding conductance peaks are not changing. Despite the fact that a comparison of the calculated gate traceto the experimental one in Fig. 3(c) reveals some limitations of the model, the essential feature, i.e., the suppression insidethe large Coulomb diamond, is reproduced. B. Mechanism of current suppression For a better understanding of the mechanism underlying the blockade, we derive analytically the differential conductancefor the island 1 at the two points (P) and (Q) marked in Fig. 6(b). For simplicity, the case θ=πis considered since qualitatively the blockade mechanism is the same in both cases. Notice that both P and Q correspond to a gate voltage such that /Delta1E G N=0, i.e., at the charge degeneracy point of the N-N+1 transition. To obtain the differential conductance, according to Eqs. ( 8) and ( 9), the transition rates /Gamma1N→N±1 ηασ are required. For simplicity, we have dropped the subscript 1 fromthe excitation energy /Delta1E G Nsince we will refer from now on always to the same island. In Fig. 7, we show the transition rates as a function of the bias Vb. To simplify the notation, we replaced /Gamma1N→N±1 ηασ → /Gamma1≷ ησ. Notice their linear dependence on the bias above a certain threshold. Thus, in that bias range one can approximate them as /Gamma1< S↓=−BS↓Vb, /Gamma1> D↓=BD↓Vb, (13) /Gamma1> D↑=AD↑+BD↑Vb, whereAD↑is a constant accounting for the threshold bias, and Bησ=2πe /planckover2pi12D01+σp 2˜gσαη|tη|2. (14) Here,D0=Dηis assumed to be independent of the lead. For the point (P) within the first plateau only the rates with σ=↓, namely /Gamma1> D↓and/Gamma1< S↓, are nonzero. Hence, according to the principle of detailed balance /Gamma1> D↓PN=/Gamma1< S↓PN+1. Imposing probability conservation we find PN=/Gamma1< S↓/(/Gamma1> D↓+/Gamma1< S↓). Thus, the stationary current equals I(P) D=−e/Gamma1> D↓PN= −e/Gamma1> D↓/Gamma1< S↓/(/Gamma1> D↓+/Gamma1< S↓)∝(1−p)2, which is suppressedfor a large spin polarization p. Here, the polarization pis assumed to be equal for both leads. At the point (Q), only one additional rate /Gamma1> D↑is con- tributing (the rate /Gamma1< S↑is zero due to the lower bound of the density of states). In this bias range, the equa-tions of detailed balance and probability conservation yieldP N=/Gamma1< S↓/(/Gamma1> D↓+/Gamma1> D↑+/Gamma1< S↓).The resulting stationary cur- rent is then I(Q) D=−e(/Gamma1> D↓+/Gamma1> D↑)PN=−e(/Gamma1> D↓+/Gamma1> D↑) /Gamma1< S↓/(/Gamma1> D↓+/Gamma1> D↑+/Gamma1< S↓)∝(1−p).Again, the current is sup- pressed for large spin polarization. Inserting Eq. ( 13) into the current expressions at the points P and Q, we find I(P) D=eBS↓BD↓ BD↓−BS↓Vb (15) and I(Q) D=e+BS↓AD↑Vb−BS↓(BD↑+BD↓)V2 b AD↑+(BD↑+BD↓−BS↓)Vb. (16) Taking the ratio of the two differential conductance plateaus, i.e., the ratio of Eqs. ( D1) and ( D2), we find dI(P) D dVb/slashbiggdI(Q) D dVb =1 αD|tD|2−αS|tS|2 ×/parenleftbigg αD|tD|2−αS|tS|2 (1−p)˜g↓ (1+p)˜g↑+(1−p)˜g↓/parenrightbigg .(17) Thus, within our simple model, the ratio Rof the height of the two plateaus is limited by αD|tS|2 αD|tD|2−αS|tS|2/lessorequalslantR/lessorequalslant1. (18) In other words, the ratio of the two differential conductance plateaus is limited in our theory, leading to some discrepancywith the experimentally observed ratio [cf. points P and Qmarked in Fig. 3(b)]. Since the parameters α ηare determined experimentally, the only possibility to change the ratio is tomodify the coupling constants |t η|. However, the increase of the coupling constants necessary to fit the experimental valuewould lead to a huge asymmetry in the stability diagramwhich is not observed experimentally. Despite the discrepancybetween Rand the experimental ratio, we think that the theory clearly suggests a mechanism which can lead to a suppressionof the conductance due to spin polarization in the frameworkof an orthodox theory of Coulomb blockade. To better fitthe experiments, a more realistic energy dependence of thedensity of states which also accounts for valence bands isnecessary. With such an energy dependence, the rates canchange their slope as a function of the bias voltage, leading toan even more pronounced bias-dependent suppression of thedifferential conductance. VII. CONCLUSION In this work, we have reported on a detailed study of the transport characteristics of nanofabricated narrow constric-tions in (Ga,Mn)As thin films. By means of a two-step electron 195432-8DOUBLE-ISLAND COULOMB BLOCKADE IN (Ga,Mn)As . . . PHYSICAL REVIEW B 91, 195432 (2015) beam lithography technique we have fabricated well-defined nanoconstrictions of different sizes. Depending on channelwidth and length, for a specific material, different low-temperature transport regimes have been identified, namely,the Ohmic regime, the single-electron tunneling regime (SET),and a completely insulating regime. In the SET, complex sta-bility diagrams with nested Coulomb diamonds and anomalousconductance suppression in the vicinity of charge degeneracypoints have been measured. In order to rationalize these observations, we proposed, for a specific nanoconstriction, a model consisting of twoferromagnetic islands coupled to ferromagnetic leads. Inparticular, the angle θbetween the leads and the islands magnetization lies in the range 1 2π<θ<3 2π. Moreover, the full polarization of one of the metallic islands is crucial. Thedata do not conclusively support a two-island model andwe can not exclude a more complex island structure. Westudied the transport characteristics of the system in termsof a modified orthodox theory of Coulomb blockade whichtakes into account the energy dependence of the density ofstates in the metallic islands. The latter represents an importantgeneralization of existing formulations and is determinant forthe qualitative understanding of the present experiments. Infact, the explicit appearance of the minority-spin band edgein the expression of the tunneling rates yields a pronouncedconductance suppression at the charge degeneracy points. Toaccount for the full suppression of conductance observed inthe experiments, the simple model used in this work shouldbe further improved. For example, the hole character of thecharge carriers and associated spin-orbit coupling effects arenot captured by our model. Furthermore, it is straightforwardto combine the present theory with microscopic models thatallow for a realistic description of the islands density ofstates. ACKNOWLEDGMENT We acknowledge for this work the financial support of the Deutsche Forschungsgemeinschaft under the researchprograms SFB 631 and SFB 689. APPENDIX A: EXPERIMENTAL DETAILS 1. Sample fabrication: Two-step EBL fabrication process Both steps are based on the standard EBL resist poly- methyl-methacrylate (PMMA). In the first step, one exposesthe resist using an extremely high line dose (approximately3×10 4pC/cm) in order to define a narrow crosslinked PMMA line. This line is very robust and does not getremoved by common organic solvents like acetone. Hence,after cleaning the sample in a bath of acetone, the crosslinkedPMMA line remains on top of the sample while the unexposedPMMA is removed from the sample surface. For the secondstep, the sample is again coated with a fresh layer of PMMAresist. This time one uses a common dose (approximately2000 pC /cm) in order to expose a second line perpendicular to the crosslinked one. After removing the exposed resistusing a standard developer solution consisting of isopropylalcohol and methyl-isobutyl-ketone (MIBK), we get the FIG. 8. (Color online) Open circles show various NC devices having different length and width fabricated from the material usedin experiment. Their corresponding room-temperature resistance, normalized to the sheet resistance R s=4k/Omega1of the (Ga,Mn)As layer, is color coded and sorted into three classes. At low temperatures,devices with R NC/Rs>15 were in most cases found to be fully insulating, samples with RNC/Rsbetween 10 and 15 showed Coulomb blockade, while nanocontacts with a relative resistance smaller than10 displayed an essential linear I-Vcharacteristic. patterned mask for the subsequent ion-beam etching, shown in Fig.1(a). 2. Size dependence of the transport characteristics As mentioned in Sec. IV A , the transport characteristics of the samples crucially depend on the dimensions of thenanoconstriction. Figure 8relates the transport behavior of the devices to the dimensions of the constriction. In particular,it shows room-temperature measurements of the devices’resistance normalized to the sheet resistance of R s=4k/Omega1, as a function of the constrictions width and length. Thewhite circles represent measurements of different samples.Their transport behavior is schematically illustrated by thebackground color, while devices in the red areas werepredominantly insulating, the ones in the green areas showedin most cases SET-like behavior, whereas in the gray areasOhmic behavior prevails. A discussion about the dependenceof the transport behavior on the constriction size is alreadygiven in Sec. IV A and is confirmed by Fig. 8. The devices shown in Fig. 8were measured at room tem- perature directly after sample fabrication without additionalannealing steps applied to the sample and before the firstcooldown. Aside from the contact size, additional annealingsteps can drastically alter the transport regime. As mentioned inthe main text, this can change an initially insulating sample toone in which SET effects prevail or even to a conducting one.Figure 9(a) displays the stability diagram of another device before annealing which is in the Coulomb-blockade regime.After annealing, a similar device (the original one broke),displayed in Fig. 9(b), shows no Coulomb blockade at all, but essentially linear behavior. 195432-9S. GEIßLER et al. PHYSICAL REVIEW B 91, 195432 (2015) FIG. 9. (Color online) (a) A device with length/width of 16 /15 nm before annealing and with RNC/Rs∼12.15 at room temperature shows pronounced SET behavior. (b) Another device with similar RNC/Rs∼12 shows after annealing at 150◦C for 3 h nearly Ohmic behavior. 3. Angular dependence of the transport characteristic Figure 10displays differential conductance stability dia- grams, measured for different directions of the magnetic fieldfor the device discussed in the main text. As already discussedthere, the data exhibit frequent vertical discontinuities. Hence,conclusions about the magnetic field dependence of themeasurements can be drawn only in undisturbed gate voltageregions. The white arrows in Fig. 10mark the position of two characteristic features in all subfigures. One clearly sees thatthe Coulomb-blockade threshold marked by (1) is shrinkingtowards lower bias voltages by rotating the magnetic fielddirection. A similar behavior is observed at position (2). Thisstrong dependence on the magnetic field directions reflects theanisotropy, typical in these systems [ 6,10]. APPENDIX B: EQUATION OF MOTION FOR A ORTHODOX THEORY OF COULOMB BLOCKADE In this Appendix, we derive an extension of the orthodox theory of Coulomb blockade for the case of spin polarizedcontacts as well as of a spin polarized metallic island. Inparticular, we will consider explicitly the lower bound of thedensity of states in the metallic island. The transport theory is based on the Liouville–von Neu- mann equation for the reduced density matrix in the interactionpicture i/planckover2pi1∂ ∂tˆρI(t)=[ˆHT,I(t),ˆρI(t)], (B1) FIG. 10. (Color online) Differential conductance stability dia- grams of the sample discussed in the main text measured for differentdirections of the magnetic field. The field strength was in all cases 1 T, the temperature 25 mK. Arrows in (a) mark the tip of two diamonds. TheirV gposition is kept fixed in (b) and (c) showing that the direction of the magnetic field changes Coulomb blockade. which we expand to second order in the tunneling Hamiltonian ˆHT.P r i o rt o t=0 the system and the leads do not interact and the density matrix can be written as a tensor product of thedensity matrices of the subsystems ˆρ=ˆρ S(0)⊗ˆρL≡ˆρS(0) ˆρL. (B2) Since the leads are considered thermal baths of noninteracting fermions, ˆ ρLreads as ˆρL=e−β(ˆHL−/summationtext ημηˆNη) ZL,G. (B3) Further, we assume that due to fast relaxation processes in the leads, the density matrix can be written as ˆ ρI(t)=ˆρred,I(t)ˆρL+ O(ˆHT), with ˆ ρred,I=TrLˆρ. Moreover, due to the independence of the two metallic islands ˆ ρred(t)=ˆρ1 red(t)ˆρ2 red(t) and each 195432-10DOUBLE-ISLAND COULOMB BLOCKADE IN (Ga,Mn)As . . . PHYSICAL REVIEW B 91, 195432 (2015) component obeys the following equation of motion: ˙ˆρα red(t)=−i /planckover2pi1/bracketleftbigˆHS,ˆρα red(t)/bracketrightbig −1 /planckover2pi12/integraldisplayt 0dt/prime/primeTrL/braceleftbig/bracketleftbigˆHT,/bracketleftbig HT,I(−t/prime/prime),ˆρα red(t)ρL/bracketrightbig/bracketrightbig/bracerightbig , (B4) where α=1,2 labels the metallic island. For the system we assume that the metallic islands are large enough to possess a quasicontinuous single-particle spectrum,but small enough that their charging energy dominates thetunneling processes that change their particle number. Fur-thermore, it is assumed that the islands will relax to a localthermal equilibrium on a time scale shorter than the inverse ofthe average electronic tunneling rate. Under these assumptions,the reduced density matrix can be written as ˆρ α red(t)=/summationdisplay NαPNαe−βˆHS,α ZNαPNα(t), (B5) withZNα=TrS{PNαe−βˆHS,α}, and PNα=/summationdisplay {ni}α /summationtext ini=Nα|{ni}α/angbracketright/angbracketleft {ni}α| (B6) is the projection operator on the Nα-particle subspace. Notice that in Eq. ( B5), due to the projector operator PNα, the only statistically relevant term of the system Hamiltonian ˆHS,α isˆhα S=/summationtext iσ/epsilon1αiσd† αiσdαiσ.T h et e r m e−β[Uα 2Nα(Nα−1)+αgeVgNα] becomes a constant and is canceling out in the density matrix. Inserting explicitly ˆHTin Eq. ( B4), we find TrS/braceleftbig PNα˙ˆρα red(t)/bracerightbig =−1 /planckover2pi12/summationdisplay ηη/prime/summationdisplay kiστ k/primei/primeσ/primeτ/primetηασuστ(θ)t∗ η/primeασ/primeu∗σ/primeτ/prime(θ)/integraldisplayt 0dt/prime/prime/braceleftbig TrS/braceleftbig PNαˆd† αiτˆdαi/primeτ/prime,I(−t/prime/prime)ˆρα red(t)/bracerightbig ×TrL{ˆcηkσˆc† η/primek/primeσ/prime,I(−t/prime/prime)ˆρL}+TrS/braceleftbig PNαˆdαiτˆd† αi/primeτ/prime,I(−t/prime/prime)ˆρα red(t)/bracerightbig TrL{ˆc† ηkσˆcη/primek/primeσ/prime,I(−t/prime/prime)ˆρL} −TrS/braceleftbigˆdαi/primeτ/prime,I(−t/prime/prime)PNαˆd† αiτρα red(t)/bracerightbig TrL/braceleftbigˆc† η/primek/primeσ/prime,I(−t/prime/prime)ˆcηkσˆρL/bracerightbig −TrS/braceleftbigˆd† αi/primeτ/prime,I(−t/prime/prime)PNαˆdαiτˆρα red(t)/bracerightbig TrL{ˆcη/primek/primeσ/prime,I(−t/prime/prime)ˆc† ηkσˆρL}+c.c./bracerightbig . (B7) In the following, we are analyzing the first term of Eq. ( B7) in more detail, the other terms can be evaluated in complete analogy. The calculation of the trace over the lead degrees of freedom gives TrL{ˆcηkσˆc† η/primek/primeσ/prime,I(−t/prime/prime)ˆρL}=ei /planckover2pi1Eηk(−t/prime/prime)f−(Eηk−μη)δkk/primeδηη/primeδσσ/prime, (B8) where the time evolution of the creation and annihilation operators of the leads is given by ˆc† ηkσ, I(t)=ei /planckover2pi1Eηktˆc† ηkσ. For the system operators, the time evolution can be carried out in a similar way, keeping in mind that the parts proportional to the total numberoperator can be factorized Tr S/braceleftbig PNαˆd† αiτˆdαi/primeτ/prime,I(−t/prime/prime)ˆρα red(t)/bracerightbig =ei /planckover2pi1[/epsilon1αi/primeσ/prime+αgeVg+U(Nα−1)]t/prime/primeTrS/braceleftbig PNαˆd† αiτˆdαi/primeτ/primeˆρα red(t)/bracerightbig . (B9) In order to perform the trace over the system degrees of freedom, another approximation is necessary. By taking the average in the grand canonical ensemble, the particle number is determined by the chemical potential and we can remove the projectionoperator: Tr S/braceleftbig PNαˆd† αiσˆdαi/primeσ/prime,I(−t/prime/prime)ˆρα red(t)/bracerightbig =TrS/braceleftbigg PNαˆd† αiσˆdαi/primeσ/prime,I(−t/prime/prime)e−βˆhα S ZNα/bracerightbigg PNα≈TrS/braceleftbigg ˆd† αiσˆdαi/primeσ/prime,I(−t/prime/prime)e−β(ˆhα S−μα,Nα) Zμα,Nα/bracerightbigg PNα.(B10) This approximation becomes exact in the limit of N→∞ . In presence of a quasicontinuous energy spectrum of the islands, we can further drop the Nαdependence of the chemical potential, for small relative variations of Nα. The trace in Eq. ( B9) can now be evaluated in the standard way and it yields Fermi functions. Inserting the results for the traces in Eq. ( B7) we obtain TrS/braceleftbig PNα˙ˆρα red(t)/bracerightbig =−1 /planckover2pi12/summationdisplay η/summationdisplay kiστ|tηασ|2|uστ(θ)|2/integraldisplayt 0dt/prime/prime/braceleftbig ei /planckover2pi1[−Eηk+/epsilon1αiτ+αgeVg+Uα(Nα−1)]t/prime/prime ×f+(/epsilon1αiτ−μα)f−(Eηk−μη)PNα(t)+e−i /planckover2pi1(−Eηk+/epsilon1αiτ+αgeVg+UNα)t/prime/primef−(/epsilon1αiτ−μα)f+(Eηk−μη)PNα(t) −ei /planckover2pi1[−Eηk+/epsilon1αiτ+αgeVg+U(Nα−1)]t/prime/primef−(/epsilon1αiτ−μα)f+(Eηk−μη)PNα−1(t) −e−i /planckover2pi1(−Eηk+/epsilon1αiτ+αgeVg+UNα)t/prime/primef+(/epsilon1αiτ−μSα)f−(Eηk−μη)PNα+1(t)+c.c./bracerightbig . (B11) Since we are only interested in the stationary solution of the master equation, we send t→∞ and use the Dirac identity /integraldisplay∞ 0dteiωt=πδ(ω)+ilim η→0Im/parenleftbiggi ω+iη/parenrightbigg (B12) 195432-11S. GEIßLER et al. PHYSICAL REVIEW B 91, 195432 (2015) to evaluate the integrals. Due to statistical averages, no coherences are possible in the master equation and the two complex- conjugated parts can be summed up. We find TrS/braceleftbig PNα˙ˆρα ∞/bracerightbig =0=−2π /planckover2pi1/summationdisplay η/summationdisplay kiστ|tηασ|2|uστ(θ)|2{δ[−Eηk+/epsilon1αiτ+αgeVg+Uα(Nα−1)]f+(/epsilon1αiτ−μα)f−(Eηk−μη)PNα +δ(−Eηk+/epsilon1αiτ+αgeVg+UNα)f−(/epsilon1αiτ−μα)f+(Eηk−μη)PNα−δ[−Eηk+/epsilon1αiτ +αgeVg+U(Nα−1)]f−(/epsilon1αiτ−μα)f+(Eηk−μη)PNα−1−δ(−Eηk+/epsilon1αiτ+αgeVg+UNα) ×f+(/epsilon1αiτ−μα)f−(Eηk−μη)PNα+1}. (B13) Further, we consider the continuum limit of the states in the quantum dot /summationdisplay i→/integraldisplay∞ −∞d/epsilon1g ατ(/epsilon1), (B14) withgατ(/epsilon1) being the energy-dependent density of states in island αwith the spin τ, defined in Eq. ( 6). For the leads /summationdisplay k→/integraldisplay∞ −∞dED ησ, (B15) where Dησis the density of states of lead ηwhich is considered in the flat-band limit. The integration over the lead degrees of freedom gives TrS/braceleftbig PNα˙ˆρα ∞/bracerightbig =0=−2π /planckover2pi1/summationdisplay ηστ|tηασ|2|uστ(θ)|2Dησ/integraldisplay d/epsilon1g ατ(/epsilon1)/braceleftbig f+/parenleftbig /epsilon1−μα/parenrightbig f−(/epsilon1+/Delta1ENα−1−μη)PNα +f−(/epsilon1−μα)f+/parenleftbig /epsilon1+/Delta1ENα−μη/parenrightbig PNα−f−(/epsilon1−μα)f+/parenleftbig /epsilon1+/Delta1ENα−1−μη/parenrightbig PNα−1 −f+(/epsilon1−μα)f−/parenleftbig /epsilon1+/Delta1ENα−μη/parenrightbig PNα+1}, (B16) where /Delta1ENα=UNα+αgeVg. In a last step we insert gατ(/epsilon1) [see Eq. ( 6) in the main text], and the remaining integral can be done by using the following identities: f+(x)f−(y)=b+(x−y)[f+(y)−f+(x)], (B17) /integraldisplay∞ −∞dx[f+(x)−f+(x+ω)]=ω, (B18) /integraldisplay∞ −∞dxf+(x+a)f−(x+b)f−(x+c) =/integraldisplay∞ −∞dxb+(a−b)(f+(x+b)−f+(x+a))f−(x+c) =b+(a−b)/braceleftbigg b+(b−c)/integraldisplay∞ −∞dx(f+(x+c)−f+(x+b))−b+(a−c)/integraldisplay∞ −∞dx(f+(x+c)−f+(x+a))/bracerightbigg =b+(a−b)(F(b−c)−F(a−c)). (B19) b±(x) andF(x) are defined in the main text just below Eq. ( 10). Using these identities yields the final result TrS/braceleftbig PNα˙ˆρα ∞/bracerightbig =0=/summationdisplay ησ/braceleftbig −/Gamma1Nα→Nα−1 ηασ PNα−/Gamma1Nα→Nα+1 ηασ PNα+/Gamma1Nα−1→N ηασ PNα−1+/Gamma1Nα+1→N ηασ PNα+1}. (B20) APPENDIX C: CURRENT Finally, we briefly outline the derivation of the current formula. The current is defined as Iη=ed dt/angbracketleftˆNη/angbracketright(t). (C1) In the interaction picture, the total particle-number operator of lead η,ˆNη, is not evolving in time since it commutes with the unperturbed part of the Hamiltonian. Therefore, the current reads as Iη=eTrS+L/braceleftbigg ˆNηd dtˆρI(t)/bracerightbigg =−i /planckover2pi1TrS+L{ˆNη[ˆHT,I(t),ˆρI(0)]}−1 /planckover2pi12/integraldisplayt 0dt/primeTrS+L{ˆNη[ˆHT,I(t),[ˆHT,I(t/prime),ˆρI(t/prime)]]}, (C2) 195432-12DOUBLE-ISLAND COULOMB BLOCKADE IN (Ga,Mn)As . . . PHYSICAL REVIEW B 91, 195432 (2015) where we expandd dtˆρI(t) up to second order in ˆHT. The first term of Eq. ( C2) vanishes since only an odd number of operators appear in the trace. In the second term, we replace ˆ ρI(t/prime)→ρI(t). Exploiting further the cyclic invariance of the trace, we find Iη=−e /planckover2pi12/integraldisplayt 0dt/primeTr{[[ˆNη,ˆHT,I(t)],ˆHT,I(t/prime)]ˆρI(t)}=−2e /planckover2pi12Re/parenleftbigg/integraldisplayt 0dt/primeTrS+L{[ˆNη,ˆHT,I(t)]ˆHT,I(t/prime)ˆρI(t)}/parenrightbigg . (C3) In the last step, we exploited the anti-Hermiticity of [ ˆNη,ˆHT,I(t)]. Following the same steps as in the derivation of the master equation, one can identify the rates, and one finds the well-known expression of the current Iη=−e/summationdisplay ασ/summationdisplay Nα/braceleftbig /Gamma1Nα→Nα+1 ηασ PNα−/Gamma1Nα→Nα−1 ηασ PNα/bracerightbig . (C4) APPENDIX D: CALCULATION OF THE DIFFERENTIAL CONDUCTANCE Differentiating Eq. ( 15) with respect to Vband inserting the definition of Eq. ( 14) yields the differential conductance of the first plateau: dIP D d(Vb)=2πe2 /planckover2pi1D0˜g↓(1−p) 2αD|tD|2αS|tS|2 αD|tD|2−αS|tS|2. (D1) To calculate the differential conductance at this point we differentiate Eq. ( 16) with respect the bias voltage and find dI(Q) D dVb=−e2αγ+2βγV b+βV2 b (γ+δVb)2, (D2) where we defined α=−BS↓AD↑,β=−BS↓(BD↑+BD↓),γ=AD↑, andδ=−BS↓+BD↑+BD↓. In order to find the value of the differential conductance plateau, we have to consider the high-bias limit and we find lim Vb→∞dI(Q) S dVb=−e2β δ=e2BS↓(BD↑+BD↓) −BS↓+BD↑+BD↓. (D3) Inserting back the physical constants, we find lim Vb→∞dI(Q) D dVb=e22π /planckover2pi1D0˜g↓(1−p) 2αS|tS|2 αD|tD|2[(1+p)˜g↑+(1−p)˜g↓] −(1−p)˜g↓αS|tS|2+[(1+p)˜g↑+(1−p)˜g↓]αD|tD|2. (D4) [1] H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, S. Katsumoto, and Y . Iye, Appl. Phys. Lett. 69,363(1996 ). [2] T. Dietl and H. Ohno, Rev. Mod. Phys. 86,187(2014 ). [3] T. Jungwirth, J. Sinova, J. Ma ˇsek, J. Ku ˇcera, and A. H. MacDonald, Rev. Mod. Phys. 78,809(2006 ). [4] K. Sato, L. Bergqvist, J. Kudrnovsk ´y, P. H. Dederichs, O. Eriksson, I. Turek, B. Sanyal, G. Bouzerar, H. Katayama-Yoshida, V . A. Dinh et al. ,Rev. Mod. Phys. 82,1633 (2010 ). [5] C. R ¨uster, T. Borzenko, C. Gould, G. Schmidt, L. W. Molenkamp, X. Liu, T. J. Wojtowicz, J. K. Furdyna, Z. G. Yu,a n dM .E .F l a t t ´e,Phys. Rev. Lett. 91,216602 (2003 ). [6] A. Giddings, M. Khalid, T. Jungwirth, J. Wunderlich, S. Yasin, R. Campion, K. Edmonds, J. Sinova, K. Ito, K.-Y . Wang et al. , Phys. Rev. Lett. 94,127202 (2005 ). [7] M. Schlapps, M. Doeppe, K. Wagner, M. Reinwald, W. Wegscheider, and D. Weiss, Phys. Status Solidi A 203,3597 (2006 ). [8] M. Ciorga, M. Schlapps, A. Einwanger, S. Geißler, J. Sadowski, W. Wegscheider, and D. Weiss, New J. Phys. 9,351(2007 ). [ 9 ]K .P a p p e r t ,S .H ¨umpfner, C. Gould, J. Wenisch, K. Brunner, G. Schmidt, and L. W. Molenkamp, Nat. Phys. 3,573(2007 ).[10] J. Wunderlich, T. Jungwirth, B. Kaestner, A. Irvine, A. Shick, N. Stone, K.-Y . Wang, U. Rana, A. Giddings, C. Foxon et al. , Phys. Rev. Lett. 97,077201 (2006 ). [11] M. Schlapps, T. Lermer, S. Geissler, D. Neumaier, J. Sadowski, D. Schuh, W. Wegscheider, and D. Weiss, Phys. Rev. B 80, 125330 (2009 ). [12] D. V . Averin and K. Likharev, J. Low Temp. Phys. 62,345 (1986 ). [13] D. V . Averin and K. K. Likharev, Mesoscopic Phenomena in Solids (Elsevier, Amsterdam, 1991). [14] H. Grabert, Z. Phys. B: Condens. Matter 85,319(1991 ). [15] Single Charge Tunneling , NATO ASI Series, edited by H. Grabert and M. H. Devoret (Springer, New York, 1992). [16] Mesoscopic Electron Transport , NATO ASI Series, edited by L. Sohn, L. Kouwenhoven, and G. Sch ¨on (Kluwer, Amsterdam, 1997). [17] J. Barna ´s and I. Weymann, J. Phys.: Condens. Matter 20,423202 (2008 ). [18] K. Edmonds, P. Boguslawski, K. Wang, R. Campion, S. Novikov, N. Farley, B. Gallagher, C. Foxon, M. Sawicki,T. Dietl et al. ,P h y s .R e v .L e t t . 92,037201 (2004 ). 195432-13

PhysRevB.93.224508.pdf

PHYSICAL REVIEW B 93, 224508 (2016) Impurity scattering effects on the superconducting properties and the tetragonal-to-orthorhombic phase transition in FeSe M. Abdel-Hafiez,1,2,*Y . J. Pu,3J. Brisbois,4R. Peng,3D. L. Feng,3D. A. Chareev,5,6A. V . Silhanek,4 C. Krellner,2A. N. Vasiliev,6,7,8and Xiao-Jia Chen1,† 1Center for High Pressure Science and Technology Advanced Research, Shanghai 201203, China 2Institute of Physics, Goethe University Frankfurt, 60438 Frankfurt am Main, Germany 3Department of Physics and Advanced Materials Laboratory, Fudan University, Shanghai 200433, China 4D´epartement de Physique, Universit ´ed eL i `ege, B-4000 Sart Tilman, Belgium 5Institute of Experimental Mineralogy, Russian Academy of Sciences, 142432 Chernogolovka, Moscow District, Russia 6Institute of Physics and Technology, Ural Federal University, 620002 Ekaternburg, Russia 7Low Temperature Physics and Superconductivity Department, Physics Faculty, M.V . Lomonosov Moscow State University, 119991 Moscow, Russia 8National University of Science and Technology “MISiS, ” Moscow 119049, Russia (Received 8 December 2015; revised manuscript received 9 May 2016; published 8 June 2016) A comprehensive study of the doping dependence of the phase diagram of FeSe-based superconductors is still required due to the lack of a clean and systematic means of doping control. Here, we report on the magneto-opticalimaging, thermodynamic and transport properties, as well as in situ angle-resolved photoemission spectroscopy (ARPES) studies of impurity scattering in stoichiometric FeSe single crystals. Co doping at the Fe site is foundto decrease the superconducting transition temperature ( T c). The upper critical field and specific heat all indicate a possible multiband superconductivity with strong coupling in the Co-doped system. A remarkable feature inFeSe is that its temperature dependent resistivity exhibits a wide hump at high temperatures, a signature ofa crossover from a semiconductinglike behavior to metallic behavior. A structural tetragonal-to-orthorhombicphase transition ( T s) (a consequence of the electronic nematicity) is suppressed by either physical or chemical pressures. Due to the reconstruction of the Fermi surface at Ts, specific heat anomalies at Tspresent /Delta1Cp/Ts≈γn, being γnthe Sommerfield coefficient at low temperature. This reflect an additional electronic instability in the FeSe 1−xSxsystem. ARPES data between 180 and 282 K indicates the existence of a chemical potential shift with increasing thermal excitations, resulting in a change of the Fermi-surface topology and exhibiting a semimetalbehavior. We found that the temperature-induced Lifshitz transition is much higher than the temperature for thenematic order. DOI: 10.1103/PhysRevB.93.224508 I. INTRODUCTION The majority of the parent and underdoped compounds of iron-based superconductors exhibit a stripe-type long-rangeantiferromagnetic order, accompanied by a nematic order[1]. Superconductivity in these materials emerges when the magnetic and nematic order are partially or completely sup-pressed by chemical doping or by the application of pressure[2,3]. This is because the interaction that drives the nematic order may also mediate the Cooper pairing. This emergenceand strengthening of antiferromagnetic order was directlyevidenced by muon rotation spectroscopy [ 4,5]. Therefore, there is a great deal of interest and excitement in understandingthe microscopic origin of nematicity in iron-based supercon-ductors. Among iron-based superconductors, FeSe exhibitsintriguing and distinctive properties, which are currently theresearch focus in the field of high temperature superconductors[6–8]. Undoped FeSe possesses a nematic order below 90 K and becomes superconducting below 8 K [ 9,10]. The most interesting property of these materials is not only the pressureor strain increasing the T c, but a giant enhancement of the superconductivity at the FeSe/SrTiO 3(STO) interface, where *m.mohamed@hpstar.ac.cn †xjchen@hpstar.ac.cnthe strain drastically changes the parameters of the magneticsubsystem in FeSe [ 11]. It seems that STO provides phonons that enhance superconductivity in single-layered FeSe [ 12,13]. Recently, the superconducting transition was enhanced inK-dosed FeSe. Although the competition between nematicityand superconductivity likely plays an important role in theenhanced superconductivity, it does not imply that this is thesingle cause behind the observed effect [ 14]. Whether the nematic order is driven by spin or orbital fluctuations is still hotly debated. The orbital fluctuationmechanism produces a sign preserving s ++-wave pairing, where the order parameters of the electron and hole pocketsdo not change their relative signs [ 15,16], while the spin fluctuation mechanism favors a sign-changing s-wave pairing, where the electron and the hole Fermi surfaces have orderparameters with opposite signs [ 17,18]. Furthermore, in the spin-fluctuation-based pairing theory the possible existenceof order-parameter nodes is reported in both singlet andtriplet superconducting states [ 19,20]. Although impurity scattering shows a pair breaking effect, there are differentopinions on the possible pairing symmetry; for instance, thesuppression of superconductivity by the Co replacement on theFe sites [ 21]. However, superconductivity suppression is much weaker than that expected in a s ±pairing, and thus supports sign-preserving s-wave pairing. In contrast, in Co-doped Ba(Fe 1−xCox)2As2, a believed s±-pairing superconductor, 2469-9950/2016/93(22)/224508(10) 224508-1 ©2016 American Physical SocietyM. ABDEL-HAFIEZ et al. PHYSICAL REVIEW B 93, 224508 (2016) increasing the Co concentration leads to an enhancement of the critical temperature Tcup to 26 K, instead of effectively suppressing the superconductivity [ 22]. Impurity scattering in high temperature superconductors is a critical parameterthat governs the electron correlations and ground states. Thishelps to understand the interplay and mechanism of differentphases and investigates rich phase diagrams. Nevertheless,due to the lack of clean and systematic means of a dopingcontrol, a comprehensive study of the doping dependence ofthe phase diagram of an FeSe-based superconductor is stilllacking. With the hope of filling this gap, we report hereon the effect of impurity scattering on superconductivity andtetragonal-to-orthorhombic phase transition in FeSe. Transportdata at higher temperatures in FeSe exhibit a wide hump witha crossover, probably from a semiconductinglike behaviorto a metallic behavior. This is supported by angle-resolvedphotoemission spectroscopy (ARPES) data at higher tem-peratures, where the data shows a change in the Fermi-surface topology and, therefore, exhibits a semimetal behavior.Additionally, the temperature-induced Lifshitz transition isfound to be much higher than the temperature for the nematicorder. II. EXPERIMENT The conductance anisotropy in layered material single crys- tals is large, so a traditional four-terminal method measuring the resistivity along the abplane, ρab, may be unreliable [ 23]. In view of this fact, we used six terminals to determine eachprincipal component of resistivity. In the latter method, the cur-rent was injected through the outermost contacts on one surfacewhereas voltages were measured across the innermost contactsof each surface. The Laplace equation was solved and invertedto find ρ ab[24]. In addition, this method allowed the sample homogeneity to be tested by permuting the electrodes whichwere used for the current and voltage [ 23,24]. We investigated selected platelike FeSe, FeSe 1−xSx, and Fe 1−xCoxSe single crystals, grown in an evacuated quartz ampoule, using theAlCl 3/KCl flux technique with a constant temperature gradient of 5◦C/cm along the ampoule length. The temperature of the hot end was kept at 427◦C, and the temperature of the cold end was about 350 −330◦C. The phase purity of the resulting crystal was checked with x-ray diffraction.The chemical compositions of the crystals were studied usinga digital scanning electronic microscope, TESCAN Vega IIXMU, with the energy dispersive microanalysis system INCAEnergy 450/XT (20 kV). The good quality of the crystals wasconfirmed by specific-heat jump, a complete superconductingvolume, and sharp superconducting transition [ 25–28]. The resistivity and thermodynamic measurements were measuredin a Quantum Design Physical Property Measurement Sys-tem with an adiabatic thermal relaxation technique. Thevisualization of the magnetic flux landscape was performedthrough the Faraday rotation of linearly polarized light in aBi-doped yttrium iron garnet with in-plane magnetic domains,a technique known as magneto-optical imaging (MOI) [ 29,30]. This technique requires planar surfaces in order to ensure goodproximity of the magneto-optical layer to the sample. To thatend, we cleaved large single crystals using a traditional scotchtape method on both sides and thus obtained flat samples onthe millimeter scale length. Our ARPES data was gathered under an ultrahigh vacuum of 1 .5×10 −11mbar, with a SPECS UVLS discharge lamp (21.2 eV He-I α) and a Scienta R4000 electron analyzer. The energy resolution is 8 meV and theangular resolution is 0.3. III. RESULTS AND DISCUSSION A. Magneto-optical imaging Figure 1summarizes the most representative results ob- tained by MOI for a FeSe crystal (upper row) and for a 9%S-doped FeSe crystal (lower row). Panels (a) and (e) showan optical microscopy image of the investigated samples.In (a) clear straight lines manifest the presence of terracesfollowing the main crystallographic axes of the crystal. Thesecond column in Fig. 1[i.e., panels (b) and (f)] shows the magnetic field landscape obtained by the MOI technique at amagnetic field H=0.3 mT, applied after cooling the sample down to 4 K. In these very weak fields, little flux penetration isobserved into the sample, which is indicative of the Meissnerphase. These images clearly illustrate that a macroscopicsuperconducting current is able to circulate in the entiresample surface and effectively screen the applied externalfield. This behavior contrasts with the field penetration inpolycrystalline FeSe tapes, where a considerable distributionofT cand weak-link features has been reported [ 31]. In the third column in Fig. 1, panels (c) and (g) show the magnetic field penetration at higher applied fields. Both samples exhibita highly inhomogeneous field penetration. Indeed, the fieldadvances into the sample by following two well-definedperpendicular directions. The fact that one of these directionsis aligned with the observed terraces in the original opticalimage leads us to believe that the magnetic flux penetration isalso aligned with the crystallographic axis of the orthorhombicstructure. This is consistent with the recent finding of vortextrapping into twin planes in stoichiometric FeSe samples[32]. It is worth noting that the observed field penetration substantially departs from the critical state model typicallyapplied for extracting the critical current density in hard typeII superconductors. As such, critical currents obtained frommacroscopic magnetization loops should be interpreted withcaution [ 33]. In the rightmost column of Fig. 1[panels (d) and (h)] the average intensity was recorded as a function oftemperature in a square area of 50 μm×50μm in the center of the sample, which was set in a remanent state after fieldcooling in H=1 mT and subsequently set H=0 mT. From these measurements it is easy to identify the superconducting-to-normal transition. The onset of this transition agrees wellwith the values obtained by other global techniques such asspecific heat, ac susceptibility, and resistivity. B. Thermodynamic and transport properties 1. Effect of Co doping Thermodynamic data of FeSe 1−xSxand Fe 1−xCoxSe are presented in Fig. 2. Figure 2(a) presents the magnetic sus- ceptibility χmeasured following zero-field-cooling (ZFC) and field-cooling procedures in an external field of 1 mTapplied along the caxis. It is obvious that introducing small amounts of Co into the Fe site leads to suppression of the 224508-2IMPURITY SCATTERING EFFECTS ON THE . . . PHYSICAL REVIEW B 93, 224508 (2016) 500 µm 500 µm(a) (b) (c) (e) (f) (g) 456789 1 0 1 1 1 20.00.20.40.60.81.0 I/I0 T (K)0.00.20.40.60.81.0 I/I0 (d) (h) FIG. 1. MOI for FeSe (upper row) and FeSe 0.91S0.09(lower row). Panels (a) and (e) show optical images of the sample. Panels (b) and (f) show the magnetic flux distribution at 4 K for an applied field H=0.3 mT, where bright (dark) areas correspond to high (low) magnetic fields. In panels (c) and (g), His further increased and the flux penetration follows the crystallographic axes of the orthorhombic structure. Tc is determined in panels (d) and (h) by tracking the average intensity IasTis increased, in a 50 ×50μm2square at the center of the sample. I is normalized by the intensity I0outside the sample. superconducting transition temperature. This contrasts with the FeSe 1−xSx[8,27], where Tcfirst increases and then decreases as shown in Figs. 2(a) and6. However, despite the suppression of superconductivity, no signatures of structuraltransitions are observed in the Co-doped samples with x= 0.04. In addition, this change of Se or Fe content not only leads to a different T c[10], but also to slight changes from the ideal 1:1 ratio in FeSe, leading to severe changes oftheir superconducting properties. For instance, the low fieldmagnetization data of various FeSe 1+δsamples showed that the strongest superconducting signal occurs for the moststoichiometric sample, whereas it has been shown that inthe FeSe 0.82case, there is no superconducting signal [ 34]. We should note that the suppression of the superconductingtransition in Co-doped FeSe suggests the strong pair breakingeffect of Co in heavily electron-doped FeSe. However, it iscurrently not certain whether Co in FeCoSe is a magnetic ornonmagnetic impurity. Although Co is generally consideredas nonmagnetic in Fe-based superconductors, it is shownthat Co may behave as magnetic impurities in overdopedBa(Fe,Co) 2As2due to the incomplete charge transfer [ 35]. As shown from our results, the electron doping of 7% inCo-doped FeSe is limited by the solubility of Co. SomeCo atoms that partially transfer electrons to FeSe may actas magnetic impurities. In addition, Co doping causes strongsingle particle scattering effects, which is also harmful to thesuperconductivity [ 36]. The temperature dependence of the specific heat as C P/T vsTin zero field is shown in Fig. 2(b). The sharp diamagnetic signal in the ZFC data and the specific heat jump confirm bulksuperconductivity in the investigated systems. In Fe 1−xCoxSe, the estimated universal parameter /Delta1Cel/γnTcof the specific heat at Tcis≈2.14, 2.05, 2.12, and 1.82 mJ /mol K2for x=0, 0.012, 0.024, and 0.04, respectively. These values are very close to the FeSe 1−xSxsystem [ 27]. The specific heat for x=0.07 does not show any indication of superconductivity. This is very convenient because we can safely ignore thespin-fluctuation contribution to the specific heat in this systemand can use it to remove the phonon contribution. However,jumps of specific heat at T cin these materials are higher than the prediction of the weak-coupling Bardeen-Cooper-Schrieer(BCS) theory ( /Delta1C el/γnTc=1.43). As the superconducting transition is relatively sharp in our single crystals, a distributioninT cor the presence of impurity phases cannot explain the large values of the normalized specific heat jump. Which may,instead, evidence the presence of a stronger-coupling strengthin Fe 1−xCoxSe. Additionally, as highlighted previously [ 27], the normalized specific heat jump reveals the presence ofstrong-coupling superconductivity in FeSe 1−xSx. Figure 2(c) presents the temperature dependence of the ac susceptibilitiesfor Fe 0.988Co0.012Se. The measurements were done in an ac field with an amplitude Hac=0.5 mT and a frequency f=1 kHz at different applied magnetic fields up to 9 T parallel tothecaxis. The transition temperature T chas been extracted from the bifurcation point between the real and imaginaryparts of the ac susceptibilities χ /primeandχ/prime/prime. In zero field, the superconducting transition is seen around 7.5 K, and shifts tolower Twhen the field is increased. Figure 2(d) summarizes the temperature dependence of the upper critical field H c2for the corientation of the 224508-3M. ABDEL-HAFIEZ et al. PHYSICAL REVIEW B 93, 224508 (2016) -1.0-0.50.0 02468 1 00255075-10 2468 1 002040 24680246810(c) (b) FeSe 0.86 S 0.15Fe0.96Co0.04Se0.96 Fe0.988Co0.012Se4πχ FeSeFeSe 0.89 S 0.11(a) T(K)Co0.07Co0.04Co0.012 T(K)Cp/T(mJ/mol K2)Co0.024 χ',χ'' (a.u.) (d)9T0T9T 0T Co = 0.012Co = 0.012 Co = 0.012 μ0Hc2(T)H| |c χac Cp WHH FIG. 2. (a) The temperature dependence of the magnetic susceptibility in an external field of 1 mT is applied along the caxis. The superconducting volume fraction ( Vfr) is reduced by increased doping by either introducing Co or S to the FeSe system. (b) Temperature dependence of C p/T vsTin zero magnetic field. The inset presents specific-heat data of Fe 0.98Co0.012Se in various applied magnetic fields up to 9 T parallel to the caxis. (c) The temperature dependence of the complex ac susceptibility components of Fe 0.98Co0.012Se measured in an ac field with an amplitude of 0.5 mT and a frequency of 1 kHz up to 9 T. The data was collected upon warming in different dc magnetic fields after cooling in a zero magnetic field. (d) Summarizes the phase diagram of Hc2vs temperature of Fe 0.98Co0.012Se for the field applied parallel to c.Tchas been estimated from an entropy-conserving construction and ac measurements. The open symbols are estimated from the ac magnetization, while the closed circles represent the specific heat. The dashed line represents the WHH model for λ=0,α=0. Fe0.98Co0.012Se sample. The small differences observed be- tween the data obtained from the specific heat and the acmagnetization for H/bardblcare not surprising because these meth- ods naturally imply different criteria for the T cdetermination. In order to determine the upper critical field Hc2for the corientation, we used the single-band Werthamer-Helfand- Hohenberg (WHH) formula [ 37] for an isotropic one-band BCS superconductor in the dirty limit. An example of WHHfitting is shown with the dashed line in Fig. 2(d). The WHH theory ( α=0,λ so=0) predicts the behavior of Hc2(Tc), where αis the Maki parameter which describes the relative strength of orbital breaking and the limit of paramagnetism,andλ sois the spin-orbit scattering constant [ 37]. Using the data in Fig. 2(d), the upper critical field value at T=0f o r the Fe 0.98Co0.012Se system was evaluated to be ≈11.5T . I t is evident that the one-band WHH model fails to satisfy theextracted H c2(0). Using an additional two-band model with s-wave-like gaps, the temperature dependence of the electronic specific-heat data in Fe 1−xCoxSe can be well described, whereas single-gap BCS theory under the weak-couplingapproach cannot describe our data (the data will be publishedelsewhere). Therefore, we believe that the observed deviationfrom the single-band WHH model is related to multiband effects in Co-doped FeSe. 2. Structural transition To investigate the nature of tetragonal-to-orthorhombic phase transition in FeSe 1−xSx, we conducted specific-heat and electrical resistivity ρ(T) measurements. Specific-heat data was collected up to 200 K for FeSe 1−xSx(x=0, 0.04, and 0.11) and is presented in Fig. 3. Clear and sharp anomalies were resolved at the structural phase transition,hinting that an electronic structure transition took place, asa consequence of the nematic electronic transition. Data forthe FeSe superconductors shows a very sharp orthorhombicphase transition at 87 K (upon heating), with a width of about2 K. Upon S doping, the structural anomaly of the parent compound gradually shifted to lower temperatures down to 81 K and 72 K, for x=0.04 and 0.11, respectively. The error in the determination of the T stransition temperatures is estimated to be around 1 K when we consider the fact thatthe peak in the first derivative of the specific heat is relativelysharp. The specific heat anomaly at T sgives/Delta1Cp/Ts≈5.57, 5.43, and 4.1 mJ /mol K2forx=0.04 and 0.11, respectively. 224508-4IMPURITY SCATTERING EFFECTS ON THE . . . PHYSICAL REVIEW B 93, 224508 (2016) 0 50 100 150 2000100200300 30030531031532065 70 75 80 85 FeSe0.89S0.11 T(K)Cp/T(mJ/mol K2) FeSe FeSe0.96S0.04 FIG. 3. Temperature dependence of specific heat with temper- atures of up to 200 K for FeSe 1−xSx(x=0, 0.04, and 0.11). The anomalies at higher temperatures which reflect the structural transition (see inset) are in agreement with the resistivity data. Interestingly, the value of these anomalies at Tsis very close to the Sommerfield coefficient, γn, at low temperature [ 27]. This can be directly linked to the reconstruction of the Fermi surfaceatT sand reflects an electronic instability in our investigated systems. The electronic instability is supported by the field dependence of the magnetotransport at 12 K, which shows anabrupt sign change, suggesting a drastic reconstruction of theFermi surfaces across the structural transition [ 38]. In addition, ARPES data at 30 K shows two holelike bands at the Mpoint, in contrast with the single holelike band seen at 120 K [ 39]. This is likely caused by the formation of electronically drivennematic states. The resistivity ρ(T) for FeSe 1−xSxis shown in Fig. 4. All compounds are metals with resistivities ρ(250 K) varying from 0.708 m /Omega1cm for the parent compound (see Fig. 5)t o 0.36, 0 .52 m/Omega1cm for xS=0.19 and xCo=0.04, respectively. This reflects the good quality of the investigated crystals.The upper inset presents the derivative of resistivity curvesforx=0.04, 0.09, and 0.11. With increasing S doping, the nematicity is shifted to lower temperature and disappears underheavy doping. Simultaneously, the resistivity shows a dropat lower temperatures and zero resistivity at optimal dopingwithT c=11.5 K, indicating the coexistence of nematicity and superconductivity. This coexistence is observed up tox=0.15 and no anomaly is associated with the nematic order forx=0.15 and 0.19. The lower inset in Fig. 4presents the temperature-dependence resistivity of FeS 0.81S0.19in various applied magnetic fields up to 3 T parallel to the caxis. Another noteworthy peculiarity is the large variability of the room temperature resistivity of our investigatedsamples as shown in Fig. 4. It can be partially explained by the large error in the geometry factor but the ratio ofresistivity of about 2.0 for samples with x=0 and x= 0.15 definitely exceeds the possible error of our calcula- tion. On the other hand, the absolute value of resistivityreported for pure FeSe differs significantly (more than threetimes over the value observed for x=0.19). For example,0 100 200 3000.00.20.40.60.8 100 200 300 0.000.050.10468 x=0 .19TsT* T* x=0 . 1 1 T(K)x=0 . 0 7x=0 . 0 4 dρab/dT(a.u.) x=0 . 1 9x=0 . 1 5x=0 . 1 1 x=0 . 0 9x=0 . 0 4 T(K)ρ(mΩcm)x=0 . 0 1 5 0T3T FIG. 4. In-plane resistivity of FeSe 1−xSxin a zero field. The upper inset presents the derivative of ρforx=0.04, 0.09, and 0.11. The arrows represent both Tsand the T∗. The lower inset shows the resistivity data for FeS 0.81S0.19in various applied magnetic fields up to 3 T, parallel to the caxis. the estimated room temperature value is changed from 0.4 m/Omega1×cm to 1.7 m /Omega1×cm [ 38]. However, the absolute value of resistivity was not precisely determined [ 38]. There- fore, it is not clear if the difference comes from systematicerrors or from the resistivity dependence on any other(unknown) parameters, as for example, iron stoichiometry,impurity level, or chemical degradation in particular crystals.Nevertheless, based on our data there is a clear trend of a 0 100 200 300 4000.00.20.40.60.81.0 100 200 300 400100 200 300 400FeSedρab/dT (a.u.) T*Fe0.96Co0.04SeFeSe0.81S0.19 Ts1.6 GPa 0G p a T(K)ρ(mΩcm) T(K) dρab/dT (a.u.) Ts FIG. 5. The temperature dependence of the in-plane resistivity of FeSe by chemical and physical pressures in zero field. Increasing the pressure leads to a suppression of the nematic order state andto further suppression of the wide hump at higher temperature. The lower inset presents the derivative of an FeSe curve, displaying a sharp minima at T sand maximum at T∗. The upper inset shows the derivative curves for xCo=0.04 and xS=0.19. 224508-5M. ABDEL-HAFIEZ et al. PHYSICAL REVIEW B 93, 224508 (2016) large resistivity decrease with sulfur substitution confirmed by presented curves and our other measurements. These resultsare very similar to those reported for the in-plane resistivity ofFeSe single crystals under pressure [ 40]. With the application of pressure, the room temperature resistivity decreases by afactor of more than 3; it reaches a minimum at 10 GPa [ 40]. It is important to note that the residual resistance ratio in our cases,FeSeS, is not adequate due to the complex shape of R(T). Nevertheless, in the case of the x=0.19 sample, for which the R(T) range is minimal, the ratio of resistivity at 10 K and 100 K is equal to 8, which is equivalent to about 25 at 300 K. However,the most important issue is the absence of any traces ofimpurity or defect scattering at low temperatures in measuredR(T) curves, which reflects the good quality of crystals. 3. Chemical and physical pressure on FeSe In order to further explore the effect of pressure and doping on the FeSe single crystal, the temperature dependence of thein-plane resistivity of FeSe, FeSe 0.81S0.19, and Fe 0.96Co0.04Se single crystals is summarized in Fig. 5. At the parent com- pound, resistivity decreases on cooling and shows an anomalyassociated with the structural phase transition at T s≈86 K and a sharp superconducting transition at Tc≈8.9K .T h i si s in agreement with the specific heat. It is obvious that the dρ/dT in FeSe at ambient pressure exhibits a remarkable feature withsharp minima at T sand a maximum at T∗associated with a wide hump at high temperature, which shows a crossover froma probable semiconductinglike behavior to metallic behavior(see the inset in Fig. 5). A similar and consistent issue concerning the wide hump has been previously reported [ 6,41]. However, the origin of this crossover at high temperature couldbe associated with a change of carrier density. The values ofT sandT∗were obtained from the features in the resistivity derivative (insets of Figs. 3and4). However, the hump phe- nomenon has been found in other iron-selenide, K xFe2−ySe2, superconductors [ 42,43], but it was not present in FeAs-based superconductors, where resistivity data for the pristine ordoped compound exhibit a metallic behavior over the entiretemperature range [ 44]. More interestingly, T c,Ts, and the maximum in dρ/dTare suppressed by increasing the Co or S doping in FeSe. Upon compression to 1.6 GPa, the structuraltransition becomes significantly suppressed with increasingthe pressure. Therefore, the structural transition in FeSe is ini-tially suppressed under applied physical pressure with a similarmanner to the chemical pressure effect of S substitution. A re-markable observation upon compression is the linear behaviorof resistivity below 400 K, which is also reported for otherFe-based superconductors [ 45]. Nuclear magnetic resonance (NMR) measurements on FeSe show that with cooling belowT sspin fluctuations exist and even increase upon applying hydrostatic pressure [ 46]. Therefore, we cannot evidence the linear behavior of resistivity in FeSe upon compression withthe strength of antiferromagnetic spin fluctuations. C. Electronic phase diagram Using the experimental results of the thermodynamic and electrical resistivity data, we summarize the evolution of thedistinct features of impurity scattering in the FeSe system.TheT c,Ts, andT∗of FeSe 1−xSxsingle crystals, as a function FIG. 6. The S concentration ( x) dependence of the superconduct- ing transition temperature ( Tc), structural transition ( Ts), and the T∗ obtained from magnetic, specific-heat, and electric resistivity data. Tsis compared to values reported in Ref [ 8]. The phase diagram highlights the suppression of Tsand the transition at T∗by increasing the S concentration. The inset summarizes the Co concentration dependence, in which the Tcdecreases upon increasing doping. of the S content, are shown in Fig. 6.B o t h Tsand the maximum of the dρ(T)/dT,T∗, are intimately linked, even for under/optimal doping. In the overdoped regime theselinked features are suppressed by doping and disappear atx=0.15. However, we shallow the area of the T ∗in the main panel of Fig. 6.T h ei n s e to fF i g . 6illustrates the electronic phase diagram of Fe 1−xCoxSe. This correlated suppression of both TsandTccould be related to the orbital fluctuation induced by either Co or S substitutions.Additionally, the nesting between electron and hole pocketsplays an important role in this suppression of structuralordering in Fe-based superconductors [ 18,47]. In our case upon S doping and once nematicity is suppressed, superconductivitystarts to decrease. Concentrations shows suppression of bothnematicity and superconductivity in FeSe. This is in contrastto Co-doped Ba(Fe 1−xCox)2As2where superconductivity is enhanced instead by the suppression [ 22]. Therefore, we think that we cannot rule out other roles of charge dopingbesides suppressing nematicity. Although the K-dosed FeSeand FeSe under high pressure both show suppressed nematicityand an enhanced T caround 40 K, they are different in several important regards: (i) The K-dosed FeSe is heavilyelectron doped with only electron Fermi surfaces, while theFeSe under pressure should be undoped with very differentFermi-surface topology, and (ii) FeSe under high pressureshows a compressed lattice and reduced anion height, dueto the external pressure, compared with K-dosed FeSe. Despite this, whether the nematic order is driven by a spin or an orbital fluctuation remains controversial. If orbital orderingis the efficient cause, the phase below the nematic breaks C 4 symmetry, and quantum fluctuations associated with this phase are nematic in character [ 48–50]. However, the resistivity data exhibits a non-Fermi-liquid-like behavior above Tc, which would suggest orbital fluctuations exist below the nematic 224508-6IMPURITY SCATTERING EFFECTS ON THE . . . PHYSICAL REVIEW B 93, 224508 (2016) -0.2 0180K 282K-0.20 -0.20 -0.20 -0.4 0.4 -0.4 0.4 -0.4 0.4 -0.4 0.4 -0.4 0.4180K 200K 225K 250K 282K k1-0.10 -0.10 -0.10 -0.4 0.4 M -0.4 0.4 M -0.4 0.4 M -0.4 0.4 M -0.4 0.4 M180K 200K 225K 250K 282K LowHigh LowHigh LowHigh(a) (b) (c) 180K 282K Intensity (arb. units)-0.2 0(d) (e)(f) (g) (h) 180K -0.2 0282K(i) 180K -0.2 0282K(j) 0.03 0 -0.03 -0.2 M 0.2180K 282K(k) Energy shift (meV)-40-200 250 200 150 T(K)(l) Intensity (arb. units)282K 250K 225K 200K 180K 282K-0.3 0.3 M(m) E-E(eV)E-E(eV) E-E(eV) E-E(eV) E-E (eV) E-E (eV) k (Å)k (Å) k (Å) k (Å) FIG. 7. The temperature dependence of the band structure of FeSe 1−xSx(x=0.055): (a) Temperature dependence of the photoemission spectra around /Gamma1, (b) the spectra divided by the energy-resolution-convoluted Fermi-Dirac function, and (c) their second derivative with respect to energy. The red dashed lines are local minimum locus to indicate the band position of α,β,a n dω. (d) The energy distribution curves (EDCs) divided by the energy-resolution-convoluted Fermi-Dirac function at /Gamma1with varied temperature. (e) The second derivative of the EDCs in panel (d), the positions of the band top of β,a n dωare obtained by tracking the local minimum locus of the EDCs. (f)–(h) are the same as (a)–(c), respectively, but around M, the red dashed curves indicate the dispersion of α,/epsilon1. (i) The same as panel (d), but at k1. The momentum position of k1 is indicated in panel (h). (j) The same as (e), but at k1. The energy positions of αat k1 is obtained by tracking the local minimum locus of the EDCs. (k) The temperature dependence of the dispersion of the /epsilon1band, which is obtained by tracking the local maximum locus of the momentum distribution curves (MDCs) at different temperatures. (l) The energy shifts as a function of temperature for the different bands.(m) The MDCs integrated near Fermi energy ( E F) over ( EF−10 meV , EF+10 meV) with a loop in temperature. order [ 49]. Additionally, there is no change of the Tsanomaly under 9 T in transport and specific-heat measurements of FeSe(not shown), which might indicate that spin fluctuations are notinvolved directly in the structural transition. However, recentsound experimental studies on the origin of the nematic phasein iron chalcogenides reach opposing conclusions and thisquestion remains highly debated. Experimental evidence ofthe existence of strong nematic fluctuations up to 200 K hasbeen reported in Ba(Fe ,Co) 2As2[51]. However, NMR mea- surements suggest the absence of spin fluctuations above Ts in the tetragonal phase and spontaneous orbital order has been invoked which explains the nematic state in FeSe [ 7,9]. In con- trast to the NMR data, recent neutron scattering measurementsreveal substantial spin fluctuations in the tetragonal phase inFeSe [ 52]. These measurements demonstrate that the absence of spin fluctuations suggested by NMR is simply due to theopening of a 2.5 meV spin gap in a quantum nematic para-magnetic state (NMR only probes very low energy spin fluc-tuations). Furthermore, very recently, Glasbrenner et al. [53] have shown that the long-range magnetic ordering in FeSe isprevented by the excitation of spin fluctuations, but allows theusual spin-driven nematic order. Additionally, the spin-drivennematic order is also accompanied by a ferro-orbital order. D. ARPES A noticeable change in the electronic properties of FeSeS is observed in our transport and thermodynamic data. This canbe consistently explained by the Fermi-surface reconstruction under sulfur substitution. The Lifshitz-type quantum transitioncan be a possible source of changes in absolute values ofresistivity and suppression of certain power terms in the lowtemperature R(T) expansion. According to our data the tran- sition can happen between 5% and 10% of sulfur substitution.Therefore, to check this possibility the microscopic propertiesof Fe(SeS) should be addressed. However, to comprehend theT ∗in the main panel of Fig. 6, it is also very interesting to comprehend the real band structure. Additionally, the hump atelevated temperatures seems to be a standard feature of anydegenerated semiconductors observed many times in varioussystems [ 54], which only reflects the crossover between semi- conducting and metallic behavior. In order to further explorethis behavior at higher temperature above structural transition,we performed ARPES measurements at different temperaturesabove the nematic transition temperature in FeSe 1−xSxsingle crystals for x=0.055. From the temperature dependence of band structure around /Gamma1shown in Figs. 7(a)–7(c), the generic features for all the different temperatures include two parabolicbands noted as αandβnearE F, and a relative flat band noted asωat high binding energy. An energy shift of bands exists with increasing temperature [Figs. 7(a)–7(c)]. At 180 K, the top of the hole band αaround /Gamma1is slightly above EFwithin 10 meV and the top of the hole band βaround /Gamma1is about 7 meV below EF[Fig. 7(c)]. At 282 K, both αandβcompletely sink below the EF[Fig. 7(c)], indicating a temperature-induced Lifshitz transition, similar to those in Ba(Fe ,Co) 2As2[55] and 224508-7M. ABDEL-HAFIEZ et al. PHYSICAL REVIEW B 93, 224508 (2016) in WTe 2[56]. Note that the band tops of αandβare within the energy scale of the thermal excitation; these bands stillcontain hole carriers at 282 K although they have shifted belowE F[55]. Quantitatively, as shown in Figs. 7(d) and7(e),t h e energy shifts of the βband and the ωband at /Gamma1are remarkably similar. It should be noted that the temperature of the Lifshitztransition is much higher than the temperature for the nematictransition. As shown in the temperature dependence of band structures around M[Figs. 7(f)–7(h)], the generic features for all the different temperatures include an electronlike band noted as/epsilon1nearE F, and a parabolic band noted as αat high binding energy. At 180 K, the bottom of /epsilon1is≈35 meV below EF [Fig. 7(h)]. In Figs. 7(f)–7(h), an energy shift of the two bands exists with increasing temperature. As shown by the energydistribution curves (EDCs) at k1 in Figs. 7(i)–7(j),αgradually shifts to higher binding energies with increasing temperature.Moreover, band /epsilon1shifts to higher binding energies rigidly [Fig. 7(k)]. As shown by the quantitative analysis of energy shifts in Fig. 7(l), all bands near /Gamma1andMshift similarly with increasing temperature, indicating a temperature-inducedchemical potential shift, i.e., a Lifshitz transition, involving achange of the Fermi-surface topology in FeSeS. The tempera-ture cycle measurement between 282 K and 180 K [Fig. 7(m) ] demonstrates that the temperature-induced chemical potentialshift is intrinsic. The temperature-induced chemical potential shift has been observed in several materials, such as Ba(Fe ,Co) 2As2[55], Ba(Fe,Ru) 2As2[57], and WTe 2[56,58]. The origin of the shift has been explained by the thermal excitations of carriersin semimetals, where the top of the hole bands and bottom ofthe electron bands are close to the chemical potential withinthe energy range of thermal broadening. As calculations showin [55,56], the numbers of hole and electron carriers both increase with increasing temperature due to thermal excitation.However, if the chemical potential μwere fixed, the increased number would have been different for hole and electroncarriers according to the calculations [ 55,56]. To avoid this, charge carriers redistribute from holes to electrons to keepthe conservation of the net charge of carriers (proportionalto the filling), resulting in the chemical potential shift and aLifshitz transition [ 55,56]. For FeSeS, the observed semimetal behavior of the electronic structure meets the prerequisite ofthe scenario proposed in Refs. [ 55,56], and can qualitatively explain the shift of chemical potential observed here. IV . CONCLUSION To summarize, from extensive thermodynamics, transport, and ARPES studies, we report on the effect of Co and Ssubstitution on the superconductivity and structural transi-tion/orbital order in FeSe. Furthermore, images of the magneticflux penetration in the whole sample show that it stronglydeparts from the Bean critical state model, often applied to hardtype-II superconductors such as iron-based superconductors.We demonstrate that /Delta1C p/Ts≈γnin S-doped systems due to the reconstruction of the Fermi surface at Ts, which reflects an electronic instability in this system. We have shown thatFeSe exhibits remarkable features with a wide hump athigh temperature, suppressed by Co or S doping or externalpressure. This hump, together with the nematic order, wassuppressed by further doping. Our ARPES data between 180 Kand 282 K indicate that chemical potential shift with increasingthermal excitations exists, resulting in a change of the Fermi-surface topology. In addition, the temperature-induced Lifshitztransition is similar to WTe 2and Ba(Fe ,Co) 2As2is observed. Our results establish the correlation between superconductivityand the nematicity. ACKNOWLEDGMENTS We are grateful to Alexander Kordyuk and Goran Kara- petrov for stimulating discussions. The work in Germany wassupported by program MO 3014/1-1 of the DFG. The work inRussia was supported in part by the Ministry of Education andScience of the Russian Federation in the framework of IncreaseCompetitiveness Program of NUST “MISiS” (K2-2015-075),and by Act 211 of the Russian Federation Government,Contract No. 02.A03.21.0006. J.B. acknowledges supportfrom F.R.S.-FNRS (Research Fellowship). The work of A.V .S.has been partially supported by the “Mandat d’ImpulsionScientifique” MIS F.4527.13 from F.R.S.-FNRS. [1] R. M. Fernandes, A. V . Chubukov, and J. Schmalian, What drives nematic order in iron-based superconductors?, Nat. Phys. 10,97(2014 ). [2] P. C. Dai, J. P. Hu, and E. Dagotto, Magnetism and its micro- scopic origin in iron-based high-temperature superconductors,Nat. Phys. 8,709(2012 ). [3] D. Parker, M. G. Vavilov, A. V . Chubukov, and I. I. 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PhysRevB.76.052402.pdf

Spin-gap behavior in the two-leg spin-ladder BiCu 2PO6 B. Koteswararao,1S. Salunke,1A. V. Mahajan,1I. Dasgupta,1and J. Bobroff2 1Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India 2Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France /H20849Received 10 July 2007; published 14 August 2007 /H20850 We present magnetic susceptibility and heat capacity data on a new S=1/2 tw o-leg spin ladder compound BiCu 2PO6. From our susceptibility analysis, we find that the leg coupling J1/kBis/H1101180 K and the ratio of the rung-to-leg coupling J2/J1/H110110.9. We present the magnetic contribution to the heat capacity of a two-leg ladder. The spin-gap /H9004/kB=34 K obtained from the heat capacity agrees very well with that obtained from the magnetic susceptibility. Significant interladder coupling is suggested from the susceptibility analysis. Thehopping integrals determined using the Nth order muffin-tin-orbital based downfolding method lead to ratios of various exchange couplings in agreement with our experimental data. Based on our band structure analysis, wefind the interladder coupling in the bcplane J 3to be about 0.75 J1placing the compound presumably close to the quantum critical limit. DOI: 10.1103/PhysRevB.76.052402 PACS number /H20849s/H20850: 75.10.Pq, 71.20. /H11002b, 75.40.Cx INTRODUCTION Following the discovery of high-temperature supercon- ductivity /H20849HTSC /H20850in the cuprates,1there has been an in- creased focus on the properties of low-dimensional antifer-romagnetic systems. This is due to the innate exoticproperties of these magnetic systems themselves and theirsupposed connection with HTSC. Significant work has takenplace recently elucidating the properties of S=1/2 and 1 Heisenberg chains and their response to impurity substitu-tions. Whereas quantum fluctuations prevent long-range or-der /H20849LRO /H20850in one-dimensional /H208491D/H20850Heisenberg systems, three-dimensional /H208493D/H20850systems exhibit conventional LRO. On the other hand, in two-dimensional /H208492D/H20850systems where the strength of magnetic interactions and quantum fluctua-tions can be comparable, one might expect competingground states and a quantum critical point separating them.Spin-ladders serve as a bridge between one-dimensional/H208491D/H20850and two-dimensional /H208492D/H20850magnetic systems and it is believed that an improved understanding of spin-ladders willlead to a better understanding of magnetism in the 2D sys-tems. A major step was taken in this direction with the pre-diction of spin gaps in even-leg ladders and their absence inodd-leg ladders, 2followed by experimental verification in SrCu 2O3/H20849two-leg ladder /H20850and Sr 2Cu3O5/H20849three-leg ladder /H20850.3 However, in spite of the large experimental effort, only a small number of gapped ladders have been synthesized andstudied. Of these, only two /H20849LaCuO 2.5and Sr 14Cu24O41/H20850 could be doped significantly with holes of which only thelatter becomes superconducting. 4Some other compounds which have been investigated are /H20849C5H12N/H208502CuBr 4/H20849Ref. 5/H20850, Cu2/H20849C5H12N2/H208502Cl4/H20849Ref. 6/H20850, and Cu 2/H20849C5H12N2/H208502Br4/H20849Ref. 7/H20850 which have substantially smaller spin gaps. There is contin- ued effort to synthesize and study new low-dimensional sys-tems since they provide a rare opportunity to elucidate thesignificance of low-dimensionality, spin gap, etc. to HTSC asalso allow one to examine impurity and doping effects in astrongly correlated cuprate. In this Brief Report, we report on the preparation and properties of a cuprate which, we demonstrate, can be mod-eled as a two-leg ladder system with significant interladdercoupling in the bcplane and negligible interplanar coupling. The spin gap, as determined from our susceptibility and heatcapacity measurements is about 34 K while the intraladderleg coupling is about 80 K. Our electronic structure calcula-tions within the framework of the Nth order muffin-tin- orbital /H20849NMTO /H20850downfolding method 8yield hopping inte- grals between various Cu atoms. Using the NMTOdownfolding method, we calculate the Wannier-like effectiveorbitals which illustrate the shape and extent of the active Cuorbitals and therefore indicate the exchange pathways whichlead to the ladder topology. From a practical standpoint, theestimated J/k B/H1101580 K provides a unique opportunity to ex- amine the excitations of the coupled ladder system at tem-peratures ranging from well above J/k Bto well below J/kB. Impurity substitutions will then allow us to probe the natureof magnetic effects thus induced, in a wide temperaturerange. CRYSTAL STRUCTURE AND MEASUREMENTS Our measurements are on single phase, polycrystalline BiCu 2PO6samples /H20849space group Pnma with lattice param- eters a=11.776 Å, b=5.1776 Å, and c=7.7903 Å /H20850. A schematic diagram of the structure is shown in Fig. 1. The unit cell contains four formula units, with two inequiva-lent Cu /H20849Cu1 and Cu2 /H20850sites and four inequivalent O /H20849O1- O4/H20850sites. The characteristic feature of the structure are CuO 5 distorted square pyramids, with a Cu2+ion at the center of the fivefold oxygen coordination. Two such pyramids sharean edge formed from a pair of basal oxygens /H20849O2/H20850to give rise to a Cu dimer with an intradimer distance of 2.8 Å.Along the baxis, each dimer connects two others by its four O1 corners resulting in a zigzag double chain /H20849ladder /H20850run- ning along the baxis /H20849see Fig. 1/H20850. The interdimer cohesion is further strengthened by PO 4tetrahedra that connect two con- secutive dimers by O2 corners. The Bi ions are positionedbetween two ladders. The Cu-O-Cu angle along the leg isabout 112° and that along the rung is about 92°. In Fig. 1, various exchange couplings /H20849J 1,J2, etc. /H20850and hopping inte- grals /H20849t1,t2, etc. /H20850have been indicated.PHYSICAL REVIEW B 76, 052402 /H208492007 /H20850 1098-0121/2007/76 /H208495/H20850/052402 /H208494/H20850 ©2007 The American Physical Society 052402-1Our results of the susceptibility /H9273meas /H20849magnetization M divided by applied field H/H20850as a function of temperature T using a vibrating sample magnetometer /H20849VSM /H20850of a physical property measurement system /H20849PPMS /H20850from Quantum De- sign are plotted in Fig. 2. As seen, /H9273meashas a broad maxi- mum around 57 K /H20849indicative of a low-dimensional magnetic system /H20850below which it drops rapidly /H20849suggestive of a spin gap/H20850. A very weak low-temperature upturn is seen below 6.5 K, likely due to extrinsic paramagnetic impurities and/ornatural chain breaks in our polycrystalline sample. We nowanalyze these data quantitatively. An analytical solution for the spin-susceptibility of two- leg ladders in the full T-range is not known. However, Johnston, based on extensive quantum Monte Carlo /H20849QMC /H20850 simulations 9has proposed an equation which accurately re- produces the QMC-determined susceptibilities at discretetemperatures. This equation /H20849not reproduced here since it is unwieldy /H20850is useful for determining the exchange couplingsby fitting the measured susceptibility data and has been used to analyze such data in the two-leg ladder SrCu 2O3.W e then fit /H20849dashed line in Fig. 2/H20850/H9273meas to/H9273o+C//H20849T−/H9258/H20850 +m/H9273ladder /H20849T/H20850where the fitting parameters are /H9273o,C,/H9258,J2/J1, J1, and m. Here /H9273ladder /H20849T/H20850is the/H9273of isolated ladders as given by Johnston.9In the absence of a generic fitting function which can take into account arbitrary interladder interac-tions, we attempt to do so using the parameter m. With mas a variable, the obtained parameters are /H9273o=/H208494.4±0.1 /H20850 /H1100310−4cm3/mol Cu, C=/H208493.0±0.2 /H20850/H1100310−4cm3K/mol Cu, /H9258/H110110K , J2/J1=0.87±0.05, J1/kB=/H2084980±2 /H20850K, and m =0.41±0.02. The value of the spin gap using9/H9004/J1 =0.4030 /H20849J1 J2/H20850+0.0989 /H20849J1 J2/H208503is about 34 K. The Curie constant corresponds to less than 0.1% of isolated S=1/2 impurities. This value is comparable to typical parasitic Curie termsfound in single crystals, indicating the very high quality ofour samples. Since the core-diamagnetic susceptibility 10/H9273core is −0.6 /H1100310−4cm3/mol, /H9273o−/H9273coreyields the Van Vleck sus- ceptibility /H9273VV=5/H1100310−4cm3/mol which is somewhat higher than/H9273VVof other cuprates. We show in Fig. 2the curve for isolated two-leg ladders /H20849with J1/kB=80 K /H20850. We also show the simulated curve for a uniform, 2D S=1/2 HAF with J/kB=80 K where the high- Tbehavior is generated using the series expansion given by Rushbrooke and Wood.11 Also, Johnston12parametrized the low- T/H20849kBT J/H333551/H20850simula- tions of Takahashi13and Makivic and Ding,14which we use. The experimental data are lower than both the 2D HAFcurve and the isolated ladder susceptibility. This behaviorpoints to the importance of a next-nearest-neighbor /H20849NNN /H20850 interaction along the leg /H20849which might be expected due to the zigzag nature of the leg /H20850which might be frustrating and might even enhance the spin gap. In a latter section, based onour band-structure calculations, we actually find significant FIG. 1. /H20849Color online /H20850A schematic of the BiCu 2PO6crystal structure is shown. It can be seen that two-leg ladders run along thecrystallographic bdirection. The two-leg ladder is separately shown for clarity. Also shown are the various significant hopping param-eters and exchange couplings between Cu atoms.FIG. 2. Magnetic susceptibility /H20849/H9273meas=M/H/H20850vs temperature T for BiCu 2PO6in an applied field of 5 kG. The open circles repre- sent the raw data and the dashed line is a fit /H20849see text /H20850. Also shown are simulated curves for the isolated ladder /H20849dark gray line /H20850and for the 2D HAF /H20849gray line /H20850. The inset shows the dependence of /H9273*on kBT/J1/H20849see text /H20850.BRIEF REPORTS PHYSICAL REVIEW B 76, 052402 /H208492007 /H20850 052402-2NNN as also interladder couplings. The absence of LRO in spite of these deviations from the isolated ladder pictureshould motivate the theorists to refine their models of suchsystems. In the inset of Fig. 2, we plot the normalized sus- ceptibility /H9273*/H20849T/H20850=/H9273spin/H20849T/H20850J1//H20849Ng2/H9262B2/H20850/H20851where /H9273spin/H20849T/H20850=/H9273meas −/H9273o−C/T/H20852as a function of kBT/J1. We find /H9273*,max/H20849i.e.,/H9273*at the broad maximum /H20850to be about 0.05 which is lower than the expected value for isolated ladders of about 0.12. To further confirm the spin-gap nature of BiCu 2PO6,w e did heat capacity Cpmeasurements. Since the lattice Cp dominates the data, it has so far not been possible to experi- mentally determine the magnetic contribution to Cpin any spin-ladder compound unambiguously. In the present case,we are fortunate to have a nonmagnetic analog of BiCu 2PO6 in BiZn 2PO6. We have then determined the magnetic heat capacity CMof BiCu 2PO6by subtracting the measured Cp of BiZn 2PO6from that of BiCu 2PO6/H20849see Fig. 3inset /H20850. The data are fit to9CM/H20849T/H20850=3 2NkB/H20849/H9004 /H9266/H9253/H208501/2/H20849/H9004 kBT/H208503/2/H208511+kBT /H9004 +0.75 /H20849/H9004 kBT/H208502/H20852exp/H20849−/H9004 kBT/H20850shown by the solid line /H20849Fig.3inset /H20850. From the fit, the spin gap/H9004 kB/H1101134 K, in excellent agreement with our susceptibility results. FIRST PRINCIPLES STUDY The local density approximation-density functional theory /H20849LDA-DFT /H20850band structure for BiCu 2PO6is calculated using the linearized-muffin-tin-orbital /H20849LMTO /H20850method based on the Stuttgart TB-LMTO-47 code.15The key feature of the non- spin-polarized electronic structure presented in Fig. 4is eight bands crossing the Fermi level which are well-separatedfrom the rest of the bands. These bands are predominantlyderived from the antibonding linear combination of Cu d x2-y2 and basal O p/H9268states in the local reference frame where the zaxis is along the shortest Cu-O bond while the xandyaxes point along the basal oxygens O1 and O2. The band structureis 2D with practically no dispersion perpendicular to the lad-der plane /H20849along /H9003X/H20850. The eight band complex is half-filled and metallic as expected in LDA. It lies above the otheroccupied Cu- d,O -p, and Bi- scharacter dominated bands. The P /H20849s,p/H20850and Bi /H20849p/H20850derived states are unoccupied and lie above the Fermi level, with the Bi- pstates having non- negligible admixture with the conduction bands. This admix-ture of the conduction band with Bi- pstates is important in mediating the Cu-Cu interladder exchange coupling. Startingfrom such a density functional input we construct a low-energy model Hamiltonian using the NMTO downfoldingtechnique. This method 8extracts energy selective Wannier- like effective orbitals by integrating out high energy degreesof freedom. The few orbital Hamiltonian is then constructedin the basis of these Wannier-like effective orbitals. Here, weshall retain only Cu d x2-y2orbital in the basis and downfold the rest. The effective Cu dx2-y2muffin-tin orbitals generated in the process will be renormalized to contain in their tail other Cu- d,O -p, Bi, and P states with weights proportional to the admixture of these states with Cu dx2-y2. Fourier trans- form in the downfolded Cu dx2-y2basis gives the desired tight-binding Hamiltonian H=/H20858/H20855i,j/H20856tij/H20849cj†ci+ci†cj/H20850in terms of the dominant Cu-Cu hopping integrals tij. This tight binding Hamiltonian will serve as the single electron part of themany-body Hubbard model relevant for this system and canbe mapped to an extended Heisenberg model with the ex- change couplings related to the LDA hoppings by J ij=4tij2 Ueffwhere Ueffis the screened onsite Coulomb interaction. The various hoppings are displayed in Table Iand indicated in Fig. 1. The intradimer /H20849rung /H20850hopping proceeds mainly via the edge sharing oxygens while the interdimer interaction/H20849leg hopping /H20850proceeds via the corner sharing oxygens with support from the PO 4complex. As anticipated in the experi- ments, we do indeed find that the ratio of the rung hopping tothe leg hopping J 2/J1/H110151. We find that the NNN coupling along the leg J4is about 0.3 J1. Depending on the relative sign of this interaction with respect to that of J1one might get significant frustration effects which should also have abearing on the ground state of the system. We also find anappreciable coupling between the ladders /H20849J 3/J1/H110150.75/H20850me- diated primarily by the unoccupied Bi- pstates. Our conclu-FIG. 3. /H20849Color online /H20850The measured heat capacity as a function ofTfor BiCu 2PO6and BiZn 2PO6. Inset: the magnetic specific heat of BiCu 2PO6along with a fit /H20849see text /H20850.-6-4-2024 Γ X S Y Γ ZEnergy (eV) Cu−Bi− d 22x− y Cu−xy O− pp Cu− d3z−r22 Cu− dxz Cu−d dyz FIG. 4. /H20849Color online /H20850LDA band dispersion of BiCu 2PO6along various symmetry directions.BRIEF REPORTS PHYSICAL REVIEW B 76, 052402 /H208492007 /H20850 052402-3sion is further supported by the plot of the corresponding Cudx2-y2Wannier function in Fig. 5. We find that each Cudx2-y2orbital in the unit cell forms strong pd/H9268antibond- ing with the neighboring O- px/O-pyorbitals resulting in the conduction band complex. The Cu ions strongly couplealong the leg as well as the rung of the ladder confirming thatthe hoppings in either direction should be comparable. TheO-p x/O-pytails bend towards the Bi atom, indicating the importance of the hybridization effect from the Bi cationsand therefore enhances the Cu-Cu interladder exchange in-teraction /H20849see Table I/H20850. CONCLUSION In conclusion, we have presented a S=1/2 two-leg ladder BiCu 2PO6. From our /H9273and CMdata we obtain a spin gap /H9004/kB/H1101134 K and a leg coupling J1/kB/H1101180 K. From our first principles LDA-DFT calculations, we find J2/J1/H110111 and a significant interladder interaction in the corrugated bcplane /H20849J3/J1/H110110.74/H20850. Considering that the uniform S=1/2 2D AF system has an ordered ground state, we feel that the strong interladder interaction in BiCu 2PO6places it close to a quan- tum critical point. The moderate value of the /H9004/kBin BiCu 2PO6will allow one to explore the magnetic properties in a large Trange, well below and well above the gap tem- perature, enabling a comparison with and refinement of the-oretical models. We feel that there might still be unantici- pated features in the physics of low-dimensional magnetsand we expect our work to motivate others to carry out nu-merical simulations and explore the phase diagram ofcoupled two-leg ladders in the presence of NNN couplingsalong the leg. We are presently considering doping and sub-stitutions in this two-leg ladder which might be able to tunethe interladder exchange and effect a quantum phase transi-tion. ACKNOWLEDGMENTS We thank the Indo-French Center for the Promotion of Advanced Research for financial support. 1J. G. Bednorz and K. A. Mueller, Z. Phys. B: Condens. Matter 64, 189 /H208491986 /H20850. 2S. Gopalan, T. M. Rice, and M. Sigrist, Phys. Rev. B 49, 8901 /H208491994 /H20850. 3M. Azuma, Z. Hiroi, M. Takano, K. Ishida, and Y. Kitaoka, Phys. Rev. Lett. 73, 3463 /H208491994 /H20850. 4E. Dagotto, Rep. Prog. Phys. 62, 1525 /H208491999 /H20850. 5B. C. Watson, V. N. Kotov, M. W. Meisel, D. W. Hall, G. E. Granroth, W. T. Montfrooij, S. E. Nagler, D. A. Jensen, R.Backov, M. A. Petruska, G. E. Fanucci, and D. R. Talham, Phys.Rev. Lett. 86, 5168 /H208492001 /H20850. 6G. Chaboussant, P. A. Crowell, L. P. Lévy, O. Piovesana, A. Madouri, and D. Mailly, Phys. Rev. B 55, 3046 /H208491997 /H20850. 7H. Deguchi, S. Sumoto, S. Yamamoto, S. Takagi, H. Nojiri, and M. Motokawa, Physica B 284-288 , 1599 /H208492000 /H20850. 8O. K. Andersen and T. Saha-Dasgupta, Phys. Rev. B 62, R16219/H208492000 /H20850. 9D. C. Johnston, M. Troyer, S. Miyahara, D. Lidsky, K. Ueda, M. Azuma, Z. Hiroi, M. Takano, M. Isobe, Y. Ueda, M. A. Korotin,V. I. Anisimov, A. V. Mahajan, and L. L. Miller, arXiv:cond-mat/0001147. 10P. W. Selwood, Magnetochemistry /H20849Interscience, New York, 1956 /H20850. 11G. S. Rushbrooke and P. J. Wood, Mol. Phys. 1, 257 /H208491958 /H20850. 12D. C. Johnston, Handbook of Magnetic Materials , edited by K. H. J. Buschow /H20849Elsevier Science Publishers, Amsterdam, 1997 /H20850. 13M. Takahashi, Phys. Rev. B 40, 2494 /H208491989 /H20850. 14M. S. Makivic and H.-Q. Ding, Phys. Rev. B 43, 3562 /H208491991 /H20850. 15O. K. Andersen, Phys. Rev. B 12, 3060 /H208491975 /H20850; O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 /H208491984 /H20850; O. K. Ander- sen and O. Jepsen, The STUTTGART TB-LMTO program, version 47 /H208492000 /H20850.TABLE I. Hopping parameters /H20849tn/H20850between various Cu’s are tabulated along with the corresponding Cu-Cu distances. The hop-ping paths are indicated in Fig. 1. Hopping pathCu-Cu distance /H20849Å/H20850t n /H20849meV /H20850Jn/J1 =/H20849tn/t1/H208502 Leg /H20849t1/H20850 3.22 155 1 Rung /H20849t2/H20850 2.90 154 1 Interladder /H20849t3/H20850 4.91 133 0.74 NNN in leg /H20849t4/H20850 5.18 91 0.34 Diagonal /H20849t5A/H20850 4.43 30 0.04 Diagonal /H20849t6A/H20850 5.81 26 0.03 Cu2Bi Cu1O1 O2Cu2 FIG. 5. /H20849Color online /H20850Effective Cu1 dx2−y2orbital with lobes of opposite signs colored as black and white. The dx2−y2orbital is defined with the choice of the local coordinate system as discussedin the text /H20849height of the isosurface= ±0.09 /H20850. The spheres represent the ions.BRIEF REPORTS PHYSICAL REVIEW B 76, 052402 /H208492007 /H20850 052402-4

PhysRevB.94.155429.pdf

PHYSICAL REVIEW B 94, 155429 (2016) Surface plasmon lifetime in metal nanoshells Arman S. Kirakosyan,1Mark I. Stockman,2and Tigran V . Shahbazyan1,* 1Department of Physics, Jackson State University, Jackson, Mississippi 39217, USA 2Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA (Received 4 March 2016; published 17 October 2016) The lifetime of localized surface plasmon plays an important role in many aspects of plasmonics and its applications. In small metal nanostructures, the dominant mechanism of plasmon decay is size-dependent Landaudamping. We performed quantum-mechanical calculations of Landau damping for the bright surface plasmonmode in a metal nanoshell with dielectric core. In contrast to the conventional model based on the electronsurface scattering, we found that the damping rate decreases as the nanoshell thickness is reduced. The originof this behavior is traced to the spatial distribution of plasmon local field in the metal shell. We also found that,due to the interference of electron scattering amplitudes from the two nanoshell metal surfaces, the damping rateexhibits pronounced quantum beats with changing shell thickness. DOI: 10.1103/PhysRevB.94.155429 I. INTRODUCTION Lifetime of localized surface plasmons (SP) in metal nanos- tructures is one of the fundamental problems in plasmonics thathas been continuously addressed for about 50 years [ 1–5]. The importance of this issue stems from one of the major objectivesof plasmonics—generation of extremely strong local fieldsat the nanoscale. The range of physical phenomena andapplications related to this goal cuts across physics, chemistry,biology, and device applications. A small sample of examplesincludes plasmon-enhanced spectroscopies of molecules orsemiconductor quantum dots near metal nanostructures, suchas surface-enhanced Raman scattering (SERS) [ 6], plasmon- enhanced fluorescence [ 7–10], plasmon-assisted fluorescence resonance energy transfer (FRET) [ 11–14], and plasmonic laser (spaser) [ 15–20]. High Ohmic losses in bulk metal due to strong electron-phonon interactions impose limitations on thequantum yield of metal-based plasmonic devices, which can,to some extent, be remedied by reducing the metal componentsize. However, at the length scale below ∼10 nm, new limitations on the SP lifetime and, consequently, on quantum yield arisedue to the quantum-size effects [ 1]. Among those, the most important is the Landau damping (LD) of SP—decay of SPinto the Fermi sea electron-hole pair [ 21–31]. This process has been recently suggested as an efficient way of hot carriersexcitation in plasmon-based photovoltaic devices [ 32–42]. Starting with the pioneering work of Kawabata and Kubo[43] for a spherical nanoparticle (NP), quantum-mechanical calculations of LD rate were performed, using random phaseapproximation (RPA) [ 43–50] or density functional theory (DFT) [ 51–58] methods, for several NP shapes. Excitation of an electron-hole pair with large optical frequency requiresmomentum relaxation to satisfy the energy and momentumconservation laws which, in small systems, can take place viathe electron surface scattering. Based on this picture, it wassuggested [ 59–63] that the SP LD rate in any small system *shahbazyan@jsums.edushould have the form γs=AvF L, (1) where vFis the electron Fermi velocity (hereafter we set /planckover2pi1=1) and Lis the effective mean free path of ballistic electrons confined in a hard-wall potential well, while the phenomenological constant A, measured in the range 0 .3−1.5 [1], accounts for surface potential, electron spillover, and dielectric environment effects. Note that, for nonsphericalNPs, the SP damping by interband excitations can complicatethe LD size dependence. For example, absorption spectra forgold nanorods [ 22,27] and nanoshells [ 23,24] show overall narrowing of the SP resonance that is redshifted away from theinterband transitions onset. At the same time, recent systematicstudies of scattering spectra of single silver nanoprisms [ 29], gold nanorods [ 30], and gold nanodisks [ 31] revealed signifi- cant discrepancies with Eq. ( 1), while no size dependence was detected for the SP resonance width of single gold nanoshells[26], implying that LD is shunted by the bulk SP damping even for relatively thin shells. There is also a physical argument that renders Eq. ( 1) invalid for nanostructures of general shape. Indeed, the rateof electron-pair excitation by the SP local field must besensitive to the field distribution in the NP. Note that for asolid sphere, the dipole SP electric field in the NP is uniformand size independent, which is the reason Eq. ( 1) holds well for spherical NPs in a very wide size range [ 1]. However, in the general case, the local field distribution depends strongly onNP size or shape, so that the simple picture implied by Eq. ( 1) fails. Below we demonstrate that the effect of field distributionleads to a drastically different size and shape dependence of theLD decay rate in a nanostructure than that implied by Eq. ( 1). In this paper, we present a quantum-mechanical calculation of the LD rate for bright SP modes in a metal nanoshell(NS) with a dielectric core. We find that, with decreasing NSthickness d, the LD rate decreases as well, in sharp contrast to the surface scattering model [ 59–63] predicting an increase of /Gamma1as the effective mean free path is reduced. Furthermore, for small overall NS sizes, the SP LD rate exhibits quantum beats as a function of shell thickness caused by the interference 2469-9950/2016/94(15)/155429(8) 155429-1 ©2016 American Physical SocietyKIRAKOSY AN, STOCKMAN, AND SHAHBAZY AN PHYSICAL REVIEW B 94, 155429 (2016) between electron scattering amplitudes from the inner and outer NS boundaries. The paper is organized as follows. In Sec. IIwe outline our approach and present a formal expression for the LD ratein terms of the SP eigenmodes. In Sec. IIIwe describe the plasmon eigenmodes in metal NS with dielectric core andevaluate the NS internal energy. In Sec. IV, we evaluate the power dissipated through electron-hole excitation by the SPeigenmodes. The calculated LD rates are discussed in Sec. V, and Sec. VIconcludes the paper. II. SURFACE PLASMON LANDAU DAMPING RATE IN COMPOSITE METAL-DIELECTRIC NANOSTRUCTURES In this section we outline our approach for calculations of the plasmon damping rate in a composite metal-dielectricstructure embedded in a dielectric medium. We assume thatthe structure is characterized by dielectric function of the formε(ω,r)=ε /prime(ω,r)+iε/prime/prime(ω,r) and the retardation effects are unimportant. In the quasistatic case, the plasmon eigenmodes,labeled by nhere, are determined by the Gauss’s law ∇·[ε /prime(ωn,r)En]=0, (2) where ωnis the eigenfrequency, En=−∇/Phi1nis the mode local field, and /Phi1nis the potential. In the following we assume that only the metal dielectric function εm(ω)=ε/prime m(ω)+iε/prime/prime m(ω)i s complex and dispersive. The decay rate of a plasmon mode isgiven by [ 64] /Gamma1 n=Qn/Un, (3) where Unis the mode energy [ 65], Un=/integraldisplaydV 16π∂(ωnε/prime) ∂ωn|En|2=ωn 16π∂ε/prime m ∂ωn/integraldisplay dVm|En|2,(4) andQnis the mode dissipated power Qn=ωn 2Im/integraldisplay dVE∗ n·Pn, (5) where Pnis the polarization vector ( Vmstands for the metal volume). In the local case, i.e., Pn=En(ε−1)/4π,Qis given by the usual expression [ 65] Qn=ωnε/prime/prime m 8π/integraldisplay dVm|En|2, (6) which, together with the mode energy ( 4), yields the standard plasmon damping rate [ 66], /Gamma1n=2ε/prime/prime m/parenleftbigg∂ε/prime m ∂ωn/parenrightbigg−1 . (7) For the Drude form of metal dielectric function, εm=εi− ω2 p/ω(ω+iγ), where εiis a weakly-dispersive interband contribution, ωpis the bulk plasmon frequency, and γis the scattering rate, one obtains γn=γfor all modes. The surface contribution Qs noriginates from the genera- tion of electron-hole pairs by the plasmon local field nearmetal-dielectric interfaces and can be included in Eq. ( 5) by relating the polarization vector P n(r) to the microscopic electron polarization operator P(ω;r,r/prime) via the induced charge density: ρ(r)=/integraltext dr/primeP(r,r/prime)/Phi1(r/prime)=−∇·P(r)[64].Integrating Eq. ( 5) by parts, we obtain Qs n=ωn 2Im/integraldisplay dVdV/prime/Phi1∗ n(r)P(ωn;r,r/prime)/Phi1n(r/prime).(8) In the first order, Qs nis obtained within RPA as [ 67] Qs n=πωn/summationdisplay αα/prime|/angbracketleftα/prime|/Phi1n|α/angbracketright|2δ(/epsilon1α−/epsilon1α/prime+ωn), (9) where /angbracketleftα/prime|/Phi1n|α/angbracketright=/integraltext dVmψ∗ α/prime/Phi1nψαis the transition matrix element between electron state ψαwith energy /epsilon1αbelow the Fermi level EFand electron state ψα/primewith energy /epsilon1α/primeabove the Fermi level under the perturbation /Phi1n(factor 2 due to the spin degeneracy is included). Note that often in the literature,the plasmon surface-assisted decay rate /Gamma1 s nis identified with the first-order transition probability rate, similar to Eq. ( 9)( u p to the factor ωn/2); it must be emphasized that, in a system with dispersive dielectric function, the accurate expression is/Gamma1 s n=Qs n/Un[64]. In the rest of this paper, this expression will be used to calculate the SP damping rate in a metal NS. III. PLASMON MODES IN METAL NANOSHELLS WITH DIELECTRIC CORE Here we collect the relevant formulas for plasmonic eigenstates in a spherical NS with inner and outer radii R1 andR2, respectively, and core dielectric constant εc,i na medium with dielectric constant εd(see inset in Fig. 1). In the quasistatic limit, the plasmonic eigenfunctions in each region have the form /Phi1LM(r)=/Phi1(i) L(r)YLM(ˆr), where rand ˆrare the magnitude and orientation of the radius vector with the origin at NS center, i=(c,m,d ) denotes core, metal, and outside dielectric regions, respectively, and YLM(ˆr) are spherical harmonics. In each region, the eigen- functions are superpositions of two independent solutionsof Laplace equation in spherical coordinates, r LYLM(ˆr) and r−L−1YLM(ˆr). The equation for eigenvalues is obtained by imposing standard boundary conditions on the radial part of potentials, /Phi1(i) L(r), and radial component of electric field, FIG. 1. Frequency of bright dipole plasmon mode in gold NS with various core and outside dielectrics is plotted vs NS aspect ratio. Inset: Electric field distribution for SiO 2/Au/H 2O NS with aspect ratioR1/R2=0.7. 155429-2SURFACE PLASMON LIFETIME IN METAL NANOSHELLS PHYSICAL REVIEW B 94, 155429 (2016) E(i) L(r)=−∂/Phi1(i) L(r)/∂r,a s ˜εcm˜εmd+L(L+1)εcmεmdκ2L+1=0, (10) where κ=R1/R2is the NS aspect ratio, and we denoted εαβ=ε/prime α−ε/prime βand ˜εαβ=Lε/prime α+(L+1)ε/prime β. The plasmon fre- quencies are obtained by solving Eq. ( 10) for the real part of metal dielectric function, ε/prime m(ωL)=−μL 2±/radicalBigg μ2 L 4−εcεd, (11) μL=(2L+1) L(L+1)˜εcd (1−κ2L+1)−εc−εd, where alternating ( ±) sign correspond to bright and dark plasmon modes, respectively. The bright plasmon spectrummatches that of a solid NP plasmon in the κ=0 limit: ˜ ε md= Lε/prime m+(L+1)εd=0. The higher frequency dark plasmon mode couples weakly to the external fields and will not beconsidered here. In Fig. 1, we show the dependence of bright plasmon mode frequency ω 1(forL=1) on aspect ratio κ=R1/R2of Au NS with several choices of core and outside dielectrics. In allnumerical calculations, the experimental dielectric function forgold as well as for core and outside dielectrics were used [ 68]. With decreasing shell thickness, after a prolonged plateau for κ up to approximately 0.5–0.7 (depending on dielectric content),the frequency develops a redshift. The inset shows electric fielddistribution for the dipole plasmon mode oscillating along thezaxis in a NS with κ=0.7. Note that, in a thin NS, the electric field of bright plasmon is mainly concentrated outside of the metal shell, in contrast to the field distribution in a solid metalNP. The normalized (dimensionless) radial eigenfunctions /Phi1 L(r) in core ( r<R 1), shell ( R1<r<R 2), and outer dielectric ( r>R 2) regions have the form /Phi1(c) L(r)=(2L+1)ε/prime mκL ˜εcm/parenleftbiggr R1/parenrightbiggL , (12) /Phi1(m) L(r)=κL/parenleftbiggr R1/parenrightbiggL +1 L+1˜/epsilon1md /epsilon1md/parenleftbiggR2 r/parenrightbiggL+1 , (13) /Phi1(d) L(r)=2L+1 L+1ε/prime m εmd/parenleftbiggR2 r/parenrightbiggL+1 , (14) and are continuous at the metal-dielectric interfaces, /Phi11L≡/Phi1(m) L(R1)=(2L+1)ε/prime mκL ˜εcm, (15) /Phi12L≡/Phi1(m) L(R2)=2L+1 L+1ε/prime m εmd. (16) The radial electric fields satisfy the standard boundary con- ditions, i.e., εαE(α) L(r) is continuous, and take the following values at the interfaces (on the metal side) E1L≡E(m) L(R1)=−L R1εc ε/primem/Phi11L, (17) E2L≡E(m) L(R2)=L+1 R2εd ε/primem/Phi12L,FIG. 2. Normalized energy of bright dipole plasmon modes in gold NS with various core and outside dielectrics is plotted vs NSaspect ratio. while their ratio at the interfaces is given by qL=E1L/E2L=−LκL−1εmdεc ˜εcmεd. (18) Note that the electric field orientations at the inner and outer interfaces (on the metal side) are opposite. Using the above eigenfunctions, the plasmon mode energy can be straightforwardly calculated from Eq. ( 4). Since the eigenfunctions are harmonic functions inside each region, theintegral in Eq. ( 4) reduces to the boundary terms, and, using the relations ( 17) between fields and potentials at the interface, we obtain U L=|ε/prime m|ωL 16π∂ε/prime m ∂ωL/bracketleftbiggR3 1 L/epsilon1cE2 1L+R3 2 (L+1)/epsilon1dE2 2L/bracketrightbigg . (19) The aspect ratio dependence of the bright dipole plasmon energy U1normalized to solid NP plasmon energy Unp 1with the same overall size is plotted in Fig. 2. The NS mode energy depends strongly on core and outside dielectrics, butis largely comparable to that for a solid NP. This is due to asomewhat similar distribution of the surface charges for brightNS plasmon and solid NP plasmon modes: In both cases,the opposite charges are located at different hemispheres sothe energy is proportional to the core-shell particle volume.In contrast, for dark modes (not shown here), the oppositecharges are located at inner and outer boundaries, so the energyvanishes as the shell thickness decreases. IV . POWER DISSIPATED BY PLASMON MODES IN NANOSHELLS We now turn to calculation of dissipated power Eq. ( 9) (we drop superscript sin the following). We represent the NS confining potential as a three-dimensional quantum wellwith hard boundaries at R 1andR2and amplitude V0:V(r)= V0θ(r−R1)θ(R2−r). The role of realistic surface potential and nonlocal effects will be discussed later. The electron wavefunctions have the form ψ nl(r)Ylm(r), where n,l, andmand electron radial, angular momentum, and magnetic numbers,respectively. Due to spherical symmetry, the angular part 155429-3KIRAKOSY AN, STOCKMAN, AND SHAHBAZY AN PHYSICAL REVIEW B 94, 155429 (2016) factorizes out and Eq. ( 9) takes the form QL=πωL/summationdisplay nn/primell/primeaL ll/prime/vextendsingle/vextendsingleML nl,n/primel/prime/vextendsingle/vextendsingle2δ(/epsilon1nl−/epsilon1n/primel/prime+ωL),(20) where ML nl,n/primel/prime=/angbracketleftnl|/Phi1L|n/primel/prime/angbracketrightis the radial transition matrix element and aL ll/prime=1 2L+1/summationdisplay Mmm/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay d/Omega1Y LMY∗ lmYl/primem/prime/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 (21) is the angular contribution. The latter is nonzero only for l= l/prime±L, and for typical l,l/prime/greatermuchLcan be approximated as aL ll/prime≈ δll/primel/2π. The matrix element /angbracketleftα|/Phi1LM|α/prime/angbracketrightin Eq. ( 9) is dominated by the surface contribution, which can be obtained byfirst commuting twice the plasmon potential /Phi1 LMwith the Hamiltonian, /angbracketleftα|/Phi1LM|α/prime/angbracketright=1 ω2 L/angbracketleftα[H[H,/Phi1 LM]]|α/prime/angbracketright ≈1 mω2 L/angbracketleftα|∇/Phi1LM·∇V|α/prime/angbracketright, (22) which, after separating out the angular part, leads to the following expression for the radial matrix element, ML nl,n/primel/prime=V0 mω2 L[ψnl(R1)ψn/primel/prime(R1)EL(R1) −ψnl(R2)ψn/primel/prime(R2)EL(R2)]. (23) Then, for infinitely high potential barrier ( V0→∞ ), match- ing the wave functions across the well boundaries gives√2mV 0ψnl(Ri)≈−ψ/prime nl(Ri) (here prime stands for the deriva- tive), and the matrix element takes the form ML nl,n/primel/prime=1 2m2ω2 L[ψ/prime nl(R1)ψ/prime n/primel/prime(R1)E1L −ψ/prime nl(R2)ψ/prime n/primel/prime(R2)E2L]. (24) The first and second terms in the r.h.s. describe excitation, by the plasmon electric field, of a Fermi sea electron-holepair accompanied by momentum transfer to the inner andouter boundaries, respectively. Correspondingly, Q Lcan be decomposed as QL=Q11 L+Q22 L−2Q12 L, where Qij L=e2 8m4ω3 LE1LE2L/summationdisplay lnn/primeψ/prime nl(Ri) ×ψ/prime n/primel(Ri)ψ/prime nl(Rj)ψ/prime n/primel(Rj)δ(/epsilon1nl−/epsilon1n/primel+ωL), (25) and we used that aL ll/prime≈δll/primel/2π. Consider first the inner surface contribution Q11 L.F o r typical electron energies /epsilon1nl∼EF, we can adopt semiclassical approximation for the electron wave functions: ψnl(r)=/radicalBigg 4m plτlsin/integraldisplayR2 rpldr,p l=/radicalbigg 2m/epsilon1−(l+1/2)2 r2, (26)where τl(/epsilon1) is the period of classical motion between two turning points. In this case, we find ψ/prime nl(R1)=−/radicalbig 4mpl(R1)/τl. (27) Since the plasmon energy ωLis much larger than the spacing /epsilon10=vF/dbetween the energy levels with adjacent n(at fixed l) in a spherical well, the sums in Eq. ( 25) can be replaced by the integrals,/summationtext n→/integraltext d/epsilon1ρl(/epsilon1) (with /epsilon1<E F,/epsilon1/prime>E F), where ρl(/epsilon1)=∂n/∂/epsilon1 nlis the partial density of states related to the classical period as ρl=τl/2π(see Appendix). The result reads Q11 L=E2 1L 2π2m2ω3 L/summationdisplay ll/integraldisplayEF EF−ωLd/epsilon1pl(/epsilon1,R 1)pl(/epsilon1+ωL,R1). (28) Note that ρlcancels out, i.e., the level spacing disappears from the result. In the energy integral, the integration variable isfirst shifted as /epsilon1→E F+/epsilon1−ωL/2, where /epsilon1now changes in the interval ( −ωL/2,ωL/2), and then rescaled to x=/epsilon1/ωL. The sum over lis replaced by the integral restricted by maximal value l∼pFR1that is determined by the condition pl(/epsilon1,R 1)/greaterorequalslant0. After the change of variables to s=l2/(pFR1)2, it contributes a factor proportional to the inner surface area.The result reads Q 11 L=E2 FR2 1 2π2ω2 LE2 1Lg(ω/E F), (29) where g(ξ)=2/integraltext1/2 −1/2dx/integraltext dsf(ξ,x,s ) with f(ξ,x,s )= [(1+ξx−s)2−ξ2/4]1/2is a dimensionless function normal- ized to g(0)=1. Turning to the outer surface term Q22 L, the main contribution into the r.h.s. of Eq. ( 25) comes from the terms with pl(/epsilon1,R 2)/greaterorequalslant 0 [otherwise ψnl(R2) are exponentially small]. In this case, we have ψ/prime nl(R2)=− (−1)n/radicalbig 4mpl(R2)/τl, (30) where the sign factor ( −1)naccounts for the parity of electron wave function with n−1 nodes between R1andR2.T h er e s t of the calculation is carried in a similar way, and the result, Q22 L=E2 FR2 2 2π2ω2 LE2 2Lg(ω/E F), (31) is proportional to the outer surface area. Finally, consider now the interference term Q12 L.U s i n g Eqs. ( 27) and ( 30), we write Q12 L=2E1LE2L m2ω3 L/summationdisplay lnn/primel(−1)n−n/prime τl(/epsilon1nl)τl(/epsilon1n/primel) ×Fl(/epsilon1nl,/epsilon1n/primel)δ(/epsilon1nl−/epsilon1n/primel+ωL), (32) withFl(/epsilon1,/epsilon1/prime)=√pl(/epsilon1,R 1)pl(/epsilon1/prime,R1)pl(/epsilon1,R 2)pl(/epsilon1/prime,R2). AsωL changes (e.g., with changing aspect ratio), the relative parity of electron and hole states, separated by energy ωL, changes too, leading to a different sequence of alternating signs inthe sum in Eq. ( 32) which, in turn, results in oscillations ofQ 12 L(quantum beats). The number of states contributing into the sum in Eq. ( 32) is large, so that the oscillations 155429-4SURFACE PLASMON LIFETIME IN METAL NANOSHELLS PHYSICAL REVIEW B 94, 155429 (2016) can be described by substituting ( −1)n−n/prime=cosπ(n−n/prime)= cos[π/integraltext/epsilon1/prime /epsilon1d/epsilon1ρl(/epsilon1)]. Then Q12 Ltakes the form Q12 L=E1LE2L 2π2m2ω3 L/summationdisplay ll/integraldisplayEF EF−ωLd/epsilon1Fl(/epsilon1,/epsilon1+ωL) ×cos/bracketleftbigg π/integraldisplay/epsilon1+ωL /epsilon1d/epsilon1/primeρl(/epsilon1/prime)/bracketrightbigg , (33) where lis restricted by the condition pl(/epsilon1,R 1)/greaterorequalslant0. Equation (33) can be brought to the form Q12 L=e2R2 1E2 F 2π2ω2 LE1LE2LG(ωL/EF), (34) where the dimensionless function G(ξ) is rather cumbersome and is given in the Appendix. For thin nanoshells, d/R 2/lessmuch1, it can be evaluated analytically (see Appendix) and the resultreads G(ξ)=− 4sinD Dsin(ξD/ 4) ξD/ 4, (35) where D=ωL//epsilon10=ωLd/vFis the ratio of plasmonic and electronic energy scales. Putting all together, we finally obtain QL=E2 FR2 2 2π2ω2 L/parenleftbig E2 2L+κ2E2 1L−2κ2E1LE2LG/parenrightbig .(36) The last term in Q12oscillates as a function of shell thickness d due to the interference of electron scattering amplitudes frominner and outer NS boundaries. These oscillations are, in fact,quantum beats caused by the change, with d, of the number of electron levels with alternating parities within the plasmonenergy ω L(i.e., the difference between numbers of even and odd states oscillates between 0 and 1). The oscillations period2πv F/ωLdepends weakly on the shell thickness through dependence of ωLonκ(see Fig. 1), and their amplitude slowly dies out with increasing d. In fact, the quantum beats of Q12have a rather general origin. Indeed, excitation of an electron-hole pair with energyωis accompanied by momentum transfer p 0∼ω/vFand occurs in a region with the size r0∼vF/ω. Therefore, oscillations of the pair excitation rate with changing D=d/r 0 reflect the nonlocality of surface-scattering mechanism of momentum relaxation. In Fig. 3, normalized dissipated power for the bright dipole plasmon mode Q1is plotted vs aspect ratio κfor overall NS sizesR2=30 nm and R2=10 nm. Numerical calculations were performed using the full expression for G(ξ) given by Eq. ( A1) in the Appendix. While for larger NS with overall sizeR2=30 nm, oscillations of Q1are relatively weak [see Fig.3(a)], they become more pronounced for smaller NS ( R2= 10 nm) [see Fig. 3(b)]. Note that, for smaller R2,t h es a m e values of κcorrespond to smaller shell thicknesses. Another striking feature is the decrease of dissipated power for κlarger than 0.4. The reason for this behavior is that, with decreasingshell thickness, the local field is pushed outside the metal shell(see inset in Fig. 1) which, in turn, leads to the reduction of the transition matrix element.FIG. 3. Normalized dissipated power by bright dipole plasmon modes in gold NS with various core and outside dielectrics is plotted vs NS aspect ratio for (a) R2=30 nm and (b) R2=10 nm. V. LANDAU DAMPING OF PLASMON MODES IN NANOSHELLS The plasmon damping rate, /Gamma1L=QL/ULwithQLandUL given by Eqs. ( 36) and ( 19), respectively, takes the form /Gamma1L=2ω2 pγL ω3 L/parenleftbigg∂ε/prime m ∂ωn/parenrightbigg−1 , (37) where γL=3vF 4R2εd(L+1) |ε/primem(ωL)|1+κ2q2 L−2κ2qLG 1+κ3q2 L(L+1)εd/Lεc(38) is the LD rate. Here qL=E1L/E2Lis the electric fields’ ratio at the interfaces given by Eq. ( 18). In deriving Eq. ( 38), we used the relation ω2 p=4πn/m =4p3 F/3πm (fore=1), where n is the electron concentration. Equations ( 37) and ( 38) represent our central result. Apart from the dimensional factor vF/R2,t h eL Dr a t e( 38)i s determined by the ratio of plasmon local fields at the metal-dielectric interfaces q L. The last factor describes the relative contribution of the NS interfaces and includes the interferencecorrection. Importantly, comparison of Eqs. ( 37) and ( 7) indicates that LD rate can be incorporated into the Drudescattering rate as γ=γ 0+γL, where γ0is the bulk scattering rate, so the full plasmon damping rate is still given by Eq. ( 7), but with modified Drude dielectric function. 155429-5KIRAKOSY AN, STOCKMAN, AND SHAHBAZY AN PHYSICAL REVIEW B 94, 155429 (2016) FIG. 4. Normalized Landau damping rate for bright dipole plasmon modes in gold NS with various core and outside dielectrics is plotted vs NS aspect ratio for (a) R2=30 nm and (b) R2=10 nm. For solid NP ( κ=0), the plasmon eigenfrequency is determined from Lεm(ωL)+(L+1)εd=0, and we re- cover the LD rate of the Lth mode in a spherical NP [ 43–47], γnp L=3L 4vF R2. (39) In thin NSs, the electric field is pushed out of the metal shell, leading to the reduction of electron-hole excitation rate. Forthin NS ( d/R 2/lessmuch1), the explicit dependence of the LD rate on the shell thickness is obtained from Eq. ( 38)a s( f o r L=1) γ1≈3 2vFd R2 2/bracketleftbigg 1−4εcεd ˜ε2 cd(1−G)/bracketrightbigg , (40) indicating a linear dependence on the shell thickness. In Fig. 4, we show the calculated LD rate γ1for the bright dipole plasmon mode in gold NSs of overall sizes R2=30 nm andR2=10 nm and several choices of core and outside dielectrics. The rate shows approximately linear decrease withincreasing κ(i.e., decreasing d), consistent with Eq. ( 40). The oscillations of γ 1are quite pronounced for smaller overall NS size ( R2=10 nm) and could be observable for typical experimental range of aspect ratios (0.6–0.8) provided thatNS overall size is sufficiently small, so that the LD is notshunted by the bulk scattering. Note that these oscillationsshould be distinguished from those observed in solid NP[51,52,57] due to size quantization of the electron energylevels in a confined nanostructure, while here they are quantum beats between electron scattering paths from different NSinterfaces. VI. CONCLUSIONS In conclusion, let us discuss the role surface potential, dielectric environment, and nonlocal effects near the metalsurface on the plasmon LD that was extensively studied in solidNPs [ 51–58]. These effects mainly affect the overall magnitude of LD rates, but play no significant role in determining the LDdependence on the nanostructure shape which, according toour findings, is mainly determined by the local field ratio atthe interfaces. Extensive theoretical and experimental studiesof spherical NPs indicate that surface effects mainly affect thephenomenological constant A[see Eq. ( 1)], but the overall 1/Rdependence of the LD rate is unchanged [ 1]. In fact, the important role of local fields in plasmon LD rate can explainthe relatively wide range of measured A(0.3–1.5 [ 1]), which raised questions about the validity of the scattering model [ 56]. Indeed, as we mentioned in Sec. IV, excitation of an e-hpair by plasmon local field takes place in a surface layer of thicknessr 0∼vF/ω.F o rvF≈1.4×106m/s in Au and Ag, we have vF/ω≈1n mf o r /planckover2pi1ω=1.0 eV , i.e., for typical plasmon frequencies in the range 1.5–3.5 eV , the layer thickness is justaf e w ˚A. In a thin surface layer, the local fields are strongly affected by the electron spillover and surface roughness effectsas well as by the dielectric environment, which can lead to largevariations of overall LD rate magnitude for different samplesand/or environments. Within our approach, the constant Acan be estimated by computing the effect of the above factors onthe local field, which is, however, out of the scope of this paper. In summary, we calculated the Landau damping rate of surface plasmons in metal nanoshells with dielectric core. Wefound that the damping rate decreases with the shell thicknessdue to the reduction of the local field magnitude inside a thinmetal shell. We also found that the Landau damping rateexhibits quantum beats caused by the interference betweenelectron scattering paths from the nanoshell inner and outermetal-dielectric interfaces. ACKNOWLEDGMENTS This work was supported in part by NSF Grants No. DMR- 1610427 and No. HRD-1547754. APPENDIX Here we analyze function G(ξ) in the interference term ( 34). After shifting integration variables in Eq. ( 33)a s/epsilon1→EF+ /epsilon1−ωL/2 and /epsilon1/prime→EF+/epsilon1+/epsilon1/primeand rescaling to x=/epsilon1/ωL ands=l2/(pFR1)2, we arrive at ( 34) with G(ξ)=2/integraldisplay1/2 −1/2dx/integraldisplay ds/radicalbig f(ξ,x,s )f(ξ,x,κ2s) ×cos/bracketleftbigg πωL/integraldisplay1/2 −1/2dx/primeρl[EF[1+ξ(x+x/prime)]]/bracketrightbigg , (A1) 155429-6SURFACE PLASMON LIFETIME IN METAL NANOSHELLS PHYSICAL REVIEW B 94, 155429 (2016) where f(ξ,x,s )=/radicalbig (1+ξx−s)2−ξ2/4, and the partial density of states is given by ρl(/epsilon1) =m π/integraldisplayR2 R1dr pl(/epsilon1,r)=R2pl(/epsilon1,R 2)−R1pl(/epsilon1,R 1) 2π/epsilon1 =R2 πvF/bracketleftbigg/radicalbig 1+ξ(x+x/prime)−κ2s−κ√1+ξ(x+x/prime)−s 1+ξ(x+x/prime)/bracketrightbigg . (A2) Forω/E F/lessmuch1, thex/primeintegrals are easily evaluated, yielding G(ξ)=2/integraldisplay1/2 −1/2dx/integraldisplay1+ξx 0ds/radicalbig f(ξ,x,s )f(ξ,x,κ2s) ×cos[w(ξ,x,s )ωL//epsilon10], (A3) where w(ξ,x,s )=(/radicalbig f(ξ,x,κ2s)−κ√f(ξ,x,s ))/(1+ξx) with f(ξ,x,s )≈1+ξx−s(here /epsilon10=vF/R2). Rescal- ingsby 1 +ξx,E q .( A3) factorizes as G(ξ)=/integraltext1/2 −1/2dx(1+ξx)2S(ξ,x), where S(ξ,x)=2/integraldisplay1 0ds/radicalbig (1−s)(1−κ2s) ×cos[a(ξ,x)(/radicalbig 1−κ2s−κ√ 1−s)],(A4)with shorthand notation a(ξ,x)=(ωL//epsilon10)/√1+ξx. With substitution s=1−1−κ2 κ2sinh2α,Sis brought to the form S(ξ,x)=4(1−κ2)2 κ3/integraldisplayα0 0dα(sinhαcoshα)2 ×cos[a(ξ,x)/radicalbig 1−κ2e−α], (A5) where sinh α0=κ/√ 1−κ2.F o ra(ξ,x)/greatermuch1, the integral is dominated by the upper limit, and for thin shells, 1 −κ/lessmuch1, corresponding to α0>1, can be evaluated as S≈− 4sin(a√ 1−κ2e−α0) a√ 1−κ2e−α0=− 4sin[a(1−κ)] a(1−κ).(A6) With the above Sand after change of variable t=√1+ξx, the expression for G(ξ) takes the form G(ξ)=−8 ξD/integraldisplayt+ t−dtt3sin(tD), (A7) where t±=√1±ξ/2, and D=(1−κ)ωL//epsilon10=ωLd/vF. Note that even though for ξ/lessmuch1 the integration interval is small, the integrand is still an oscillating function sinceD/greatermuch1, and so the product Dξcan be arbitrary. In this case, a straightforward evaluation yields Eq. ( 35). [1] U. Kreibig and M. V ollmer, Optical Properties of Metal Clusters (Springer, Berlin, 1995). [2] W. P. Halperin, Rev. Mod. Phys. 58,533(1986 ). [3] V . V . Kresin, Phys. 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PhysRevB.102.075419.pdf

PHYSICAL REVIEW B 102, 075419 (2020) Magnetization generated by microwave-induced Rashba interaction O. Entin-Wohlman,1,*R. I. Shekhter,2M. Jonson,2and A. Aharony1 1School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel 2Department of Physics, University of Gothenburg, SE-412 96 Göteborg, Sweden (Received 13 July 2020; accepted 21 July 2020; published 10 August 2020) We show that a controllable dc magnetization is accumulated in a junction comprising a quantum dot coupled to nonmagnetic reservoirs if the junction is subjected to a time-dependent spin-orbit interaction. The latter isinduced by an ac electric field generated by microwave irradiation of the gated junction. The magnetizationis caused by inelastic spin-flip scattering of electrons that tunnel through the junction, and depends on thepolarization of the electric field: a circularly polarized field leads to the maximal effect, while there is noeffect in a linearly polarized field. Furthermore, the magnetization increases as a step function (smoothened bytemperature) as the microwave photon energy becomes larger than the absolute value of the difference betweenthe single energy level on the quantum dot and the common chemical potential in the leads. DOI: 10.1103/PhysRevB.102.075419 I. INTRODUCTION The possibility to create and manipulate magnetic order confined to the nanometer length scale is currently attractinginterest because of possible implications for magnetic devicesand material developments [ 1]. Such a confined magnetization is seldom achieved by applying an external magnetic field,due to practical difficulties encountered when attempting tospatially localize the field. It can, however, be realized bymodulating the exchange-interaction strength, for instance,along a depth-profile variation of certain alloys’ constituents[2]. In contrast to external magnetic fields, electrical currents can be localized quite easily when injected from nanometer-size electric weak links (e.g., quantum point contacts). In casesuch currents are spin polarized, as happens for electronsinjected from magnetic materials, they lead to the creationof magnetic torques that can be exploited to manipulate andcontrol the local magnetization of a ferromagnet [ 3]. Spin injection of ac and dc currents from ferromagnetic materialswas indeed detected and imaged [ 4]. Yet another tool for efficient manipulation of magnetic order in nanoscale devicesdepends on the interplay between charge and spin broughtabout by the spin-orbit interaction [ 5] which couples the spin and the momentum of the electrons. This is the so-called“spin-charge conversion” or the Edelstein-Rashba effect [ 6,7], which occurs at interfaces where the Rashba spin-orbit inter-action is active [ 8,9]. The phenomenon of spin-charge conversion at an interface with broken inversion symmetry has also been achieved byshining light on the sample [ 10,11]. In these configurations the radiated field couples equally to both spin components,and the spin selectivity needed for the spin-charge conversionis procured by the presence of a (static) Rashba interaction atthe irradiated interface. We propose in this paper a differentscenario: the possibility to magnetize initially spin-inactive *orawohlman@gmail.comconducting nanostructures through a Rashba interaction in- duced by an ac electric field generated by external microwave radiation. Put differently, the generated electric field couplesthe momenta of the electrons with their spins. Employing anac electric field to induce the Rashba interaction on nanos-tructures modifies qualitatively and profoundly the electrons’kinematics in them. The inelastic transitions of electrons thattunnel through the junction acquire a spin dependence dueto a correlation between photon absorption and emissionprocesses and distinct spin-flip transitions. This paves a way tomagnetize a spin-inactive material in the absence of externalmagnetic fields. Once the Rashba interaction is established in the junction, the tunneling amplitudes are augmented by the Aharonov-Casher [ 12] phase factors which in turn render the tunneling to be accompanied by spin flips [ 13]. Namely, the Aharonov- Casher factors can be considered as unitary rotations of themagnetic moment. This by itself is insufficient to producespin selectivity, as follows from considerations based on time-reversal symmetry [ 14]. However, the ac electric field gener- ates a Rashba interaction which depends on time, thus breakstime-reversal symmetry and makes spin-selective tunnelingpossible. We have recently observed that such time-dependenttunneling can result in the appearance of a dc electromotiveforce on the junction [ 15]. In this paper we show that spin- selective transport between nonmagnetic conductors is createdwhen the Rashba interaction is induced by an oscillatingelectric field, and leads to the accumulation of a dc magneticorder, even when the junction is unbiased. The magnitude ofthe induced magnetization depends on the polarization of theelectric field, and reaches its maximal value for a circularlypolarized field. Accordingly, a totally nonmagnetic conductorcan be magnetized when subjected to a rotating electric field. The paper is divided into two parts. We first analyze in Sec. IIthe simplest possible junction, which comprises a quantum dot coupled to a single metal reservoir, as shown inFig. 1. We derive there the dc magnetization on the dot and the rate by which a magnetic order is built up in the lead. The total 2469-9950/2020/102(7)/075419(9) 075419-1 ©2020 American Physical SocietyO. ENTIN-WOHLMAN et al. PHYSICAL REVIEW B 102, 075419 (2020) FIG. 1. A quantum dot, represented by a single localized en- ergy level, is attached to a nonmagnetic metal lead by a weak link along the xaxis. The four plates represent the application of microwave-induced ac gate voltages, vy(t)a n d vz(t), which create time-dependent electric fields along the ˆyand ˆzdirections, respec- tively. The resulting total electric field along the vector ˆn(t) can be made to rotate in the y-zplane by introducing a phase shift between the oscillating gate voltages. The electric field induces a Rashba interaction in the weak link, that is represented by the effective magnetic field BL(t), which is perpendicular to both ˆxand ˆn(t). magnetization in the junction is not expected to be conserved when a time-dependent Rashba interaction is active. However,when an electron moves via the spin-orbit-active link from thedot to the reservoir, its magnetization rotates by the Aharonov-Casher factor to a new direction. Therefore (as we show inSec. II), the sum of the time derivatives of the magnetization in the dot along an arbitrary direction ˆ/lscript, and that of the magnetization in the lead along the direction ˆ/lscript /prime/prime L(t), obtained from ˆ/lscriptafter rotating it by the Aharonov-Casher factors, is zero, namely, the two magnetization rates cancel one another. In the second part of the paper, Sec. III, we consider a configuration where the dot is coupled to two reservoirs (seeFig. 4). These can be kept at different chemical potentials (or temperatures), which provide another tool for controlling thesystem. Not surprisingly (in view of the results in Sec. II), the magnetization accumulated on the dot in this case de-pends on electron tunneling from both leads. It hinges on thechemical potential and temperature of each lead via the Fermidistribution there. Note, though, that its existence does notnecessitate a chemical potential difference, or a temperaturedifference, between the two leads. The dc rate of change ofthe magnetization in each of the leads, however, is modifiedqualitatively as compared to the one found in Sec. IIfor a dot connected to a single lead: a voltage bias across thejunction, or a temperature difference between the two leads,allows for an “extra” dc magnetization in one lead, at theexpense of the other lead. Similar to the findings in Sec. II, the total magnetization in the system is not conserved, but themagnetization rates along appropriate rotated directions canadd up to zero. Technical details of the calculation are relegated to the Appendix. There, calculations are carried out for the secondconfiguration, depicted in Fig. 4, since it is straightforward to infer from those the relations needed for the first configura-tion, depicted in Fig. 1. For this reason, our notations in Sec. II assign the letter Lto the physical characteristics of the single lead.II. SPIN IN A SINGLE-LEAD JUNCTION We begin by considering a quantum dot coupled to just a single, nonmagnetic, metal lead by a weak link, as depicted inFig. 1. This, the simplest configuration of interest here, serves to demonstrate the building up of a magnetic moment in thedot and in the lead under the effect of a rotating electric field. By applying microwave-induced time-dependent gate volt- ages as indicated in Fig. 1, an ac electric field is exerted on the weak link. The field is oriented along the vector ˆn(t), which rotates with the microwave frequency /Omega1in the y-zplane: ˆn(t)=ˆzcos(/Omega1t)−γˆysin(/Omega1t). (1) Here, γis the parameter that measures the deviation from perfectly circular polarization: for γ=1( o r γ=− 1) the electric field is circularly polarized, rotating in a clockwise (oranticlockwise) direction with respect to the positive xdirec- tion. For γ=0 the field is linearly polarized. The significance ofγis elucidated below. In a weak link with broken inversion symmetry [ 6], the electric field creates a time-dependent Rashba interaction[16], which manifests itself in the form of a phase factor superimposed on the tunneling amplitude. This phase factor,arising from the Aharonov-Casher effect [ 12], reads as V L(t)=exp[iksodL׈n(t)·σ], (2) where dL=−dLˆxis the radius vector from the dot to the lead (see Fig. 1). In Eq. ( 2),σ=[σx,σy,σz] is the vector of the Pauli matrices, and ksorepresents the strength of the Rashba spin-orbit interaction (in inverse-length units), whichis proportional to the ac electric field associated with the mi-crowave radiation. The tunneling Hamiltonian that describestransitions between electronic states in the lead (given bythe operator c † kσthat creates an electron of energy /epsilon1k,w a v e vector k, and spin index σ) and those on the dot (given by the operator d† σ/primethat creates an electron of energy /epsilon1with spin indexσ/prime)i s HL tun(t)=JL/summationdisplay σ,σ/prime[V∗ L(t)]σσ/prime/summationdisplay kd† σ/primeckσ+H.c. ∼JL/summationdisplay σ,σ/prime([1−|BL(t)|2/2]δσ,σ/prime −i[σ·BL(t)]σ/primeσ)/summationdisplay kd† σ/primeckσ+H.c., (3) up to second order in the spin-orbit coupling αL=ksodL(JLis the tunneling energy scale). The spin-orbit interaction appearsas a dimensionless effective magnetic field oscillating withfrequency /Omega1, B L(t)=ei/Omega1tB− L+e−i/Omega1tB+ L, B± L=(αL/2)[ˆy±iγˆz], (4) that is perpendicular to the direction of the weak link (see Fig. 1). To this order in αL, one identifies two processes in Eq. ( 3). The first conserves the electronic spin during tunneling, whilethe second, the effective Zeeman term, involves spin flipsaccompanied by the absorption or emission of an energyquantum /Omega1from the electric field [ 17], as manifested in 075419-2MAGNETIZATION GENERATED BY MICROW A VE-INDUCED … PHYSICAL REVIEW B 102, 075419 (2020) Eqs. ( 4), using ¯ h=1. At very low temperatures the absorption transitions dominate (for both the d† σ/primeckσterm and its Hermi- tian conjugate) in which case Eq. ( 3) simplifies. In particular, σ·BL(t)≈e−i/Omega1tσ·B+ L=e−i/Omega1t(αL/2)(σy+iγσz). Note that σy+iγσz=σ+(1+γ)/2+σ−(1−γ)/2, where σ±=σy±iσzare operators that increase ( +) and lower ( −) the spin projection in the ˆxdirection. We may now infer that in a circularly polarized electric field, rotating inthe clockwise direction ( γ=+ 1), absorption transitions lead to an accumulation on the dot of spins whose projections ontheˆxaxis are positive (spin up), while if the electric field rotates in the anticlockwise direction ( γ=− 1) absorption transitions lead to an accumulation of spins whose projectionson the ˆxaxis are negative (spin down). In a linearly polarized field (γ=0) there is no preference for either spin projection and no net spin is accumulated. Obviously, these qualitativearguments will have to be verified by a detailed calculation,which is carried out in the following. Quite generally, the magnetization on the dot, given by the (dimensionless) vector M d(t) (in units of −gμB/2, where gis thegfactor of the electron and μBis the Bohr magneton), is a priori time dependent, Md(t)=/summationdisplay σ,σ/prime/angbracketleftd† σ(t)[σ]σσ/primedσ/prime(t)/angbracketright, (5) and the angular brackets denote quantum averaging with respect to the Hamiltonian of the junction H(t)=H0+HL tun(t). (6) The time-independent Hamiltonian H0pertains to the decou- pled system H0=/summationdisplay σ/epsilon1d† σdσ+/summationdisplay k,σ/epsilon1kc† kσckσ, (7) with the first term describing the decoupled dot and the second the decoupled electronic reservoir, assumed to consistof nonpolarized free electrons; H L tun(t)i sg i v e ni nE q .( 3). The quantum average in Eq. ( 5) is related to the lesser Keldysh Green’s function on the dot at equal times, defined as [G< dd,L(t,t)]σ/primeσ≡i/angbracketleftd† σ(t)dσ/prime(t)/angbracketright. (8) This Green’s function is derived [ 18] in the Appendix, exploiting the Keldysh technique. Inserting Eq. ( A8) into the definition ( 5), one finds Md(t)=2/Gamma1L/integraldisplaydω 2πfL(ω)Tr{WL(t,ω)σ}, (9) where the trace is carried out in spin space. Here, /Gamma1Lis the width of the Breit-Wigner resonance formed on the dot dueto the coupling with the lead [ 18], and f L(ω) is the Fermi distribution in the lead. The matrix WL(t,ω) represents the correlation of the Aharonov-Casher phase factors at different times, WL(t,ω)=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay t dt1ei(ω−/epsilon1+i/Gamma1L)(t−t1)V† L(t1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 , (10)FIG. 2. The dimensionless function FL(/Omega1) [Eq. ( 12)] for several values of /epsilon1measured with respect to the chemical potential on the lead, for /Gamma1LβL=10, where βLis the inverse temperature of the metal lead. and is calculated in the Appendix. The dc spin accumulation on the dot results from the corresponding dc part of WLwhich involves the effective Zeeman interaction, i.e., from the lastterm on the right-hand side of Eq. ( A13), M dc d=2ˆxγα2 LFL(/Omega1), (11) where FL(/Omega1) is an odd function of /Omega1, FL(/Omega1)=/Gamma1L/integraldisplaydω 2πfL(ω)[|D(ω+/Omega1)|2−|D(ω−/Omega1)|2], (12) with [ 19]|D(ω)|2=|ω−/epsilon1+i/Gamma1L|−2. This function is de- picted in Fig. 2; as seen, the integrand (for /epsilon1> 0) is dominated by the resonance of D(ω+/Omega1) since the Fermi function (at low temperatures) is nonzero only for the negative ω/primes. In Eq. ( 11)w eh a v eu s e dE q s .( 4) to obtain 2 iB− L×B+ L= −ˆxα2 Lγ. The magnetization accumulated on the dot is indeed along ˆx, as implied by the heuristic argument above. The probability to magnetize the dot is determined by the po-larization of the time-dependent electric field. For a linearlypolarized electric field ( γ=0) the effective magnetic field for the absorption process is parallel to that of the emissionB − L/bardblB+ L, leading to a vanishing magnetic order. In contrast, for circular or elliptic polarization ( γ/negationslash=0) there appears a dc magnetization on the dot, which is linear in γ. Evidently [see Eq. ( 9)], the magnetic order built on the dot has also an ac component which oscillates with the fre-quencies /Omega1and 2/Omega1[see Eq. ( A14)]. This component gives the temporal variation of the spin polarization on the dot.In the following, we add to this component the rate by whichthe magnetic order is established on the lead, thus examiningthe total time dependence of the spin population in the entiresystem. The magnetization rate in the metal lead ˙M L(t)i s defined as ˙ML(t)=d dt/summationdisplay k/summationdisplay σ,σ/prime/angbracketleftc† kσ(t)ckσ/prime(t)/angbracketrightσσσ/prime, (13) where the time derivative and the quantum average are with respect to the Hamiltonian ( 6). This rate can be expressed 075419-3O. ENTIN-WOHLMAN et al. PHYSICAL REVIEW B 102, 075419 (2020) in terms of lesser Green’s functions G< LdandG< dL, defined in Eqs. ( A2): d dt/summationdisplay k/angbracketleftc† kσ(t)ckσ/prime(t)/angbracketright =JL[G< Ld(t,t)V† L(t)−VL(t)G< dL(t,t)]σ/primeσ. (14) By solving the corresponding Dyson equations [Eqs. ( A4)], one obtains this magnetization rate ˙ML(t)=Tr{XL(t)V† L(t)σVL(t)}. (15) Here, we have introduced the matrix XL(t)=idG< dd,L(t,t) dt =− 2/Gamma1L/integraldisplaydω 2πfL(ω)∂WL(t,ω) ∂t. (16) [The derivation is contained in Eqs. ( A19)–(A21).] Compar- ing Eqs. ( 9) and ( 15), we find that while the (oscillating) rate of change of the magnetic moment on the dot is ˙Md(t)=− Tr{XL(t)σ}, (17) that in the lead, Eq. ( 15), in addition to a sign difference, requires a rotation of σby the Aharonov-Casher phase factors σ→V† L(t)σVL(t). (18) The total rate of the spin population in the junction is ˙ML(t)+˙Md(t)=Tr{XL(t)[V† L(t)σVL(t)−σ]}, (19) and it vanishes only if there is no rotation, i.e., VL(t)=1. Put differently, the total rate of the spin population along anarbitrary direction ˆ/lscriptis ˆ/lscript·[˙M L(t)+˙Md(t)] =2/Gamma1L/integraldisplaydω 2πfL(ω)Tr/braceleftbigg∂WL(t,ω) ∂tσ·[ˆ/lscript−ˆ/lscript/prime L(t)]/bracerightbigg ,(20) where ˆ/lscript/prime L(t) is the direction obtained upon rotating ˆ/lscriptby the Aharonov-Casher phase factors σ·ˆ/lscript/prime L(t)=V† L(t)σ·ˆ/lscriptVL(t). (21) The deviation of ˆ/lscript/prime L(t) away from ˆ/lscriptdetermines the amount by which the magnetization in the entire system is not conservedfor a fixed direction ˆ/lscript. Interestingly, the nonconservation has a dc component. Up to second order in the spin-orbit coupling, it sufficesto consider the rotation to linear order in the spin-orbitcoupling [ 20] ˆ/lscript /primeL(t)∼ˆ/lscript+2[B+ Le−i/Omega1t+B− Lei/Omega1t]׈/lscript. (22) Introducing this expression into Eq. ( 15) [and making use of Eqs. ( 21) and ( A14)], one finds that the total rate in the junction includes two contributions: an oscillating part,which exists in both the lead and in the dot, and a dc part,which exists only in the lead (since the non-oscillating dotFIG. 3. The dimensionless function /tildewideFL(/Omega1) [Eq. ( 24)] for several values of /epsilon1measured with respect to the chemical potential on the lead, for /Gamma1LβL=10, where βLis the inverse temperature of the metal lead. magnetization is constant in time), along the ˆxaxis, ˙ML(t)|dc=2ˆxγα2 L/Gamma1L/tildewideFL(/Omega1), (23) where /tildewideFL(/Omega1)=4/Gamma1L/Omega12/integraldisplaydω 2πfL(ω)|D(ω)|2 ×[|D(ω+/Omega1)|2−|D(ω−/Omega1)|2]. (24) This function is plotted in Fig. 3. As seen, this dc component of the rate is along the ˆxaxis, just like the dc magnetization on the dot [Eq. ( 11)], both quantities being odd in the microwave frequency /Omega1. The total magnetization in the system along a fixed (in time) direction ˆ/lscriptis not conserved. However, one may examine possible cancellations of the magnetization rates. Adding themagnetization rate in the dot along ˆ/lscript, to that in the lead along a time-dependent vector given by ˆ/lscript /prime/prime L(t), σ·ˆ/lscript/prime/prime L(t)=VL(t)σ·ˆ/lscriptV† L(t), (25) results in Tr{XL(t)V† L(t)σ·ˆ/lscript/prime/prime L(t)VL(t)}+ ˙Md(t)·ˆ/lscript=0, (26) which implies that the sum of the spin currents along these specific directions vanishes. The sum of the dot magnetizationalong ˆ/lscriptand of the lead magnetization along ˆ/lscript /prime/prime L(t) is con- served. This is physically understood: an electron magnetiza-tion along ˆ/lscriptin the dot rotates by the Aharonov-Casher factor to be along ˆ/lscript /prime/prime L(t) in the lead. III. A DOT COUPLED TO TWO METAL RESERVOIRS The main reason for extending our scheme to a dot coupled to more than a single lead (see Fig. 4) is to explore the possibility that the induced spin-orbit interaction in, say, theleft weak link, will generate a magnetic moment in the rightlead. In other words, we wish to find out how the existenceof one lead affects the accumulated spin magnetization in theother. Consider the magnetization rate in the left lead ˙M L(t), as defined in Eqs. ( 13) and ( 14), when applied to the 075419-4MAGNETIZATION GENERATED BY MICROW A VE-INDUCED … PHYSICAL REVIEW B 102, 075419 (2020) FIG. 4. Illustration of a junction comprising a quantum dot, attached by two weak links lying along the ˆxaxis to two reser- voirs, denoted Land R. As in Fig. 1, the four plates mark the application of microwave-induced ac gate voltages vy(t)a n d vz(t). These give rise to time-dependent spin-orbit interactions in the weak links. two-terminal junction depicted in Fig. 4. It is again convenient to express this quantity in terms of the (matrix) function XL(t) [cf. Eq. ( 15)]. However, in contrast to the configuration dealt with in Sec. II, in the case where the dot is coupled to two leads, XL(t) takes the form XL(t)=2/integraldisplaydω 2π/parenleftbigg −/Gamma1LfL(ω)∂WL(t,ω) ∂t +2/Gamma1L/Gamma1R[fR(ω)WR(t,ω)−fL(ω)WL(t,ω)]/parenrightbigg .(27) [This expression results upon inserting Eq. ( A8)f o r Gdd,L, and the corresponding one for Gdd,R, into Eq. ( A21).] The analo- gous function XR(t) is obtained from Eq. ( 27) by replacing L with R. The detailed calculation of the rate ˙ML(t) is carried out in the Appendix [see Eq. ( A22) there]. The dc component is presented here, ˙Mdc L=ˆx/parenleftbig γα2 L[2/Gamma1/tildewideFL(/Omega1)−4/Gamma1RFL(/Omega1)] +γα2 R4/Gamma1LFR(/Omega1)−γαLFLR(/Omega1)/parenrightbig , (28)where FL(/Omega1) is defined in Eq. ( 12),FR(/Omega1) is derived from the same equation by replacing Lwith R,/tildewideFL(/Omega1) is defined in Eq. ( 24), and FLR(/Omega1)=8/Gamma1L/Gamma1R/integraldisplaydω 2π[αRfR(ω) +αLfL(ω)]2 Re[ D3(ω)], (29) with 2R e [ D3(ω)]=4|D(ω−/Omega1)D(ω+/Omega1)|2 ×/Omega1(ω−/epsilon1)[1−/Omega12|D(ω)|2]. (30) Adding the rate of change of the magnetization in the left lead [using Eqs. ( 15) and the analogous one for the right lead] to the analogous one for the rate of change of themagnetization in the right lead ˙M Ryields [˙ML(t)+˙MR(t)]·ˆ/lscript =Tr{XL(t)ˆ/lscript/prime L(t)·σ}+XR(t)ˆ/lscript/prime R(t)·σ}, (31) where ˆ/lscriptis again an arbitrary direction, and ˆ/lscript/prime L(t), defined in Eq. ( 21), is the direction reached upon rotating ˆ/lscriptby the (time-dependent) Aharonov-Casher factors of the left link.Similarly, ˆ/lscript /prime R(t) is the direction reached by the rotation with the Aharonov-Casher factors of the right link. The rate ofchange of the magnetization in the dot ˙M d(t) comprises contributions from the coupling with the left reservoir and theright one (see Appendix). The first is given in Eq. ( 9), and the second is obtained from it by replacing Lwith R. Thus, its rate of change is ˙M d(t)·ˆ/lscript=2/integraldisplaydω 2πTr/braceleftbigg/parenleftbigg /Gamma1LfL(ω)dWL(t,ω) dt +/Gamma1RfR(ω)dWR(t,ω) dt/parenrightbigg σ·ˆ/lscript/bracerightbigg . (32) Adding together Eqs. ( 31) and ( 32) [using Eq. ( 27) and the analogous one for XR(t)] gives the total rate of change of the magnetization in the two-terminal junction along an arbitrarydirection ˆ/lscript: [˙Md(t)+˙ML(t)+˙MR(t)]·ˆ/lscript=2T r/braceleftbigg/integraldisplaydω 2π/parenleftbigg /Gamma1LfL(ω)dWL(t,ω) dtσ·[ˆ/lscript−ˆ/lscript/prime L(t)]+/Gamma1RfR(ω)dWR(t,ω) dtσ·[ˆ/lscript−ˆ/lscript/prime R(t)]/parenrightbigg/bracerightbigg +4/Gamma1L/Gamma1RTr/braceleftbigg/integraldisplaydω 2π[fL(ω)WL(t,ω)−fR(ω)WR(t,ω)][ˆ/lscript/prime R(t)−ˆ/lscript/prime L(t)]·σ/bracerightbigg . (33) As found in Sec. II, the total magnetization would have been conserved had the rotations of the spin on their way betweenthe dot and the leads been ignored. The amount by whichthe total magnetization is not conserved when measuredalong a fixed (time-independent) direction ˆ/lscriptis determined by the rotations of this direction from the dot to the leftlead and to the right one. Thus, the time-dependent spin-orbit coupling generates a time-independent magnetization,and the amount by which it is not conserved has also a dcpart.IV . DISCUSSION We propose that inelastic tunneling of electrons through a weak link, accompanied by spin flips generated by a spin-orbit coupling caused by a rotating electric field, is capableof producing a net spin population in a nonmagnetic device;the field can be induced by microwave radiation as indicatedin Fig. 1. The origin of this effect is the correlation between emission and absorption of photons by tunneling electrons andspecific spin flips (from spin down to spin up or from spin up 075419-5O. ENTIN-WOHLMAN et al. PHYSICAL REVIEW B 102, 075419 (2020) to spin down). Our conjecture was verified in Sec. IIfor a single-level quantum dot coupled to a nonmagnetic reservoirof electrons, in the particular case when the dot energy level/epsilon1is situated above the Fermi energy /epsilon1 Fof the reservoir and hence is unoccupied at zero temperature. However, one caneasily convince oneself that the effect is the same if the dotlevel is situated below the the Fermi energy /epsilon1</epsilon1 F, so that the dot level is doubly occupied at zero temperature. We would like to remind the reader that our calculation is carried out in the weak electron tunneling limit. Thismeans that the probability for double occupancy of an ini-tially empty dot (the case discussed above) due to inelastictunneling is negligibly small. Therefore, the intradot Coulombrepulsion energy Uin a doubly occupied dot does not enter the calculation. By the same argument, if the dot is initiallydoubly occupied only one electron can be removed due toinelastic tunneling in the weak tunneling limit. In this case,the Coulomb energy Udoes play a role since the energy cost of removing one electron from the dot is /epsilon1 F−(/epsilon1+U) (compared to the energy /epsilon1−/epsilon1Frequired to add one electron to an empty dot). Except for this difference, the role of theinteraction energy is trivial. As discussed in Sec. II, photon absorption processes dom- inate at low temperatures. For a circularly polarized electricfield rotating in, say, the clockwise direction (in the sensedefined in Sec. II) the requirement that spin angular momen- tum is conserved then only allows spins to flip from “down”to “up”. For an unoccupied dot /epsilon1>/epsilon1 F, this means that only transitions from an occupied electron state with spin down inthe reservoir to the spin-up state in the dot are allowed. If,on the other hand /epsilon1</epsilon1 F, only transitions from an occupied spin-down state in the dot to an unoccupied electron spin-upstate in the reservoir are allowed, leaving an uncompensatedspin-up electron on the dot. Consequently, inelastic transitionsbetween electron states in the lead and both occupied ( /epsilon1</epsilon1 F) and unoccupied ( /epsilon1>/epsilon1 F) dot states result in the same spin state on the dot. This allows one to expect that if the dotcontains several energy levels that can be involved in photon-assisted spin-flip transitions, the amount of spin accumulationon the dot can be augmented compared to when the dot hasonly one level. Driving the electron spin dynamics by a rotating electric field as suggested in this paper represents only one of severaloptions for achieving a time-dependent spin-orbit couplingin nanodevices. Another possibility is to use a mechanicaldrive by temporally modulating the geometry of the device[21]. A related recent theoretical idea [ 22] proposes to exploit externally excited chiral phonon modes in graphene (whichcause the carbon atoms to rotate and hence the spin-orbitinteraction to be time dependent) to accumulate spin andgenerate magnetization. The Keldysh Green’s function for the dot, defined in Eq. ( 8), can be viewed as the spin density matrix of a spin q-bit. Its quantum coherent dynamics is fully determined by the time dependence of the spin-orbit interaction, which isinduced by the ac gate voltages [see Eq. ( A20)]. Hence, driving the device by microwaves as envisaged here offers thepossibility to create and manipulate a spin q-bit by applying appropriate microwave pulses as is well known from the fieldof quantum computing.The results presented in this paper open the possibility to use microwave radiation to activate a magnetic pattern atthe surface of a conductor. An array of quantum dots couldbe deposited on the surface, each dot individually coupledto the conductor by spin-orbit-active tunnel junctions. Themagnetization of each dot could in principle be controlledlocally by electrostatic gates or by mechanical deformationsof the tunneling weak links. In this way, one might be ableto create a multiple q-bit structure in which communication between the dots would be governed by spin currents flowingbetween the dots and the common reservoir. A study ofsuch possibilities is well beyond the scope of this paper, butmight serve as a motivation for further investigations of thepossibility to create static magnetization by irradiation withmicrowaves. ACKNOWLEDGMENTS This research was partially supported by the Israel Science Foundation (ISF), by the infrastructure program of IsraelMinistry of Science and Technology under contract 3-11173,and by the Pazy Foundation. We acknowledge the hospitalityof the PCS at IBS, Daejeon, Korea [where part of this workwas supported by IBS funding number (IBS-R024-D1)], andZhejiang University, Hangzhou, China. APPENDIX: TECHNICAL DETAILS 1. Green’s functions in the time domain. For a dot coupled to two leads (Fig. 4), the Hamiltonian ( 7) is augmented by a term describing the right lead/summationtext p,σ/epsilon1pc† pσcpσ. In addi- tion, the tunneling Hamiltonian in Eq. ( 6) includes a term yielding the tunneling between the dot and the right leadwhich takes the same form as in Eq. ( 3), with kreplaced bypand LbyR. [Note that d R=ˆxdRand consequently BR(t)/αR=−BL(t)/αL.] The Dyson equation for the Green’s function on the dot Gdd(t,t/prime) (in matrix notations in spin space) reads as Gdd(t,t/prime)=gd(t,t/prime)+/integraldisplay dt1gd(t,t1)[JLV† L(t1)GLd(t1,t/prime) +JRV† R(t1)GRd(t1,t/prime)]. (A1) The first term in the square brackets results from the tunnel coupling with the left lead [see Eq. ( 3)], and the second comes from the tunnel coupling with the right lead. The two otherGreen’s functions introduced in Eq. ( A1)a r e G dL(t,t/prime)=/summationdisplay kGdk(t,t/prime), GLd(t,t/prime)=/summationdisplay kGkd(t,t/prime)( A 2 ) (with analogous definitions for GdRandGRd). The Dyson’s equation ( A1), as all other encountered below, refer to all three Keldysh Green’s functions, the lesser (superscript <), the retarded (superscript r), and the advanced (superscript a) [23,24]. In Eq. ( A1),gd(t,t/prime) is the Green’s function of the isolated dot; its retarded and advanced forms are gr(a) d(t,t/prime)=∓i/Theta1(±t∓t/prime)e x p [−i/epsilon1(t−t/prime)], (A3) 075419-6MAGNETIZATION GENERATED BY MICROW A VE-INDUCED … PHYSICAL REVIEW B 102, 075419 (2020) while the lesser function is zero since the isolated dot is a s s u m e dt ob ee m p t y . The Dyson’s equations for the Green’s functions ( A2) read (in matrix notations in spin space) GLd(t,t/prime)=JL/integraldisplay dt1gL(t,t1)VL(t1)Gdd(t1,t/prime), GdL(t,t/prime)=JL/integraldisplay dt1Gdd(t,t1)V† L(t1)gL(t1,t/prime), (A4) where gL(t,t/prime) is Green’s function of the decoupled left lead. Within the wide-band approximation [ 25], the retarded, ad- vanced, and lesser functions of the latter are gr(a) L(t,t/prime)=∓iπNLδ(t−t/prime)( A 5 ) and g< L(t,t/prime)=i/summationdisplay ke−i/epsilon1k(t−t/prime)fL(/epsilon1k) =2πiNL/integraldisplaydω 2πe−iω(t−t/prime)fL(ω). (A6) The density of states of the left lead at the Fermi energy is denoted NL, and fL(/epsilon1k) is the Fermi function there. The physical quantities studied in the main text involve the lesser Green’s functions at equal times. Straightforwardmanipulations of Eqs. ( A1) and ( A4) yield that G < dd(t,t) comprises contributions from the coupling of the dot to theleft and right leads: G < dd(t,t)=G< dd,L(t,t)+G< dd,R(t,t), (A7) where G< dd,L(t,t)=2i/Gamma1L/integraldisplaydω 2πfL(ω)WL(t,ω). (A8) (For more details, see Refs. [ 26,27].) Here, /Gamma1L=2πJ2 LNLis the partial width of the resonance on the dot, created by thetunnel coupling with the left lead. An analogous expressionpertains for G < dd,R(t,t). The total resonance width on the dot is/Gamma1=/Gamma1L+/Gamma1R. The key player in our scheme is the 2 ×2 matrix in spin space WL(t,ω), defined in Eq. ( 10). Exploiting the expression forV† L(t) valid for a weak spin-orbit coupling [see Eq. ( 3)], we find /integraldisplayt dt1ei(ω−/epsilon1+i/Gamma1)(t−t1)[1−|BL(t1)|2/2−iσ·BL(t1)] =D(ω)/bracketleftbig 1−(1+γ2)α2 L/4/bracketrightbig −[(1−γ2)α2 L/4]F2(t,ω) −iσ·[B− Lei/Omega1tD(ω−/Omega1)+B+ Le−i/Omega1tD(ω+/Omega1)],(A9) where D(ω)i s[ 19] D(ω)=i/[ω−/epsilon1+i/Gamma1] (A10) and F2(t,ω)=1 2[ei2/Omega1tD(ω−2/Omega1)+e−i2/Omega1tD(ω+2/Omega1)].(A11) Using the result ( A9)i nE q .( 10), one finds that WL(t,ω)=Wdc L(ω)+Wac L(t,ω), (A12)where Wdc L(ω) does not depend on time, Wdc L(ω)=|/tildewideD(ω)|2+iD1(ω)σ·B− L×B+ L, (A13) andWac L(t,ω) oscillates with frequencies /Omega1and 2/Omega1, Wac L(t,ω)=B− L·B− Le2i/Omega1tD2(ω)+c.c. +iσ·[B+ Le−i/Omega1tD3(ω)−c.c.]. (A14) The function |/tildewideD(ω)|2in Eq. ( A13), |/tildewideD(ω)|2=|D(ω)|2−(1+γ2)/parenleftbig α2 L/2/parenrightbig ×(|D(ω)|2−[|D(ω−/Omega1)|2+|D(ω+/Omega1)|2]/2), (A15) is the correction (due to the spin-orbit coupling) of the Breit-Wigner resonance on the dot. The other functions inEqs. ( A13) and ( A14)a r e D 1(ω)=|D(ω−/Omega1)|2−|D(ω+/Omega1)|2, D2(ω)=[(ω−/epsilon1)2−(/Omega1−i/Gamma1)2]−1 −[1+4i/Gamma1/Omega1|D(ω)|2][(ω−/epsilon1)2−(2/Omega1−i/Gamma1)2]−1, D3(ω)=|D(ω)|2[2/Omega1(ω−/epsilon1)][(ω−/epsilon1)2−(/Omega1+i/Gamma1)2]−1, (A16) and they all vanish when /Omega1=0. 2. Magnetization rates in the leads. By solving the Dyson’s equations ( A4), the magnetization rate in the left lead, given in Eqs. ( 13) and ( 14), can be expressed in terms of the Green’s functions on the dot [ 27]: d dt/summationdisplay k/angbracketleftc† kσ(t)ckσ/prime(t)/angbracketright =− 2i/Gamma1L/parenleftbigg [VL(t)G< dd(t,t)V† L(t)]σ/primeσ−/integraldisplaydω 2πfL(ω) ×/integraldisplay dt1[e−iω(t−t1)VL(t1)Ga dd(t1,t)V† L(t)−H.c.]σ/primeσ/parenrightbigg . (A17) This expression is conveniently written in the form d dt/summationdisplay k/angbracketleftc† kσ(t)ckσ/prime(t)/angbracketright=[VL(t)XL(t)V† L(t)]σ/primeσ, (A18) where XL(t)=− 2i/Gamma1LG< dd(t,t)+2i/Gamma1L/integraldisplaydω 2πfL(ω) ×/integraldisplay dt1[e−iω(t−t1)V† L(t)VL(t1)Ga dd(t1,t)−H.c.]. (A19) 075419-7O. ENTIN-WOHLMAN et al. PHYSICAL REVIEW B 102, 075419 (2020) The advantage of this representation is revealed when Eqs. ( A8) and ( 10) are used to find d dtG< dd,L(t,t)=− 2/Gamma1G< dd,L(t,t)+2/Gamma1L/integraldisplaydω 2πfL(ω) ×/integraldisplay dt1[V† L(t)e−iω(t−t1)VL(t1)Ga dd(t1,t) −H.c.]. (A20) It then follows thatXL(t)=idG< dd,L(t,t)/dt+2i[/Gamma1RG< dd,L(t,t) −/Gamma1LG< dd,R(t,t)]. (A21) For the single-lead junction, considered in Sec. II,/Gamma1R=0, and therefore only the first term on the right-hand side ofEq. ( A21) survives. The corresponding expression for the two-terminal junction is obtained upon inserting Eqs. ( A7) and ( A8)i nE q .( A21); this yields Eq. ( 27) in the main text. The explicit expression for the magnetization rate in the left lead is obtained by using Eq. ( 27)i nE q .( 15). Denoting for brevity ˆ/lscript /primeL(t)=V† L(t)ˆ/lscriptVL(t), and using Eqs. ( A13) and ( A14), we find ˙ML(t)·ˆ/lscript=− 4i/Gamma1L/integraldisplaydω 2πfL(ω)ˆ/lscript/primeL(t)·d dt[B+ Le−i/Omega1tD3(ω)−c.c.] +8i/Gamma1L/Gamma1R/integraldisplaydω 2πD1(ω)[B− R×B+ RfR(ω)−B− L×B+ LfL(ω)]·ˆ/lscript/primeL(t) −8i/Gamma1L/Gamma1R/integraldisplaydω 2π/parenleftBig fR(ω)[B+ Re−i/Omega1tD3(ω)−c.c.]−fL(ω)[B+ Le−i/Omega1tD3(ω)−c.c.]/parenrightBig ·ˆ/lscript/primeL(t), (A22) where the functions D1(ω) and D3(ω) are defined in Eqs. ( A16). The dc magnetization rate (to second order in the spin-orbit coupling) is ˙Mdc L·ˆ/lscript=− 8i/Gamma1L/Omega1B+ L×B− L·ˆ/lscript/integraldisplaydω 2πfL(ω)2 Im[ D3(ω)] +8i/Gamma1L/Gamma1R/integraldisplaydω 2πD1(ω)[B− R×B+ RfR(ω)−B− L×B+ LfL(ω)]·ˆ/lscript +16i/Gamma1L/Gamma1R/parenleftbigg B+ R×B− L·ˆ/lscript/integraldisplaydω 2πfR(ω)D3(ω)+B+ L×B− R·ˆ/lscript/integraldisplaydω 2πfR(ω)D∗ 3(ω)/parenrightbigg −16i/Gamma1L/Gamma1RB+ L×B− L·ˆ/lscript/integraldisplaydω 2πfL(ω)2 Re[ D3(ω)]. (A23) AsB± L/αL=−B± R/αR, and by Eqs. ( A16) 2R e [ D3(ω)]=4|D(ω−/Omega1)D(ω+/Omega1)|2/Omega1(ω−/epsilon1)[1−/Omega12|D(ω)|2], 2I m [ D3(ω)]=2/Gamma1/Omega1|D(ω)|2[|D(ω−/Omega1)|2−|D(ω+/Omega1)|2], (A24) the rate ˙Mdc Ltakes the form given in Eq. ( 28). [1] D. Sander, S. O. Valenzuela, D. Makarov, C. H. Marrows, E. E. Fullerton, P. Fischer, J. McCord, P. Vavassori, S. Mangin, P.Pirro, B. Hillebrands, A. D. Kent, T. Jungwirth, O. Gutfleisch,C. G. Kim, and A. Berger, The 2017 magnetism roadmap,J. Phys. 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Oppeneer, Orbitally dominated Rashba-Edelstein effect in noncentrosym-metric antiferromagnets, Nat. Commun. 10, 5381 (2019) . [10] J. Puebla, F. Auvray, N. Yamaguchi, M. Xu, S. Zulkarnaen Bisri, Y . Iwasa, F. Ishii, and Y . Otani, Photoinduced RashbaSpin-to-Charge Conversion via Interfacial Unoccupied State,P h y s .R e v .L e t t . 122, 256401 (2019) . [11] D. Hernangómez-Pérez, J. D. Torres, and A. López, Photoin- duced electronic and spin properties of quantum Hall systemswith Rashba spin-orbit coupling, arXiv:2005.05450 . [12] Y . Aharonov and A. Casher, Topological quantum Effects for Neutral Particles, P h y s .R e v .L e t t . 53, 319 (1984) . [13] R. I. Shekhter, O. Entin-Wohlman, and A. Aharony, Suspended Nanowires as Mechanically-Controlled Rashba Spin-Splitters,P h y s .R e v .L e t t . 111, 176602 (2013) . [14] J. H. Bardarson, A proof of the Kramers degeneracy of trans- mission eigenvalues from antisymmetry of the scattering ma-trix, J. Phys. A: Math. 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Murakami, Conversion between electron spin and microscopic atomic rotation, Phys. Rev. Res. 2, 023275 (2020) . [23] D. C. Langreth, Linear and nonlinear response theory with applications, in Linear and Nonlinear Electron Transport in Solids , edited by J. T. Devreese and E. van Boren (Plenum, New York, 1976). [24] A. P. Jauho, Nonequilibrium Green function modeling of trans- port in mesoscopic systems, in Progress in Nonequilibrium Green’s Functions II , edited by M. Bonitz and D. Semkat (World Scientific, Singapore, 2003). [25] A.-P. Jauho, N. S. Wingreen, and Y . Meir, Time-dependent transport in interacting and noninteracting resonant-tunnelingsystems, P h y s .R e v .B 50, 5528 (1994) . [26] M. M. Odashima and C. H. Lewenkopf, Time-dependent reso- nant tunneling transport: Keldysh and Kadanoff-Baym nonequi-librium Green’s functions in an analytically soluble problem,Phys. Rev. B 95, 104301 (2017) . [27] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.102.075419 for details of the Keldysh Green’s functions’ calculation (see Ref. [ 15]). 075419-9

PhysRevB.96.184506.pdf

PHYSICAL REVIEW B 96, 184506 (2017) Interface between Sr 2RuO 4and Ru-metal inclusion: Implications for its superconductivity Soham S. Ghosh,1,2Yan Xin,2Zhiqiang Mao,3and Efstratios Manousakis1,2,4,* 1Department of Physics, Florida State University, Tallahassee, Florida 32306-4350, USA 2National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306-4350, USA 3Department of Physics, 5024 Percival Stern Hall, Tulane University, New Orleans, Louisiana 70118 USA 4Department of Physics, University of Athens, Panepistimioupolis, Zografos, 157 84 Athens, Greece (Received 28 November 2016; revised manuscript received 30 August 2017; published 8 November 2017) Under various conditions of the growth process, when the presumably unconventional superconductor Sr 2RuO 4 (SRO) contains microinclusions of Ru metal, the superconducting critical temperature increases significantly. An atomic resolution high-angle annular-dark-field scanning transmission electron microscopy study shows a sharpinterface geometry which allows crystals of SRO and of Ru metal to grow side by side by forming a commensuratesuperlattice structure at the interface. In an attempt to shed light on why this happens, we investigated the atomicstructure and electronic properties of the interface between the oxide and the metal microinclusions using densityfunctional theory calculations. Our results support the observed structure, indicating that it is energetically favoredover other types of Ru-metal/SRO interfaces. We find that t 2g−egorbital mixing occurs at the interface with significantly enhanced magnetic moments. Based on our findings, we argue that an inclusion-mediated interlayercoupling reduces phase fluctuations of the superconducting order parameter, which could explain the observedenhancement of the superconducting critical temperature in SRO samples containing microinclusions. DOI: 10.1103/PhysRevB.96.184506 I. INTRODUCTION The superconducting state of Sr 2RuO 4(SRO) has been intensively studied [ 1] since its discovery in 1994 [ 2]. SRO is a layered perovskite oxide sharing the same structureas La 2CuO 4, one of the parent compounds of the cuprate superconductors, and it is believed to be a p-wave supercon- ductor with odd spin-triplet pairing [ 1,3–8]. A chiral orbital order parameter of the form px+ipyhas been suggested by time-reversal symmetry-breaking experiments [ 9,10]. The role of strong ferromagnetic spin fluctuations in mediating super-conductivity has been pointed out [ 11]. Theoretical [ 12,13] and experimental investigations which were carried out before2002 have been reviewed [ 1]. Quantum oscillation experiments [14] indicate that the normal state can be understood as a two-dimensional Fermi liquid [ 1]. The electronic properties have been studied by a number of methods which are summarized in Ref. [ 1], including the local-density approximation (LDA) method [ 15,16] and the generalized gradient approximation (GGA) [ 17,18] method. Depending on the exchange correlation functional used, GGApredicts either a nonmagnetic state [ 18] or an antiferro- magnetic (AF) state [ 17] with ferromagnetically ordered RuO 2basal planes. There is evidence for incommensurate antiferromagnetic spin fluctuations [ 19,20], and it has been shown that ferromagnetic and antiferromagnetic fluctuationscoexist in this oxide [ 21]. In more recent studies, superconductivity in bulk SRO is found to be enhanced under uniaxial /angbracketleft001/angbracketright[22],/angbracketleft100/angbracketright, and /angbracketleft110/angbracketrightstrains, where in the latter two cases strain-driven asym- metry of the lattice is believed to cause a change in symmetryof the superconducting order parameter [ 23].T cis also found to be enhanced due to dislocations [ 24] and in a system where there is an interface of W/Sr 2RuO 4point contacts [ 25]. *manousakis@magnet.fsu.eduThe unexpected enhancement of Tcfrom 1.5 to almost 3 K, when microsized ruthenium-metal inclusions are embeddedinside SRO in the eutectic system during crystal growth [ 26], is a very interesting and unexplained phenomenon. Thereis evidence [ 27] that the “3 K” superconducting phase is unconventional with the presence of a hysterisis loop [ 28]. In this 3 K phase, the ruthenium microplatelets are notuniquely oriented with respect to the SRO lattice. This,together with large diamagnetic shielding, could suggestthe existence of interface superconductivity at the Ru-SROinterface, residing primarily in SRO [ 26]. Sigrist and Monien’s [29] phenomenological analysis postulates a superconducting state with different symmetry and higher T cthan in the bulk. High-angle annular-dark-field (HAADF) scanning trans- mission electron microscopy (STEM) images of a represen-tative interface were presented in Ref. [ 30] and are given in the present paper with higher resolution, as shown in Fig. 1. Here, we explore the atomic structure, stability, and electronicproperties of this interface using density functional theory(DFT) calculations. We notice that this stable interface isperpendicular to the SRO bulk crystallographic aaxis and has alternating intact meandering Ru-O octahedra, whichcan be conceived as continuations of the bulk SRO RuO 2 planes. Alternating pairs of Ru columns (which, as we show,are coordinated by oxygen atoms which are not visible inthe HAADF-STEM images) fill in the gaps created by themeandering interface at the same periodicity as the SRO unitcell along the crystallographic caxis. We show that these pair columns of Ru atoms have a coordination number differentfrom that in the metal phase and that in the SRO phase. A nearlyperfect hcp crystal of Ru metal grows from the next metal layerwith a small lattice mismatch that eventually relaxes as onemoves away from the interface and into the inclusion. In the lowest-energy interface the Ru-metal grows its first layer commensurate with the SRO interface at a wavelengthwhich nearly corresponds to 11 Ru-metal atoms for every twoperiods of the SRO crystal along its own caxis. Therefore, 2469-9950/2017/96(18)/184506(8) 184506-1 ©2017 American Physical SocietyGHOSH, XIN, MAO, AND MANOUSAKIS PHYSICAL REVIEW B 96, 184506 (2017) FIG. 1. (a) Scanning electron microscope image of a cleaved (001) Sr 2RuO 4bulk crystal containing parallel Ru microplatelets (brighter contrasted short lines). (b) HAADF-STEM image of the Sr2RuO 4/Ru interface at lower magnification showing an atomically straight and sharp superstructured interface, where the brighter regionon the left is the Ru-metallic inclusion and on the right is the Sr 2RuO 4 phase. The brightest spots are Ru, followed by Sr. O atoms are toofaint to be observed. There is no concentration gradient of SRO orRu on either side of the interface. The inset gives a close-up view of the interface at higher magnification, where the atomic structure is clearly revealed. The HAADF-STEM images were taken with aprobe of 0.078 nm and a convergence semiangle of 21 mrad and inner collection angle of 78 mrad. Brighter-contrast atoms are Ru atoms, while Sr atoms are less bright. the inclusions connect RuO 2planes through a commensurate interface which is almost perfectly ordered at the atomic scale. We show that this interface is stable against phase sepa- ration, and it is more stable than other conceivable interfacesbetween SRO and Ru metal. A spin-polarized GGA calculationyields significant magnetic moments of the Ru atoms in theSRO phase near the interface. Our study establishes a clearpicture of the stable Ru-Sr 2RuO 4interface, which is important in understanding the unconventional 3 K phase.We also argue that our findings, that the Ru inclusions form a nearly atomically perfect interface with the SRO crystal,imply the emergence of a significant interlayer coupling whichcould give rise to reduction of phase fluctuations of thesuperconducting order parameter characterizing the variousRuO 2planes. This is expected to lead to an enhancement of the superconducting critical temperature, as observed in theSRO crystals with Ru inclusions. The paper is structured as follows. In Sec. II, we present our experimental results and the computational details of our DFTcalculations of the observed interface structure and stabilityconsideration of various terminations. In Sec. III, we analyze the experimentally observed interface in light of the DFT-basedcalculations and also discuss our detailed results obtained fromthese calculations. In Sec. IV, we discuss the possible causes of increased T cbased on our findings. Last, in Sec. V,w e highlight our conclusions and the implications of this work. II. EXPERIMENTAL AND DFT STUDY OF THE INTERFACE STRUCTURE The platelet inclusions are the result of excess Ru (more RuO 2in the mixture than needed to make stoichiometric SRO) in the initial mixture. We will show by means of DFTcalculations that the interface seen in our HAADF-STEMimages is low enough in energy to be preferable to macroscopicphase separation of SRO and Ru-metal inclusions. In addition,we will show that other related interfaces are energeticallyhigher than the one shown in Fig. 1. Sr 2RuO 4occurs in bulk in the body-centered tetragonal structure like the high-temperature superconductor La 2CuO 4 [31]. Ru, the brightest atoms in the HAADF-STEM images in the SRO phase, and the planar O(1) atoms form a two-dimensional square lattice, with Ru-O bond length being1.95 ˚A, which is less than the sum of the ionic radii of Ru 4+ and O2−, suggesting planar hybridization [ 16]. The apical O(2) height above and below is larger at 2.06 ˚A. A. Analysis of the observed interface First, we notice in Fig. 1and in Fig. 1of Ref. [ 30] that the interface is perpendicular to the bulk [100] SRO direction andis terminated at alternating intact meandering Ru-O octahedra,which are continuations of the bulk SRO RuO 2planes, and alternating pairs of Ru columns filling in the gaps created bythe meandering interface. Notice that, as seen in Fig. 1,t h eR u ions of the metallic interface, without any significant changeof the unit-cell size, form an interface at a commensuratewavelength nearly twice that of the SRO unit-cell size inthe [001] direction (corresponding to 11 unit cells of pureRu crystal). The commensurate growth leaves a small latticemismatch between the SRO layers near the interface andthose deeper in the bulk. This causes a strain that growswith system size until it becomes energetically favorable toproduce dislocations [ 24] relieving the strain. As seen in the transmission electron microscopy (TEM) images in Ref. [ 24], the dislocations are complex structures which occur, for a sharpand flat interface, over a length scale longer than that visiblein Fig. 1. These structures are beyond the scope of our present ab initio calculations, and we focus here on the microscopic length scale at which the interface is free of dislocations. 184506-2INTERFACE BETWEEN Sr 2RuO 4AND Ru-METAL . . . PHYSICAL REVIEW B 96, 184506 (2017) TABLE I. The average atomic column intensity and EELS- measured sample thickness in the SRO phase, in the interface pair columns, in the first hcp layer (closest to the interface), and in the third hcp layer. The intensity is calculated as the sum of pixel intensitiesof one atom column within a window size of 15 ×17 pixels in units of 10 23electrons. Atom and location Sr Ru Ru Ru Ru SRO SRO interface 1st hcp layer 3rd hcp layer Intensity (1023e) 3.78 4.26 4.90 5.05 5.33 Error ( ±) 0.10 0.10 0.08 0.08 0.08 Thickness (nm) 43.2 43.2 47.09 47.09 49.63 Error ( ±) 0.54 0.54 0.91 0.91 0.68 To verify that the interface columns are Ru atoms, in Table Iwe provide the average HAADF-STEM intensity of several atomic columns (along the SRO [010] direction),as well as the sample thickness at those column locationsmeasured with electron-energy-loss spectroscopy (EELS). TheTEM sample preparation of such a heterostructural interfacecreates a sudden thickness change within a few nanometersat the interface between two such dissimilar materials thathave different milling rates. The atomic column intensity inthe HAADF-STEM image scales proportionally with sample thickness and is affected by crystal orientation. As we shall see in subsequent sections, the average nearest-neighbordistance along the beam direction for the interface Ru pairatoms is 3.9 ˚A, the same as that in SRO. The intensity is marginally higher for the interface Ru atoms because theinterface columns are slightly thicker than the SRO bulk region(see Table I). In addition, there is higher background noise at the interface due to contamination and scattering fromadjacent Ru atoms in the Ru platelet. Given that the columnsin the hcp platelet have more Ru atoms than those in theSRO phase and those in the interface columns, we find thecolumn intensity in the hcp platelet to be somewhat lower thanexpected. This is due to the fact that the intensity of the Ruatoms in the platelet is more diffused than in SRO, as can beclearly seen from Fig. 1. Here, the Ru atom columns are not lined up precisely along the beam direction, and therefore, thedynamical scattering effect is less in the platelet. We confirmedthis by changing the window size of the intensity measurement,and we found that the largest change is in the intensity of theplatelet columns. The small misalignment angle ( <1 ◦), over a column several hundred atoms thick, is enough to reducethe intensity noticeably. It is, however, too small to be takeninto account in our DFT calculations. Namely, taking sucha small misalignment into account would require a unit cellcontaining several tens of times more atoms than the unit cellpresently used. Therefore, we have assumed perfect alignmentbetween the SRO and the hcp phase in this direction in ourcomputational work as we do not expect such a small deviationfrom the perfect alignment to yield a significant effect. The HAADF-STEM image provides direct information on the interface structure but leaves ambiguous the location of anyoxygen atom near the interface and the Ru atom positions in theSRO [010] direction, which is perpendicular to the HAADF-STEM field of view. This is the first item we address by means FIG. 2. Sr 2RuO 4-Ru supercell used for spin-GGA calculations of the heterojunction as generated by the VESTA software package. There are three different Ru-O bond lengths: The Ru-O bond lengths in the interface columns are the shortest, followed by the in-plane Ru-O(1) distances, followed by the Ru-O(2) bond lengths. The interface Ruatom columns mediate between the SRO phase and the Ru-metallic phase. We show part of the next repeated image for clarity. SRO lattice vectors are shown. of our DFT calculations, the technical details of which are discussed later. By examining various possibilities, we findthat in their optimum positions, the Ru pair columns bridgediagonal oxygen atoms of the terminating SRO layer. On oneside of the interface these Ru pair columns form a bond tooxygen atoms similar to the 1.95- ˚A Ru-O bond in the rutile phase of RuO 2, and on the other side they have an hcp Ru-metal environment. This arrangement leaves the meandering SROtermination layer unchanged, subject to ionic relaxations, andis consistent with the experimental image in Fig. 1. Therefore, we believe that this part of our DFT study complements theHAADF-STEM image, and we now have a complete pictureof the structure of the interface. B. Stability of the interface The implementation of the observed structure by DFT calculations ideally requires at least 11 bulk unit cells of the hcpRu-metal phase, commensurate with two conventional bulkunit cells of the SRO along the SRO [001] axis, and at leastsix bulk unit cells of the hcp Ru, commensurate with four bulkunits cells of the SRO along the SRO [010] direction. Such asupercell proved computationally unfeasible, particularly dueto the large number of ruthenium atoms. Instead, to keep thesupercell size reasonable, we used a supercell geometry witha stretched unit-cell length of 13.52 ˚A along the SRO [001] direction and double the SRO bulk unit-cell length in the SRO[010] direction. In this reduced supercell containing 128 atoms, shown in Fig. 2, one conventional bulk SRO unit cell is commensurate with six bulk hcp Ru unit cells along SRO [001], and two bulkSRO unit cells are commensurate with three bulk metal unitcells in the SRO [010] direction. The atoms have been relaxedto their final positions. This geometry stretches the SRO caxis by 5% and compresses the aandbaxes of the hcp Ru by 4% each. The apical O(2) height is increased from 2.06 to 2 .09˚A. In Fig. 3we show the SRO b−cplane of the two layers of the 184506-3GHOSH, XIN, MAO, AND MANOUSAKIS PHYSICAL REVIEW B 96, 184506 (2017) FIG. 3. The SRO b−cplane showing the two layers of the SRO-Ru interface, formed by the interface Ru pair columns and the terminating layer of the hcp metal. The unit-cell lengths are slightly different from ideal bulk values (see details in the text). The triangularlattice of the Ru-metal layer and the rectangular lattice of the interface Ru pairs are commensurate with 7 .81˚A×13.52˚A wavelengths in the SRO [010] and [001] directions, respectively. The Ru columns are situated in the low-energy valleys of the triangular lattice. The experimental wavelength is at least twice in each direction with anincreased fraction of misalignment where some interface columns are located away from the hcp valleys, but the idea of commensuration between a rectangular lattice and a triangular lattice is preserved. SRO-Ru interface formed by the interface Ru pair columns and the terminating layer of the hcp metal. Even though wewill use this smaller supercell in our DFT implementation, bycarrying out various types of optimization calculations, it willbecome reasonably clear that the observed structure is, indeed,the most energetically favorable among various other plausibleatomic configurations. Table IIsummarizes our stability analysis results. We describe the procedure and the notations below. First, we find that the energy needed for breaking apart the interface of our reduced structure to create the two constituentslabs, SRO with [100] surfaces and Ru metal with [001]surfaces (see Fig. 4; denoted by “slabs” in Table II), is 13.16 eV , a significant amount of energy. TABLE II. Total energy of various geometries per unit interface area using the conventional unit cell of Sr 2RuO 4as one unit. Etotis defined relative to the most stable state, namely, the interface with meanders. For details about the geometries, see Sec. II B. Configuration Slabs Flat Aligned Bulk Etot(eV) 3.29 2.09 1.01 0.43 FIG. 4. (a) The 2.5-layer-thick Sr 2RuO 4slab with equivalent (010) surfaces used to compute the energy of the (001) linear strain changing the c-lattice vector length from 12.9 to 13.52 ˚Aw h i l e keeping the aandbvectors the same as in the bulk. (b) The 1.5-layer-thick ruthenium metal slabs in the hcp structure with (001) surfaces used to compute the energy of planar compression. The a andblattice vectors were each reduced from 2.73 to 2.61 ˚A. In both calculations we used at least 15 ˚A of vacuum between repeated images. Second, we compare the energy of our reduced structure to the energy of a similar supercell but with a flat SRO [100]-Ru[001] interface (shown in Fig. 5and denoted by “flat” in Table II) which we compute by bringing the SRO and Ru phases closer in small steps and relaxing the ions around theirpositions to find the minimum of the energy. We find thatthis supercell is higher in energy by 8.37 eV compared to ourreduced meandering geometry in Fig. 2. Thus, the meandering octahedra and the interface ruthenium columns are necessaryto stabilize the interface. In both the above calculations, thereare 128 atoms on either side of the equation, and we accountfor the balance of atoms in the nonmeandering structuresby putting the extra Ru pair columns of the interface in a FIG. 5. A hypothetical interface between SRO [100] and Ru hcp [001] surfaces, created from the slabs in Fig. 4by bringing them together gradually and allowing the atoms to relax until the optimuminterlayer distance at the interface is reached. This geometry is similar to that in Fig. 2except for the absence of the Ru interface columns. The hcp caxis is parallel to the SRO [100] direction. Part of the next repeated image is shown for clarity. 184506-4INTERFACE BETWEEN Sr 2RuO 4AND Ru-METAL . . . PHYSICAL REVIEW B 96, 184506 (2017) ruthenium bulk phase, which is the most stable ruthenium phase. To describe why the meandering Ru interface columnsare necessary, consider the two interface layers ( b−cplane of SRO) in Fig. 3. We note that the interface Ru columns form a rectangular planar lattice (which is commensurate with thesame periodicity of our reduced interface geometry) adjacentto the triangular lattice of the terminating Ru-metal plane.Furthermore, the interface Ru pairs to a large extent are situatedin potential valleys where the atoms of the next Ru hcp layerwould have been in the absence of an interface. Third, having established the necessity of the meanders, we argue that the larger interface with 11 unit cells of Rumetal in the SRO [001] direction and 6 Ru-metal unit cellsin the SRO [010] direction is more stable than our computedreduced interface as follows. The reduced geometry suffers from artificial planar com- pression of its Ru-metal phase and uniaxial strain of its SROphase, but it accommodates the interface Ru pair columns inthe grooves created by the terminating hcp Ru layer. By slidingthe Ru-metal phase across the interface in Fig. 2, we find that this is, indeed, an energy minimum. The larger experimentallyobserved periodicity breaks this symmetry, and at least someof the Ru column pairs are misaligned with respect to the hcplayer. This leads to frustration in the terminating hcp layerof the metal, as can be observed directly in Fig. 1.T oafi r s t approximation, the price of lattice length manipulation can be calculated by computing the sum of the individual energy losses caused by separately compressing an ideal Ru-metalslab and stretching an ideal SRO slab (Fig. 4) to their respective values necessary to create the reduced supercell shown inFig. 2. We find this energy to be 8.54 eV . On the other hand, the energy loss due to nonalignment of the Ru interfacepair columns in the experimentally observed geometry can beupper bounded by sliding the Ru-metal layers of the reducedgeometry along the interface, thus moving the Ru pair columnsaway from the potential valleys of the hcp layer, until we reachan energy maximum. The maximum cost of misalignment of the interface Ru pair columns is 4.49 eV , less than the cost of stretching andcompressing the constituent phases. It follows that in anyinterface between the SRO (100) surface and Ru (001) surface,meandering Ru interface columns with the experimentallyobserved periodicity are the most stable structure. We denotethis configuration in which the SRO slab is compressed andRu metal is stretched but, as a result, the interface is alignedas “aligned” in Table II. As the final part of our stability argument, we consider the possibility that there could be other types of interfaces withoutthese interface Ru columns which are more stable than ours.In particular, the (001) surface of SRO can be cleaved witha terminating SrO or RuO 2layer, and its interface with a ruthenium hcp layer can be conceived. We show in Fig. 6 a commensurate geometry at the interface in which a 5 ×6 RuO 2superstructure (defined along the SRO aandbaxes, respectively) is commensurate with a 7 ×5 superstructure of the metal after accounting for a 0.8% uniaxial strain on the Rumetal. Most of the atoms of each phase in this superlattice arerandomly oriented with those of the other phase, and therefore,each layer at the interface will experience the laterally averagedpotential of the other layer. This is arguably a small energy gain FIG. 6. A possible interface between (001) SRO and hcp Ru metal with a 19.52 ˚A×23.42 ˚A superlattice, where after a 0.8% strain on the hcp plane, five planar unit cells of RuO 2are commensurate with seven unit cells of the Ru metal in the SRO [100] direction and six unit cells of RuO 2are commensurate with five unit cells of the Ru metal in the SRO [010] direction. Most of the atoms at the interfaceare frustrated and experience the laterally averaged potential, leading to very little energy gain. Both SRO and Ru-metal lattice vectors are shown. of the order of a few tens of meV for each SRO formula unit. In fact, any interface without the periodic meanders observed inour experiment is likely to be disfavored for the same reason.A straightforward GGA computation shows that our reducedstructure is 1.72 eV lower in energy than the sum of energiesof a Ru-metal slab similar to that shown in Fig. 4(b) and the appropriate amount of bulk SRO (the sum is denoted by “bulk”in Table II), a surprising result. The experimentally observed superlattice should be even more stable. We conclude thereforethat the observed meandering interface is favored even overphase separation of the eutectic mixture between bulk SRO andRu. Therefore, we cannot think of any other interface whichcan compete with the one observed by our HAADF-STEMstudy. Last, the following question arises. Since we find that the observed interface lowers the energy with respect to bulk SROand a semi-infinite Ru metal, why does the system not try tocreate more such interfaces and instead grow mesoscopic-sizeinclusions? Indeed, our findings indicate that the lowest-energystate of such a system of SRO with excess Ru metal shouldbe a state with a high density of such interfaces separatedby a microscopic-size length. However, the combined systemwas created under nonequilibrium conditions of the eutecticmixture which do not allow the system to search for a globallowest-energy state. First, at a relatively short time scale thefree energy is only locally minimized, and then, the systemfreezes in a macroscopic state of domains which require anoverwhelmingly large amount of time to find the state which isthe global minimum. More specifically, once a few layers of Ru 184506-5GHOSH, XIN, MAO, AND MANOUSAKIS PHYSICAL REVIEW B 96, 184506 (2017) metal have grown, it becomes locally energetically favorable for more Ru atoms from the excess of Ru to attach themselvesto those existing Ru-metal layers. Forming another interfacewhich combines a simultaneous and coherent arrangement ofmany atoms is a much slower process (i.e., a low-entropy state)than simply adding to the existing Ru-metal layer an additionalsingle Ru atom. This means that the path to the actual groundstate is “narrow” and requires a very slow process, and asa result, the system gets stuck in other metastable localfree-energy minima. Thus, although a high density of suchinterfaces is preferred by taking into consideration just theenergy of the system, the meandering termination layer ofSRO with intact RuO 2octahedra and interface Ru columns are long-range phenomena generally suppressed by the largeentropy present in the high-temperature eutectic mixture. C. Computational details Spin-GGA computations were performed using a plane- wave basis set (cutoff of 540 eV) with the projector augmented-wave methodology [ 32] used to describe the wave functions of the electrons as implemented in the V ASP package [ 33– 36] using the Perdew-Burke-Ernzerhof exchange correlation functional [ 37]. The 4 s,4p, and 5 selectrons of strontium; the 5 s,4d,4p, and 4 selectrons of ruthenium; and the 2 sand 2pelectrons of oxygen were treated as valence electrons. The Brillouin zones of the 128-atom supercell with the meanderinginterface geometry (Fig. 2) and the 124-atom supercell with flat interfaces (Fig. 5) were sampled with a 1 ×6×4k-point grid, and 60 kpoints were used to compute the electronic density of states (DOS). Increasing the k-point grid from 1 × 6×4t o1 ×7×5 leads to a negligibly small change in the total energy of the meandering interface geometry by 0.04 eVand an increase of the moments of the magnetic rutheniumatoms by 0 .08μ B. The 70-atom SRO slab [Fig. 4(a)]i s2 . 5 layers thick with symmetric 2 ×1b−csurfaces. We used a tetragonal geometry with b=3.90˚A and c=12.90˚Af o r them and sampled the Brillouin zone with a 1 ×8×4k-point grid. For the 54-atom Ru-metal slab [Fig. 4(b)], which is 1.5 layers thick with a 6 ×3×1.5 structure, we used an hcp unit- cell length of 2.739 ˚A and sampled the Brillouin zone with a 4×8×1k-point grid. For both slabs, we used a vacuum layer at least 15 ˚A thick. All the supercells were structurally relaxed while keeping the cell shape and cell volume fixed until theforces were converged to less than 10 meV /˚A for each ion. III. ELECTRONIC PROPERTIES Our computed geometry (Fig. 2)s h o w sa1 .8◦rotation of the RuO 6octahedron on the RuO 2planes along with small amounts of buckling. The number of atoms and electrons inthe supercell prevent us from sampling the Brillouin zonewith enough k-point accuracy to compare various magnetic orderings. Various magnetic orderings differ from each otherby only a few meV , and their accurate study requires anextremely dense sampling of kpoints [ 20,38]. Therefore, we limit ourselves to the q=0 state, which we find to be lower in energy than the nonmagnetic state. We find no magneticmoments in the interface columns of Ru atoms and metallic-phase Ru atoms but strong magnetic moments in the Ru-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Energy (eV)00.10.20.3Orbital Projected DOS (states/ eV)dxy dyz dz2 dxz dx2-y2 FIG. 7. Ylm-projected density of states for a Ru atom from the interface column pair. The atom has an O 2environment like in the rutile RuO 2phase with a bond angle of 115◦and a Ru-O bond length of 1.83 ˚A. On the other hand, it also neighbors the Ru-metal closed-pack layer at the interface. Both these environments influence its electronicstructure. atoms ( MRu=1.532μB) in the SRO phase. GGA calculations have previously predicted [ 39] surface ferromagnetism in SRO where it was stabilized by a large (9◦) surface octahedra rotation and consequent band narrowing. To ensure that thisis not purely an effect of the (001) strain, we performedspin-GGA calculations of bulk SRO with stretched caxis values. We find in such a system the octahedra rotationsare absent and the ground state is ferromagnetic but withmuch smaller magnetic moments ( M Ru=0.220μB). Since we consider only translationally invariant q=0 states, the possibility of complex nonzero qstates cannot be ruled out. Furthermore, GGA does not correctly account for correlations,so the magnetic picture must be taken with caution. In the interface Ru-atom columns, there are two different types of RuO 2bonds in each pair. One has a bond angle close to 90◦and a Ru-O bond length close to the planar Ru-O(1) bond length, whereas the other has a bond angle of ∼115◦ and a Ru-O bond length of ∼1.83˚A. Both Ru atoms lack the planar square-lattice coordination and are expected to havedifferent orbital structure compared to those in SRO. We findthe interface Ru atoms in a +3 valence state, consistent with a Bader [ 40] charge analysis. We also see significant t 2g−eg mixing in each of them. Figure 7shows the density of states of the Ru atom which has a RuO 2bond angle of ∼115◦.B o t h thedx2−y2anddz2states are pulled down below the Fermi level and mixed with t2gorbitals. The t2g−egmixing was found at the well-studied SrTiO 3/LaAlO 3interface [ 41], where an eg splitting was caused by an oxygen vacancy and gave rise to magnetic order. Here, it is caused by severe Ru-O hybridizationand nonplanar RuO 2geometry. IV . IMPLICATIONS FOR SUPERCONDUCTIVITY Superconductivity in Sr 2RuO 4is believed to arise from pairing within the RuO 2layers. The interlayer coupling between RuO 2planes is very weak, which is expected to lead to large-amplitude phase fluctuations of the order parameter, 184506-6INTERFACE BETWEEN Sr 2RuO 4AND Ru-METAL . . . PHYSICAL REVIEW B 96, 184506 (2017) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Energy (eV)-2.5-2-1.5-1-0.500.511.52Orbital Projected DOS (states/ eV)dxy dyz dz2 dxz dx2-y2 FIG. 8. Ylm-projected density of states for a Ru atom in the SRO phase. The egorbitals are unoccupied and pushed away from the t2gorbitals due to the crystal field. The magnetic moments ( MRu= 1.532μB)a r ear e s u l to ft h e t2gspin split. Spin up is above the horizontal axis, and spin down is below the axis. The Fermi level is at zero. like in the case of cuprate superconductivity [ 42]. These phase fluctuations lower the value of the superconductingtransition temperature. In the case where inclusions arepresent, a remarkably ordered interface geometry betweenthe tetragonal unconventional superconductor Sr 2RuO 4and hcp metal Ru has been discovered, as illustrated in ourHAADF-STEM images in Fig. 1and justified by means of our DFT calculations (Fig. 2), which also reveal the structure along the perpendicular direction, as illustrated in ourderived highly ordered structure in Fig. 3(a direction which is hidden from any HAADF-STEM study). These interfaceswith a metallic inclusion clearly should lead to an effectiveinterlayer Josephson junction coupling of the superconductingorder parameter which reduces these phase fluctuations. Thecoupling produced by these ordered inclusions should thus lead -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Energy (eV)00.020.040.060.08 Orbital Projected DOS (states/eV)py pz px FIG. 9. Ylm-projected density of states for an oxygen atom bonded to an interface Ru. The Fermi level (placed at zero) is occupied by all three porbitals. A Lorentzian smear of 0.2 eV has been applied to the DOS data.to an enhanced Tc, as observed in the 3 K phase. We believe that the reason for the enhancement of Tcdue to the inclusions is different from Sigrist and Monien’s [ 29] phenomenological model and from the cause of the strain-driven increase of Tcin pure bulk SRO seen in Ref. [ 23]. In the latter case the increase is due to the fact that the strain affects the symmetry character(i.e.,p x+ipy) of the superconducting order parameter, as argued in Ref. [ 23]. It has been shown experimentally [ 26] that inclusions increase the interlayer coherence length and significantlyreduce the anisotropy of superconductivity: ξ ab(0)/ξc(0)=3.6 in the 3 K phase with metal inclusions compared to 20 for the1.5 K phase SRO. As discussed in Sec. III, an accurate study of the magnetic ordering of an interface such as ours with a large number ofelectrons is beyond present computational capacity. However,if we take seriously (a) our finding, i.e., that the Ru-atommagnetic moments near the interface are increased comparedto the bulk, and (b) the suggested pairing mechanism dueto paramagnon exchange [ 11], we might not exclude the possibility that assumption (a) leads to an increased electron-paramagnon coupling and, as a consequence, to a T cenhance- ment in the spin-triplet pairing channel. While this scenariois possible, we believe that our observation and calculationsdiscussed in the previous two paragraphs are more likely to bethe cause of the enhancement of T cdue to the inclusions. V . CONCLUSIONS AND IMPLICATIONS A remarkably ordered interface geometry between the tetragonal unconventional superconductor Sr 2RuO 4and hcp metal Ru formed as metallic inclusions during the growthprocess of the superconductor has been revealed by HAADF-STEM studies and has been investigated and understood in thepresent paper using DFT. The heterojunction is characterizedby regular columns of Ru pairs in the SRO [001] directionand clean octahedra terminations of the ruthenate oxide.Using DFT, we have correctly reproduced the experimentalstructure, including the positions of interface oxygen atomsand along directions hidden from any HAADF-STEM study,and investigated the electronic structure of the interface. Wehave found rotated octahedra, modified Ru dorbitals, and enhanced magnetic moments near the interface in the SROphase. Application of GGA to magnetism should be done withcaution since it does not correctly account for correlations, butgiven the proximity to Stoner instability, it is possible that theinterface is in or energetically very close to a ferromagneticground state. Our study provides a possible explanation of the enhance- ment of the superconducting T cwhen Ru-metal inclusions are present. We find that these inclusions form microscopicallywell ordered interfaces and structure. The interfaces act as a“ladder” which couples the superconducting order parameterof a large number of RuO 2SRO layers over a micrometer- size length. These inclusions should lead to an interlayercoupling which can significantly reduce the superconductingorder parameter phase fluctuations, thereby increasing thesuperconducting critical temperature. This observation opens up exciting prospects for when a similar growth process is applied to the case of the cuprate 184506-7GHOSH, XIN, MAO, AND MANOUSAKIS PHYSICAL REVIEW B 96, 184506 (2017) superconductors. If these inclusions introduce an interlayer Josephson-junction-type coupling, we should expect a signifi-cant enhancement of the superconducting critical temperature. ACKNOWLEDGMENT This work was supported in part by the U.S. National High Magnetic Field Laboratory, which is partially funded by NSFGrant No. DMR-1157490 and the state of Florida. APPENDIX: PARTIAL DOS In contrast to a ruthenium atom in the interface column, the Ru atoms in the SRO phase are located at the centerof an oxygen octahedra. Due to crystal-field symmetry, they have a clear t2g−egsplit, as can be seen in Fig. 8.T h e t2g orbitals are clearly spin split, causing the significant magnetic moments discussed in the main text. This picture is in contrastto the SrTiO 3/LaAlO 3interface [ 41], where an egsplitting is associated with magnetic order. Figure 9shows the partial density of states of an oxygen atom bonded to one of the interface column rutheniumatoms, showing the Fermi level contribution of all three p orbitals. The orbital character is determined by the uniquecoordination number of the Ru atoms at the interface and ismarked by the absence of octahedral symmetry and enhancedhybridization. [1] A. P. Mackenzie and Y . Maeno, Rev. Mod. Phys. 75,657(2003 ). [2] Y . Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. 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PhysRevB.75.155315.pdf

Laser cooling of semiconductor quantum wells: Theoretical framework and strategy for deep optical refrigeration by luminescence upconversion Jianzhong Li * NanoScience Solutions, Inc., Cupertino, California 95014, USA /H20849Received 24 November 2005; revised manuscript received 8 February 2007; published 12 April 2007 /H20850 Optical refrigeration has great potential as a viable solution to thermal management for semiconductor devices and microsystems. We have developed a first-principles-based theory that describes the evolution ofthermodynamics—i.e., thermokinetics—of a semiconductor quantum well under laser pumping. This thermo-kinetic theory partitions a well into three subsystems: interacting electron-hole pairs /H20849carriers /H20850within the well, the lattice /H20849thermal phonons /H20850, and the ambient /H20849a thermal reservoir /H20850. We start from the Boltzmann kinetic equations and derive the equations of motion for carrier density and temperature, and lattice temperature, underthe adiabatic approximation. A simplification is possible as a result of ultrafast energy exchange between thecarriers and phonons in semiconductors: a single-temperature equation is sufficient for them, whereas thelattice cooling is ultimately driven by the much slower radiative recombination /H20849upconverted luminescence /H20850 process. Our theory microscopically incorporates photogeneration and radiative recombination of the interact-ing electron-hole pairs. We verify that Kubo-Martin-Schwinger relation holds for our treatment, as a necessarycondition for consistency in treatment. The current theory supports steady-state solutions and allows studies ofcooling strategies and thermodynamics. We show by numerical investigation of an exemplary GaAs quantumwell that higher power cools better when the laser is detuned from the band edge between a critical negative value and the ambient thermal energy. We argue for the existence of such a counterintuitive lower bound. Mostimportantly, we show that there exists an actual detuning, 3 meV above the band edge in the simulated free-carrier case and expected to be pinned at the excitonlike absorption peak owing to Coulomb many-bodyeffects, for optimal laser cooling. Significant improvement in cooling efficacy and theoretical possibility ofdeep refrigeration are verified with such a fixed optimal actual detuning. In essence, this work provides a consistent microscopic framework and an optimization strategy for achieving net deep cooling of semiconduc-tor quantum wells and related microsystems. DOI: 10.1103/PhysRevB.75.155315 PACS number /H20849s/H20850: 78.55.Cr, 32.80.Pj, 78.67.De, 78.20.Bh I. INTRODUCTION Recent advances in laser cooling of rare-earth-doped glasses and dye-added fluids1–3foreshadows the current pur- suit with semiconductors. The cooling mechanism is lumi-nescence upconversion, which was proposed for optical re-frigeration of matter decades ago. 4It removes heat from material under refrigeration in three steps, as schematicallyshown in Fig. 1:/H20849i/H20850A laser beam illuminates the material and creates cold electron-hole /H20849e-h/H20850pairs, /H20849ii/H20850the pairs experi- ence energy exchange through inelastic collision and therebyabsorb thermal lattice vibrations, and then /H20849iii/H20850the pairs con- vert into blueshifted, incoherent photons through radiativerecombination and carry heat away from the material. As aresult, the material is cooled down by the laser beam. 5How- ever, other processes can interfere and degrade the coolingoperation. For instance, some carriers could lose their energythrough nonradiative recombination, instead of the desiredradiative kind, and dump the absorbed laser photon energy asheat inside the material. Also, it is highly likely that thephotoluminescence /H20849PL/H20850generated in step /H20849iii/H20850cannot escape from the material because of internal reflection /H20849semiconduc- tors have typical refractive indices around 3.4, relative toair’s unit value, correspnding to a critical angle of 17.1° fortotal internal reflection /H20850, but recycles in the material. Even- tually, the recycled photons either get away or get absorbedand generate new e-h pairs. This recycling process couldrepeat itself, if without intervention. Ongoing effort mitigatesthis recycling problem by introducing an antireflective coat- ing through materials engineering and/or by liquid immer-sion approach that effectively increases the refractive indexof the surrounding medium. Whereas optical refrigeration isexpected to offer a viable solution to active thermal manage-ment of semiconductor devices and microsystems that isnonmechanical /H20849thus vibration-free and noiseless /H20850, reliable, and integratable, the phenomenon has yet to be successfullydemonstrated in semiconductors. Photon recycling and car-rier trappings that lead to nonradiative recombination areshown to be the reasons. 6,7 FIG. 1. /H20849Color online /H20850Schematic of luminescence upconversion for laser cooling of semiconductors. Ultrafast carrier-longitudinaloptical phonon collision aids efficient thermalization /H208492/H20850of photo- generated cold carriers /H208491/H20850and the sequential luminescence upcon- version /H208493/H20850. Cooling is achieved as a result of heat extraction by the thermal photoluminescence.PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 1098-0121/2007/75 /H2084915/H20850/155315 /H2084910/H20850 ©2007 The American Physical Society 155315-1On the theoretical front, effort8–13is limited and focuses on bulk materials. Phenomenologically, Bowman8analyzed heat generation by an optical pumping and cooling effect byanti-Stokes spontaneous emission /H20849luminescence upconver- sion /H20850in a solid-state laser system, thus proposed the “radia- tion balanced laser” to achieve high output power; Mungan 9 quantified the steady-state efficiency of such a doped fiberlaser from thermodynamic laws; Sheik-Bahae and Epstein 10 investigated the effects of external efficiency and photon re-cycling on net optical cooling in GaAs. In addition, a rate-equation approach 13for carrier and photon densities includ- ing photon recycling was used recently to quantify thecooling efficiency, but without temperature dependence. Onthe other hand, a microscopic description is rare: the resultsof Huang et al. 11,12were the only known results to the author at the time of submission of this contribution. The studiesadopted a nonlocal energy-balance equation for carrier tem-peratures, without consideration of carrier density. Theseearly studies addressed such critical issues as experimentalconditions, efficiency, and limitations, but not to a satisfac-tory level as no understanding regarding the intrinsic anddynamic behaviors is achieved. The main reason is that acomplete and consistent theoretical framework is lacking.Such a framework would allow for a critical examination ofthe issues, both intrinsic and extrinsic, that prevent us froman ultimate demonstration of laser cooling of semiconduc-tors. Particularly, such a framework would facilitate simula-tions of the cooling operations, generate sufficient and nec-essary understanding and insights, and provide indispensableguidance for achieving net cooling of semiconductors. To this end, we present a first-principles-based, consistentthermokinetic theory for the description of laser cooling ofquantum wells /H20849QWs /H20850in this work. We focus on intrinsic processes and thus the fundamental limit of laser cooling,and ignore photon recycling and carrier trappings, as well asAuger process. Incorporation of these physical processes intothe current framework is straightforward and shall be ex-plored in further studies. The remainder of the article is organized as follows: First, we set up the physics foundation for the thermokinetic theoryand lay out the starting equations in Sec. II. This is followedin Sec. III by a derivation of the main results of this work:the equations of motion for carrier density, carrier and pho-non thermal energy, and a conversion to the equation forlattice temperature. Then we treat photogeneration and radia-tive recombination microscopically in Sec. IV and producethe necessary radiative contributions to the equations of mo-tion, including noninteracting results for illustration. A deri-vation of the Kubo-Martin-Schwinger relation is conductedin Sec. V to demonstrate the consistency of the theoreticalframework, as well as analytical expressions for photoge-neration and radiative recombination with many-body ef-fects. To illustrate the implications and utility of the devel-oped framework we present in Sec. VI simulation results andexplore the cooling strategy. In Sec. VII technical issues andopen questions pertinent to the theoretical framework arediscussed, as well as Coulomb many-body effects. This con-tribution is concluded with a summary in Sec. VIII.II. MODEL AND BASIC EQUATIONS The thermokinetic theory is developed from the kinetic equations of the statistical distributions of elementary exci-tations in a semiconductor quantum well system. We parti-tion a semiconductor quantum well under laser pumping intothree subsystems: /H20849i/H20850carriers that are generated by a laser beam, /H20849ii/H20850thermal longitudinal optical /H20849LO/H20850phonons that are the elementary excitations associated with the underlying lat-tice, and /H20849iii/H20850the ambient. The carriers are electrons and holes, and are confined in the quantum well. Phonons consistof acoustic and optical kinds, of both longitudinal and trans-verse modes. In particular, the LO phonons are the mosteffective ones that exchange energy with the carriers throughinelastic collision. Due to their vast difference from the oth-ers, in terms of the energy exchange time scale with thecarriers, the LO phonons /H20849in subpicoseconds /H20850are singled out as the only constituent of the second subsystem, whereas theacoustic ones and transverse optical modes /H20849with time scales in nanoseconds to microseconds /H20850are relegated to be part of the ambient. The ambient, treated as a thermal reservoir, ba-sically includes all forms of elementary excitations that in-teract with the first two subsystems on a slower time scaleand render them incoherent ultimately. To account for theenergetics that drives the system into a steady state underlaser pumping, our theory microscopically incorporates thephotogeneration and radiative recombination of e-h pairs.The steady state is a consequence of thermokinetic balanceamong the subsystems. First, the laser beam generates non-thermal e-h pairs in the QW through optical excitation. For amonochromatic wave, as considered here, these pairs are dis-tributed along an isoenergetic contour in their joint phasespace because of energy and momentum conservation. Next,carriers and LO phonons thermalize as a result of carrier-carrier and carrier–LO-phonon collisions. We assume thatquasiequilibrium is sustained for carriers and LO phonons by the ultrafast carrier-carrier and carrier–LO-phonon collisionsso that the carriers are well described with Fermi-Dirac distribution 14,15and the phonons are well described by Bose- Einstein distribution throughout the cooling process. Finally, thermal exchange with the ambient, termed thermal loading,by the lattice pumps heat into the lattice and checks the run-away cooling of the carriers and the lattice. We start from the Boltzmann kinetic equations for carrier and phonon distributions /H20849f kc,nQ/H20850in momentum space /H20851in- plane wave vector kfor carriers and three-dimensional /H208493D/H20850 wave vector Qfor phonons /H20852: f˙ kc/H20849t/H20850=/H20841f˙ kc/H20849t/H20850/H20841abs−/H20841f˙ kc/H20849t/H20850/H20841rad+/H20841f˙ kc/H20849t/H20850/H20841c-p, /H208491/H20850 n˙Q/H20849t/H20850=/H20841n˙Q/H20849t/H20850/H20841e-p+/H20841n˙Q/H20849t/H20850/H20841h-p+/H20841n˙Q/H20849t/H20850/H20841amb, /H208492/H20850 where the evolution of carrier /H20849c=e,h/H20850distributions, repre- sented by its time derivative denoted by f˙ kc/H20849t/H20850, is driven by optical absorption /H20849/H20841abs/H20850, which generates carriers, radiative recombination /H20849/H20841rad/H20850, which annihilates carriers, and carrier- phonon collision /H20849/H20841c-p/H20850, which redistributes carriers. The change in phonon population is caused by carrier-phonon collision, which emits or absorbs phonons, and thermal ex-change with the ambient /H20849/H20841 amb/H20850, which alters lattice tempera-JIANZHONG LI PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-2ture, and thus phonon population. The source terms on the right-hand side in the above equations, except the ambientcontribution, are systematically obtained by a quantum ki-netic treatment from a Hamiltonian that takes into accountcarriers, LO phonons, photons, and their interactions as theBoltzmann kinetic equations themselves are derived whenspatial gradients are neglected for a uniform system. Thederivation is nontrivially involving and invokes the well-known Markovian approximation and semiclassical limit ofthe operator equations. Interested readers can consult Refs.16–19for technical details. For completeness, we list the results used in this work as follows: /H20841f˙ kh/H20849t/H20850/H20841abs/H110132 /H6036Im/H20853d/H9260/H20648*E/H20849t/H20850pk/H20849/H9260/H20648/H20850/H20854, /H208493/H20850 /H20841f˙ k+/H9260/H20648e/H20849t/H20850/H20841abs=/H20841f˙ kh/H20849t/H20850/H20841abs, /H208494/H20850 /H20841f˙ ke/H20849t/H20850/H20841rad/H110132 /H6036Re/H20877/H20858 qEquQW,q*dq/H20648*/H20855bq†pˆk−q/H20648/H20849q/H20648/H20850/H20856/H20878, /H208495/H20850 /H20841f˙ kh/H20849t/H20850/H20841rad/H110132 /H6036Re/H20877/H20858 qEquQW,q*dq/H20648*/H20855bq†pˆk/H20849q/H20648/H20850/H20856/H20878, /H208496/H20850 where d/H9260/H20648is the interband dipole matrix element, E/H20849t/H20850is the pumping laser electric field with in-plane wave vector /H9260/H20648, Vk−k/H11032sis the screened Coulomb potential, and pk/H20849/H9260/H20648/H20850is the interband polarization; Eq=/H20881/H6036/H9275q/2/H92800/H9280bis the vacuum-field amplitude for an optical mode with circular frequency /H9275q =c/H20841q/H20841/nand wave vector q/H11013/H20849q/H20648,q/H11036/H20850in the background me- dium /H20849here the barrier material /H20850with dielectric constant /H9280b and refractive index n=/H20881/H9280b,uQW,qis the overlap integral of the QW confinement wave functions and the optical mode function with wave vector q,bq†is the photon creation opera- tor, and pˆk/H20849q/H20648/H20850is the interband polarization operator between hole state /H20841v−k/H20856and electron state /H20841ck+q/H20648/H20856. The parallel /H20849/H20648/H20850, or in-plane, component of a vector lies in the QW plane, and the vertical /H20849/H11036/H20850one is perpendicular to the plane. The quan- tity /H20855bq†pˆk/H20849q/H20648/H20850/H20856, termed the three-point correlation function, measures the creation amplitude of a photon with an in-plane momentum /H6036q/H20648when an electron-hole pair with the same center-of-mass momentum recombines. The remaining nota-tions have the standard mathematical and physics meaning. Note that the equalities /H20841/H20858 kf˙ ke/H20849t/H20850/H20841abs=/H20841/H20858kf˙ kh/H20849t/H20850/H20841absand /H20841/H20858kf˙ ke/H20849t/H20850/H20841rad=/H20841/H20858kf˙ kh/H20849t/H20850/H20841radare always valid, as expected for pairwise processes, where the sum over kimplicitly accounts for spin degree of freedom of the carriers. Photogeneration and luminescence are treated indepen- dently in this work. As known, photons need to be secondquantized in order to describe luminescence. As a result, boththe photon operator and interband polarization operator ap-pear in the theoretical description of luminescence, and stan-dard quantum statistical ensemble averaging /H20855¯/H20856has to be taken, whereas for absorption, the laser field is semiclassical,and only the polarization operator undergoes the averaging process, which results in its value p k/H20849/H9260/H20648/H20850/H11013/H20855 pˆk/H20849/H9260/H20648/H20850/H20856in Eq. /H208493/H20850. Specifically, the interband polarization is described bythe semiconductor Bloch equations,17,18and the three-point correlation function for spontaneous emission is described bythe semiconductor luminescence equations, 18as briefly de- scribed in Sec. IV. Furthermore, the carrier–LO-phonon col-lision is not explicitly used in the final thermokinetic theory,as a result of the merger of the carrier subsystem and phononsubsystem, which is delineated in the upcoming section.Therefore, we do not list the related expressions here, butarguments in support of the merger are given. For complete-ness, interested readers should consult the author’s earlierwork, Refs. 15and20. Finally, our theory considers thermal exchange between the ambient and the QW through thermalradiation. Doubtlessly, thermal transport occurs as a result oflocal cooling of the QW with respect to the background me-dium. Whereas transport is driven by the temperature gradi-ent, thermal radiation is proportional to the difference in tem-peratures with a fourth power, as well known for theblackbody radiation law. Without much compromise in rigor,we neglect thermal transport processes with a weaker depen-dence on temperature difference and assume that thermal ex-change through radiation dominates. Interested readers canrefer to the latest work of Huang et al. 12where thermal trans- port across the QW is considered. III. THERMOKINETIC EQUATIONS Computationally, it is daunting and unrealistic to simulate laser cooling thermokinetics with Eqs. /H208491/H20850and /H208492/H20850. Further simplification is necessary. Fortunately, such a task is pos-sible and straightforward. The underlying reason is that thereexist several ultrafast collision processes within the sub-systems and between the subsystems. In particular, we havementioned carrier-carrier and carrier–LO-phonon collisions,which all occur on a subpicosecond time scale. Because ofthese ultrafast collision processes, each individual subsystemsustains a quasithermal distribution such that a thermokineticdescription is possible and sufficient. In other words, Eqs. /H208491/H20850 and /H208492/H20850overdescribe the laser cooling process. This line of physical analysis is in general termed the adiabatic approxi- mation . Thanks to this approximation and a few observa- tions, it is algebraically straightforward to obtain the equa-tions of motion for the carrier density /H20849N/H20850, carrier thermal energy /H20849E c/H20850, and LO-phonon thermal energy /H20849Ep/H20850, by inte- grating Eq. /H208491/H20850over k, Eq. /H208491/H20850over kweighted with e-h kinetic energy, and Eq. /H208492/H20850over Qweighted with LO-phonon energy /H6036/H9275Q, respectively, and the results are as follows: N˙=aF/H20849t/H20850−BN2, /H208497/H20850 E˙c=aEF/H20849t/H20850−BEN2+/H20841E˙c/H20841e-p+/H20841E˙c/H20841h-p, /H208498/H20850 E˙p=/H9268/H20849Tamb4−Tp4/H20850−/H20841E˙c/H20841e-p−/H20841E˙c/H20841h-p. /H208499/H20850 Whereas ais the QW absorbance, aEdescribes the carrier kinetic /H20849thermal /H20850energy gain per unit photon flux. Bis the radiative recombination coefficient, while BEis its counter- part for carrier thermal energy. /H20841E˙c/H20841e-pand /H20841E˙c/H20841h-pare the energy exchange rates due to electron–LO-phonon and hole–LASER COOLING OF SEMICONDUCTOR QUANTUM WELLS: … PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-3LO-phonon collisions. Thermal loading by the ambient at temperature Tambis approximated by the blackbody radiation law:11/H9268is the Stefan-Boltzmann constant. The photon flux F/H20849t/H20850relates to laser power and electric field by P/H20849t/H20850 =/H6036/H9275F/H20849t/H20850=cn/H92800/H20841E/H20849t/H20850/H208412/2. Note that aand aEare functions of detuning /H20849difference between photon energy /H6036/H9275and QW band gap, as defined later /H20850,N, and Tp/H20849lattice temperature /H20850;B and BEdepend on Nand Tpexplicitly. Their expressions are given in the next two sections. The observations made whenderiving the above equations are that /H20849i/H20850electrons and holes are fermions so that independent equations are needed fortheir number /H20849density /H20850and energy /H20849temperature /H20850, whereas phonons are bosons so that only one equation is needed, herefor their energy /H20849temperature /H20850;/H20849ii/H20850a carrier-phonon collision only facilitates energy exchange between these two sub-systems, and the carrier density is conserved during the pro-cess; and /H20849iii/H20850photogeneration depends linearly on photon flux when the laser power is reasonably low so that funda-mental nonlinear optical processes can be neglected /H20849thus the linear form /H20850. At this point, it is possible to close the thermokinetic equations. However, two insights are instrumental that resultin further simplification, in terms of computational budgetand algebra. First, equilibration between the e-h subsystemand the LO-phonon subsystem is characterized by a subpico-second temperature relaxation time scale thanks to efficientenergy exchange between the carriers and the LO phonons. 20 In comparison, a pumping-dependent but much slower den-sity relaxation rate, proportional to BNand typically in nano- seconds, characterizes the carrier density kinetics. Secondand equally important, the heat capacity of the e-h subsystemis orders of magnitude smaller than that of the phonon sub-system, even at liquid nitrogen temperature. These two in-sights dictate that Eqs. /H208498/H20850and /H208499/H20850can be combined into a single one: E˙ p=/H9268/H20849Tamb4−Tp4/H20850+aEF/H20849t/H20850−BEN2. /H2084910/H20850 As a result, the subpicosecond thermokinetics due to carrier– LO-phonon collisions does not show in the final form of ourtheory, represented by Eqs. /H208497/H20850and /H2084910/H20850. We note that Eq. /H2084910/H20850here is formally analogous to Eq. /H208495/H20850in Ref. 11.But only in combination with Eq. /H208497/H20850does a consistent descrip- tion of the thermokinetics of laser cooling become complete.Finally, we close Eq. /H2084910/H20850for LO phonons using E p =/H20858QnQ/H6036/H9275Qto relate the LO-phonon thermal energy Epto the phonon temperature Tp. As mentioned, the other phonon modes are treated as part of the ambient and thermokineti-cally ignored in the current theory. In essence, Eqs. /H208497/H20850and /H2084910/H20850allow us to simulate laser cooling thermokinetics with the knowledge of photogeneration and radiative recombina-tion rates. Before proceeding to the derivation of the explicit form of the rates for formal closure, it would be auxiliary to empha-size that the final form of our theory cannot be characterizedas phenomenological rate equations for the thermokineticvariables Nand T p. Even though the equations are cast de- ceivingly in a familiar form in order to bring out some trans-parency in the relevant physics, all the coefficients /H20849a,B,a E,and BE/H20850are variable dependent and incorporate relevant mi- croscopic physics /H20849refer to the following two sections for details /H20850. In particular, since the direct process of radiative recombination is microscopically binary and has been con-ventionally described by a square power law, its rates arewritten in the same form here. For these reasons, it would beerroneous to take the theory as a phenomenological one. IV. TREATMENT OF PHOTOGENERATION AND PHOTOLUMINESCENCE We now supplement our theory with a microscopic treat- ment of photogeneration and photoluminescence in semicon-ductor quantum wells so as to derive the photogeneration andradiative recombination rates, thus the a,B,a E, and BEco- efficients formally. The treatment is at the screened Hartree-Fock level, and correlation effects are phenomenologicallymodeled under the dephasing rate /H20849 /H9253pbelow /H20850approximation. For brevity, we drop the explicit time dependence of carrierdistributions hereafter. First of all, optical absorption via in-terband process is described by the semiconductor Blochequation 17,18 /H20849/H9255k+/H9260/H20648e+/H9255−kh−i/H9253p/H20850pk/H20849/H9260/H20648/H20850−i/H6036p˙k/H20849/H9260/H20648/H20850 =/H208491−fk+/H9260/H20648e−f−kh/H20850/H20873d/H9260/H20648E/H20849t/H20850+/H20858 k/H11032Vk−k/H11032spk/H11032/H20849/H9260/H20648/H20850/H20874,/H2084911/H20850 where /H9255kc/H20849c=e,h/H20850is the renormalized carrier energy17,18rela- tive to the conduction band edge. For a monochromatic pumping laser beam E/H20849t/H20850=E0/H20851exp /H20849−i/H9275t/H20850+c.c. /H20852with a circular frequency /H9275=c/H20841/H9260/H20841/nand an in-plane wave vector component /H9260/H20648, a driven /H20849resonant /H20850solution for the polarization can be sought by invoking the ansatz pk/H20849/H9260/H20648/H20850=p˜k/H20849/H9260/H20648/H20850e−i/H9275t, while ig- noring the counterrotating portion /H20849the rotating-wave ap- proximation /H20850and an envelope part associated with the slow change in the carrier distributions. Note that for a laser beamwith a slowly varying power profile, the same treatment ap-plies. By defining the k-resolved susceptibility according to p ˜k/H20849/H9260/H20648/H20850/H11013/H9273k/H20849/H9260/H20648/H20850E0, the QW optical susceptibility is obtained via/H9273/H20849/H9275/H20850/H11013/H20858 kdk*/H9273k/H20849/H9260/H20648/H20850//H92800and the QW absorbance can be found using a=/H9275Im/H20853/H9273/H20849/H9275/H20850/H20854/cn. The thermal energy gain can be calculated with a corresponding summation weighted by the carrier kinetic energies. Thereby, formal expressions forthe QW absorbance and the thermal energy gain due to op-tical absorption can be written as follows: a= /H9275 cn/H92800/H20858 kIm/H20853d/H9260/H20648*/H9273k/H20849/H9260/H20648/H20850/H20854, /H2084912/H20850 aE=/H9275 cn/H92800/H20858 k/H20849/H9280k+/H9260/H20648e+/H9280kh/H20850Im/H20853d/H9260/H20648*/H9273k/H20849/H9260/H20648/H20850/H20854, /H2084913/H20850 where /H9280kc=/H60362k2/2mcis the kinetic energy of a carrier with effective mass mc. If Coulomb many-body effects /H20849refer to Sec. V for many-body results /H20850are neglected, the free-carrier result /H20849denoted by the superscript 0 throughout this work /H20850is given asJIANZHONG LI PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-4/H9273k0/H20849/H9260/H20648/H20850=/H208491−fk+/H9260/H20648e−f−kh/H20850d/H9260/H20648 /H9280k+/H9260/H20648e+/H9280−kh−/H9254−i/H9253p, /H2084914/H20850 where the detuning of the laser beam /H9254=/H6036/H9275−EgQW/H20849Tp/H20850has been used, with EgQW/H20849Tp/H20850=Eg/H20849Tp/H20850+/H9280ze+/H9280zhbeing the temperature-dependent QW band gap, Eg/H20849Tp/H20850being the temperature-dependent bulk band gap of the well semicon- ductor, and /H9280zcbeing the carrier /H20849c=e,h/H20850confinement energy perpendicular to the QW. We now turn to the direct radiative recombination pro- cess, or photoluminescence. To determine the radiative re-combination rates, a three-point correlation function isneeded that is described by the semiconductor luminescenceequation 18 /H20849/H9255k+q/H20648e+/H9255−kh−/H6036/H9275q−i/H9253p/H20850/H20855bq†pˆk/H20849q/H20648/H20850/H20856−i/H6036/H11509t/H20855bq†pˆk/H20849q/H20648/H20850/H20856 =/H208491−fk+q/H20648e−f−kh/H20850/H20858 k/H11032Vk−k/H11032s/H20855bq†pˆk/H11032/H20849q/H20648/H20850/H20856 −iEquQW,qdq/H20648fk+q/H20648ef−kh, /H2084915/H20850 where /H11509tdenotes the time derivative. We have assumed the same dephasing rate for the radiative recombination /H20849photo- luminescence /H20850process as for the optical absorption /H20849photo- generation /H20850process. In the same vein as with the absorbance case, we ignore the slow change in carrier distributions andseek an adiabatic solution to the above equation for thethree-point correlation function. Because of the nonlocal na-ture of the Coulomb interaction, only a formal solution ex-ists, which can be found by defining /H9264k/H20849q/H20850, according to /H20855bq†pˆk/H20849q/H20648/H20850/H20856/H11013/H9264k/H20849q/H20850Eq, termed k-resolved luminescence sus- ceptibility. Given this quantity, an explicit form of the re- combination coefficients in Eqs. /H208497/H20850and /H2084910/H20850is readily ob- tained as B=2 /H6036N2/H20858 k,qRe/H20853Eq2uQW,q*dq/H20648*/H9264k/H20849q/H20850/H20854, /H2084916/H20850 BE=2 /H6036N2/H20858 k,q/H20849/H9280k+q/H20648e+/H9280kh/H20850Re/H20853Eq2uQW,q*dq/H20648*/H9264k/H20849q/H20850/H20854. /H2084917/H20850 Similar to the absorbance case, if Coulomb many-body ef- fects are neglected, the free-carrier k-resolved luminescence susceptibility is readily available from Eq. /H2084915/H20850: /H9264k0/H20849q/H20850=−iuQW,qdq/H20648fk+q/H20648ef−kh /H9280k+q/H20648e+/H9280−kh−/H9254q−i/H9253p, /H2084918/H20850 where we have used /H9254q=/H6036/H9275q−EgQW/H20849Tp/H20850. Then, the free- carrier recombination coefficient is B=1 /H92800/H9280bN2/H20858 k,qIm/H20877/H9275q/H20841uQW,q/H208412/H20841dq/H20648/H208412fk+q/H20648ef−kh /H9280k+q/H20648e+/H9280−kh−/H9254q−i/H9253p/H20878, /H2084919/H20850 and the counterpart coefficient for the carrier thermal energy, BE, can be found correspondingly with a weighted summa- tion by the carrier kinetic energies.In general, the time-resolved PL spectrum with photon energy /H6036/H9024can be calculated from the k-resolved lumines- cence susceptibility according to IPL/H20849/H9024/H20850=2/H20858 k,q/H9254/H20849/H6036/H9024−/H6036/H9275q/H20850Re/H20853Eq2uQW,q*dq/H20648*/H9264k/H20849q/H20850/H20854,/H2084920/H20850 where /H9254/H20849x/H20850is the Dirac delta function. Note that an integra- tion over the photon energy of the time-resolved photolumi- nescence spectrum equals BN2, as each e-h pair recombines to emit one PL photon. Explicitly, we have BN2 =/H20848d/H9024IPL/H20849/H9024/H20850. The free-carrier PL result can be found by plugging Eq. /H2084918/H20850into the above equation. Furthermore, the angle-resolved PL spectrum can be obtained using Eq. /H2084920/H20850 by integrating over the radial coordinate of qonly, whereas the time-integrated PL spectrum integrating over time, ofcourse. V. KUBO-MARTIN-SCHWINGER RELATION Incoherent radiative recombination—i.e., thermal photoluminescence—as treated above, can also be obtainedvia the Kubo-Martin-Schwinger relation 21,22from the optical absorption spectrum. On the other hand, the relation exhibitsone of the connections between stimulated and spontaneousoptical processes, which are required for thermodynamicconsistency. We now prove that our theory agrees with thefamous relation by following Refs. 17and18. It is insightful to note that Eq. /H2084911/H20850is analogous to Eq. /H2084915/H20850after invoking therotating-wave ansatz that ignores the slowly varying en- velope part. They take the form of a hydrogen-atom-likeproblem in Fourier space: /H20858 k/H11032Mkk/H11032/H20849q/H20648/H20850Ak/H11032/H20849q/H20850−/H20851E/H20849q/H20850+i/H9253p/H20852Ak/H20849q/H20850=Sk/H20849q/H20850,/H2084921/H20850 where the nondiagonal matrix elements are given by Mkk/H11032/H20849q/H20648/H20850/H11013/H20849/H9255k+q/H20648e+/H9255−kh/H20850/H9254k,k/H11032+/H20849fk+q/H20648e+f−kh−1/H20850Vk−k/H11032s, /H2084922/H20850 with/H9254k,k/H11032being the Levi-Cività delta notation. For photoge- neration, Ak/H20849q/H20850,E/H20849q/H20850, and Sk/H20849q/H20850correspond to p˜k/H20849q/H20648/H20850,/H6036/H9275, and /H208491−fk+q/H20648e−f−kh/H20850dq/H20648E0. For photoluminescence, they corre- spond to /H20855bq†pˆk/H20849q/H20648/H20850/H20856,/H6036/H9275q, and − iEquQW,qdq/H20648fk+q/H20648ef−kh. The stan- dard Green’s function method applies to Eq. /H2084921/H20850, which pro- duces the following formal solution: Ak/H20849q/H20850=/H20858 /H9263/H9274/H9263k/H20858 k/H11032/H9274/H9263k/H11032*Sk/H11032/H20849q/H20850 E/H9263−/H20851E/H20849q/H20850+i/H9253p/H20852, /H2084923/H20850 where E/H9263and/H9274/H9263ksatisfy /H20858k/H11032Mkk/H11032/H20849q/H20648/H20850/H9274/H9263k/H11032=E/H9263/H9274/H9263k. Note that the eigenvalue set /H20853E/H9263/H20854includes discrete /H20849excitonic /H20850and con- tinuous /H20849continuum /H20850entries. Plugging in the corresponding expressions, we have /H9273k/H20849/H9260/H20648/H20850=/H20858 /H9263k/H11032/H9274/H9263k/H9274/H9263k/H11032*/H208491−fk/H11032+/H9260/H20648e−f−k/H11032h/H20850d/H9260/H20648 E/H9263−/H20849/H6036/H9275+i/H9253p/H20850, /H2084924/H20850 for the k-resolved susceptibility, andLASER COOLING OF SEMICONDUCTOR QUANTUM WELLS: … PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-5/H9264k/H20849q/H20850=−i/H20858 /H9263k/H11032/H9274/H9263k/H9274/H9263k/H11032*uQW,qdq/H20648fk/H11032+q/H20648ef−k/H11032h E/H9263−/H20849/H6036/H9275q+i/H9253p/H20850, /H2084925/H20850 for the k-resolved luminescence susceptibility. By taking the limit /H9253p→0 such that Im /H208531//H20851E/H9263−/H20849/H6036/H9275+i/H9253p/H20850/H20852/H20854=/H9266/H9254/H20849E/H9263−/H6036/H9275/H20850 and ignoring the weak kdependence in dk/H20849denoted as dcv hereafter /H20850, the QW absorbance is found to be a=/H9266/H9275 cn/H92800/H20841dcv/H208412/H9274/H9263¯/H208490/H20850/H20858 k/H11032/H9274/H9263¯k/H11032*/H208491−fk/H11032+/H9260/H20648e−f−k/H11032h/H20850, /H2084926/H20850 where /H9263¯denotes the subset of eigenstates that satisfy E/H9263¯ =/H6036/H9275and/H9274/H9263¯/H208490/H20850=/H20858k/H9274/H9263¯kis the Fourier transform at position r=0, the origin of the QW. In the same fashion, the PL spectrum is derived as follows: IPL/H20849/H9024/H20850=/H9266/H9024 /H92800/H9280bD/H20849/H9024/H20850/H20841u¯QW,Qdcv/H208412/H9274/H9270¯/H208490/H20850/H20858 k/H11032/H9274/H9270¯k/H11032*fk/H11032+Q/H20648ef−k/H11032h, /H2084927/H20850 where /H9024=c/H20841Q/H20841/n,D/H20849/H9024/H20850=/H20858q/H9254/H20849/H9024−/H9275q/H20850=n3/H90242/2/H92662c3is the photon mode density, u¯QW,Qis the averaged overlap integral uQW,Qover a 4 /H9266solid angle, and /H9270¯denotes the subset of eigenstates that satisfy E/H9270¯=/H6036/H9024. Given that the carriers fol- low the Fermi-Dirac distribution, we have fk+Q/H20648ef−kh=g„/H9255k/H20849Q/H20850…/H208491−fk+Q/H20648e−f−kh/H20850, /H2084928/H20850 g/H20849/H9255k/H20849Q/H20850/H20850=1 exp/H20873/H9255k+Q/H20648e+/H9255−kh−/H20849/H9262e+/H9262h/H20850 kBTp/H20874+1, /H2084929/H20850 where function g(/H9255k/H20849Q/H20850)denotes a Bose-Einstein distribution with an aggregate carrier energy /H9255k/H20849Q/H20850=/H9255k+Q/H20648e+/H9255−khand an aggregate chemical potential for electrons and holes. Finally, by realizing that the resonant condition23dictates /H9255k/H20849Q/H20850 =/H6036/H9275, we arrive at the desired Kubo-Martin-Schwinger rela- tion IPL/H20849/H9275/H20850=c nD/H20849/H9275/H20850/H20841u¯QW,Q/H208412g/H20849/H6036/H9275/H20850a, /H2084930/H20850 where Qsatisfies dispersion relation /H6036/H9275=c/H20841Q/H20841/n. As a side note, it is easy to check that the generalized 2D Elliot formula17is recovered with phase-space filling effect from Eq. /H2084924/H20850when the weak kdependence in dkis ignored. To recover from these many-body expressions the free-carrier results presented in the preceding section, it sufficesto use two observations: /H20849i/H20850the free-carrier eigenvalues E /H9263 are all degenerate, and /H20849ii/H20850/H20858/H9263/H9274/H9263k/H9274/H9263k/H11032*=/H9254k,k/H11032is the condition of completeness. In addition, it is well known that an Ein- stein relation holds between the stimulated and spontaneousemissions for two-level systems, which applies to the presentcase as an extended example. We do not intend to verify thatin this article, but we mention that using the same approachin this section, a formal verification can be achieved.VI. NUMERICAL RESULTS In this section, we demonstrate the utility of the devel- oped theoretical framework by studying laser coolingthermokinetics of an exemplary quantum well system nu-merically. Most importantly, we put forth an adaptive coolingstrategy for deep optical refrigeration of semiconductor QWsystems. To this end, we select a 15-nm GaAs QW at ambi-ent temperature T amb=295 K. At such an elevated tempera- ture the Coulomb many-body effects still play a significantrole, as discussed in Sec. VII together with aspects of theirimportance at low temperature /H20849Ref.24appeared as this con- tribution is under review /H20850. Nevertheless, we ignore them in our simulations for the sake of numerical efficiency. Table I lists the material parameters used. The bulk GaAs band gap 25 follows Eg/H20849Tp/H20850=Eg/H208490/H20850−0.5405 Tp2//H20849Tp+204 /H20850, and the quan- tum confinement energies for electrons and holes are /H9280ze =16.8 and /H9280zh=2.5 meV, respectively. An adaptive Runge- Kutta algorithm is used to integrate Eqs. /H208497/H20850and /H2084910/H20850. Pho- togeneration terms /H20849i.e., aand aE/H20850are calculated on the fly from the semiconductor Bloch equations /H20849SBEs /H20850with band structure determined by k·pmethod.26Recombination terms use fitted expressions of Nand Tpto calculated emission rates from the semiconductor luminescence equations /H20849SLEs /H20850 /H20849see Fig. 2/H20850. As expected, B/H11008Tp−1/H20849see, for example, Ref. 27/H20850 and BEisTpindependent, nearly; the average energy ex- tracted per photon /H20849BE/B/H20850iskBTpfor nondegenerate elec- trons. The initial condition for integration is an unexcited intrinsic QW. The cw photon flux is ramped up according toF/H20849t/H20850=F l/H208531−exp /H20851−/H20849t//H9270/H208502/2/H20852/H20854with a judicious time constantTABLE I. GaAs parameters used: m0, free-electron mass; /H9280’s, dielectric constants; aL, lattice constant; P, Kane’s parameter; /H9253p, dephasing rate; g, mass density. me /H92800 Eg/H208490/H20850 P /H9253p 0.067 m0 13.1 1519 meV 25 eV 5 meV mh /H9280/H11009 aL /H6036/H9275LO g 0.45 m0 10.9 5.6533 Å 35 meV 5317.6 kg/m3 FIG. 2. /H20849Color online /H20850Microscopic radiative recombination rates for carrier density /H20849a/H20850and thermal energy /H20849b/H20850used in thermo- kinetic study of a 15-nm GaAs QW. Upper /H20849b/H20850inset shows their ratio in meV. Lower inset shows semilog arithmic PL spectra for 1,5, 10/H1100310 10cm−2at 295 K with energy in meV.JIANZHONG LI PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-6/H9270=50 ns. The laser power is given by Pl=h/H9263Fl. Figure 3presents pumping thermokinetics at a series of nominal detunings /H9254˜=h/H9263−/H20851Eg/H20849Tamb/H20850+/H9280ze+/H9280zh/H20852for low-power /H20851120 W/cm2,/H20849a/H20850and /H20849c/H20850/H20852and high-power /H2085112 KW/cm2,/H20849b/H20850 and /H20849d/H20850/H20852cases. Note that /H20849i/H20850the change in band gap during cooling effectively alters the actual detuning /H9254=h/H9263 −/H20851Eg/H20849Tp/H20850+/H9280ze+/H9280zh/H20852/H20849thus nominal /H20850and /H20849ii/H20850it takes about 3 /H9270 =150 ns to reach peak laser pumping value. Clearly, our model supports a conditional cooled steady state /H20849we delay discussion about heating /H20850. The low-power case features two phases: /H20849i/H20850a generation-dominated ramp-up phase and /H20849ii/H20850a recombination-compensating cooling phase. Carrier genera-tion collapses onto a single curve at positive detuning owingto the constant QW density of states. Optical nonlinearityfrom the phase space filling effect is negligible in the studied pumping range. Maximum cooling is achieved around /H9254˜ =12 meV. /H20849The optimal detuning is actually at 3 meV, as shown later. /H20850At high power, the cooling thermokinetics de- pends much on detuning. Near the band gap, the behavior issimilar to the low-power case, except that faster, deeper cool-ing is observed, obviously due to power increase. Then oc- curs a transition around /H9254˜=12 meV. As can be seen, /H20849i/H20850the detuning for maximum cooling is blueshifted and /H20849ii/H20850cooling accelerates before the transition. The blueshift is due to theband gap increase as cooling continues. As for the transition,when correlating data in /H20849b/H20850and /H20849d/H20850, we see that it starts when the carrier density approaches saturation. More reveal-ingly, the cooling behavior at positive detuning strikingly resembles the near-gap case /H20849 /H9254˜=0 meV curve /H20850after the tran- sition, which indicates that the actual detuning after the tran- siton tends to lock into a near-gap value. Indeed, take the /H9254˜ =24 meV case for example: When cooled to 240 K where the transition kicks in, Eg/H20849Tp/H20850=1451.37 meV; the gap has increased by 24.11 meV from 1427.26 meV at 295 K. Cor- respondingly, the carrier density ceases to grow. Therefore,the initial phase relates to a positive actual detuning, whichsettles into a near-gap value after transition. In essence, cool- ing is the most effective right before the transition. As the cooling process is driven by thermal energy extrac- tion via luminescence upconversion, the amount of thermalenergy taken away is usually determined by BN 2/H20849kBTp−/H9254/H20850, where kBis the Boltzmann constant, without Coulomb many- body effects. Whereas the initial cooling is dominated by an increase in carrier density, the second phase appears thanksto the slower growth in N/H20849near balance /H20850and in the magni- tude of /H9254due to the further blueshift of the band gap. In addition, the cooling acceleration in the first phase is an in-dication that there exists an optimal actual detuning for cool- ing. Furthermore, the crossovers for different detuning cases in/H20849d/H20850result from an interplay of Nand /H9254: Pumping above the gap increases photogeneration /H20849BN2/H20850, but reduces the aver- age net energy extracted per photon /H20849kBTp−/H9254/H20850vice versa. However, the increase in carrier density pays off eventually /H20849thus acceleration /H20850owing to the quadratic dependence. We turn now to discuss heating at large negative detuning. Thiscounterintuitive observation deserves a close reexaminationof the photogeneration process in our model, since the re-combination process leaves no ambiguity. We argue that theactual detuning is nota proper measure for the average car- rier energy in the present case. Pumping below the edgecreates carriers via absorption in the Lorentzian tail. In con-trast with a resonant pumping case where carriers are createdwith energies of a spread characterized by the linewidth/H20849 /H9253p=1 meV here /H20850, far-off-resonant cases with a Lorentzian line shape, as considered now, feature a divergent spread.When the detuning is too far off the band gap, this energyspread will surpass the thermal energy scale k BTp. As a re- sult, there exists a critical negative detuning, which is appar- ently independent of pumping level, where the average en-ergy of the photogenerated carriers equals that of therecombined carriers. Passing it, instead of generating coldcarriers that cool the QW, the laser beam heats up the QW. Next, we show steady-state solutions to Eqs. /H208497/H20850and /H2084910/H20850 as a function of nominal detuning and laser power in Fig. 4. First of all, laser cooling of the GaAs QW is observed be-tween a critical negative detuning and k BTamb. Second, the higher the power, the lower the lattice temperature. Third, thecritical negative detuning is around −9 meV and indepen- FIG. 3. /H20849Color online /H20850Laser cooling thermokinetics of the GaAs QW at ambient temperature Tamb=295K for indicated nominal de- tunings at low /H20849a/H20850,/H20849c/H20850and high /H20849b/H20850,/H20849d/H20850power. /H20849a/H20850Inset: absorbances at 100, 200, 300 K for 1, 10 /H110031010cm−2. High power cools better and its behavior reveals an optimal actual detuning for laser cool- ing. See text for simulation details. FIG. 4. /H20849Color online /H20850Thermokinetic steady-state solutions vs laser power and nominal detuning. Laser cooling is observed from a critical detuning value of 9 meV below to kBTambabove the QW band gap /H20849x=0 dashed line /H20850. Higher power cools deeper and blue- shifts the optimal cooling detuning.LASER COOLING OF SEMICONDUCTOR QUANTUM WELLS: … PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-7dent of pumping level, indeed. At low power, the carrier density approximately follows the absorbance spectrum andthe optimal detuning for cooling is near the band edge. Aspower increases, the steady state supports a larger carrierdensity and deeper cooling. Moreover, the optimal nominal detuning for cooling moves toward k BTamb, as the QW band gap moves up with further cooling. On the other hand, heat-ing below the critical detuning increases with laser power as well. As shown, an increase in laser power leads to deeper re- frigeration with the optimal nominal detuning for cooling shifting with the band edge. However, the obvious drawbackis the high demand on laser pumping power. A weaker powerlevel would help mitigate issues that are not addressed in thiswork, such as carrier trapping and photon recycling, whichwill become worse as the pumping level /H20849carrier density /H20850 increases. A strategy to reduce the pumping level arises fromthe following insight: There is an optimal actual detuning for cooling , as mentioned while discussing the behavior transi- tion in the high-power case in Fig. 4. Direct evidence is shown in Fig. 5/H20849d/H20850where the laser frequency is tuned to fix theactual detuning during cooling. As can be seen, the op- timal detuning lies 3 meV above the band edge and does notdepend on laser power. Whereas there seemed to be a lowerbound in Fig. 4/H20849b/H20850, sub-100-K cooling is now observed. Fur- thermore, we show in /H20849a/H20850and /H20849b/H20850the thermokinetics for fixed actual detuning at 12 kW cm −2. The distinction is clear: /H20849i/H20850 By trailing the band edge, carrier generation and cooling arethe most efficient; /H20849ii/H20850the second phase vanishes; and /H20849iii/H20850 the coolest temperature jumps from 235 down to 100 K. Thestrategy works wonders: It creates much more carriers andreduces the power level by orders of magnitude to cool theQW below 225 K. VII. TECHNICAL REMARKS Some comments on the technical limitations and issues with the current theoretical framework are in order. First ofall, our theory only considers fundamental linear optical pro- cess, and optical nonlinearity arises as a consequence of thepumping-induced phase-space filling effect. Therefore, it islimited to a moderate laser power level. Second, we haveused the dephasing rate approximation in the description ofthe optical processes. The resulting Lorentzian line shape hasbeen known to largely overestimate the absorption below theband edge, which is supposed to follow the well-known “Ur-bach tail” behavior at low temperature. It is possible to over-come this shortcoming within the framework of our theoryby considering the actual dephasing processes, which re-quires a systematic treatment of the Coulomb correlation ef-fects within the second or self-consistent Born approxima-tion and is mathematically involving and computationallycostly. In view of the present transparent and insightful for-mulation, the approximation is a “necessary sin.” Third, in-clusion of the other phonon branches, as well as their corre-sponding interactions with carriers in the theory, is straightforward and amounts to increasing the QW heat ca-pacity at elevated temperature when the LO phonons play adominant role, which would accordingly reduce the coolingspeed and adversely increase the computing time. Neverthe-less, such a treatment does not affect the steady state solutionand much of the thermokinetic characteristics. Fourth, ourtheory serves as a local description for the laser cooling ofsemiconductor QWs. Thermal transport and carrier diffusionshall modify the thermokinetics to a limited degree at thequantitative level. Last, we note that cooling into the cryo-genic regime /H20849below 77 K /H20850requires careful treatment of Coulomb many-body effects, as they largely alter the char-acteristics of photogeneration and radiative recombination/H20849see, for example, Refs. 17–19,22, and 24/H20850. Inclusion of Coulomb many-body effects normally renders Eqs. /H2084911/H20850and /H2084915/H20850intractable analytically. However, both equations can be numerically solved by the matrix inversion method. Never-theless, note that Ref. 22provides some approximate analyti- cal results on many-body effects for optical absorption andphotoluminescence, as well as Refs. 17and18. Besides Cou- lomb many-body effects, acoustic phonons play a more im-portant role, as the number of optical phonons dwindles. Inother words, the optical refrigeration physics at cryogenictemperature becomes fundamentally different from that ofambient cooling. The theoretical framework presented in thisarticle serves as a good starting point for such studies. We now take the liberty to reckon with possible quantita- tive modifications by Coulomb many-body effects, at theHartree-Fock level, to the simulated free-carrier results pre-sented in the preceding section before the conclusion of thisarticle. In contrast to their fundamental qualitative impact atlow temperature, such as the existence of discrete excitonicstates, Coulomb many-body effects are known for two majorreasons at elevated temperature: /H20849i/H20850band gap renormalization /H20849BGR /H20850, which shifts the spectroscopic response of the many- body system, and /H20849ii/H20850Coulomb enhancement /H20849excitonic ef- fect /H20850, which alters spectroscopic features around the van Hove singularities, such as the band edge. The former in-creases roughly proportional to the carrier density while thelatter is weakened. Depicted in Fig. 6is a telltale demonstra- tion of the effects in the 15-nm GaAs QW. Clearly, the ef-fects are drastic, even at room temperature, and lead to a FIG. 5. /H20849Color online /H20850Laser cooling thermokinetics /H20849a/H20850,/H20849b/H20850and efficacy /H20849c/H20850,/H20849d/H20850vs laser power and actual detuning. /H20849b/H20850Far deeper cooling is achieved at 12 kW cm−2with fixed actual detuning than fixed nominal detuning /H20851Fig.3/H20849d/H20850/H20852./H20849d/H20850Optimal actual detuning is 3 meV above QW band gap.JIANZHONG LI PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-8large condensation /H20849spectral narrowing /H20850of the oscillator strength onto the excitonlike peak which is redshifted from the free-carrier band edge. Within the relevant density range,the excitonic effect dominates over that of BGR, as evi-denced from the insensitive dependence of the peaks on den-sity. Because of the spectral narrowing and maximal feature,optical pumping becomes more efficient and faster aroundthe band edge, which will lead to a higher steady-state carrierdensity as well as a lower steady-state lattice temperature.Therefore, one expects that the excitonlike absorption peakpins down the optimal actual detuning near the peak. On the other hand, the detuning range for cooling is affected insig-nificantly for the lower bound is determined by the Lorentz-ian tail and the upper bound by the LO phonon energy. Assuch, Coulomb many-body effects are expected to /H20849i/H20850draw the optimal actual detuning to the band-edge absorption peak and /H20849ii/H20850redshift the spectral response of the QW relative to the free-carrier case. As for the adaptive cooling strategy, it isexpected to work as before, subject to a minor spectral shiftand slight change in the optimal actual detuning thanks to the aforediscussed Coulomb many-body effects, and to fol-low the much more significant band-gap shrinkage resulting from lattice cooling. Finally, we note that Coulomb many-body effects become more pronounced for more polar semi-conductors, such as II-VI compounds, and narrower QWs ornanostructured materials at a lower dimension. Hence, it isimperative to take into account the effects and even exploitthem for more efficient cooling schemes and engineering. VIII. SUMMARY In summary, we have presented a first-principles-based thermokinetic theory for laser cooling of semiconductorquantum wells. The theory reduces the Boltzmann kineticequations under the adiabatic approximation to a set of thermokinetic equations for carrier density and lattice tem-perature. Photogeneration and radiative recombination ofcarriers that are in quasiequilibrium with thermal LOphonons are microscopically taken into account. In particu-lar, semiconductor Bloch equations and semiconductor lumi-nescence equations are used to derive the generation andrecombination rates. We verify that the Kubo-Martin-Schwinger relations are valid within the current theory. Onthe numerical end, we show that stronger laser pumpingleads to deeper optical refrigeration within the detuningrange demarcated by a critial value and the ambient thermal energy. Of particular interest, we prove for the free-carriercase the existence of a pumping-independent optimal actual detuning for cooling above the band gap /H20849Coulomb many- body effects are expected to pin the optimal detuning at theexcitonlike absorption peak /H20850. This revelation leads to a cool- ing strategy that reduces the laser power drastically. Thiswork provides a consistent theoretical framework to modellaser cooling operations, may be invaluable to achieving la-ser cooling of semiconductors as a thermal management so-lution, and serves to inspire further work in this emergingtechnologically important research area. ACKNOWLEDGMENTS The author acknowledges helpful discussions with Yong- Hang Zhang /H20849Arizona State University /H20850who suggested the adaptive cooling scheme, Danhong Huang /H20849Air Force Re- search Lab /H20850for introduction to optical refrigeration of semi- conductors, and Cun-Zheng Ning /H20849NASA Ames Research Center /H20850for proofreading part of the manuscript. *Electronic address: jianzhng@netzero.com 1R. I. Epstein, M. I. Buchwald, B. C. Edwards, T. R. Gosnell, and C. E. Mungan, Nature /H20849London /H20850377, 500 /H208491995 /H20850. 2J. L. Clark and G. Rumbles, Phys. Rev. Lett. 76, 2037 /H208491996 /H20850. 3C. W. Hoyt, M. Sheik-Bahae, R. I. Epstein, B. C. Edwards, and J. E. Anderson, Phys. 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Coulomb enhancement and band-gap renormalization lead to spectral shifts and excitonlikefeatures in optical absorption /H20849left panel /H20850and photoluminescence /H20849right panel /H20850within the Hartree-Fock /H20849HF/H20850treatment at a constant dephasing rate of 5 meV. Free-carrier results are shown in compari-son with dashed lines /H20849denoted as SP /H20850. Also visible is the phase- space filling effect in the absorption spectra as carrier densityincreases.LASER COOLING OF SEMICONDUCTOR QUANTUM WELLS: … PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-912D. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, Phys. Rev. B 72, 195308 /H208492005 /H20850. 13J.-B. Wang, S. R. Johnson, D. Ding, S.-Q. Yu, and Y.-H. Zhang, J. Appl. Phys. 100, 043502 /H208492006 /H20850. 14H. Haug and S. W. Koch, Phys. Rev. A 39, 1887 /H208491989 /H20850. 15J. Li and C. Z. Ning, Phys. Rev. A 66, 023802 /H208492002 /H20850. 16M. 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As only the linear polarization induced by the pumplaser and the subsequent e-h pair generation are considered inthe kinetic description, no many-body effects are directly takeninto account in the distribution functions, except inherentlythose originated from dielectric response. Therefore, it is consis-tent within the Hartree-Fock approximation and linear responsetheory to replace the aggregate carrier energy with the photonenergy. 24G. Rupper, N. H. Kwong, and R. Binder, Phys. Rev. Lett. 97, 117401 /H208492006 /H20850. 25Landolt-Börnstein, Numerical Data and Functional Relationships in Science and Technology /H20849Springer-Verlag, Berlin, 1982 /H20850. 26S. L. Chuang, Physics of Optoelectronic Devices /H20849Wiley, New York, 1995 /H20850. 27W. W. Chow, S. W. Koch, and M. Sargent, III, Semiconductor Lasers Physcis /H20849Springer-Verlag, Berlin, 1994 /H20850.JIANZHONG LI PHYSICAL REVIEW B 75, 155315 /H208492007 /H20850 155315-10

PhysRevB.103.115424.pdf

PHYSICAL REVIEW B 103, 115424 (2021) Spin-thermoelectric transport in nonuniform strained zigzag graphene nanoribbons Fereshte Ildarabadi and Rouhollah Farghadan* Department of Physics, University of Kashan, Kashan 87317-53153, Iran (Received 25 December 2020; revised 14 February 2021; accepted 4 March 2021; published 16 March 2021) We study a nonuniform strain in zigzag graphene nanoribbons for producing the spin-thermoelectric effects, using the mean-field Hubbard approximation and a Green’s function approach. Our theoretical results showthat a sinusoidal-shaped inhomogeneous strain with electron-electron interaction could induce a different effecton each edge of zigzag nanoribbons and finally generate a spin semiconductor with a tunable spin-dependentband gap. The strength of strain also controls the magnitude of magnetization in each edge. Interestingly, purespin current and a giant spin Seebeck coefficient can be produced even at low values of strain by applying athermal gradient and without magnetic elements. These results pave a practical way toward improved design forspin-thermoelectric applications through strain engineering. DOI: 10.1103/PhysRevB.103.115424 I. INTRODUCTION Strain as a possible effective way to tailor electronic properties of two-dimensional materials has been studied sig-nificantly [ 1–6]. Strain induces new features via changing atomic bond length, angle, and strength [ 7]. Experimentally, some methods to induce strain in 2D materials are epitaxial in-teraction, thermal expansion mismatch, patterning substrates,and stretching of flexible substrates [ 8,9]. Substrate surface topography modification and piezoelectric substrate actuationare also used to apply strain in the sample [ 7]. Graphene is verified to be one of the most persistent 2D materials exposed to strain [ 10]. Its unit cell can endure elastic tensile strain more than 20% without breaking [ 10]. Thus, graphene is a feasible candidate for straintronics [ 4,11–13] due to its high mechanical stability [ 10]. Based on previous studies, inducing strain in graphene gives rise to producing an effective scalar potential [ 14] and a useful gauge potential in the system [ 15,16]. The induced pseudomagnetic field has opposite signs for two valleys,while the external magnetic field has the same effect on allelectrons in the lattice [ 8]. Also, an in-plane electric field may be generated in regions with different stretches due tothe various local electron densities [ 8]. Strain by changing the features of the Dirac fermions reveals new fascinatingtransport properties. For instance, the Dirac cones are shiftedaway from previous positions by uniaxial strain below thecritical strain value in graphene around 20% and becomemassive in strain values above the critical value [ 17,18]. Opening a band gap by uniaxial strain has been studied a lotin graphene [ 19,20]. Contrarily, biaxial strain only changes the Fermi velocity by varying the slope of the Dirac cones[17]. Moreover, some strain distributions by inducing strong uniform pseudomagnetic field give rise to the appearance of *rfarghadan@kashanu.ac.ira pseudo-quantum-Hall effect in the absence of real magneticfield [ 4,21,22]. Changing the spin polarization and the local magnetization is another exciting performance of strained graphene [ 21,23– 25] for spintronic devices. Enhancing the magnetic order at edges of graphene nanoribbons (GNRs) and quantum dots waspredicted by Viana-Gomes et al. [26]. Then it was extensively studied, and we mention some of the studies here. Lu andGuo indicated that strain changes the spin polarization at theedges of GNRs in the presence of the Hubbard interactionand, in addition to that, adjusts the band gap [ 19]. Kou et al. revealed a ferromagnetic ground state for strained graphenewith line defects [ 24]. Inducing a magnetic effect in rippled graphene by vacancies is also explored in [ 23]. Chang et al. assumed an arc-bend strained GNR with Hubbard repulsionand showed that a measurable polarization difference be-tween two sublattices is induced and generates nonuniformmagnetic field [ 21]. In addition, Yang et al. illustrated that Coulomb interaction induces a ferromagnetic-like behaviorunder a proper strain, and strain could control graphene mag-netism [ 27]. Other researchers investigated edge magnetism of graphene quantum dots [ 28,29] and spin-resolved trans- port of strained with external exchange field [ 30]. Recently, transistors using strained graphene have been introducedtoo [ 31]. On the other hand, strain increases the mismatch between phonon vibrations, leading to suppressing thermal conductiv-ity [ 32,33]. Hence, the strained graphene could be a good choice for devices with thermal transport. Thermal conduc-tivity of both strained and unstrained graphene sheets [ 33] and graphene nanoribbons [ 34] has been calculated. Moreover, the band gap opened in graphene by strain, owing to the modifiedorbital hybridization [ 35,36], has added another superiority for graphene-based thermoelectric devices. First-principlesstudies investigated the thermoelectric properties of strainedgraphene nanoribbons [ 37]. Mani and Benjamin also showed that strained graphene acts as a highly efficient quantum heatengine operating at maximum power [ 38,39]. Also, thermal 2469-9950/2021/103(11)/115424(8) 115424-1 ©2021 American Physical SocietyFERESHTE ILDARABADI AND ROUHOLLAH FARGHADAN PHYSICAL REVIEW B 103, 115424 (2021) FIG. 1. Sinusoidal strain applied along the width of ZGNR. λandhmdenote the wavelength and amplitude of the wave, respectively. The periodic structure of the ZGNR is along the xaxis, and the sinusoidal strain is extended in the ydirection perpendicular to the nanoribbon. transport in the strain-induced rippled graphene is studied in Ref. [ 40]. Very recently, Banerjee et al. disused a periodic strain on a graphene sheet that created a strong periodic pseudo-gauge-field varying at the sample’s length scale and studied its Diracbehavior electrons [ 8]. The inhomogeneous strain gradient creates nonuniform pseudomagnetic and electric fields result-ing in new transport properties [ 8]. Furthermore, electronic properties of rippled graphene are affected by height, wave-length, and a number of these deformations [ 8,41]. Ripples are also detected in suspended graphene, in addition to grapheneon a substrate [ 42]. However, several theoretical proposals on various shapes of rippled graphene and experimental attemptshave been performed that achieved exotic findings which canbe found in [ 43–47]. Our idea in this work is assuming a sinusoidal (strain based on [ 8]) transverse to a zigzag graphene nanoribbon (ZGNR) in the presence of e-e interaction. Our choice is dueto the magnetic features of the zigzag edge GNRs contraryto the armchair GNRs [ 48]. Then we apply the temperature gradient nanoribbon. This nonuniform strain induces differenthopping amplitudes across the ribbon resulting in impressiveband structure beside the Hubbard interaction. The presenceof Hubbard interaction causes local magnetism resulting instrong spin polarization. Consequently, applying a tempera-ture gradient could make a pure thermospin current and largeSeebeck effect. The proposed system could be a practicalcandidate for new transport device generation through strainengineering. II. METHOD We applied a sinusoidal strain with out-of-plane displace- ments along the width of the ZGNR as depicted in Fig. 1. Dislocation of each atom in the strained ZGNR (SZGNR)relative to its position in the ZGNR is determined by h(y)= h msin(2πy/λ) with λandhmbeing the wavelength and am- plitude of the sinusoidal function, respectively. This shape ofstrain has been introduced on the graphene sheet by Banerjeeet al. in [8] along with an in-plane displacement. Here, we consider zero in-plane stretchings, while all atoms in a specificyare displaced in the zdirection by the value h(y). In the presence of such strain, the carbon-carbon bond length aroundthe peaks of the sinusoid is shorter compared with middlelocations. It is worth noting that such a nonlinear out-of-planestrain profile can be rescaled to an in-plane strain tensorby the Foppl–von Karman model [ 49,50]. Especially, whenour out-of-plane sinusoidal strain wavelength is very large analogously to the atomic bond length, the in-plane strainapproximation by the Foppl–von Karman model is extremelyclose to our strain profile in every point of the ribbon. Theaccuracy of the Foppl–von Karman model for graphene isdiscussed in [ 51]. To survey the electronic structure of the proposed device, we use the tight-binding approximation in the framework ofthe mean-field Hubbard model [ 52,53], and then utilize coher- ent transport formalism [ 52] to calculate the spin-dependent thermocurrent. The Hamiltonian of the system is representedby [52] H=−/summationdisplay /angbracketlefti,j/angbracketright,σt/prime ijc† iσcjσ +U/summationdisplay i(ni,↑/angbracketleftni,↓/angbracketright+ni,↓/angbracketleftni,↑/angbracketright−/angbracketleft ni,↑/angbracketright/angbracketleftni,↓/angbracketright).(1) The first term indicates the tight-binding Hamiltonian of the SZGNR, where c† iσandcjσare the creation and annihilation operators for an electron with the spin index σ=↑,↓at sites i and j, respectively. t/prime ijin this term is the hopping parameter between the nearest-neighbor atoms at sites iand jof the strained system and is a function of atomic bond length ras [18] t/prime=t0exp/bracketleftBig −β/parenleftBigr r0−1/parenrightBig/bracketrightBig . (2) Here, t0=2.7 eV and r0=1.42 Å [ 18] are the hopping pa- rameter and bond length of unstrained graphene, respectively.βis the decay rate and is assumed to be 4.45 based on first- principles calculations on similar structures in [ 8], although other values are reported too [ 18]. The second term in Eq. ( 1) expresses the e-e Hubbard interaction in the framework of the mean-field approximation. Uis the on-site Coulomb energy assumed to be equal to t 0 [54–56].ni,σis the particle number operator for an electron with the spin σat the site i, and/angbracketleftni,σ/angbracketrightis the mean value of the number operator calculated self-consistently as [ 57] /angbracketleftni,↑/angbracketright=/summationdisplay ↑bandsa 2π/integraldisplayπ/a −π/aψ∗ i(/epsilon1k)ψi(/epsilon1k)f(/epsilon1k−μ)dk, /angbracketleftni,↓/angbracketright=/summationdisplay ↓bandsa 2π/integraldisplayπ/a −π/aψ∗ i+N(/epsilon1k)ψi+N(/epsilon1k)f(/epsilon1k−μ)dk, (3) 115424-2SPIN-THERMOELECTRIC TRANSPORT IN NONUNIFORM … PHYSICAL REVIEW B 103, 115424 (2021) where a,N, and i=1,..., Ndetermine the lattice con- stant, the number of atoms, and the atomic site in the unitcell, respectively. Here, indices iandi+Na r ei nr e g a r dt o wave function components corresponding to spin-up and spin-down, respectively. f(/epsilon1 k−μ) is the Fermi-Dirac distribution function where μis the chemical potential and /epsilon1kis the energy in the wave number k. All band structure calculations are performed at room temperature. ψα(/epsilon1k) is the wave function of the Hamiltonian in Eq. ( 1)[58]. Consistently, the magneti- zation at site iis defined by ρi=(/angbracketleftni,↑/angbracketright−/angbracketleft ni,↓/angbracketright)μBwhere μB is the Bohr magneton. As concerns matching Bloch wave functions in contacts and the scattering region, we first calculate total transmis- sion and spin transmission by T(/epsilon1)=/summationtextMf α=1/summationtextN i=1ψ† αiψαiand Ts(/epsilon1)=/summationtextMf α=1/summationtextN i=1ψ† αiσzψαi, respectively [ 57], where ψαiis the two-component part of the eigenfunction ψαandσzis thezcomponent of the Pauli matrices. Then, spin-resolved transmission is obtained by T↑(↓)(/epsilon1)=[T(/epsilon1)±Ts(/epsilon1)]/2[57]. Proceeding from the linear response regime, we compute the thermospin current and Seebeck coefficient under ap-plied temperature gradient /Delta1T. The temperature difference between two sides of the strained ZGNR changes the Fermidistribution of the two contacts giving rise to the flow current.The spin-resolved current in Landauer-Büttiker formalism isattained by [ 59] I σ=e h/integraldisplay+∞ −∞Tσ(/epsilon1)[fL(/epsilon1,μ L,TL)−fR(/epsilon1,μ R,TR)]d/epsilon1,(4) where LandRspecify left and right contacts. Total charge and spin currents are also obtained by IC=I↑+I↓andIS= I↑−I↓, respectively [ 52]. In the following, we determine the spin-dependent Seebeck coefficient using [ 52,53] Sσ(μ,T)=−1 |e|TL1σ(μ,T) L0σ(μ,T), (5) where intermediate function Ln,σ(μ,T) in defined as Ln,σ(μ,T)=−1 h/integraldisplay+∞ −∞(/epsilon1−μ)n∂f(/epsilon1,μ, T) ∂/epsilon1Tσ(/epsilon1)d/epsilon1. Ultimately, charge and spin Seebeck coefficients are given by Sc=(S↑+S↓)/2 and Ss=(S↑−S↓), respectively [ 52]. III. RESULTS AND DISCUSSION The proposed device in this paper is a ZGNR with a si- nusoidal strain across the ribbon. The applied strain inducessmaller atomic bond lengths close to the top and bottom ofsinusoidal ZGNR rather than middle ones. So the density ofatoms around the peaks is higher than the middle region. TheZGNR is along the xdirection and our proposed supercell con- tains a whole or part of a wavelength that we investigate here.Therefore, the density of atoms along the supercell varieslocally, creates nonuniform charge distribution, and inducesan electric field [ 8]. On the other hand, strain gradient can cre- ate spatially variable pseudomagnetic field [ 8,16,60]. These factors give rise to different band structures and hence revealnew transport properties in the system. Indeed, calculationscan be based on a strained tight-binding Hamiltonian (as donehere) or on an unstrained one with pseudofields. A comparison between the two calculation methods has been done in [ 8] and shows similar results. To proceed further, we added an e-e Hubbard interaction to our sinusoidal SZGNR and found fascinating results. In-terestingly, a gap opens, and high spin polarization appearsin specific widths of our designed structure in the presenceof the e-e interaction. Also, the edge magnetization varies bythe strength of strain that we will discuss later. We plot thespin-resolved low-energy spectrum of our SZGNR in Fig. 2 forλ=20 nm, h m=1 nm, and widths W=λ,λ/ 2,λ/4, and λ/5. These amounts of λandhmcreate the maximum value of strain 5% in atomic bond lengths of defined strain. Fig-ure2(a) shows the low-energy band structure of the SZGNR with W=λin the absence of the Hubbard interaction. As we expect, there is no gap around the Fermi level and no splittingbetween the two types of spin. This result is similar to reportsof previous works on strain in zigzag nanoribbons [ 27]. The importance of the e-e interaction is revealed in the four di-agrams shown in Figs. 2(b)–2(e) . A band gap is opened by assuming the existence of the e-e interaction in the system, asseen in Figs. 2(b)–2(e) . Figures 2(b) and2(c) are related to the SZGNR with complete and half wavelength in the width of theribbon, respectively. In the two later cases, up and down spinstates are degenerate due to the same magnitude of strain neartwo edges of the nanoribbon. It is worth mentioning that if anyother nonuniform strain profile induces different stretchingaround the two edges, spin splitting may occur in the bandstructure; meanwhile an out-of-plane sinusoidal strain is achoice consistent with experiments [ 8] and accompanied by more efficient results. Indeed, the A and B sublattices near thetwo edges have similar bond lengths, and nonuniform straincould not stretch two zigzag edges differently. Therefore,edge magnetization does not change in widths W=λ,λ/ 2 by applying sinusoidal strain. These results for edge statesare the same as constant strain along the ydirection. Any- way, high spin splitting is observed for the SZGNR with W=λ/4 and λ/5i nF i g s . 2(d) and 2(e). This splitting be- tween spin states is in a way that the bands of one type ofspin move upward, while the other type moves downwardin the band structure similarly to the spin semiconductingphase. Indeed, by ignoring the interaction between electrons,no gap appears even by applying nonuniform strain in thesystem. Our calculation shows that the highest spin polarization occurs for W=λ/4, and it is reduced for larger or smaller widths. This is owing to the most difference between amountsof strain in two edges of the ribbon. Each atomic sublatticenear two edges of the ribbon experiences a different strain thatcauses spin-polarized edge states. Experimentally, periodicripples with various wavelengths from 2 nm to several tensof nanometers are observed in the graphene sheet [ 61]. How- ever, here we investigated sinusoidal modulation across thenanoribbon. Moreover, when W=λenergy bands of spin-up and spin-down are degenerate, while W=λ/4 corresponds to the highest spin polarization, so W=N(λ+λ/4) could cause approximately the same behavior in the band structure withN=1,2,.... This can also be used in experiment. Furthermore, if an in-plane strain is taken into account sim- ilar to [ 8], the bond lengths near the edge located at the crest 115424-3FERESHTE ILDARABADI AND ROUHOLLAH FARGHADAN PHYSICAL REVIEW B 103, 115424 (2021) FIG. 2. Low-energy band structure of SZGNR with zero Hubbard interaction in (a) width W=λand nonzero Hubbard interaction in (b)W=λ,( c )W=λ/2, (d) W=λ/4, (e) W=λ/5. Here, λ=20 nm, hm=1 nm equivalent to maximum strain ε=5%. would be larger than the other edge when W=λ/4. So again, different strain profiles near the edges lead to the spin splittingin the band structure. Anyway, such nonuniform strain couldinduce remarkable spin splitting, although the spin polariza-tion usually happens by applying an electric or magnetic fieldon the system [ 62,63]. For example, the in-plane transverse electric field induces different spin-dependent band gaps fortwo types of spin [ 62,64]. However, in our defined struc- ture, spin-dependent bands move oppositely in the edge bandstructure. Hence, their band gap is the same, approximately.Furthermore, by comparing Fig. 2(b) to Fig. 2(e), it has been found that the spin gap is increased by decreasing the widthof the SZGNR from W=λtoW=λ/5 in a particular wave- length. Therefore, the e-e interaction with inhomogeneousstrain in the ZGNR generates a spin semiconductor phase with a tunable spin-dependent band gap. For more clarification, we plot the low-energy band struc- ture of the SZGNR for different sets of Wandλin Fig. 3. Figures 3(a)–3(c) besides Fig. 2(b) correspond to a given width W=20 nm and maximum strain /epsilon1=5% but differ- ent values of λ. Figures 3(d) and 3(e) are in regard to the different strain wavelengths when W=λ/4 and/epsilon1=5%. Fig- ures 3(b),3(d) and 3(e) along with Fig. 2(d) show slight increase in spin splitting of the band structure by enhancing λ when the maximum strain is constant /epsilon1=5% and W=λ/4. On the other hand, these band structures illustrate the effect ofthe SZGNR width in constant strain when W=λ/4. Here, increasing the ribbon width is equivalent to increasing the FIG. 3. (a)–(c) Low-energy band structures of SZGNR with indicated values of λand for given ribbon width W=20 nm and /epsilon1=5%. They correspond to hm=2,4,5 nm, respectively. (d), (e) Band structure of SZGNR with different ribbon widths and strain wavelengths when W=λ/4a n d/epsilon1=5%, corresponding to hm=2,3 nm, respectively. In all parts U=t0. 115424-4SPIN-THERMOELECTRIC TRANSPORT IN NONUNIFORM … PHYSICAL REVIEW B 103, 115424 (2021) FIG. 4. The effect of strain magnitude on low-energy band structure of SZGNR and corresponding magnetizations with λ=20 nm, width W=λ/4, and U=t0. (a) Maximum strain ε=5%, (b) ε=10%, (c) ε=18%, and (d) ε=0. They are corresponding to equivalent values hm=1 nm, hm=1.5 nm, hm=2 nm, and hm=0, respectively. strain wavelength and leads to less band gap. Indeed, reducing the ribbon width enhances the coupling between edge states,or in other words raises the e-e interaction effect [ 65]. This effect also occurs in other sets of Wandλthat can be seen by comparing plots Fig. 2(c) and Fig. 3(a) forW=λ/2o r Fig. 2(e) and Fig. 3(c) forW=λ/5. In the rest of this paper, we perform our calculation on the case of W=λ/4 due to its most spin polarization compared to other widths of SZGNRs.We investigate the low-energy band structure and spin magne-tization of the designed system under different magnitudes ofstrain in the following. To this purpose, we assume λ=20 nm andW=λ/4 in the SZGNR. Figure 4shows the effect of increasing strain on edge states. First, we apply the sinusoidalstrain with h m=1n mi nF i g . 4(a), which produces maximum strainε=5% in bond lengths. As seen, the spin degeneracy is broken in the presence of our defined nonuniform strain andHubbard interaction. Besides, the two edges show differentmagnetization, and therefore the system exhibits a small spinpolarization. The changes are in a way that one edge showslarger magnetization rather than another one. The spin densitydecays from the ribbon’s edges to the middle, and the edgeshave more contribution relative to the bulk. Second, by increasing maximum strain in bond lengths to ε=10% (this is equivalent to h m=1.5 nm) in Fig. 4(b),spin polarization is enhanced. In contrast, the spin band gap decreases. Then, we raise the strain to ε=18% and see similar changes to those seen in Fig. 4(c). Again, the spin gap decreased, and spin polarization increases. Also, a sub-stantial difference between the spin distribution of two edges,aboutδρ=0.06μ B, is observed in the case of Fig. 4(c) where ρ=0.26μBat the upper edge that is related to down spins. This difference for Figs. 4(a) and 4(b) isδρ=0.02μBand δρ=0.04μB, respectively. Ultimately, we plot the edge states of the ZGNR with W=5 nm in the absence of strain in Fig. 4(d). As we expected there is no spin polarization in this case. As a result, the defined strain profile could induce spinsplitting and finite magnetization in SZGNRs. Also, compar-ing magnetization in panels (a)–(c) in Fig. 4shows that there is more variation of edge polarization in the higher valuesof strain. So, enhancing strain could increase the magneticeffects of the system and decreases the band gap. To progress further, we apply a thermal gradient between two sides of our designed SZGNR in this section to studythe thermoelectric current and Seebeck effect. The proposedstructure for this purpose is the SZGNR with λ=20 nm and W=λ/4 with the three strain values given in Figs. 4(a)–4(c) . The corresponding spin-polarized thermocurrents are plottedin Fig. 5as a function of temperature. Interestingly, the pure FIG. 5. Spin-resolved, charge, and spin currents in SZGNR with λ=20 nm, width W=λ/4, and U=t0for (a) ε=5%, (b) ε=10%, and (c) ε=18%, under applied /Delta1T=20 K. The currents are in regard to the band structures of panels (a)–(c) in Fig. 4, respectively. 115424-5FERESHTE ILDARABADI AND ROUHOLLAH FARGHADAN PHYSICAL REVIEW B 103, 115424 (2021) FIG. 6. Charge and spin Seebeck coefficient against chemical potential in SZGNR with λ=20 nm, W=λ/4, and U=t0for (a) ε=5%, (b)ε=10%, and (c) ε=18% under applied /Delta1T=20 K. The currents are in regard to the band structures of panels (a)–(c) in Fig. 4, respectively. spin current is found in a wide range of temperatures for these three systems with different strain values. The bandstructure is symmetric for inversion of electrons and holes inthe energy window around the Fermi energy correspondingto the considered temperatures. Here, the parts of bands thatparticipate in the current show symmetry features. This en-ergy range is about −0.25 eV to 0.25 eV at T=300 K by considering the variation of the Fermi-Dirac distribution. Asseen in Fig. 5(a), in the case of ε=5% up and down spin currents appear from T=60 K and have the same magnitude and opposite directions to 300 K in temperature difference/Delta1T=20 K. This leads to zero charge current and also a high spin current that flows from left to right of the sample.At room temperature, the induced pure thermospin currentreaches 60 nA. Here, spin-up carriers are holes that flowfrom left to right, and electrons carry spin-down that moveinversely. They are equal, so there is no net charge current inthe system. By increasing the strain in Figs. 5(b) and5(c),w e observe that a pure spin current exists, and it begins in lowertemperatures. Smaller band gaps around the Fermi level couldproduce current at lower temperatures. This is owing to theexistence of involved bands in low temperatures (due to theeffect of Fermi distribution). Figure 5(c) shows that pure spin current goes up to 160 nA at room temperature. Generally, byincreasing the strain’s strength, the spin gap decreases, andtherefore the spin current increases. This magnitude is morethan ten times larger than previous works on GNRs and carbonnanotubes with vacancy [ 59,66]. Figure 6shows spin and charge Seebeck coefficients against chemical potential μatT=300 K corresponding to band structures in Fig. 4, respectively. In all strains, the spin Seebeck coefficient has a peak at μ=0, while the charge Seebeck is zero at this point. This peak is sharper and hasa larger value for cases with a higher strain than seen inFigs. 6(a)–6(c) . This peak for the case of ε=5% has the value S s=450μV/K that is relatively large and comparable to other reported values for ZGNRs [ 67–69] and carbon chains[53]. The value of the spin Seebeck is dependent on the energy gap, and by increasing the strength of the strain, the spin gapdecreases, and therefore the spin Seebeck slightly decreases,but the flat plateau increases. In this regard, the SZGNRwithε=18% shows an approximately flat plateau around μ=0 similarly to armchair and zigzag graphene nanoribbons with structural defects [ 59,66]. Moreover, by increasing or decreasing the chemical potential, S sis reduced and Schas nonvanishing values. IV . CONCLUSION In summary, we applied a sinusoidal-shaped inhomoge- neous strain across a zigzag edge graphene nanoribbon. Wedescribed its effect on the low-energy band structure of theGNR using the tight-binding method and mean-field Kane-Mele Hubbard model. Then we calculated the thermoelectricproperties of the GNR in the presence of a temperature gradi-ent in the framework of the linear response regime. The resultsshowed that the simultaneous presence of Hubbard interactionand nonuniform strain (that are barely considered in studies)in the GNR could lift the spin degeneracy and produce spin-semiconducting behavior with a tunable spin gap. Generally,for creating the spin effect by strain, the nonuniform straincould induce different effects on both edges of the zigzagnanoribbon. In detail, two different sublattices near the zigzagedges feel different strains that cause a spin filtering effectin the presence of e-e interaction. Furthermore, as we expect,the larger strain produces higher spin polarization and, finally,higher values for spin current. Therefore, the SZGNR exhibitsspin-up and spin-down currents with opposite flow directionsand, therefore, pure spin current in a wide range of temper-atures. Besides, high values of the spin current and Seebeckcoefficient are found at room temperature, even in low valuesof strain compared to other reported values for ZGNRs. Thesefindings will be useful for designing stretchable electronicsand spin-based thermoelectric devices. [1] E. Han, J. Yu, E. Annevelink, J. Son, D. A. Kang, K. Watanabe, T. Taniguchi, E. Ertekin, P. Y . Huang, and A. M. van der Zande,Nat. 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PhysRevB.88.224109.pdf

PHYSICAL REVIEW B 88, 224109 (2013) Local structural displacements across the structural phase transition in IrTe 2: Order-disorder of dimers and role of Ir-Te correlations B. Joseph,1M. Bendele,1L. Simonelli,2L. Maugeri,1S. Pyon,3K. Kudo,3M. Nohara,3T. Mizokawa,4and N. L. Saini1 1Dipartimento di Fisica, Universit ´a di Roma “La Sapienza” - P . le Aldo Moro 2, 00185 Roma, Italy 2European Synchrotron Radiation Facility, BP220, F-38043 Grenoble Cedex, France 3Department of Physics, Okayama University, Kita-ku, Okayama 700-8530, Japan 4Department of Complexity Science and Engineering, University of Tokyo, 5-1-5 Kashiwanoha, Chiba 277-8561, Japan (Received 30 October 2013; published 31 December 2013) We have studied local structure of IrTe 2by IrL3-edge extended x-ray absorption fine structure (EXAFS) measurements as a function of temperature to investigate origin of the observed structural phase transition atT s∼270 K. The EXAFS results show an appearance of longer Ir-Te bond length ( /Delta1R∼0.05˚A) atT< T s.W e have found Ir-Ir dimerization, characterized by distinct Ir-Ir bond lengths ( /Delta1R∼0.13˚A), existing both above and below Ts. The results suggest that the phase transition in IrTe 2should be an order-disorder-like transition of Ir-Ir dimers assisted by Ir-Te bond correlations, thus indicating important role of the interaction between the Ir5dand Te 5 porbitals in this transition. DOI: 10.1103/PhysRevB.88.224109 PACS number(s): 74 .70.Xa,74.81.−g,61.05.cj,74.62.Bf Recently, the layered 5 d-transition-metal dichalcogenide IrTe 2has been in the limelight after the observation of superconductivity in Ir 1−xPtxTe2withTcof about 3 K.1 This observation was followed by several studies revealing superconductivity due to intercalation addition and/or sub-stitution by different metals in the parent IrTe 2.2–4It has been argued that the superconductivity might be inducedby Ir vacancies or excess Te in the sample. 5On the other hand, the parent IrTe 2exhibits a first-order structural phase transition5–8atTs∼270 K from the trigonal ( P-3m1) at T> T sto a lower-symmetry phase at T< T s, accompanied by an anomalous electrical and magnetic transport.1Since the superconductivity is induced with concomitant suppression ofstructural phase transition, the origin of this transition in IrTe 2 remains one of the intriguing physical problems.1–5,8–11 The structural phase transition in IrTe 2has been argued to be similar to the one observed in the spinel-type CuIr 2S4,12 showing Ir-Ir dimerization. A recent angle-resolved pho-toemission spectroscopy (ARPES) study 13has shown that, in the trigonal phase ( T> T s), the Fermi surface consists of a flower-shaped outer part and six connected bead-likeinner parts, consistent with the band-structure calculations.In the low-temperature phase ( T< T s) the flower shape of the outer Fermi surface does not change appreciably;however, the topology of the inner Fermi surface revealsstraight portions, suggesting possible Fermi-surface nesting. 13 The perfect or partial nesting of Fermi surface can induce acharge density wave (CDW) and hence a superstructure, asobserved by electron diffraction below the structural phasetransition. 2However, the gap opening expected for a CDW was not observed,14suggesting that the structural transition in IrTe 2may not be of conventional CDW type but could involve Te 5 porbitals.5Therefore, the two main issues to be addressed are (i) is the structural phase transition drivenby some kind of dimerization? and (ii) what is the roleof Ir and Te bonding (i.e., Ir 5 dand Te 5 phybridization) in the phase transition? To affront these questions we havestudied the local structure of IrTe 2by IrL3-edge extended x-ray absorption fine structure (EXAFS) measurements as afunction of temperature, providing direct information on thefirst-order atomic correlation functions across the structural phase transition. We have found clear evidence of dimerizationin IrTe 2characterized by two different Ir-Ir distances separated by∼0.13 ˚A. The dimerization survives even above the phase transition temperature; however, two distinct Ir-Te distances(/Delta1R∼0.05˚A) at low temperature merge in a single distance above the transition, highlighting the importance of the (Ir5d)-(Te 5 p) hybridization. The IrTe 2samples were prepared by solid-state reac- tion starting from stoichiometric amounts of Ir and Te.Details on the sample preparation and characterization aregiven elsewhere. 1Temperature-dependent Ir L3-edge ( E= 11 215 eV) x-ray absorption measurements, in the range 20to 300 K, on powder samples of IrTe 2, were performed in transmission mode at the beamline BM26A15of the European Synchrotron Radiation Facility (ESRF), Grenoble (France).The synchrotron radiation emitted by a bending magnetsource was monochromatized using a double crystal Si(111)monochromator. Several scans were collected at a given tem-perature to ensure the spectral reproducibility. The EXAFS os-cillations were extracted using the standard procedure based onthe spline fit to the pre-edge subtracted absorption spectrum. 16 Figure 1shows EXAFS oscillations (multiplied by k2), extracted from Ir L3-edge x-ray absorption spectra, measured on IrTe 2at several temperatures. The oscillations are clear up to high krange even at high temperature. Apart from the thermal damping, the EXAFS oscillations reveal someapparent changes across the structural phase transition tem-perature of about 270 K. For example, ∼13˚A −1the double peak structure becomes single-peak like, different from anexpected thermal damping. Similarly, around 14 ˚A −1the asymmetric shape of the peak becomes more symmetric aboveT s. The differences can be further seen in the partial atomic distribution function around the photoabsorbing atoms, givenby the Fourier transform (FT) of the EXAFS oscillations. Figure 2shows the FT magnitudes of the EXAFS os- cillations. The FTs are performed in the krange of 3 to 17˚A −1using a Gaussian window. The main peak in the FTs is due to six near-neighbor Te atoms at ∼2.6 ˚A. The next-nearest neighbors are the Ir atoms (at ∼3.8˚A) and their 1098-0121/2013/88(22)/224109(4) 224109-1 ©2013 American Physical SocietyB. JOSEPH et al. PHYSICAL REVIEW B 88, 224109 (2013) 2 4 6 8 10 12 14 16 k (Å-1)k2(k) 20 K100 K150 K200 K225 K275 K280 K290 K300 K FIG. 1. (Color online) Ir L3-edge EXAFS oscillations (weighted byk2) at several temperatures. Apart from a temperature-dependent damping, some small changes can be seen across T=270 K (see, for example, karound 14 ˚A−1). contributions appear around ∼3.5–4.5 ˚A. The weak signal of the latter indicates that the system is characterized by a largeintrinsic disorder, consistent with polymeric networks. 9The large intrinsic disorder is also apparent from highly dampedFT intensity of higher shells. The local structure parameters as a function of temperature are determined by standard EXAFS modeling based on thesingle-scattering approximation. 16In the present case, we used a model with two Ir-Te and two Ir-Ir bond lengths, similar to thelow-temperature-diffraction observations. 2,11,17The EXCURVE 9.275 code (with calculated backscattering amplitudes and phase-shift functions) was used for the EXAFS model fits.18 The radial distances Riand the corresponding Debye–Waller factors σ2 iwere allowed to vary in the least-squares fits. The coordination number for the near neighbors N iwere obtained by a constrained refinement with the total number of nearneighbors being equal to known values from diffraction studiesdescribing the long-range structure. For the present analysis, 234520 K|FT(k2(k))| R (Å)100 K150 K200 K225 K275 K280 K290 K300 K Ir-Te Ir-Ir FIG. 2. (Color online) Fourier transform magnitudes of the EX- AFS oscillations (weighted by k2) at several temperatures (symbols) together with model fits (solid lines). The fit range in the Rspace is indicated by a bracket on the top.2.632.652.672.69R Ir-Te (Å)IrTe2 Ir-Te 2 Ir-Te 1Ir-Te m 0.00.10.2 0 40 80 120 160 200 240 280 320Probability (Ir-Te 2) T (K) FIG. 3. (Color online) Temperature dependence of two Ir-Te distances (Ir-Te 1and Ir-Te 2) as a function of temperature (upper panel). The mean bond length Ir-Te mis also included. The relative probability of the Ir-Te 2bond length is shown in the lower panel. The dashed lines are to guide the eyes and show a clear transition at∼270 K. The inset shows the temperature dependence of the Ir-Te bond length MSRD. The solid line in the inset is a Einstein model fit with the θ E=231±20 K for T< T s. the number of independent data points, Nind∼(2/Delta1k/Delta1R )/π16 were about 25 ( /Delta1k=13˚A−1and/Delta1R=3˚A), for a maximum of twelve parameters fits. The uncertainties in the localstructure parameters were determined by creating correlationmaps and analyzing five different scans. The model fits in realspace using four shells are shown in Fig. 2as solid lines. Figure 3shows the temperature dependence of local Ir-Te bond lengths (see, e.g., the upper panel of Fig. 3). There are two Ir-Te distances ( ∼2.69 ˚A and ∼2.64 ˚A) below the structural phase transition temperature T s∼270 K, that appear to merge in a single distance ( ∼2.65 ˚A) above Ts. The relative probability of the longer distance, which is 20% to 30% of thetotal number of Ir-Te bonds below the transition temperature,drops down to zero while heating across the transition (seelower panel of Fig. 3). This observation is in fair agreement with diffraction studies that show almost one third of the(two out of six) Ir-Te bonds to be longer. 2,11,17The longer Ir-Te distance appears consistent with the recent structuralstudies 11,17where the deformation of the IrTe 6octahedron is identified with about 2% variation of the Ir-Te distance acrossthe structural phase transition. The Debye–Waller factors measured by EXAFS describe mean-square relative displacements (MSRD) unlike the one 224109-2LOCAL STRUCTURAL DISPLACEMENTS ACROSS THE . . . PHYSICAL REVIEW B 88, 224109 (2013) 3.803.904.00R Ir-Ir (Å)IrTe2 Ir-Ir 1Ir-Ir 2 0.0020.0040.0060.0080.010 0 40 80 120 160 200 240 280 320Ir-Ir 1 Ir-Ir 2 T (K)2 (Å2) FIG. 4. (Color online) The Ir-Ir distances (Ir-Ir 1and Ir-Ir 2)a r e shown as a function of temperature (upper panel). The MSRD for thetwo Ir-Ir bond lengths are shown in the lower panel. The solid lines show Einstein-model fits with θ E=200 K. The MSRD of the Ir-Ir 1 bond length shows an anomalous drop around the structural phase transition. measured in diffraction studies, providing information on mean-square displacements (MSDs). Therefore, MSRD pro-vides information on the bond-length fluctuations and has beenparticularly useful to understand local atomic displacementsacross phase transitions (see, for example, Refs. 19–22). The measured MSRD is sum of temperature independent andtemperature dependent parts [i.e., σ 2 i=σ2 0+σ2(T)]. In the present case, the MSRD of the two Ir-Te distances are similar,given by the Einstein model 23with the θE=231±20 K. If one distance model is used, the MSRD of Ir-Te bond-length(Ir-Te m) shows anomalous temperature dependence across the phase transition (see, e.g., inset of Fig. 3). Therefore, the Ir-Te bond-length correlations (shown by /Delta1R, Ir-Te 2probability and the MSRD anomaly) are directly tied to the first-orderstructural phase transition in IrTe 2. Figure 4shows the temperature evolution of Ir-Ir bond lengths. Unlike Ir-Te, the two Ir-Ir bond-lengths remain almosttemperature independent and the splitting persists even aboveT s. The presence of two different Ir-Ir distances is consistent with the diffraction studies at low temperature;1,6,8however, unlike the diffraction studies, distinct Ir-Ir bond lengths atthe local scale persist even above the transition temperature. Here, we should recall a recent systematic diffraction study byCao et al. 8showing the low-temperature symmetry of IrTe 2 being triclinic rather than the earlier proposed monoclinic.7 In addition, the newly established low-temperature diffraction structure8shows that the probability of the short Ir-Ir bond length is about 1 /5; that is, in excellent agreement with the present results revealing the probability for the shorter bonds(Ir-Ir 1∼3.83˚A) to be ∼20%. Interestingly, the MSRD of the shorter bond length shows an anomalous change around Ts(Fig. 4). Indeed, the MSRD of the two Ir-Ir bond lengths can be described by the correlatedEinstein model 23with a similar Einstein temperature ( θE= 200±15 K) up to the structural phase transition, albeit with different σ2 0(∼0.002 and ∼0.005, respectively). Close to Ts the MSRD of the shorter Ir-Ir bond length (Ir-Ir 1) shows an abrupt decrease down to the values similar to the one for thelonger (Ir-Ir 2) bond. This abrupt change in the MSRD provides an indication of some electronic topological change associatedwith the transition. It is worth noting that the ARPES studiesreveal a clear change in the Fermi-surface topology across thetransition. 13Nevertheless, the results reveal that there are local Ir-Ir dimers in IrTe 2, and they have a clear role in the structural phase transition, further supported by the unusual temperaturedependence of the MSRD of the Ir-Ir 1bond lengths. Let us discuss the implication of the present findings in relation to the highly debated question on the origin of the structural phase transition in IrTe 2. In the beginning, the first- order structural phase transition was thought to be similar tothat in spinel CuIr 2S4, in which the structural transition occurs from cubic to tetragonal due to the orbitally induced Peierlsstate below T s.1,14Actually, a superstructure was observed at low temperature and it was argued that the structural phasetransition in IrTe 2is of CDW type.2However, the absence of a CDW gap across the phase transition does not supportthe conventional density-wave-like Fermi-surface instabilitybeing responsible for the structural phase transition, 14but the crystal-field effect should have a prominent role in splittingthe Te p xyand Te pzenergy levels and reduction of the kinetic energy of the electronic system.5Similarly, it was also suggested that the interlayer and intralayer hybridizations playimportant roles in the structural phase transition, rather than theinstability of Ir t 2gorbitals. A weak interlayer orbital hybridiza- tion causes the phase transition while stronger hybridizationsuppresses it. 3Furthermore, the first-order structural phase transition has been related with polymeric networks of covalentTe-Te bonds in the adjacent Te layers, going under a reversibledepolymerization below the transition temperature. 9These arguments are consistent with the studies on pyrite-type IrTe 2 as a function of pressure, underlining the importance of Te-Tebond polymerization in the structural phase transition. 24On the basis of a detailed study of low-temperature structure andfirst-principles calculations, it has been shown that a localbonding instability associated with the Te 5 pstates is likely the origin of the structural phase transition in IrTe 2.8Yet, more pressure-effect studies on IrTe 2are found to be analogous to those of spinel CuIr 2S4,10again keeping the debate wide open on the roles of Te 5 pversus Ir 5 dorbitals to drive the structural phase transition. In this context, our findings provide importantinformation on the origin of the phase transition. We find that 224109-3B. JOSEPH et al. PHYSICAL REVIEW B 88, 224109 (2013) both Ir-Te and Ir-Ir bonds are active players in driving the structural phase transition in IrTe 2. Indeed, the experimental results suggest that Ir-Ir dimers are present across the structuralphase transition while Ir-Te bond lengths split only below thetransition temperature. Since the charge superstructure appearsbelow the transition temperature, 2it should be related to the Ir- Ir dimers which are getting ordered. Therefore, it appears thatthe structural transition should be of order-disorder-like transi-tion of Ir-Ir dimers driven by the Fermi-surface nesting 14or by local singlet formation, where the ordering of dimers is assistedby the interaction between the Ir 5 dand Te 5 porbitals. This situation is somewhat similar to the metal-insulator transitionin VO 2, where V-V dimer formation is assisted by the Jahn– Teller-like interaction between the V 3 dand O 2 porbitals.25 In summary, the local structure investigation of the IrTe 2 system provides detailed insight into the nature of the first- order phase transition occurring in the system at ∼270 K. The results reveal that the Ir-Te distances are split in two at low tem-perature and merge in a single distance above the transition. We also find clear existence of distinct Ir-Ir bond lengths, indicat-ing Ir-Ir dimerization. The local dimers survive even above thephase transition temperature. However, at the phase transition,the static disorder of Ir-Ir anomalously drops down. Therefore,the present results provide clear evidence of the importance ofboth Ir 5 dand Te 5 porbitals in the structural phase transition. Combining the present results with earlier experiments, 5,14it appears that, in the first-order phase transition of the IrTe 2sys- tem, the Ir-Ir dimers undergo an order-disorder-like transitiondriven by the interaction between the Ir 5 dand Te 5 porbitals. We thank Sergey Nikitenko of BM26A, ESRF, Grenoble for support in the EXAFS data collection. The work is a part of thebilateral agreement between the Sapienza University of Romeand the University of Tokyo. One of us (M.B.) acknowledgesthe financial support of the Swiss National Science Foundation,Project No. PBZHP2_143495. 1S. Pyon, K. Kudo, and M. Nohara, J. Phys. Soc. Jpn. 81,053701 (2012 ). 2J. J. Yang, Y . J. Choi, Y . S. Oh, A. Hogan, Y . Horibe, K. Kim, B. I. Min, and S.W. Cheong, P h y s .R e v .L e t t . 108,116402 (2012 ). 3M. Kamitani, M. S. Bahramy, R. Arita, S. Seki, T. Arima, Y . Tokura, and S. Ishiwata, P h y s .R e v .B 87,180501 (2013 ). 4K. Kudo, M. Kobayashi, S. Pyon, and M. Nohara, J. Phys. Soc. 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PhysRevB.90.115441.pdf

PHYSICAL REVIEW B 90, 115441 (2014) Hydrogenation-induced ferromagnetism on graphite surfaces Mohammed Moaied,1,2,*J. V . Alvarez,3,†and J. J. Palacios3,‡ 1Departamento de F ´ısica de la Materia Condensada, Universidad Aut ´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain 2Department of Physics, Faculty of Science, Zagazig University, 44519 Zagazig, Egypt 3Departamento de F ´ısica de la Materia Condensada, Instituto Nicol ´as Cabrera (INC), and Condensed Matter Physics Center (IFIMAC), Universidad Aut ´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain (Received 30 April 2014; revised manuscript received 23 June 2014; published 30 September 2014) We calculate the electronic structure and magnetic properties of hydrogenated graphite surfaces using van der Waals density functional theory (DFT) and model Hamiltonians. We find, as previously reported, thatthe interaction between hydrogen atoms on graphene favors adsorption on different sublattices along with anantiferromagnetic coupling of the induced magnetic moments. On the contrary, when hydrogenation takes placeon the surface of graphene multilayers or graphite (Bernal stacking), the interaction between hydrogen atomscompetes with the different adsorption energies of the two sublattices. This competition may result in all hydrogenatoms adsorbed on the same sublattice and, thereby, in a ferromagnetic state for low concentrations. Based onthe exchange couplings obtained from the DFT calculations, we have also evaluated the Curie temperature bymapping this system onto an Ising-like model with randomly located spins. Remarkably, the long-range natureof the magnetic coupling in these systems makes the Curie temperature size dependent and larger than roomtemperature for typical concentrations and sizes. DOI: 10.1103/PhysRevB.90.115441 PACS number(s): 73 .22.Pr,75.30.Hx,71.15.Mb I. INTRODUCTION Hydrogenation of carbon nanostructures is recently at- tracting a lot of interest as a methodology that allows forthe tuning of their mechanical, electronic, and magneticproperties. In contrast to direct manipulation of the carbonatoms, e.g., creating vacancies [ 1,2] or reshaping edges [ 3], hydrogenation can effectively affect the electronic propertiesin a similar manner with the advantage that is a reversibleprocess. For instance, hydrogenation of graphene was found,both theoretically and experimentally, to be a way to turngraphene from a gapless semiconductor into a gapful one witha tunable band gap [ 4–8]. It has also been predicted that partial hydrogenation may induce interesting magnetic propertiesin graphene with potential applications in spintronics [ 9]. For instance, H-induced ferromagnetism is expected undersome very particular conditions [ 10]. Recent experiments on hydrogenated or fluorinated graphene, however, do notshow evidence of ferromagnetism, but rather of paramagneticbehavior [ 11–13]. Many calculations related to the adsorption of H atoms on graphene have been reported in the literature, mostlybeing based on first-principles or density functional theory(DFT). All the reports coincide in that adsorptive carbonatoms are puckered and, most importantly, that the covalentbond between carbon and H leads to magnetic moments onneighboring carbon atoms totalling 1.0 μ B[14–17]. Such spin polarization is mainly localized around the adsorptive carbonatom. The magnetic coupling between H atoms adsorbed ongraphene has also been studied and basically follows the rulesexpected from Lieb’s theorem [ 18]. Graphene is a single layer of carbon atoms bonded together in a bipartite honeycomb *moaied5@yahoo.com †jv.alvarez@uam.es ‡juanjose.palacios@uam.esstructure. It is thus formed by two interpenetrating triangularsublattices, A and B, such that the nearest neighbors ofan atom A belong to the sublattice B and vice versa [ 19]. Three different magnetic states can be triggered by a H pair,namely, nonmagnetic, ferromagnetic, and antiferromagnetic.The most energetically stable configuration corresponds tohaving both H atoms adsorbed on two nearest-neighbor carbonatoms, leading to a nonmagnetic ground state [ 9,15–17]. When both H atoms are on the same sublattice they are coupledferromagnetically with total spin S=1. When the pair of H atoms is adsorbed on different sublattices, but sufficiently faraway from each other, they induce magnetic moments thatcouple antiferromagnetically ( S=0) [9]. As we show here, for similar distances between the H atoms the ferromagneticcoupling is always favored over the antiferromagnetic one. Pre-vious calculations for vacancy-induced magnetism in graphenehave shown similar results as long as the vacancies do notreconstruct [ 20,21]. Concerning graphite, a few experimental studies, not free from controversy, have reported changes in the magneticproperties produced by irradiation of the graphite sample.The results show that graphite can become ferromagnetic atroom temperature out of an originally nonmagnetic sample.The ferromagnetic state appears at low concentration of theimpurities induced by the irradiation and is independent of theirradiation ion type used [ 22,23]. Unlike the case of graphene, not many theoretical studies have been reported in the literatureon the magnetic properties of irradiated or hydrogenatedgraphite. Yazyev [ 24], for instance, has studied the magnetic properties of hydrogenated graphite using a combination ofmean-field Hubbard model and first-principles calculations.He obtained, as expected, that the sublattices are inequivalent(approx. 0.16 eV) for hydrogenation in bulk. Graphite is asemimetal composed of stacked graphene layers. The typicalBernal stacking of these planes effectively breaks sublatticesymmetry: A atoms (for instance) are located exactly above and below the atoms of neighboring planes ( αatoms from now 1098-0121/2014/90(11)/115441(12) 115441-1 ©2014 American Physical SocietyMOHAMMED MOAIED, J. V . ALV AREZ, AND J. J. PALACIOS PHYSICAL REVIEW B 90, 115441 (2014) on), while B atoms are located at the center of the hexagonal rings of the neighboring planes ( βatoms) [ 25]. Here we are concerned with hydrogenation of the surface of graphite. First, through DFT calculations, we revisit theenergetics of a H pair on graphene. We confirm previousresults and, by considering very large supercells, we find theexpected antiferromagnetic state when H atoms are adsorbedsufficiently far apart from each other on different sublattices.Next we present results for the adsorption energies on differentsublattices for bilayer and multilayer graphene. Both sets ofresults are then combined to estimate the maximum averageconcentration for which all H atoms may occupy the samesublattice and, thereby, will be coupled ferromagnetically. Wealso compute the exchange coupling constants as a functionof the relative distance between H atoms. Finally, we presenta study of the Curie temperature in this system based on anIsing model constructed with the DFT coupling constants. Ourresults support the possible existence of surface sublattice-polarized hydrogenation and concomitant ferromagnetism. II. ATOMIC, ELECTRONIC, AND MAGNETIC STRUCTURE OF H ATOMS ON GRAPHENE AND GRAPHENE MULTILAYERS A. Computational details Our calculations are based on the DFT framework [ 26,27] as implemented in the SIESTA code [ 28,29]. We are mostly interested here in multilayer graphene and graphite wheredispersion (van der Waals) forces due to long-range electroncorrelation effects play a key role in the binding of thegraphene layers. Therefore, we use the exchange and corre-lation nonlocal van der Waals density functional (vdW-DF)of Dion et al. [30] as implemented by Rom ´an-P ´erez and Soler [ 31,32]. To describe the interaction between the valence and core electrons we used norm-conserved Troullier-Martinspseudopotentials [ 33]. To expand the wave functions of the valence electrons a double- ζplus polarization (DZP) basis set was used [ 34]. We experimented with a variety of basis sets and found that, for both graphene and graphite, the DZP producedhigh-quality results. The plane-wave cutoff energy for the wavefunctions was set to 500 Ry. For the Brillouin zone samplingwe use 4 ×4×2 Monkhorst-Pack (MP) kmesh for the largest 12 ×12×1 single-layer and for the bilayer graphene supercells. We have also checked that the results are wellconverged with respect to the real space grid. Regarding theatomic structure, the atoms are allowed to relax down to a forcetolerance of 0.005 eV /˚A. All supercells are large enough to ensure that the vacuum space is at least 25 ˚A so that the interac- tion between functionalized graphene layers and their periodicimages is safely avoided. Spin polarization was included inthe calculations because, as discussed in the Introduction,hydrogenation is known to induce magnetism in single-layerand, possibly, also in bilayer and multilayer graphene. B. Preliminary checks We begin our study by optimizing the geometric structures of the monolayer, bilayer graphene, and graphite in theirTABLE I. Atomic structure parameters of monolayer, bilayer graphene, and graphite. C-C bond ( ˚A) a(˚A) c(˚A) d(˚A) Graphene 1.419 2.458 25 Bilayer 1.420 2.459 25 3.353 Graphite 1.417 2.455 6.709 3.354Experimental 2.456 [ 35] 6.696 [ 35] natural nonmagnetic state. The C-C bond lengths and cell parameters ( aandc) and the interlayer distances between the layers ( d) are listed in Table I. The accuracy of our procedure is very satisfactory when these magnitudes are contrastedagainst experimental values. For completeness, we present theatomic structures of single-layer and bilayer graphene in Fig. 1. Different colors are used to stress different sublattices. In Fig. 2we show the electronic band structure for monolayer, bilayer, five-layer graphene, and graphite alongthe high-symmetry points M/Gamma1KM . The well-known case of graphene is shown in Fig. 2(a), being the result similar to that found by many others (see, for instance, Refs. [ 36–38]). Since there are two basis atoms in graphene there is one pairofππ ∗bands of pzcharacter, which are degenerate at the Kpoint or Dirac point, coinciding with the Fermi level. We have considered bilayer graphene in Bernal stacking, as for atypical graphite arrangement. Since the basis consists now offour atoms, there are two pairs of ππ ∗bands and there are four sets of pz-derived bands close to the Kpoint as shown in Fig. 2(b). Due to the interaction between the graphene layers these bands split apart. Consistent with previous theoreticalworks [ 39,40], we find that, similar to monolayer graphene, the bilayer graphene is also a zero-gap semiconductor with apair of the ππ ∗bands being degenerate at the Kpoint. On the other hand, there is an energy gap of 0.8 eV between theother pair of ππ ∗bands. The band structure for five-layer graphene is shown in Fig. 2(c) which already anticipates the characteristic band structure of graphite. For instance, atthe/Gamma1point, five bands closely packed in energy manifest the emerging dispersion stemming from the perpendicularinterlayer coupling. Finally, the bands of graphite are shownin Fig. 2(d). The results are also in agreement with previous works (see, e.g., Ref. [ 41]), exhibiting a bandwidth of 1.41 eV at theKpoint [ 42]. A sublattice atoms B sublattice atomsα site atoms β site atoms 2nd layer atoms(a) (b) FIG. 1. (Color online) Atomic structure of (a) single-layer and (b) bilayer graphene for a 3 ×3×1 supercell. 115441-2HYDROGENATION-INDUCED FERROMAGNETISM ON . . . PHYSICAL REVIEW B 90, 115441 (2014) -20-15-10-505101520-20-15-10-505101520 -20-15-10-505101520-20-15-10-505101520E - E f (eV)Graphite (d) E - E f (eV)Single layer graphene (a) E - E f (eV)Five layers graphene (c) E - E f (eV)Bilayer graphene (b) M Γ KM M Γ KMM Γ KM M Γ KM FIG. 2. Electronic band structure of monolayer, bilayer, five-layer graphene, and graphite calculated with a 100 ×100×2M Pg r i df o ra 1×1×1 supercell. C. Hydrogen atoms on monolayer graphene 1. One hydrogen atom We revisit now, for the sake of completeness, the atomic, electronic, and magnetic structure changes induced on mono-layer graphene by the adsorption of a single H atom. In Fig. 3 we present a view of the atomic structure resulting after theadsorption. This can only occur when the substrate is allowedto relax. In the stable configuration the H atom is covalentlybonded to one carbon atom and is located right above thisatom, as shown in Fig. 3(a). The carbon atom in the adsorption site extrudes out of the graphene plane, displaying the typical dH-C= 1.13 Å(a) (b) Puckering dc-c=1.50Å FIG. 3. (Color online) Atomic structure of H on graphene. (a) Top and (b) side view for a 3 ×3×1 supercell.sp3hybridization to form the σC-H bond [see Fig. 3(b)]. For all supercell sizes we found that the bond lengths between the adsorptive carbon atom and its nearest neighbors increase up to1.50 ˚A (which is to be compared to the bond length in graphene of 1.42 ˚A). The other bond lengths are practically unaffected and the C-H distance is always found to be 1.13 ˚A, regardless of the supercell size. Table IIcontains a detailed account of our results compared to those found in the literature for this system.It may be worth noticing that the largest differences appearwhen contrasting results (including ours) obtained from codesthat use localized orbitals basis sets against those obtainedwith codes based on plane-wave basis sets. These differencesdecrease as the supercell size increases so, regardless of theorigin of the discrepancies for small cells, all results are fairlyconsistent within the DFT framework. The adsorption energy E afor a H atom on graphene is calculated as usual, Ea=Egraphene +H−(Egraphene +EH), (1) where Egraphene +Hdenotes the total energy of the complete system and Egraphene andEHdenote the energies of the isolated graphene and H atom, respectively. We have found that thebinding energy between the H atom and a graphene monolayerincreases with increasing supercell size. A linear fit as a 115441-3MOHAMMED MOAIED, J. V . ALV AREZ, AND J. J. PALACIOS PHYSICAL REVIEW B 90, 115441 (2014) TABLE II. Equilibrium height of the adsorptive carbon atom above the surface ( dpuck) and adsorption energies ( Ea) for different supercell sizes and corresponding H concentration C. All the carbon atoms are allowed to relax along with the H atom. dpuck(˚A) dpuck(˚A) Ea(eV) Ea(eV) Unit cell C This work Other works This work Other works 2×2 0.125 0.359 0.36 [ 43], 0.36 [ 16], 0.36 [ 44]−0.909 −0.67 [43],−0.75 [16],−0.83 [45],−0.85 [44] 3×3 0.056 0.476 0.41 [ 46], 0.42 [ 16], 0.51 [ 44]−0.915 −0.76 [46],−0.77 [16],−0.84 [44] 4×4 0.031 0.485 0.48 [ 16], 0.49 [ 47], 0.58 [ 44]−0.946 −0.76 [48],−0.85 [49],−0.79 [16],−0.89 [47],−0.89 [44] 5×5 0.020 0.500 0.59 [ 16], 0.63 [ 44] −0.950 −0.82 [50],−0.84 [16],−0.94 [44] 6×6 0.014 0.531 0.66 [ 44] −0.956 −0.96 [44] ∞×∞ 0.0 −0.98 function of the inverse supercell size can be done for the cal- culated points which shows that the adsorption energy is about−0.98 eV in the limit of zero concentration of H atoms (infinite supercell size). Obviously, for a given supercell size, the bind-ing energy of the H atom sublattice A is equal to the bindingenergy of the H atom on sublattice B [ E a(A)=Ea(B)]. In agreement with previous studies we also find that the adsorption of H leads to the appearance of a staggeredmagnetization on neighboring carbon atoms amounting toexactly 1 μ B/cell. Such spin density is mainly localized around the adsorptive carbon atom as shown in Fig. 4.I nF i g . 5we show the total density of states (DOS) for the 6 ×6 H-graphene equilibrium structure. The H adsorption causes the appearanceof peak in the DOS at the Fermi level which spin-splits dueto electron-electron interactions. Remarkably, this result iscompatible with Lieb’s theorem for the Hubbard model onbipartite lattices [ 18]. According to such theorem, the removal of a single site in the bipartite lattice should give rise to aground state with S=1/2. The covalent bond between the H atom and the C atom underneath effectively suppresses the“site” (the p zorbital), creating a vacancy in the underlying low-energy Hamiltonian. It is worth noticing how this resultcontrasts with that obtained for a vacancy. As discussed inRef. [ 51], vacancies could in principle give rise to similar magnetic states. The difference with respect to the case of H ad-sorption is that vacancies tend to reconstruct and the magneticmoment generated actually vanishes for low concentrations. 2. Two hydrogen atoms To investigate the electronic and magnetic structure induced on graphene by two adsorbed H atoms we need to use FIG. 4. (Color online) Relaxed atomic structure and spin polar- ization around an adsorbed H atom. Magnetic moments with oppositeorientations are depicted by blue and red arrows for clarity.a1 2×12×1 supercell. Figure 6shows an example and illustrates the required size of the supercell. The use ofsuch a large supercell is essential in order to minimize theinfluence of neighboring supercells on the pairwise propertiesdue to the relative long-range interaction between the magneticclouds induced by the H atoms. The relative extension ofthe magnetic clouds with respect to the supercell size isillustrated in Figs. 7(a) and7(b). Test calculations show that using larger supercells essentially gives similar results. Wecalculate the energetics for the two fundamentally differentadsorption configurations: one in which the two H atoms aresitting on the same sublattice (AA) and the other where theyare sitting on different sublattices (AB). The formation oradsorption energies for pairs of H atoms at various relativedistances for some AA and AB configurations are shown inFig. 8. (In order to see the influence of the H atoms on each other, we have subtracted twice the adsorption energy of asingle H atom.) We have not explored all possibilities, showingonly some representative ones. Since the magnetic cloud orlocalized state associated to each H atom is not isotropic, anangular dependence is expected. The overall result remainsvalid though. First, we can see that the “interaction” energy between atoms is always negative, i.e., the H atoms “attract” each otherregardless of the relative adsorption sublattices. The energygain is the largest by placing the atoms near each other (barelynoticeable for the AA cases, but clearly appreciable below DOS (states/eV/spin) E-E f (eV)Spin up Spin down ---- FIG. 5. (Color online) Total density of states for a H atom on single-layer graphene calculated with a 6 ×6×1 supercell. 115441-4HYDROGENATION-INDUCED FERROMAGNETISM ON . . . PHYSICAL REVIEW B 90, 115441 (2014) AB FIG. 6. (Color online) Atomic view of a pair of H atoms on a graphene monolayer for a 12 ×12×1 supercell. 1 nm for the AB ones). This can be understood in simple terms by noticing that the H adsorption creates a localized state at theFermi energy occupied by a single electron. When two statesare created on different sublattices, these hybridize creating Formation energy (eV) H-H distance over graphene (Å)AA-Ferromagnetic (S=1) AB-Nonmagnetic (S=0) AB-Antiferromagnetic (S=0) FIG. 8. (Color online) Total energy for a pair of H atoms on a graphene monolayer (12 ×12×1 supercell) relative to twice the adsorption energy of a single atom. Both AA and AB cases areshown. (b) ( b ) (a) DOS (states/eV/spin) E-E f (eV)Spin up Spin down(d) DOS (states/eV/spin) E-E f (eV)Spin up Spin down(c) 0.10 0.05 0.0 -0.05 -0.10 -0.15 μ /a.u.20.15 μ /a.u.2 0.15 μ /a.u.2 0.10 0.05 0.0 -0.05 -0.10 -0.15 μ /a.u.2 FIG. 7. (Color online) Spin density (blue indicates up and red down) and spin-resolved total DOS for graphene monolayer with 2 H atoms sitting on AA [(a) and (c), respectively] and AB [(b) and (d), respectively] sublattices at far distances calculated with a 4 ×4×2M Pg r i df o r a1 2×12×1 supercell. 115441-5MOHAMMED MOAIED, J. V . ALV AREZ, AND J. J. PALACIOS PHYSICAL REVIEW B 90, 115441 (2014) (a) (b) FIG. 9. (Color online) Top view of the atomic structure of H on bilayer graphene for (a) αand (b) βsites on a 4 ×4×1 supercell. a bonding state that is now occupied by the two electrons forming a singlet state [ 21]. This is essentially the reason why magnetic solutions only appear at long distances in the ABcases. As Fig. 6shows, only for the longest possible calculated distance the H atoms retain their magnetic clouds. There thecoupling is antiferromagnetic ( S=0), as expected from Lieb’s theorem. Figures 7(b) and 7(d) show the spin-density and the spin-resolved DOS, respectively, in this case. The latterexhibits magnetic splitting near the Fermi energy althoughthe DOS for both spin species are identical. On the contrary,when both atoms are on the same sublattice (AA cases) thesolution is always ferromagnetic ( S=1) regardless of distance [see Figs. 7(a) and7(c)], but the energy gain with decreasing distance is very small since the localized states induced by theH atoms belong to the same sublattice and cannot hybridize. D. Hydrogen atoms on bilayer graphene 1. One hydrogen atom The main focus of this work is actually to elucidate how the interactions of the graphene layers underlying thesurface monolayer that hosts the adsorbed H atoms changesthe well-established results presented in the previous section.As we know, the most stable structure for bilayer graphene,multilayer graphene, and bulk graphite consists of stackedgraphene monolayers following what is called Bernal stacking.In Fig. 9we present a top view of the obtained atomic structure for the adsorption of a single H atom on a graphene bilayer.Here the upper layer is allowed to relax while the carbon atomsin the lower layer were fixed at their equilibrium position.The adsorption geometry of a H atom on a bilayer graphenesurface is very similar to that for graphene monolayer. Due tothe interaction between layers, however, in the bilayergraphene case (and surface graphite as shown below) thesublattices are not equivalent which translates into differentadsorption energies [ E a(α)>E a(β)]. (In order to make clear that the surface sublattices are not equivalent anymore, wechange the labels A and B to αandβfrom now on.) In Fig. 10 we show the H adsorption energy difference between αand βsites [/Delta1E=E a(α)−Ea(β)] for different supercell sizes of the graphene bilayer. /Delta1E increases linearly with the supercell size, extrapolating to ≈85 meV for infinitely large supercells. Importantly, the induced magnetic moment is not affected bythe presence of the second graphene layer. 2. Two hydrogen atoms As shown in previous sections, to properly investigate the interaction between two adsorbed H atoms, one requires Eα - E β (eV) 1/(H-H separation) over bilayer graphene cells (1/Å)Eα - Eβ Linear fit FIG. 10. (Color online) Adsorption energy difference between the two sites ( αandβ) of bilayer graphene against different cell sizes. very large supercells. A similar study in the bilayer case is computationally prohibited. Here we adopt a differentapproach. We assume that the attractive interaction between Hatoms is not affected by the underlying graphene layer. This isnot a strong assumption since the interaction between layers ismainly of van der Waals type while the origin of the magneticstructure changes induced on graphene by the adsorbed H areof kinetic and exchange type. We now simply shift the AApair energy shown in Fig. 8by the energy difference between αandβadsorption sites, /Delta1E. There are two possibilities here. One is to use the value of /Delta1E obtained in the limit of infinitely large supercells. The other is to use a value of/Delta1E that changes with the distance between H atoms. This can be estimated from the calculation for a given supercellsize that approximately corresponds to such distance. Eitherchoice obviously favors adsorption on the same sublattice ( β in this case) when the H atoms are sufficiently far apart and theintralayer interactions are weakened. There are not significantdifferences between both choices and the result for the secondone is shown in Fig. 11. As can be seen, the pairs of H atoms prefer to sit on the same sublattice for distances longer that≈1 nm, thus favoring a ferromagnetic state on the surface layer for a maximum coverage Cof around 0.05. E. Hydrogen atoms on the surface of graphite We have mentioned in passing that the magnetic moment induced in a single graphene monolayer by the H adsorption survives when a second layer is added to form a bilayer. This result is not necessarily obvious; neither is the fact that Hadsorbed on a graphite surface may induce a magnetic momentas well. As discussed in Ref. [ 51], vacancies tend to lose the magnetic moment because the electron-hole symmetry isseverely broken and the localized state hosting the unpairedelectron is not exactly placed at the Fermi energy. A similareffect could take place here. To discard this possibility wehave evaluated the atomic and magnetic structures of a Hatom adsorbed on graphite (represented by up to a five-layergraphene structure). In Fig. 12we present the atomic structure 115441-6HYDROGENATION-INDUCED FERROMAGNETISM ON . . . PHYSICAL REVIEW B 90, 115441 (2014)Formation energy (eV) H-H distance over bilayer graphene (Å)ββ-FM (S=1) αβ-NM (S=0) αβ-AF (S=0) FIG. 11. (Color online) Total energy for two H atoms on bilayer graphene as a function of distance. determined after the adsorption of a H atom on the surface. Here, also, the upper layer is allowed to relax while the carbonatoms in the underlying layers were fixed at their equilibriumposition. The adsorption of the H atom leads to the formation ofa spin density on neighboring carbon atoms, again amountingto exactly 1 μ B/cell. Such spin density is mainly localized on the adsorptive layer, as shown in Fig. 12. Due to the stacking order in the multilayer graphene structure, the sublatticesare, again, inequivalent [ E a(α)>E a(β)] for adsorption. In Table IIIwe show the adsorption energy difference /Delta1E for a5×5 supercell size against different numbers of graphene layers. This converges very quickly with the number of layersso that the results obtained in the previous section remainvalid here: H atoms adsorbed on a graphite surface prefer tolocate themselves on the same sublattice when sufficiently farapart from each other and induce a ferromagnetic state on thesurface. The Curie temperature of this ferromagnet is analyzedin the following section. III. CURIE TEMPERATURE If we consider the H atoms as noninteracting, a very simple statistical estimate shows that the difference of 85 meV FIG. 12. (Color online) Relaxed atomic structure and spin polar- ization around an adsorbed H atom at βsite on a four-layer graphene surface. Magnetic moments are depicted by blue(red) arrows forspin-up(spin-down) for clarity.TABLE III. Energy difference ( /Delta1E) between αandβadsorption sites for a 5 ×5 supercell size against different numbers of graphene layers. No. of layers /Delta1E=Ea(α)−Ea(β) (eV) 1 0.000 00 2 0.039 303 0.037 98 4 0.038 66 5 0.038 576 0.038 71 in the adsorption energies between sublattices favors the selective population of one of them ( ≈95% even at room temperature). Although the thermodynamically most stableconfiguration is when H atoms approach one another formingpairs or clusters, the attraction between H atoms may becounteracted by the diffusion barriers, particularly at lowtemperatures [ 45]. Understanding the dynamics resulting from diffusion processes (and desorption ones for that matter) is ofgreat importance to determine actual hydrogenation patterns,but this lies beyond the scope of this work. A very simplecalculation based on the standard Arrhenius law shows thatthe jump rate is of the order of seconds at room temperature.Similar simple estimates indicate that desorption processes arealso active at room temperature, but at smaller rates than thosefor migration. Preliminary kinetic Monte Carlo studies [ 52] indicate that metastable states where all H atoms stay, at leasttemporarily, adsorbed on the same sublattice are possible witha lifetime that can be as long as needed for sufficiently lowtemperatures. Keeping the discussion of the previous paragraph in mind, we will assume in what follows a 100% sublattice selectivepopulation of H atoms. Despite the spin densities aroundthe H atoms being associated with the conduction electrons,we refrain from considering this magnetism as “itinerant” (seeRefs. [ 53,54]) since, in the diluted regime, the extension of the polarization cloud may be considerably smaller than themean distance between H atoms. In fact, graphene becomes aninsulator at the Dirac point upon H adsorption (see Ref. [ 4]) which renders the term itinerant not truly applicable. Thus, tostudy the collective magnetic properties of the system, we willuse the following Ising-like model Hamiltonian: H=−1 2/summationdisplay ijJijpiSipjSj, (2) where SiandSjare two discrete spin variables ( ±1) at sites iandjof a given sublattice (say β) of the graphite surface. The random variables piandpjrepresent the occupation of one carbon atom with a H atom. These can take the values 1(occupied) or 0 (unoccupied). These discrete random variablesare drawn from a probability density function: ρ(p)=(1−c)δ(p)+cδ(p−1), (3) where cin [0,1] is related to the graphene lattice coverage by C=c/2. The maximum coverage in our case is thus C=0.5 although, as explained above, it is only meaningful for C< 0.05. The adimensional concentration parameter cdefines a 115441-7MOHAMMED MOAIED, J. V . ALV AREZ, AND J. J. PALACIOS PHYSICAL REVIEW B 90, 115441 (2014)Magnetic couplings (eV) 1/(H-H separation) at β sites on graphite surface (Å)Magnetic couplings Linear fit FIG. 13. (Color online) Exchange energy for a pair of H atoms adsorbed on the same sublattice. mean distance between H atoms /lscript=1√cin units of the lattice parameter a.Jijis the magnetic coupling constant between two magnetic moments at sites iandj. The coupling constant is defined as the total energy difference between the antiparallel(AFM) and parallel (FM) alignment of an AA pair: J ij=(EFM)ij−(EAFM)ij. (4) In Fig. 13we show Jijas obtained from our DFT calculations for the configurations shown in Fig. 8. The exchange coupling presents a slow linear decrease with the inverse of the H-Hpair separation, J ij=J0a rij, where J0=0.0576 eV and rijis the distance between H atoms at sites iandj. As expected, it extrapolates to 0 eV in the infinite separation limit. We canonly expect quantitative changes if different functionals areused in the DFT calculations [ 55]. The long-ranged nature of the exchange coupling constants is intimately linked to thezero-energy state created by the H adsorption which slowlydecays as 1 /r. A change in the type of functional may onlyaffect the actual value of J 0and therefore the final value of the prefactor in Eq. ( 8), but not the decay law. To study the magnetic ordering in this system we have used a Monte Carlo (MC) algorithm [ 56]. We have simulated very diluted triangular lattices with L×Lcells with L in the range L=80–1280. Considering that a 1 /rcou- pling has always longer range than the size of the system,we have decided to apply open boundary conditions. Tomake contact with realistic experimental realizations [ 57], we have performed simulations at very low concentrationsC=0.0005,0.0006,0.0007,0.0008,0.0009, and 0 .0010 ( C= 1 means full coverage of the graphite surface with H atoms).Note also that our simulations are performed in the range L /lscript/greatermuch1. The thermal averaging took 50000 MC measurements, after allowing 1000 steps for thermalization. Average over50 random realizations of the H distribution was taken. In Fig. 14we show the thermal average of the magnetization absolute value |M|for two concentrations ( C=0.0005 and C=0.0010) and cell sizes of L=24.6n m( L=100 super- cell units), 49.2 nm, 73.8 nm, 98.4 nm, and 123 nm ( L=500 supercell units). The abrupt supression of |M|signals the approximate value of the ordering or Curie temperature T C. However, this ordering temperature seems to increase withthe system size. In the thermodynamic limit this behaviorextrapolates to an infinite value (i.e., a finite magnetizationat any finite temperature). We discuss now that this is an intrinsic property of the system, a consequence of the long-range coupling betweenthe induced magnetic moments. To study this we computethe Binder cumulant, used conventionally for an accuratedetermination of the critical temperature in MC simulationsof statistical systems. The Binder cumulant is the fourth-ordercumulant of the order parameter distribution [ 58,59], which is defined as U L(T)=1 2/bracketleftbigg 3−/angbracketleft¯M4/angbracketright /angbracketleft¯M2/angbracketright2/bracketrightbigg , (5) where /angbracketleft¯M2/angbracketrightand/angbracketleft¯M4/angbracketrightare the second and fourth moments of the magnetization distribution, with the brackets /angbracketleft ···/angbracketright and the bar denoting thermal and sample averaging.|Magnetization| Temperature ( eV)La= 24.6 nm La= 49.2 nm La= 73.8 nm La= 98.4 nm La= 123 nm (a) |Magnetization| Temperature ( eV)La= 24.6 nm La= 49.2 nm La= 73.8 nm La= 98.4 nm La= 123 nm (b) C=0.0005 C=0.0010 FIG. 14. (Color online) Absolute magnetization per spin for supercells sizes in the range L=24.6n m( L=100 supercell units) and 123 nm ( L=500 supercell units), using concentrations (a) C=0.0005 and (b) 0.0010. 115441-8HYDROGENATION-INDUCED FERROMAGNETISM ON . . . PHYSICAL REVIEW B 90, 115441 (2014)Binder cumulant Temperature ( eV)L= 100 L= 200 L= 300 L= 400 L= 500(a) Binder cumulant Temperature ( eV)L= 100 L= 200 L= 300 L= 400 L= 500(b) C=0.0005 C=0.0010 FIG. 15. (Color online) Fourth-order cumulant for supercells sizes in the range L=24.6n m( L=100 supercell units), and 123 nm (L=500 supercell units), using concentrations (a) C=0.0005 and (b) 0.0010. The finite-size scaling argument states that, close to a critical point, a thermal average of a generic quantity scales as /angbracketleftO/angbracketright=LμgO(L/ξ), (6) where Lis the system size, μa critical exponent, and ξis the temperature dependent correlation length which can be considered adimensional or in units of a.C l o s e to the critical point, it scales as ξ(T)∼(T−TC)−ν.I ti s well known that several physical properties have importantfinite size corrections which make the determination of T C difficult. However, if we specifically consider the scaling of the moments of the order parameter, /angbracketleftM2n/angbracketright=L2nβνgM2n(L/ξ), (7) and substitute in the Binder parameter expression of Eq. ( 5) we get UL(T)=U(L/ξ(T)), which is size independent at the critical point. At large temperatures the histogram ofthe magnetization is expected to be a Gaussian distributionand therefore U L(T→∞ )=0. On the other hand, in the zero temperature limit, the magnetization distribution functionreduces to two δpeaks at opposite values of the saturation magnetization and hence U L(T→0)=1. If a system has a well-defined second-order phase transition at a finite temper-ature, the finite-size analysis of the Binder parameter U L(T) will show a family of decreasing functions of the temperature,all of them crossing, to a very good approximation, at T C. In our case the Binder cumulant curves do not cross at a given point (see Fig. 15) which makes it impossible to define a critical temperature. However, we have realized that if weplot the Binder cumulant against the temperature for eachvalue of the product of the size and the concentration LC, then we obtain a crossing point (see Fig. 16). From this we obtain a relation between the Curie temperature T CandLC (see Fig. 17): TC=(0.77±0.01)LC (eV). (8) Strictly speaking the concept of Curie temperature should be used with caution since the ordering temperature in thethermodynamic limit is not well defined in this model.However, our numerical simulations show clearly a measur- able ordering temperature in any finite lattice. The expres- sion ( 8) and the Binder cumulant analysis of Figs. 16and17 admit a simple interpretation: if the system is going to have awell-defined critical temperature in the thermodynamic limitand the Binder cumulant analysis is going to be an accuratemethod to determine it, the coupling constant has to be rescaledwith the size of system J /prime 0(LC)=J0 LC, (9) which redefines a coupling constant with units of energy. Without such rescaling the Binder cumulant analysis results inno crossing points (see Fig. 15). This behavior is very common in systems with long-range couplings. A very illustrating example is the infinite-rangeIsing model (see for instance [ 60]), where the coupling con- stant has to be rescaled with the total number of spins to achievea well-defined critical temperature in the thermodynamic limit.In our model an equally simple scaling argument can beoffered to justify the rescaling implicit in Eq. ( 9). The effective coupling of a single spin connected by a 1 /rinteraction to the rest of the spins in the system is /angbracketleftJ/angbracketright∼C a2/integraldisplayLa 0J0a rrd rd θ ∼J0LC. (10) In other words, the effective coupling of the system increases linearly as its size increases. This is in contrast with asystem with a finite coordination number where /angbracketleftJ/angbracketrightis size independent. Here we have assumed the continuum limit, acircular sample, and we have replaced the stochastic variablep jby its mean value C. We can remove these assumptions by evaluating numerically the effective coupling /angbracketleftJ/angbracketrightin the triangular discrete lattice with a random population ofhydrogen atoms distributed with the probability density ( 3): /angbracketleftJ i/angbracketright=/summationdisplay jJijpj, (11) 115441-9MOHAMMED MOAIED, J. V . ALV AREZ, AND J. J. PALACIOS PHYSICAL REVIEW B 90, 115441 (2014)Binder cumulant T (eV)L= 640 L= 320 L= 160 L= 80 (b) LC= 1 Binder cumulant T (eV)L= 1280 L= 640 L= 320 L= 160 (c) LC= 2 Binder cumulant T (eV)L= 1280 L= 640 L= 320 L= 160 (d) LC= 4 Binder cumulant T (eV)L= 640 L= 320 L= 160 L= 80 (a) LC= 0.2 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 FIG. 16. (Color online) Fourth-order cumulant for (a) LC=0.2, (b) LC=1.0, (c) LC=2.0, and (d) LC=4, using supercell sizes L=80, 160, 320, 460, and 1280.Tc (eV) LC FIG. 17. Critical temperature against LC. M2 T/T cL=300, C=0.0005 L=400, C=0.0005 L=500, C=0.0005 L=300, C=0.0006 L=400, C=0.0006 L=500, C=0.0006 L=300, C=0.0007 L=400, C=0.0007 L=500, C=0.0007 Mean field - FIG. 18. (Color online) Magnetization square vs temperature over critical temperature computed with Monte Carlo and comparedwith the mean-field result for various concentrations and sizes. 115441-10HYDROGENATION-INDUCED FERROMAGNETISM ON . . . PHYSICAL REVIEW B 90, 115441 (2014) which, averaged over all the sites iof the lattice, also scales as LCin the limit of large cell sizeL /lscript/greatermuch1. Finally, we have compared the MC simulations with the mean-field approximation (see Fig. 18). In ordered long- ranged/high-coordination systems this approximation can evenbe exact (see Ref. [ 61] and references therein). In our model the agreement is very good. IV . CONCLUSIONS Through extensive DFT calculations we have found that the interaction between H atoms on graphene favors adsorption ondifferent sublattices along with an antiferromagnetic couplingof the induced magnetic moments. On the contrary, when hy-drogenation takes place on the surface of graphite or graphenemultilayers (in Bernal stacking), the difference in adsorptionenergies takes over the interaction between H atoms and mayresult in all atoms adsorbed on the same sublattice and, thereby,in a ferromagnetic state for low concentrations. Based onthe exchange couplings obtained from the DFT calculations,we have also evaluated the Curie temperature by mapping this system onto an Ising-like model with randomly locatedspins. The long-range nature of the magnetic coupling makesthe Curie temperature size dependent and larger than roomtemperature for typical concentrations and sizes. Obviously,a value of the Curie temperature that is higher than thetemperature for which desorption processes of H are activeare meaningless. Therefore, room temperature sets the upperlimit. ACKNOWLEDGMENTS This work was supported by MICINN under Grants No. FIS2010-21883, No. CONSOLIDER CSD2007-0010, and No.F1S2009-12721, MINECO under Grant No. FIS2012-37549,and by Generalitat Valenciana under Grant No. PROME-TEO/2012/011. We also acknowledge computational supportfrom the CCC of the Universidad Aut ´onoma de Madrid. We thank F. Yndur ´ain for discussions and I. Brihuega for sharing with us preliminary experimental data. [1] C. G ´omez-Navarro, P. 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PhysRevB.70.121304.pdf

Defect-induced perturbation on Si 1114ˆ1-In: Period-doubling modulation and its origin Geunseop Lee,1,*Sang-Yong Yu,2Hanchul Kim,2and Ja-Yong Koo2 1Inha University, Inchon 402-751, Korea 2Korea Research Institute of Standards and Science, P.O. Box 102, Yuseong, Daejeon 305-600, Korea (Received 21 June 2004; published 14 September 2004 ) A one-dimensional defect-induced local period-doubling s32dmodulation was observed with scanning tunneling microscopy on a Si s111d431-In surface at room temperature. The 32 modulated region remains metallic in contrast to the low-temperature 4 32 ground-state phase. First-principles calculations predict the lowest-energy state with exactly the same features as observed by H adsorption, but fundamentally differentfrom the experimentally observed insulating ground state at low temperature.This suggests that the true groundstate (i.e., the low-temperature phase )is not a band insulator, and is stabilized by many-body interaction. DOI: 10.1103/PhysRevB.70.121304 PACS number (s): 68.35. 2p, 68.37.Ef, 68.43.Bc Systems with reduced dimensionality frequently show phenomena that may not be present in the related three-dimensional systems due to the change in interactions. Oneof the examples is a Peierls instability in one-dimensional(1D)system, driven by a perfect Fermi nesting and a large electron-phonon coupling. 1AMott insulator may be realized when an electron-electron interaction becomes strong com-pared with the kinetic energy, which decreases with reduceddimensionality. 2In recent years, a number of surface systems have attracted considerable attention by exhibiting suchphenomena. 3–8 The role of defects become more important as the dimen- sionality is reduced.9Defects not only create local structural changes, but also affect the phase transition by suppressingthe fluctuations that are larger in lower dimensions. In doingso, the defects usually stabilize or mimic the low-temperature (LT)phase in their vicinity at temperatures higher than the transition temperature. Examples include, adefect-induced dimer buckling on Si s100d231 at room tem- perature (RT), 10–12and defects in Sn/Ge s111d˛33˛3 medi- ating the condensation of the LT 3 33 phase.13–15Recently an example emerged for the In/Si s111dwith 1 ML of In. While the system undergoes a phase transition from a metal- lic 4 31 at RT to an insulating 4 32o r8 32 phase (we will refer to this ground-state LT phase as a GS-4 32 throughout this paper, for simplicity )below 100 K,3Na adsorbates were reported to pin the GS-4 32 phase even at RT.16For the nature of the GS-4 32, a charge-density wave (CDW )con- densate induced by the Peierls instability was proposed,3al- though it was challenged by subsequent studies.17,18 In this paper, we report a very different case, where the defect-induced low-symmetry structure differs in nature fromthe LT phase. On the Si s111d431-In surface at RT, highly 1D and period-doubling s32dmodulations in scanning tun- neling microscopy (STM )images are observed near a variety of defects. We demonstrate that this defect-induced modula-tion is metallic and thus not a manifestation of the insulatingGS-4 32 phase. This contrasts to the Na adsorption on the same surface. Using density-functional theory (DFT )calcu- lations within the generalized gradient approximation(GGA ), the 32 structural distortion away from the defects has been calculated. These calculations show tha t a H impu-rity stabilizes a metallic 4 32 phase, which is the DFT-GGA ground state sGGA-4 32d. The GGA-4 32 is fundamentally different from the experimentally observed insulating GS-4 32. This indicates that the stability of the GS-4 32 and its insulating nature need explanation by mechanisms other thanthe simple Peierls instability inducing CDW. STM experiments were performed in an ultrahigh vacuum chamber with the base pressure below 1.2 310 −10Torr. The preparation of the Si s111d431-In surface was described elsewhere.19Defects were spotted either by searching for the region on the clean surface or by dosing H 2and O2gases. Gas dosing was done at RT by backfilling the chamber withan ion-gauge filament on for cracking into atoms. The STMmeasurements were made at RT. Figure 1 shows RT STM images of the Si s111d431-In surface with five different types of defects. The defects FIG. 1. STM images (Vs=+0.2 V, It=0.1 nA )with various types of defects: vacancy (V), step edge (SE), hydrogen adsorption (H), phase-shift boundary (PSB), and oxygen adsorption (O)or edge vacancy created by oxygen-induced etching (EV). Regardless of the defect types, there are 32 modulations in the vicinity of defects.PHYSICAL REVIEW B 70, 121304 (R)(2004 )RAPID COMMUNICATIONS 1098-0121/2004/70 (12)/121304 (4)/$22.50 ©2004 The American Physical Society 70121304-1shown in Fig. 1 are a vacancy, a step edge of an island (or stepped terraces ), a phase boundary on a single terrace, H atom adsorption, and O atom adsorption. All these defectsinduce a 32 modulation near the defects, superimposed with the atomic corrugation in STM images. The perturbation in-duced by the defects is highly 1D (we will discuss it later ). The amplitude of the modulation decays with distance alongthe row from the defect, extending up to 10–12 lattice units.This range is much longer than those of the quasi-two-dimensional Sn/Ge s111dand Si (100)cases observed at RT. This also reflects the 1D nature of this perturbation. STM images of the Si s111d431-In surface containing H adsorbates taken at various bias voltages are shown in Fig. 2. The characteristic bias-dependent STM features of thedefect-free 4 31 are the same as those previously reported. 19 The modulation near the defects is prominent at low bias voltages. With increasing bias voltage of both polarities, themodulation becomes weaker. Diminution of the 32 modula- tion is faster for the filled states sV s,0Vdthan for the empty states sVs.0Vd. This bias-voltage dependence of the modulation observed for the H adsorbate is representative, and shared by all other defects presented in Fig. 1. Differ-ently from the modulation amplitude, the period of themodulation s32ddoes not depend on the bias as shown in the bottom panel of Fig. 2. It is noted that the appearance of the32 modulation is quite different from those reported for the GS-4 32 phase 3and the Na-induced 32 modulation.16The defect-induced modulation in the STM images of the Sis111d431-In surface appear quite similar to those of the electron screening reported in a number of surfaces.20The metallic Si s111d431-In surface has three surface-state bands crossing the Fermi level, one of which has a Fermi wave vector of kF,G0/2, withG0being the zone boundary in the row s31ddirection.3,18The observed 32 modulation period is apparently consistent with the wavelength of the electron screening corresponding to this kF. It is noted that the wavelength of the electron standing wave is to be energydependent. 21The observed bias independency of the modu- lation period is incompatible with a simple electron standingwave picture of screening, and requires a considerationwhich goes beyond a simple plane-wave response. The lat-tice response to the electron screening must be included. If the lattice response is important, the obvious possibility for the origin of the defect-induced modulation is the localmanifestation of the GS-4 32 phase. The GS-4 32 phase was reported to have a significant lattice distortion from the431 structure as determined by a surface x-ray diffraction (SXRD ). 17Previously, Na adsorption was reported to induce this GS-4 32 at RT, based on close similarities in spectro- scopic data and in STM appearance.16For both the GS-4 32 and the Na-induced 32 phase at RT, the existing elec- tron spectroscopy data show a band gap or significant reduc-tion in density of states (DOS )atE F,NsEFd.3,16,22However, the case of the defects in our study is different. The 32 modulated region in our case remains metallic in contrast tothe gap opening s0.1–0.15 eV dreported in the Na case. This is clearly demonstrated in the current-voltage sI-Vdcharac- teristics near zero bias in Fig. 3 (a). There is virtually no reduction in NsE Fd. Stable STM imaging of the 32 area was even possible at a bias as low as ±0.01 V [see Fig. 3 (b)], which is well within the band gap of the Na-induced phaseeven considering the thermal energy of RT s,0.025 eV d. Therefore, we infer that the defect-induced 32 modulation observed in this work is different from the GS-4 32. The dissimilar appearances in the STM images also support theirdifference. Figure 3 (b)also clearly shows that the defect- induced perturbation is highly 1D, being confined only in therow where the defect is located. We have performed ab initiopseudopotential calculations within DFT-GGA. 23First, we calculated the electronic band FIG. 2. (top)STM images of the same region showing the bias- voltage dependence near H adsorbates indicated by arrow heads.(bottom )Line scans between two crosses in the −0.2 V image. The position next to H and the 32 distances are marked by a solid line and dashed lines, respectively. FIG. 3. (a)I-Vdata measured at points of unperturbed 31 and perturbed 32 rows. (b)An STM image acquired with Vs=+10 mV and It=0.17 nA.LEEet al. PHYSICAL REVIEW B 70, 121304 (R)(2004 )RAPID COMMUNICATIONS 121304-2structure of the GS-4 32 phase.Although the structure of the GS-4 32 phase is not fully established, we referred to the existing SXRD data of the lattice structure.17Without relax- ation, our calculations did not find either a gap opening or significant reduction in NsEFd, agreeing with a previous calculation.18With relaxation, we found a new lowest-energy 432 structure sGGA-4 32dthat is still metallic. The GGA- 432 possesses a slight lattice distortion from the 4 31, dif- ferentiating from the observed GS-4 32 phase. This newly found GGA-4 32 differs from the RT-4 31 and the previ- ously calculated 4 32,18too. The discrepancy with the experiments3,22is an indication that the ground-state GS-4 32 phase is not a simple band insulator. We now turn to the case with defects to elucidate the origin of the observed defect-induced modulation. H atomadsorption was chosen as a test case that is simple and trac-table. H atoms are found to be preferably inserted into In-Sibonds on the surface layer.We adopted a 4 321 supercell (21 lattice units along the row )to simulate the isolatedH adsorp- tion. While detailed theoretical results will be presentedelsewhere, 24the simulated STM images are shown in Fig. 4 with the relaxed atomic geometry. The agreement betweenthe theoretical and experimental images (Fig. 2 )is remark- able. The simulations reproduce not only the characteristic Hfeature, but also the 32 modulation of the In chains along the row, the bias dependence of the modulation, and the ob-servedI-Vcharacteristics. 25The fast disappearance of the modulation with increasing bias voltages indicates that thedefect-induced perturbation in the electronic charge profile(that is, local DOS )is mainly concentrated in the states near E F. However, the change in NsEFdis not significant. Upon adsorption, an H atom breaks one of the surface In-Si bonds and forms a Si-H bond [Si(0)-H in Fig. 4 (d)], making the reacted In atom [In(0)in Fig. 4 (d)]significantly displaced from the unperturbed position. This perturbationpropagates far enough to induce structural modification in the region away from the defect site. According to our cal-culations, the structural modification is present throughoutthe whole 4 321 cell, consistent with the observed perturba- tion range of up to ten 31 lattice units from the H adsorbate on the row. The structural change at locations away from theH atom is characterized by the pattern shown in Fig. 4 (d). This lattice distortion takes a symmetry of the normal modeat the zone boundary of 4 31 structure. Distant from the H adsorbate in our 4 321 supercell, the magnitude of the atomic displacement is only about ,0.03 Å. This is an order of magnitude smaller compared with both the SXRDs0.65 Å d 17and the previous calculation by Cho et al. s0.28 Å d.18 Based on the same appearance of the observed 32 modu- lations for a variety of defects and their resemblance to theGGA-4 32 geometry, it is inferred that the defect-induced 432 modulation commonly adopts the 4 32 geometry of the defect-free surface, GGA-4 32. This 4 32 geometry is nearly degenerate with the 4 31(the energy lowering is only about ,1 meV/4 31, well within the accuracy of DFT cal- culations ). Due to the negligible energy difference found in our calculations, it is difficult to tell which one is the lowest-energy state in the framework of the DFT-GGA. 26However, it is plausible that the presence of defects may tilt the energybalance. We believe that this happens on the Si (111)-In sur- face. The defects stabilize the GGA-4 32 geometry in their vicinity, inducing the 32 modulation on the 4 31-In surface at RT. In many surfaces showing the temperature-driven struc- tural phase transition, a symmetry breaking defect locallystabilizes the LT phase or eventually leads to it via interfer-ence with other defects, at temperatures above T c. Such ex- amples include the Cdefect or C 2H2on Si (100),11,12the vacancy or Ge-substitutional defects in Sn/Ge s111d,14and the Na adsorbate on In/Si s111d-431.16Contrasting with these examples, our work constitutes an exceptional case, where the local reduced-symmetry structure does not takethat of the LT phase. Our findings and the inferences drawn from the compari- son with past studies can be summarized as (1)There are two different 4 32 structures :GS-432 and GGA -432. The GS- 432 is an experimentally observed LT phase having large lattice distortion. While this was proposed to be a CDWcondensate based on the Fermi nesting observed in the pho-toemission spectroscopy data, 3the calculations cast some doubts on the origin of its insulating property. The GGA-432 is the lowest-energy state found in the present set of DFT calculations, for which the slight energy lowering isachieved through the charge redistribution accompanying theweak 32 lattice distortion. (2)Defects (studied in this work) on the 4 31 surface at RT stabilize the GGA -432 structure predicted by the DFT-GGA theory . It is obvious from this work that the GGA-4 32 state which is stabilized by our defects is different from the GS-4 32 LT phase. This is an unusual observation, indicating that the phase diagram of thissystem could be more complex than anticipated. An intriguing but hypothesized scenario for the temperature-driven phase transition of the defect-free surfaceand the influence of the defects is proposed. At RT, the sur- FIG. 4. (a)–(c)Simulated STM images of the 4 31-In surface with an H adsorbate in the 4 321 supercell (parallelogram ). The inset in (c)is the experimental counterpart of the H adsorbate. (d) Relaxed geometry around the H atom. A schematic in-plane relax-ation pattern is represented by arrows. The interatomic distancesbetween In atoms projected on the row direction is designated in Å.The projected inter-In distance is 3.86 Å for the unperturbed 4 31 surface.DEFECT-INDUCED PERTURBATION ON Si s111d-s431d-In: PHYSICAL REVIEW B 70, 121304 (R)(2004 )RAPID COMMUNICATIONS 121304-3face has a 4 31 structure seen by low-energy electron dif- fraction (LEED )and STM. Upon cooling, the RT-4 31 phase becomes subject to a structural phase transition into a low-symmetry phase. The DFT calculations predict an order- disorder transition to the GGA-4 32 structure, where the dynamical fluctuation is suppressed. However, it seems that athirdphase (the true ground-state GS-4 32)starts to develop aroundT c. Thus, in reality, the temperature-driven phase transition becomes a displacive type.The GS-4 32 phase has a structural distortion much larger than the GGA-4 32, and is insulating. Although it is likely that the structural distor-tion is driven by Peierls instability, the insulating property is not compatible with a simple Peierls CDW picture as previ-ously proposed 3and needs an explanation which goes be-yond it.At RT, the role of defects in our study is to stabilize the DFT-predicted GGA-4 32 structure. At present, the reason why different defects (the defects in our work and Na )stabilize different phases (GGA-4 32 pre- dicted in theory and ground-state GS-4 32, respectively )at RT is unknown. Further work is needed to unveil the mecha-nisms behind this difference. This work was supported by MOST of Korea through “The National R&D Project for Nano Science and Technol-ogy” and “The Creative Research Initiative,” and partly byGrant No. R02-2004-000-10262-0 from the Basic ResearchProgram of the Korea Science and Engineering Foundation.H.K. acknowledges support from the KISTI under “The FifthStrategic Supercomputing Support Program.” *Author to whom correspondence should be addressed. Electronic mail: glee@inha.ac.kr 1R. E. Peierls, Quantum Theory of Solids (Clarendon, Oxford, 1964 ). 2N. F. Mott, Metal-Insulator Transitions , 2nd ed. (Taylor and Fran- cis, New York, 1990 ). 3H. W. Yeom, S. Takeda, E. Rotenberg, I. Matsuda, K. Horikoshi, J. Schaefer, C. M. Lee, S. D. Kevan, T. Ohta, T. Nagao, and S.Hasegawa, Phys. Rev. Lett. 82, 4898 (1999 ). 4T. Nakagawa, G. I. Boishin, H. Fujioka, H. W. Yeom, I. Matsuda, N. Takagi, M. Nishijima, and T. Aruga1, Phys. Rev. Lett. 86, 854(2001 ). 5K. Swamy, A. Menzel, R. Beer, and E. Bertel, Phys. Rev. Lett. 86, 1299 (2001 ). 6J. M. Carpinelli, H. H. Weitering, E. W. Plummer, and R. Stumpf, Nature (London )381, 398 (1996 ). 7N. J. DiNardo, T. M. Wong, and E. W. Plummer, Phys. Rev. Lett. 65, 2177 (1990 ). 8H. H. Weitering, X. Shi, P. D. Johnson, J. Chen, N. J. DiNardo, and K. Kempa, Phys. Rev. Lett. 78, 1331 (1997 ). 9D. R. Nelson, Phase Transitions and Critical Phenomena , edited by C. Domb and J. L. Lebowitz (Academic, London, 1983 ), Vol. 7,1. 10R. M. Tromp, R. J. Hamers, and J. E. Demuth, Phys. Rev. Lett. 55, 1303 (1985 ). 11R. J. Hamers and U. K. Kohler, J. Vac. Sci. Technol. A 7, 2854 (1989 ). 12W. Kim, H. Kim, G. Lee, Y.-K. Hong, K. Lee, C. Hwang, D.-H. Kim, and J.-Y. Koo, Phys. Rev. B 64, 193313 (2001 ). 13H. H. Weitering, J. M. Carpinelli, A. V. Melechko, J. Zhang, M. Bartkowiak, and E. W. Plummer, Science 285, 2107 (1999 ). 14A. V. Melechko, J. Braun, H. H. Weitering, and E. W. Plummer, Phys. Rev. Lett. 83, 999 (1999 ).15T. E. Kidd, T. Miller, M. Y. Chou, and T.-C. Chiang, Phys. Rev. Lett.85, 3684 (2000 ). 16S. S. Lee, J. R. Ahn, N. D. Kim, J. H. Min, C. G. Hwang, J. W. Chung, H. W. Yeom, S. V. Ryjkov, and S. Hasegawa, Phys. Rev.Lett.88, 196401 (2002 ). 17C. Kumpf, O. Bunk, J. H. Zeysing, Y. Su, M. Nielsen, R. L. Johnson, R. Feidenhans’l, and K. Bechgaard, Phys. Rev. Lett. 85, 4916 (2000 ). 18J.-H. Cho, D.-H. Oh, K. S. Kim, and L. Kleinman, Phys. Rev. B 64, 235302 (2001 ). 19G. Lee, S.-Y. Yu, H. Kim, J.-Y. Koo, H.-I. Lee, and D. W. Moon, Phys. Rev. B 67, 035327 (2003 ). 20M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature (London ) 363, 524 (1993 ); Y. Hasegawa, and Ph. Avouris, Phys. Rev. Lett.71, 1071 (1993 ); P. T. Sprunger, L. Petersen, E. W. Plum- mer, E. Lægsgaard, and F. Besenbacher, Science 275, 1764 (1997 ). 21The bias-dependent modulation wavelength can appear not only indI/dVimages but also in topographic images, especially for empty-state bias, where the tunneling is most weighed to thehighest states of the applied voltage. Refer to an example [T. Yokoyama, M. Okamoto, and K. Takayanagi, Phys. Rev. Lett. 81, 3423 (1998 )]. 22H. W. Yeom, K. Horikoshi, H. M. Zhang, K. Ono, and R. I. G. Uhrberg, Phys. Rev. B 65, 241307 (2002 ). 23G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11 169 (1996 ). 24H. Kim (unpublished ). 25Calculations using a 8 39 supercell show that the perturbation is confined only along the row without affecting the neighboringrows, confirming the experimental finding. 26In our calculations, the DFT-GGA geometry reported by Cho et al.(Ref. 18 )has a higher energy than the GGA-4 32 by about 20 meV/4 31, and is unstable.LEEet al. PHYSICAL REVIEW B 70, 121304 (R)(2004 )RAPID COMMUNICATIONS 121304-4

PhysRevB.73.195301.pdf

Towards full counting statistics for the Anderson impurity model A. O. Gogolin1and A. Komnik2,3 1Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom 2Service de Physique Théorique, CEA Saclay, F–91191 Gif-sur-Yvette, France 3Physikalisches Institut, Albert-Ludwigs-Universität, D–79104 Freiburg, Germany /H20849Received 9 December 2005; published 1 May 2006 /H20850 We analyze the full counting statistics /H20849FCS /H20850of the charge transport through the Anderson impurity model /H20849AIM /H20850and similar systems with a single conducting channel. The object of principal interest is the generating function for the cumulants of charge current distribution. We derive an exact analytic formula relating the FCSgenerating function to the self energy of the system in the presence of the measuring field. We first check thatour approach reproduces correctly known results in simple limits, such as the FCS of the resonant level system/H20849AIM without Coulomb interaction /H20850. We then proceed to study the FCS for the AIM both perturbatively in the Coulomb interaction and in the Kondo regime at the Toulouse point /H20849we also study a related model of a spinless single-site quantum dot coupled to two half-infinite metallic leads in the Luttinger liquid phase at aspecial interaction strength /H20850. At zero temperature the FCS turns out to be binomial for small voltages. For the generic case of arbitrary energy scales the FCS is shown to be captured very well by generalizations of theLevitov-Lesovik type formula. Surprisingly, the FCS for the AIM indicates a presence of coherent electron pairtunnelling in addition to conventional single-particle processes. By means of perturbative expansions aroundthe Toulouse point we succeeded in showing the universality of the binomial FCS at zero temperature in linearresponse. Based on our general formula for the FCS we then argue for a more general binomial theorem statingthat the linear response zero-temperature FCS for any interacting single-channel setup is always binomial. DOI: 10.1103/PhysRevB.73.195301 PACS number /H20849s/H20850: 71.10.Pm, 72.10.Fk, 73.63. /H11002b, 73.21.La I. INTRODUCTION The Anderson impurity model is one of the best studied models in condensed matter theory.1,2Despite being exactly solvable by means of the Bethe ansatz /H20849BA /H20850method in the wide range of equilibrium parameters,3–5its nonequilibrium properties are not yet fully understood. Notable exceptionsare the works on the nonlinear I-Vcharacteristics. 6–8It was first realized by Schottky9that the current autocorrelation spectrum /H20849sometimes also called noise /H20850carries information about the charge of particles participating in transport. Theinvestigation of these properties has been startedrecently. 10–12However, the current-voltage characteristics and noise spectra are only the lowest order moments of thefull current distribution function, which is needed to com-pletely characterize the transport properties of the system.Although it is still quite challenging to access even the noisecorrelations in experiments, in recent years it became pos-sible to measure the third irreducible moment /H20849third cumu- lant /H20850of the current distribution function. 13It turned out to carry information about the influence of the electromagneticenvironment on the transport through the system underconsideration. 14Moreover, it has been argued that the third cumulant is more suited for measuring the charge of currentcarrying excitations than the noise correlations. 15In order to meet future experimental needs it is therefore natural to ana-lyze the full counting statistics of the AIM. The principal question is, what are the effects of the electron-electron interactions on the FCS? Is it possible togain insight into the properties of a strongly correlated elec-tron system by studying its FCS distribution function? Weprovide at least a partial answer in this paper. The answerturns out to be on the negative side, though it is a construc-tive one: we find that the binomial statistics is universal in the low temperature linear response limit. /H20849Interactions only affect the magnitude of the effective transmission coeffi-cient. /H20850At high voltage /H20849temperature /H20850the effects of the inter- actions are indeed profound /H20849see main text /H20850if more model dependent. The AIM model is characterised by a number of different parameters: the electronic tunnelling amplitude /H9253between the impurity level /H20849which we also shall sometimes call “dot” later /H20850and the external electrodes, its energy /H90040, and the strength of the Coulomb interaction on the dot U. There are three different transport regimes: /H20849i/H20850resonant level case, when Uis zero, /H20849ii/H20850Kondo dot regime, when Uis large and when the dot level lies deep below the Fermi energies in theelectrodes, /H20849iii/H20850mixed valence regime, which comprises all other possibilities. The most interesting situation is /H20849ii/H20850when the dot is permanently populated by a single electron. Theso-called Kondo-resonance /H20849also known as Abrikosov-Suhl resonance /H20850in the local density of states leads to a significant increase of conductivity, which has recently been observed inexperiments on ultra-small quantum dots. 16,17This phenom- enon is a signature of the Kondo effect and is a result of the exchange interaction between the local spin degree of free-dom on the dot with those in the leads. One important featureof this effect is the fact that it is growing stronger as thetemperature is lowered. From the mathematical point of viewthat means that the exchange interaction is a relevant opera-tor in the renormalization group /H20849RG /H20850sense, resulting in a new ground state where the local spin is absorbed, the leadsare coherent and the conductance is maximal /H20849perfect /H20850. The equilibrium Kondo model, being one special case of AIM model, is integrable by means of the BA technique. 5,18 While it is possible to infer the nonlinear I-Vfrom thePHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 1098-0121/2006/73 /H2084919/H20850/195301 /H2084919/H20850 ©2006 The American Physical Society 195301-1knowledge of scattering matrix, it is not yet clear whether an extraction of noise or any other higher order correlations ofcurrent is feasible. It has been pointed out by Toulouse, 19that the Kondo model allows a trivial diagonalization for onespecial parameter constellation, when the whole Hamiltonianbecomes quadratic in fermion fields. In addition to this veryuseful feature the Kondo model at the Toulouse point turnsout to be representative for the low energy behavior of ageneric Kondo model, 20reproducing all essential details of the latter in the low energy sector. For the Kondo dot asimilar procedure has been developed by Emery and Kivel-son in Ref. 21 and refined by Schiller and Hershfield 8in order to access the nonequilibrium current-voltage as well asnoise properties. As has been shown in Ref. 22, this approachcan be applied to access the FCS as well. Contrary to the single-channel Kondo model, which maps on the conventional noninteracting resonant level /H20849RL /H20850 model at the Toulouse point, the Kondo dot under the sameconditions is described by the Majorana RL Hamiltonian. 20It has been demonstrated in Refs. 23 and 24, that an RLcoupled to two half-infinite Luttinger liquids /H20849LL/H20850at the spe- cial interaction parameter g=1/2 can be rewritten in terms of a Majorana RL model as well. In this way one can obtain theexact FCS for a genuine interacting system. Related systemshave been analysed before, see Refs. 25 and 26 /H20849“Coulomb blockade” dots /H20850. Below we investigate how the two models are related. The outline of the paper is as follows. In Sec. II we present a further development of the Levitov-Reznikov 15ap- proach to the FCS calculation in tunnelling setups. In orderto test our Hamiltonian formalism we perform an explicitcalculation of the generating function for the FCS of asimple tunnelling contact between two metallic electrodes,see Sec. II B. Section III starts with a reproduction of theknown FCS for the RL model, which is the spinless versionthe of AIM setup without Coulomb interaction. Next we de-rive Eq. /H2084920/H20850, which is the general formula for the FCS of an interacting system and the main result of this paper. We thenproceed to evaluate the perturbative corrections in U, see Sec. III D, and investigate linear response FCS on generalgrounds in Sec. III E. The opposite case of large U, when the system is in the Kondo regime, is the subject of Sec. III F.We not only present analytical results at the Toulouse point,but also analyse the change in the statistics around it in Sec.III F 2. Subsequently we discuss the relation of the KondoFCS to that of a RL setup between two LL at g=1/2 and establish connections to existing results. Some conclusionsare offered in Sec. IV. There are several appendixes contain-ing technical details of some of the lengthier derivations. II. KELDYSH METHOD FOR THE CALCULATION OF CURRENT STATISTICS A. General considerations The cumulants of a given distribution function are known to define the latter in the unique way.27For practical reasons it is usually more convenient to calculate the so-called gen-erating function /H9273/H20849/H9261/H20850, which in case of charge transport is given by/H9273/H20849/H9261/H20850=/H20858qeiq/H9261Pq, where Pqis the probability for thecharge qto be transferred through the system during the measuring time T. The parameter /H9261here is referred to as the measuring field. The cumulants /H20855/H9254nq/H20856/H20849which are nothing else but the irreducible moments of Pq/H20850can then be found for according to the prescription /H20855/H9254nq/H20856=/H20849−i/H20850n/H20879/H11509n /H11509/H9261nln/H9273/H20849/H9261/H20850/H20879 /H9261=0. The measurement of the charge transmitted through a system is usually accomplished by a coupling to a “measuring de-vice.” In the original work by Levitov and Lesovik it is afictitious spin-1/2 galvanometer coupled to the current. 28,29 The transmitted charge is then proportional to the net changeof the spin phase. As has been shown by Nazarov, 30the counting of charge can in general be done by coupling thesystem to a fictitious field and calculating the nonlinear re-sponse, which leads, of course, to exactly the same results. According to Ref. 15 the generating function is given by the following average: /H9273/H20849/H9261/H20850=/H20883TCexp/H20875−i/H20885 CT/H9261/H20849t/H20850dt/H20876/H20884, /H208491/H20850 where C is the Keldysh contour, TCis the contour ordering operator,/H9261/H20849t/H20850is the measuring field which is nonzero only during the measuring time T:/H9261/H20849t/H20850=/H9261/H9258/H20849t/H20850/H9258/H20849T−t/H20850on the for- ward path and /H9261/H20849t/H20850=−/H9261/H9258/H20849t/H20850/H9258/H20849T−t/H20850on the backward path. Introducing the operator transferring an electron through the system in the positive direction /H20849i.e., in the direction of the current /H20850TR, and its counterpart TLwe can write T/H9261=ei/H9261/H20849t/H20850/2TR+e−i/H9261/H20849t/H20850/2TL. /H208492/H20850 We note in passing that TR†=TLin any system. Consequently, writing out Eq. /H208491/H20850explicitly in terms of the time-ordered and anti-time-ordered products, one arrives at the conjuga-tion property /H9273*/H20849/H9261/H20850=/H9273/H20849−/H9261/H20850. /H208493/H20850 We now allow/H9261/H20849t/H20850to be an arbitrary function on the Keldysh contour,/H9261/H11007/H20849t/H20850on the forward/backward path. Then a gener- alized counterpart of Eq. /H208491/H20850can be defined as /H9273/H20851/H9261−/H20849t/H20850,/H9261+/H20849t/H20850/H20852=/H20883TCe−i/H20885 CdtT/H9261/H20849t/H20850/H20884. /H208494/H20850 Next we assume that the measuring field changes only very slowly in time. Then up to the switching terms /H20849which are known to be proportional to ln T/H20850 /H9273/H20851/H9261−/H20849t/H20850,/H9261+/H20849t/H20850/H20852= exp/H20875−i/H20885 0T U/H20851/H9261−/H20849t/H20850,/H9261+/H20849t/H20850/H20852dt/H20876, where U/H20849/H9261−,/H9261+/H20850is the adiabatic potential. Once the adiabatic potential is computed, the statistics is recovered from ln/H9273/H20849/H9261/H20850=−iTU/H20849/H9261,−/H9261/H20850. Alternatively we can level off the /H9261±functions in Eq. /H208494/H20850to different constants asA. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-2/H9273/H20851/H9261−/H20849t/H20850,/H9261+/H20849t/H20850/H20852→/H9273/H20849/H9261−,/H9261+/H20850, then/H9273/H20849/H9261/H20850=/H9273/H20849/H9261,−/H9261/H20850. Note that the conjugation property /H208493/H20850 now generalizes to /H9273*/H20849/H9261−,/H9261+/H20850=/H9273/H20849/H9261+,/H9261−/H20850 or U*/H20849/H9261−,/H9261+/H20850=−U/H20849/H9261+,/H9261−/H20850. To calculate the adiabatic potential we observe that ac- cording to the nonequilibrium version of the Feynman-Hellmann theorem 31–33 /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=/H20883/H11509T/H9261/H20849t/H20850 /H11509/H9261−/H20884 /H9261, /H208495/H20850 where we use notation /H20855A/H20849t/H20850/H20856/H9261=1 /H9273/H20849/H9261−,/H9261+/H20850/H20883TC/H20877A/H20849t/H20850e−i/H20885 CT/H9261/H20849t/H20850dt/H20878/H20884 /H20849and similarly for multipoint averages /H20850where/H9261’s are under- stood to be different constants on the two time branches.After the calculation they have to be set equal up to theprefactor. This originates from the analysis of a couplingHamiltonian for an ideal passive charge detector without in-ternal dynamics, see Refs. 34 and 29. Note that the above one-point averages depend on the branch the time tis on /H20849though not on the value of ton that branch /H20850: /H20855A/H20849t −/H20850/H20856/H9261/HS11005/H20855A/H20849t+/H20850/H20856/H9261. Therefore the average in Eq. /H208495/H20850must be taken on the for- ward branch of the Keldysh contour. One immediate advan-tage of our Hamiltonian approach is the fact that the calcu-lation of the adiabatic potential Uamounts to a calculation of some well defined Green’s function /H20849GF /H20850, even though a nonequilibrium one. So we can use the whole power of thediagram technique and connect to many known resultswithin this method without being restricted to scattering for-malism as in Refs. 15 and 29. B. FCS of a tunneling junction In order to illustrate the procedure we calculate the FCS of the tunneling junction between two metallic electrodes,denoted by RandL, which we model by the wide flat band Hamiltonians H 0/H20851/H9274R,L/H20852. Their chemical potentials are as- sumed to be/H9262R,L=±V/2, where Vis the voltage applied across the junction /H20849we set e=m=/H6036=kB=1 and the Fermi energy EF=0 throughout /H20850. The coupling between the elec- trodes is supposed to be the conventional pointlike tunnelingwith the amplitude /H9253, so that /H20849for simplicity we assume spin- less electrons /H20850 H=/H20858 i=R,LH0/H20851/H9274i/H20852+/H9253/H20851/H9274R†/H208490/H20850/H9274L/H208490/H20850+ H.c. /H20852. The unperturbed GFs /H20849for/H9253=0/H20850can be easily evaluated, see, e.g., Ref. 35 /H20849i=R,L/H20850,gi−−/H20849/H9275/H20850=gi++/H20849/H9275/H20850=i2/H9266/H92670/H20851ni− 1/2 /H20852, gi−+/H20849/H9275/H20850=i2/H9266/H92670ni, gi+−/H20849/H9275/H20850=−i2/H9266/H92670/H208511−ni/H20852, /H208496/H20850 where/H92670is the density of states in the electrodes in the vicinity of EF. Here nR,L=nF/H20849/H9275±V/2/H20850where nFis the Fermi distribution function. We use the original notation of Keldysh for the GFs, where the superscripts stand for the position ofthe time arguments on the contour Crather than the far more widespread language in terms of retarded /H20849advanced /H20850and thermodynamic components /H20849Refs. 36 and 37 vs Refs. 38 and 39 /H20850. The reason for that is the fact that due to the pres- ence of two different fields /H9261 ±the fundamental relation con- necting the four Keldysh GFs G−−+G++=G−++G+−, does not hold any more. Therefore in the present situation there areindeed four independent GFs. For obvious reasons the T /H9261operator /H208492/H20850is given by15 T/H9261=/H9253/H20851ei/H9261/H9274R†/H208490/H20850/H9274L/H208490/H20850+e−i/H9261/H9274L†/H208490/H20850/H9274R/H208490/H20850/H20852, so that we have to evaluate /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=i/H9253/H20855ei/H9261−/H9274R†/H9274L−e−i/H9261−/H9274L†/H9274R/H20856/H9261. /H208497/H20850 Defining the mixed GFs, GRL/H20849t,t/H11032/H20850=−i/H20855TC/H9274R/H20849t/H20850/H9274L†/H20849t/H11032/H20850/H20856/H9261, GLR/H20849t,t/H11032/H20850=−i/H20855TC/H9274L/H20849t/H20850/H9274R†/H20849t/H11032/H20850/H20856/H9261, we can rewrite Eq. /H208497/H20850as /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850 = lim /H9280→0+/H20885d/H9275 2/H9266ei/H9280/H9275/H20851/H9253*ei/H9261−GLR−−/H20849/H9275/H20850−/H9253e−i/H9261−GRL−−/H20849/H9275/H20850/H20852. /H208498/H20850 The calculation of the GFs GLR/H20849RL/H20850−−/H20849/H9275/H20850is most elegantly ac- complished using functional integration. To that end we in- troduce the matrix of GFs according to gˆ=/H20900gRR−−gRR−+gRL−−gRL−+ gRR+−gRR++gRL+−gRL++ gLR−−gLR−+gLL−−gLL−+ gLR+−gLR++gLL+−gLL++/H20901. Using Eq. /H208496/H20850one easily constructs the corresponding matrix gˆ0without tunnelling. The GFs for /H9253/HS110050 are found from the equation Gˆ−1=gˆ0−1−T/H9018ˆ, /H208499/H20850 whereT/H9018ˆis the self-energy due to tunnelling. It is found to have only four nonzero componentsTOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-3T/H9018ˆ RL−−=/H9253ei/H9261−,T/H9018ˆ RL++=−/H9253ei/H9261+ T/H9018ˆ LR−−=/H9253*e−i/H9261−,T/H9018ˆ LR++=−/H9253*e−i/H9261+. Solving Eq. /H208499/H20850results in Det /H20849gˆ0−1−T/H9018ˆ/H20850GRL−−=−ei/H9261−/H9253 /H20849/H9266/H92670/H208502/H208511+/H9003+2 /H20849nR−nL/H20850/H20852, Det /H20849gˆ0−1−T/H9018ˆ/H20850GLR−− =−e−i/H9261−/H9253* /H20849/H9266/H92670/H208502/H208511+/H9003+2 /H208491−2 ei/H9261¯/H20850/H20849nR−nL/H20850/H20852, where/H9261¯=/H9261−−/H9261+and/H9003=/H20849/H9266/H92670/H9253/H208502is the dimensionless con- tact transparency. The determinant is found to be given by Det /H20849gˆ0−1−T/H9018ˆ/H20850=/H20849/H9266/H92670/H20850−4/H20853/H208491+/H9003/H208502+4/H9003/H20851/H20849ei/H9261¯−1 /H20850nL/H208491−nR/H20850 +/H20849e−i/H9261¯−1 /H20850nR/H208491−nL/H20850/H20852/H20854. Inserting this outcome into Eq. /H208498/H20850, integrating over /H9261−and setting/H9261−=−/H9261+=/H9261we immediately arrive at the Levitov– Lesovik formula ln/H92730/H20849/H9261;V;/H20853T/H20849/H9275/H20850/H20854/H20850=T/H20885d/H9275 2/H9266ln/H208531+T/H20849/H9275/H20850/H20851nL/H208491−nR/H20850/H20849ei/H9261−1 /H20850 +nR/H208491−nL/H20850/H20849e−i/H9261−1 /H20850/H20852/H20854, /H2084910/H20850 where the transmission coefficient is given by T/H20849/H9275/H20850=4/H9003/ /H208491+/H9003/H208502for the particular case of the tunneling junction setup. Equation /H2084910/H20850holds of course, for any noninteracting system, with known transmission coefficient, coupled to twononinteracting reservoirs described by filling factors n R,L. The generating function /H2084910/H20850leads at zero temperature and small voltage to the conventional binomial distributionfunction /H9273/H20849/H9261/H20850=/H208511−T/H208490/H20850+T/H208490/H20850ei/H9261/H20852N, where N=TV/2/H9266=TVe2/his the number of incoming par- ticles during the waiting time /H20849also known as “number of attempts” /H20850and 1− T/H208490/H20850andT/H208490/H20850are their probabilities to be reflected or transmitted, respectively. Generally, the terms proportional to /H20849eim/H9261−1/H20850may be interpreted as describing the tunneling processes of particles with the elementary charge me.28Negative mcorrespond then to transport in direction opposite to that of the applied voltage. Due to the detailedbalance principle, such terms do not contribute at T=0. III. FCS OF THE ANDERSON IMPURITY PROBLEM A. Preliminaries Now we are in a position to proceed to more complicated models. The Hamiltonian of the AIM model consists of threecontributions H=H 0+HT+HC. The kinetic partH0=/H20858 /H9268H0/H20851/H9274R/L,/H9268/H20852+/H20858 /H9268/H20849/H90040+/H9268h/H20850d/H9268†d/H9268, describes a single fermionic level /H20849which we shall also call “dot” /H20850with electron creation operators d/H9268†/H20849/H9268is the spin in- dex /H20850, energy/H90040and subject to a local magnetic field h.T w o noninteracting metallic leads i=R,Lare modeled as in the previous section. The leads and the dot are coupled via tun-neling, H T=/H20858 /H9268/H20851/H9253Lei/H9261/H20849t/H20850/2d/H9268†/H9274L/H9268+/H9253Rd/H9268/H9274R/H9268†+ H.c. /H20852, with different amplitudes /H9253R,L. For convenience we already included the counting field into the Hamiltonian. Notice thatsince the transfer of a physical electron through the device isa two-stage process /H20849left lead →dot→right lead or the other way round /H20850the measuring field is halved. For the sake of simplicity we incorporate the counting field only into theleft junction. Doing that at both junctions /H20849of course with the correction/H9261/2→/H9261/4/H20850leads to exactly the same results due to the gauge symmetry of the Hamiltonian. Finally, we in-clude the Coulomb repulsion on the dot H C=Un↑n↓, where n/H9268=d/H9268†d/H9268. The applied voltage is incorporated into the full Hamiltonian as in the previous section, /H9262L−/H9262R=V/H333560. We start with the definition of two auxiliary GFs F/H9261/H20849t,t/H11032/H20850=−i/H20855TC/H20853/H9274L/H20849t/H20850d†/H20849t/H11032/H20850/H20854/H20856/H9261, F˜/H9261/H20849t,t/H11032/H20850=−i/H20855TC/H20853d/H20849t/H20850/H9274L†/H20849t/H11032/H20850/H20854/H20856/H9261. Hence the derivative of the adiabatic potential is given by /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850 =/H9253L 2lim /H9280→0+/H20885d/H9275 2/H9266ei/H9280/H9275/H20851ei/H9261−/2F/H9261−−/H20849/H9275/H20850−e−i/H9261−/2F˜ /H9261−−/H20849/H9275/H20850/H20852. /H2084911/H20850 Similar to the situation of the tunnelling junction these mixed GFs can be written as combinations of bare lead GFs andexact impurity GF D/H20849t,t/H20850, F˜/H9261/H20849t,t/H11032/H20850=/H20885 Cdt/H11033gL/H20849t−t/H11033/H20850e−i/H9261/H20849t/H11033/H20850D/H20849t/H11033,t/H11032/H20850, F/H9261/H20849t,t/H11032/H20850=/H20885 Cdt/H11033D/H20849t,t/H11032/H20850e−i/H9261/H20849t/H11033/H20850gL/H20849t/H11033−t/H11032/H20850. Performing the Keldysh disentanglement and plugging the result back into Eq. /H2084911/H20850one obtains /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=/H9253L2 2/H20885d/H9275 2/H9266/H20851e−i/H9261¯/2D−+gL+−−ei/H9261¯/2gL−+D+−/H20852, /H2084912/H20850 where again/H9261¯=/H9261−−/H9261+. Hence, the whole problem is now reduced to calculation of the impurity GF. The most compactA. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-4way to access it is using the self-energy formalism. Accord- ing to Ref. 37 the self-energy /H9018ˆ/H20849/H9275/H20850for a nonequilibrium system can be defined in very much the same way as in the traditional diagram technique via Dˆ/H20849/H9275/H20850=dˆ0/H20849/H9275/H20850+Dˆ/H20849/H9275/H20850/H9018ˆ/H20849/H9275/H20850dˆ0/H20849/H9275/H20850, /H2084913/H20850 where the unperturbed dot GF is dˆ 0−1=/H20875/H9275−/H90040 0 0 −/H9275+/H90040/H20876. /H2084914/H20850 Then triviallyDˆ−1/H20849/H9275/H20850=dˆ 0−1/H20849/H9275/H20850−/H9018ˆ/H20849/H9275/H20850. Therefore our goal now is the evaluation of the self-energy. B. The U=0 case: Resonant level model We shall elaborate on the formula /H2084912/H20850, which is still valid for the interacting case, in the following subsection. Forpedagogical reasons we pause here to deal with U=0 case, when His trivially diagonalizable. This situation is referred to as the resonant level /H20849RL /H20850model. The corresponding self-energy is /H20849we neglect the spin in- dex here as GFs are diagonal in /H9268and independent of it, the subscript “0” distinguishes the U=0 quantities /H20850: /H9018ˆ0/H20849/H9275/H20850=/H20875/H9253L2gL−−+/H9253R2gR−−−ei/H9261¯/2/H9253L2gL−+−/H9253R2gR−+ −e−i/H9261¯/2/H9253L2gL+−−/H9253R2gR+−/H9253L2gL+++/H92532gR++/H20876=/H20875i/H9003L/H208492nL−1 /H20850+i/H9003R/H208492nR−1 /H20850 −2iei/H9261¯/2/H9003LnL−2i/H9003RnR 2ie−i/H9261¯/2/H9003L/H208491−nL/H20850+2i/H9003R/H208491−nR/H20850i/H9003L/H208492nL−1 /H20850+i/H9003R/H208492nR−1 /H20850/H20876, where, in order to unburden the notation, we set /H9003R,L=/H20849/H9266/H92670/H9253R,L/H208502. Consequently Dˆ 0−1/H20849/H9275/H20850=/H20875/H9275−/H90040−i/H9003L/H208492nL−1 /H20850−i/H9003R/H208492nR−1 /H20850 2iei/H9261¯/2/H9003LnL+2i/H9003RnR −2ie−i/H9261¯/2/H9003L/H208491−nL/H20850−2i/H9003R/H208491−nR/H20850−/H9275+/H90040−i/H9003L/H208492nL−1 /H20850−i/H9003R/H208492nR−1 /H20850/H20876. /H2084915/H20850 Inversion of this results in Dˆ0/H20849/H9275/H20850=1 D0/H20849/H9275/H20850/H20875/H9275−/H90040+i/H9003L/H208492nL−1 /H20850+i/H9003R/H208492nR−1 /H20850 2iei/H9261¯/2/H9003LnL+2i/H9003RnR −2ie−i/H9261¯/2/H9003L/H208491−nL/H20850−2i/H9003R/H208491−nR/H20850−/H9275+/H90040+i/H9003L/H208492nL−1 /H20850+i/H9003R/H208492nR−1 /H20850/H20876, /H2084916/H20850 where /H20849/H9003=/H9003R+/H9003L/H20850 D0/H20849/H9275/H20850=/H20849/H9275−/H90040/H208502+/H90032+4/H9003L/H9003R/H20851nL/H208491−nR/H20850/H20849ei/H9261¯/2−1 /H20850+nR/H208491−nL/H20850/H20849e−i/H9261¯/2−1 /H20850/H20852. /H2084917/H20850 Inserting these results back into Eq. /H2084912/H20850yields an equation /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=−2/H9003L/H9003R/H20885 −/H11009/H11009d/H9275 2/H9266nL/H208491−nR/H20850ei/H9261¯/2−nR/H208491−nL/H20850e−i/H9261¯/2 D0/H20849/H9275/H20850. /H2084918/H20850 Performing the integration over /H9261−and constructing the generating function we again find the formula /H2084910/H20850with the Breit- Wigner transmission coefficient T/H20849/H9275/H20850=4/H9003L/H9003R /H20849/H9275−/H90040/H208502+/H90032, as expected for the RL setups.40,41 C. The general formula The GFs /H2084916/H20850can be used to construct the consistent expansion of the FCS to all powers inU, opening the road to perturbative as well as nonperturbative studies of the FCS. From now on under /H9018ˆwe shall understand the self-energy due to the Coulomb interaction /H20849tunnelling terms are incorporated into the bare GFs /H20850. Equation /H2084915/H20850thus changes to Dˆ−1/H20849/H9275/H20850=/H20875/H9275−/H90040−i/H9003L/H208492nL−1 /H20850−i/H9003R/H208492nR−1 /H20850−/H9018−−2iei/H9261¯/2/H9003LnL+2i/H9003RnR−/H9018−+ −2ie−i/H9261¯/2/H9003L/H208491−nL/H20850−2i/H9003R/H208491−nR/H20850−/H9018+− −/H9275+/H90040−i/H9003L/H208492nL−1 /H20850−i/H9003R/H208492nR−1 /H20850−/H9018++/H20876. /H2084919/H20850 After the inversion of this matrix and insertion it into Eq. /H2084912/H20850one gets /H20851D/H20849/H9275/H20850is the corresponding counterpart to Eq. /H2084917/H20850/H20852TOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-5/H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=−/H9003L/H20885 −/H11009/H11009d/H9275 2/H9266D/H20849/H9275/H20850/H208532/H9003R/H20851ei/H9261¯/2nL/H208491−nR/H20850 −e−i/H9261¯/2nR/H208491−nL/H20850/H20852−i/H20851ei/H9261¯/2nL/H9018+− +e−i/H9261¯/2/H208491−nL/H20850/H9018−+/H20852/H20854, /H2084920/H20850 which is a general formula for the statistics in interacting systems. Here D/H20849/H9275/H20850is the,/H9261-dependent, determinant of the matrix given by Eq. /H2084919/H20850. For/H9261¯=0 the right-hand side of this relation is proportional to the current through the device.Moreover, as expected, in this particular case Eq. /H2084920/H20850can be brought into the form derived by Meir-Wingreen, 42when the transport is defined solely by the retarded dot level GF aftera symmetrization procedure. The presence of the counting field does not allow a similar reduction for arbitrary /H9261 though. Clearly formula /H2084920/H20850is not restricted to the AIM as such but is applicable for any similar one-channel impuritysetup /H20849including, e.g., electron-phonon interaction on the dot or a double dot /H20850. D. Perturbative expansion in the Coulomb interaction The obvious way to proceed is to calculate the lowest- order contributions to the self-energy. From now on we con-sider a symmetrically coupled system /H9003 R=/H9003L=/H9003/2 at zero temperature in order to simplify the algebra. In the timedomain we immediately obtain /H20849we imply U//H9003as the expansion parameter /H20850 /H9018ˆ/H20849t/H20850=/H20875−iU/H9254/H20849t/H20850D0−−/H208490/H20850+U2/H20851D0−−/H20849t/H20850/H208522D0−−/H20849−t/H20850 −U2/H20851D0−+/H20849t/H20850/H208522D0+−/H20849−t/H20850 −U2/H20851D0+−/H20849t/H20850/H208522D0−+/H20849−t/H20850 iU/H9254/H20849t/H20850D0++/H208490/H20850+U2/H20851D0++/H20849t/H20850/H208522D0++/H20849−t/H20850/H20876. The linear in Upart is diagonal and is essentially a remnant of the occupation probability of the dot level /H20855d†d/H20856. It is most conveniently evaluated in the following way: −iUD0−−/H208490/H20850=−iUD0−+/H208490/H20850=−iU/H20885d/H9275 2/H9266D0−+/H20849/H9275/H20850=Un/H9261, where the object n/H9261=1 2/H9266/H20877/H208491+ei/H9261/H20850/H20875/H9266 2− tan−1/H20873/H90040+V/2 /H9003/H20874/H20876 +ei/H9261/2/H20858 ±± tan−1/H20875/H20849/H90040±V/2/H20850e−i/H9261/2 /H9003 /H20876/H20878, is, in general,/H9261dependent. For D0−+see Eq. /H2084916/H20850. Here n0 simply gives the dot occupation probability. Plugging this result into Eq. /H2084919/H20850and proceeding to Eq. /H2084920/H20850we find the result identical to Eq. /H2084918/H20850up to the denominator Eq. /H2084917/H20850, where the bare level energy /H90040now gets renormalized /H90040 →/H90040+Un/H9261. Subsequent expansion in Uand integration over energy results in a well controlled contribution which van-ishes for the case of the symmetric Anderson model /H9004 0= −U/2, to which case the following considerations are re- stricted. We concentrate now on the correction at the second order inU. One way to access the self-energies is through the evaluation of the corresponding susceptibilities. We definethem as /H20851note the subscript “0”, not to confuse with the gen- erating function /H9273/H20849/H9261/H20850/H20852/H9273ˆ0/H20849/H9024/H20850 =i/H20885 −/H11009/H11009d/H9275 2/H9266/H20875D0−−/H20849/H9275+/H9024/H20850D0−−/H20849/H9275/H20850D0−+/H20849/H9275+/H9024/H20850D0+−/H20849/H9275/H20850 D0+−/H20849/H9275+/H9024/H20850D0−+/H20849/H9275/H20850D0++/H20849/H9275+/H9024/H20850D0++/H20849/H9275/H20850/H20876. The respective self-energy can be extracted from /H9018ˆ/H20849/H9275/H20850=i/H20885 −/H11009/H11009d/H9024 2/H9266/H20875D0−−/H20849/H9275−/H9024/H20850/H92730−−/H20849/H9024/H20850D0−+/H20849/H9275−/H9024/H20850/H92730+−/H20849/H9024/H20850 D0+−/H20849/H9275−/H9024/H20850/H92730−+/H20849/H9024/H20850D0++/H20849/H9275−/H9024/H20850/H92730++/H20849/H9024/H20850/H20876. The equilibrium results have been originally presented in famous series of papers by Yosida-Yamada43–45/H20849we set T =0 for simplicity /H20850, /H9018ˆeq/H20849/H9275/H20850=/H208491−/H9273e/H20850/H9275/H2087510 0− 1/H20876−i/H9273o2 2/H9003/H92752/H20875sign /H20849/H9275/H208502/H9258/H20849−/H9275/H20850 −2/H9258/H20849/H9275/H20850sign /H20849/H9275/H20850/H20876, where the exact even-odd susceptibilities possess the follow- ing expansions in powers of U: /H9273e=1+/H208733−/H92662 4/H20874U2 /H92662/H90032+¯,/H9273o=−U /H9266/H9003. For finite Vand at the second order in U, there are three distinct energy regions contributing to Eq. /H2084920/H20850:−V/2/H11021/H9275 /H11021V/2,V/2/H11021/H9275/H110213V/2, and −3 V/2/H11021/H9275/H11021−V/2. The low- energy expansion /H20849not only small Ubut small Vas well /H20850in presence of/H9261one finds in the region − V/2/H11021/H9275/H11021V/2:A. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-6/H9018ˆ/H20849/H9275/H20850=/H208491−/H9273e/H20850/H9275/H2087510 0− 1/H20876−iU2 8/H92662/H90033/H209006/H9275V e−i/H9261/H208733V 2−/H9275/H208742 +3/H20873V 2−/H9275/H208742 −e−2i/H9261/H208733V 2+/H9275/H208742 −3e−i/H9261/H20873V 2+/H9275/H208742 6/H9275V /H20901. Needless to say, these relations are consistent with the non- equilibrium calculation by Oguri.46On the other hand, for /H9275/H11022V/2 one obtains /H9018−+/H20849/H9275/H20850=−ie−i/H9261U2 8/H92662/H90033/H208733V 2−/H9275/H208742 /H9258/H208733V 2−/H9275/H20874, while for/H9275/H11021V/2 the relation /H9018+−/H20849/H9275/H20850=ie−2i/H9261U2 8/H92662/H90033/H208733V 2+/H9275/H208742 /H9258/H208733V 2+/H9275/H20874 holds. These self-energies, being incorporated into Eq. /H2084920/H20850, yield the following generating function for the FCS: ln/H9273/H20849/H9261/H20850=l n/H92730/H20849/H9261/H20850+TU2V3 24/H92663/H90034/H20849e−i/H9261−1 /H20850 +TU2V3 12/H92663/H90034/H20849e−2i/H9261−1 /H20850+O/H20849U4/H20850, where ln/H92730/H20849/H9261/H20850=T/H20885 −V/2V/2d/H9275 2/H9266ln/H208751+/H90032 /H9273e2/H92752+/H90032/H20849ei/H9261−1 /H20850/H20876 still contains U. Performing the expansion around the perfect transmission /H20849hence the sign change of /H9261in the following formulas /H20850we see that in terms of susceptibilities ln/H9273/H20849/H9261/H20850=N/H20877i/H9261+V2 3/H90032/H20875/H9273e2+/H9273o2 4/H20849e−i/H9261−1 /H20850+/H9273o2 2/H20849e−2i/H9261−1 /H20850/H20876/H20878, /H2084921/H20850 where N=TV//H9266is the number of incoming particles during the measuring time slice. This is, of course, only valid at theorder U 2. We speculate that the general formula for the full FCS could be written in terms of the equilibrium suscepti-bilities /H9273o,eonly. One possibility is the generating function of the form ln/H9273/H20849/H9261/H20850=Nln/H208751+/H208731−/H9273e2+3/H9273o2 12/H90032V2/H20874/H20849ei/H9261−1 /H20850 +/H9273o2 6/H90032V2/H20849e−i/H9261−1 /H20850/H20876, /H2084922/H20850 as this expression reproduces the expansion Eq. /H2084921/H20850.W e stress again that so far we have only shown that Eq. /H2084922/H20850 holds at the second order in Uand beyond that it is a mere hypothesis. It is tempting to interpret the appearance of the double exponential terms as an indication of a coherent tunneling ofelectron pairs /H20849caution: similar terms would also appear for the noninteracting RL model due to the energy dependenceof the transmission coefficient /H20850. In the Toulouse limit calcu- lation below we find further evidence for such interpretation. E. Linear response FCS Here we would like to take a closer look onto the general formula /H2084920/H20850at zero temperature and vanishing applied volt- age. In order to arrive at correct results one has to bear inmind that the limits V→0 and /H9275→0 do not commute in the presence of the counting field. Indeed, calculating theKeldysh determinant in both limits we see that lim /H9275→0lim V→0D0/H20849/H9275,V,/H9261/H20850=/H900402+/H90032/H2084923/H20850 but lim V→0lim /H9275→0D0/H20849/H9275,V,/H9261/H20850=/H900402+/H90032+4/H9003L/H9003R/H20849ei/H9261−1 /H20850. /H2084924/H20850 In fact, it is the second scheme we have to implement analysing the first term in Eq. /H2084920/H20850. This leads to a transmis- sion coefficient type contribution to the generating function. On the contrary, in the second term in Eq. /H2084920/H20850, which is produced by the self-energy, not even the integration over /H9275 is restricted to /H208510,V/H20852. As a matter of fact, due to Auger type effects47,48one expects that there are contributions to the current /H20849and FCS /H20850at all energies. This effect is itself propor- tional to the applied voltage though, and results therefore innonlinear corrections to the FCS. Hence the energy integra-tion can be regarded to be restricted to /H208510,V/H20852even in the second term in Eq. /H2084920/H20850. Moreover, since the self-energy does not have external lines and all the internal frequencieshave to be integrated over, the limits V→0 and /H9275→0 in this case commute. That means that for the evaluation of theself-energy to the lowest order in Vone is allowed to use the equilibrium GFs, calculated in presence of the counting field/H9261, i.e., Eq. /H2084916/H20850with n R=nL=nFand with the corresponding Keldysh denominator /H2084923/H20850. Therefore all diagonal Keldysh GFs are equal to those in the equilibrium and all off-diagonalones are simply proportional to the same diagrams as inequilibrium. Since any given off-diagonal self-energy dia-gram describes an inelastic process, it should vanish for /H9275 →0 and we arrive at a conclusion that lim /H9275→0/H9018ˆ/H20849/H9275/H20850=R e/H9018R/H208490/H20850/H2087510 0− 1/H20876 even at finite/H9261. Equation /H2084920/H20850thus leads to the fundamental resultTOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-7ln/H9273/H20849/H9261/H20850=Nln/H208771+/H90032 /H20851Re/H9018/H20849R/H20850/H208490/H20850/H208522+/H90032/H20849ei/H9261−1 /H20850/H20878,/H2084925/H20850 or to ln/H9273/H20849/H9261/H20850=i/H9261Nfor the symmetric Anderson impurity model. In case of the asymmetrically coupled impurity, /H9003R /HS11005/H9003Lthe numerator of Eq. /H2084925/H20850modifies to/H9003R/H9003Lwhile the denominator contains /H20849/H9003R+/H9003L/H20850/2 instead of/H9003. The result /H2084925/H20850allows simple generalizations to asymmet- ric systems in a magnetic field h. According to Refs. 43–45 the real part of the self-energy is given by Re/H9018/H9268/H20849R/H20850/H208490/H20850=/H9273c/H9260+/H9268/H9273sh, where/H9273c/sare exact charge/spin susceptibilities /H20849combina- tions of even/odd /H20850and/H9260/H11011/H90040+U/2 is a particle-hole sym- metry breaking field. Consequently ln/H9273/H20849/H9261/H20850=N 2ln/H20877/H208751+/H90032 /H20851/H9273c/H9260+/H9273sh/H208522+/H90032/H20849ei/H9261−1 /H20850/H20876/H20876/H20877 /H11003/H208751+/H90032 /H20851/H9273c/H9260−/H9273sh/H208522+/H90032/H20849ei/H9261−1 /H20850/H20876/H20878. /H2084926/H20850 The enormous advantage of this formula is the fact, that the susceptibilities can be calculated exactly for any system pa-rameters with the help of the Bethe-ansatz results. 3–5 Let us stress that the result /H2084925/H20850is not limited to the AIM but will hold for any similar model, hence the binomial theo-rem. It is clear in hindsight that all the nonelastic processesfall out in the T=0 linear response limit. Still it is a remark- able result that allmoments have a simple expression in terms of a single number: the effective transmission coeffi-cient. The binomial distribution is universal. /H20851For a multi- channel system modifications will be required as is obviousfrom looking at Eq. /H2084926/H20850/H20852. F. The Kondo regime The way to proceed further is to consider the case of very deep/H90040and strong Coulomb repulsion. In this limiting case the system is in the Kondo regime and the dot can in goodapproximation be considered to be permanently populated bya single electron. It has been shown in Ref. 49, that theconventional Schrieffer-Wolf transformation, 50which maps the Anderson impurity Hamiltonian onto that of the Kondoproblem, also works out of equilibrium. The result is thetwo-channel Kondo Hamiltonian H=H 0+HJ+HV+HM, where, with/H9274/H9251,/H9268are the electron field operators in the /H9251 =R,Lelectrodes, H0=i/H20858 /H9251=R,L/H20858 /H9268=↑,↓/H20885dx/H9274/H9251/H9268†/H20849x/H20850/H11509x/H9274/H9251/H9268/H20849x/H20850, HJ=/H20858 /H9251,/H9252=R,L/H20858 /H9263=x,y,zJ/H9263/H9251/H9252s/H9251/H9252/H9263/H9270/H9263, HV=/H20849V/2/H20850/H20858 /H9268/H20885dx/H20849/H9274L/H9268†/H9274L/H9268−/H9274R/H9268†/H9274R/H9268/H20850,HM=−/H9262Bgih/H9270z=−/H9004/H9270z. /H2084927/H20850 /H9262Bis the Bohr’s magneton, githe gyromagnetic ratio, and h denotes the local magnetic field, which is applied to the im-purity spin. Here /H9270/H9263=x,y,zare the Pauli matrices for the impu- rity spin and s/H9251/H9252/H9263=/H20858 /H9268,/H9268/H11032/H9274/H9251/H9268†/H208490/H20850/H9268/H9268/H9268/H11032/H9263/H9274/H9252/H9268/H11032/H208490/H20850, are the components of the electron spin densities in /H20849or across /H20850the leads, biased by a finite voltage V. The last term in Eq. /H2084927/H20850stands for the magnetic field /H9004=/H9262Bgih. We fol- low Ref. 8 and assume Jx/H9251/H9252=Jy/H9251/H9252=J/H11036/H9251/H9252,Jz±=/H20849JzLL±JzRR/H20850/2, and JzLR=JzRL=0. The only transport process then allowed is the spin-flip tunnelling /H20849sometimes also called “exchange cotun- nelling” /H20850, so that we obtain for the T/H9261operator T/H9261=J/H11036RL 2/H20849/H9270+ei/H9261/H20849t/H20850/2/H9274R↓†/H9274L↑+/H9270−ei/H9261/H20849t/H20850/2/H9274R↑†/H9274L↓ +/H9270+e−i/H9261/H20849t/H20850/2/H9274L↓+/H9274R↑+/H9270−e−i/H9261/H20849t/H20850/2/H9274L↑†/H9274R↓/H20850. Of course, there is also a regular elastic co-tunneling term, which couples the leads directly. However, it can be rigor-ously shown, 49that these processes are subleading in the low energy sector in comparison to spin-flip tunneling. That iswhy we keep only the latter contributions to the Hamil-tonian. We proceed by bosonization, Emery-Kivelson rota-tion, and refermionization. 8,20,21We obtain then with J± =/H20849J/H11036LL±J/H11036RR/H20850//H208812/H9266a0,J/H11036=J/H11036RL//H208812/H9266a0/H20849a0is the lattice con- stant of the underlying lattice model /H20850 H=i/H20858 /H9263=c,s,cf,sf/H20885dx/H9274/H9263†/H20849x/H20850/H11509x/H9274/H9263/H20849x/H20850+J+ 2/H20851/H9274sf†/H208490/H20850+/H9274sf/H208490/H20850/H20852 /H11003/H20849−i2/H9270x/H20850+J− 2/H20851/H9274sf†/H208490/H20850−/H9274sf/H208490/H20850/H20852/H20849−2/H9270y/H20850 −/H20851/H20849Jz+−2/H9266/H20850:/H9274s†/H208490/H20850/H9274s/H208490/H20850:+Jz−:/H9274sf†/H208490/H20850/H9274sf/H208490/H20850:/H20852/H9270z −/H9004/H9270z+V/H20885dx/H9274cf†/H20849x/H20850/H9274cf/H20849x/H20850, /H2084928/H20850 where now four fermionic channels are present: /H20849c/H20850total charge density channel for the sum of particle densities in both electrodes, /H20849cf/H20850charge flavor channel for the difference in densities. The channel-symmetric spin density channel /H20849s/H20850 and channel-antisymmetric /H20849or spin flavor channel /H20850/H20849sf/H20850/H20849see details in Refs. 8 and 20 /H20850are defined in analogy to their charge counterparts. A considerable simplification of thetheory is achieved by introduction of the Majorana compo-nents of the continuum fields /H9257/H9263=/H20849/H9274/H9263†+/H9274/H9263/H20850//H208812,/H9264/H9263=i/H20849/H9274/H9263†−/H9274/H9263/H20850//H208812, /H2084929/H20850 and of the impurity spin /H9270x=−a//H208812 and/H9270y=b//H208812. As a re- sult, the model simplifies to /H20849for convenience we kept /H9274s,sf operators in the terms quartic in fermions /H20850 H=H/H11032+H/H11033, /H2084930/H20850 H/H11032=H0/H11032−i/H20849J−b/H9264sf+J+a/H9257sf/H20850−i/H9004ab+T/H9261,A. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-8H/H11033=i/H20851v1:/H9274s†/H208490/H20850/H9274s/H208490/H20850:+Jz−:/H9274sf†/H208490/H20850/H9274sf/H208490/H20850:/H20852ab, where v1=Jz+−2/H9266and the counting term is given by T/H9261=−iJ/H11036b/H20851/H9264cfcos /H20849/H9261/2/H20850−/H9257cfsin/H20849/H9261/2/H20850/H20852. /H2084931/H20850 The fields/H9257sfand/H9264sfin the spin-flavor sector are equilibrium Majorana fields, whereas /H9257cfand/H9264cfin the charge-flavor sector are biased by V/H20849from now on we omit the CFindex and denote SFbyf/H20850, H0/H11032=i/H20885dx/H20851/H9257f/H20849x/H20850/H11509x/H9257f/H20849x/H20850+/H9264f/H20849x/H20850/H11509x/H9264f/H20849x/H20850+/H9257/H20849x/H20850/H11509x/H9257/H20849x/H20850 +/H9264/H20849x/H20850/H11509x/H9264/H20849x/H20850+V/H9264/H20849x/H20850/H9257/H20849x/H20850/H20852, where we drop the candschannels as they decouple from the impurity completely /H20849at the Toulouse point /H20850. The evalua- tion of the adiabatic potential can now be performed alongthe lines of Sec. II, /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=−J/H11036 2/H20885d/H9275 2/H9266/H20851sin/H20849/H9261−/2/H20850Gb/H9264−−/H20849/H9275/H20850 + cos /H20849/H9261−/2/H20850Gb/H9257−−/H20849/H9275/H20850/H20852, /H2084932/H20850 where we again define the mixed GFs according to the pre- scription Gb/H9264/H20849t,t/H11032/H20850=−i/H20855TCb/H20849t/H20850/H9264/H20849t/H20850/H20856/H9261,Gb/H9257/H20849t,t/H11032/H20850=−i/H20855TCb/H20849t/H20850/H9257/H20849t/H20850/H20856/H9261. /H2084933/H20850 1. The FCS at the Toulouse point For realistic systems it is reasonable to assume Jz−=0. The only remaining term which is still quartic in fermionicfields is then zero at the so-called Toulouse pointJ z+=2/H9266.19,20In this situation the Hamiltonian is quadratic in fermionic fields. The mixed GFs /H2084933/H20850are related to the exact impurity GFs Dbb/H20849t,t/H11032/H20850=−i/H20855TCb/H20849t/H20850b/H20849t/H11032/H20850/H20856/H9261and to bare GFs /H20849calculated for all Ji=0/H20850for the Majorana fields24/H20851notice that in the present situation we have to double the applied voltagein comparison to Eq. /H208496/H20850, for details see Ref. 24 /H20852, g /H9264/H9264=g/H9257/H9257=i 2/H20875nR+nL−1 nR+nL nR+nL−2 nR+nL−1/H20876, g/H9257/H9264=nL−nR 2/H2087511 11/H20876, /H2084934/H20850 in the following way: Gb/H9264/H20849t,t/H11032/H20850=iJ/H11036/H20885 Cdt/H11033Dbb/H20849t,t/H11033/H20850/H20853− cos /H20851/H9261/H20849t/H11033/H20850/2/H20852g/H9264/H9264/H20849t/H11033−t/H11032/H20850 + sin /H20851/H9261/H20849t/H11033/H20850/2/H20852g/H9257/H9264/H20849t/H11033−t/H11032/H20850/H20854, Gb/H9257/H20849t,t/H11032/H20850=iJ/H11036/H20885 Cdt/H11033Dbb/H20849t,t/H11033/H20850/H20853− cos /H20851/H9261/H20849t/H11033/H20850/2/H20852g/H9264/H9257/H20849t/H11033−t/H11032/H20850 + sin /H20851/H9261/H20849t/H11033/H20850/2/H20852g/H9257/H9257/H20849t/H11033−t/H11032/H20850/H20854. After the Keldysh disentanglement and using Eq. /H2084934/H20850 /H11509 /H11509/H9261−U/H20849/H9261−,/H9261+/H20850=iJ/H110362 4/H20885d/H9275 2/H9266/H20853Dbb−−/H20849/H9275/H20850/H20849nR−nL/H20850+Dbb−+/H20849/H9275/H20850 /H11003/H20851ei/H9261¯/2/H208491−nR/H20850−e−i/H9261¯/2/H208491−nL/H20850/H20852/H20854. /H2084935/H20850 Evaluation of the impurity GF is accomplished by the calcu- lation of the corresponding self-energy and inversion of the emerging matrix dˆ 0−1−/H9018ˆKwhere in the absence of the mag- netic field/H9004=0 and J+=0 /H20849we discuss the general case later /H20850 /H9018ˆK=/H20875J−2g/H9264/H9264/H208490/H20850−−+J/H110362g/H9264/H9264−−−J−2g/H9264/H9264/H208490/H20850−+−J/H110362/H20851cg/H9264/H9264−+−sg/H9257/H9264−+/H20852 −J−2g/H9264/H9264/H208490/H20850+−−J/H110362/H20851cg/H9264/H9264+−+sg/H9257/H9264+−/H20852 J−2g/H9264/H9264/H208490/H20850+++J/H110362g/H9264/H9264++/H20876, /H2084936/H20850 where the superscript /H208490/H20850distinguishes the equilibrium GFs for V=0 and c=cos /H20851/H20849/H9261−−/H9261+/H20850/2/H20852,s=sin /H20851/H20849/H9261−−/H9261+/H20850/2/H20852anddˆ−1is given by Ref /H2084914/H20850with/H90040=0. Using Eq. /H2084934/H20850and new definitions /H9003i=Ji2/2 we obtain /H9018ˆK=i/H20875/H9003−/H208492nF−1 /H20850+/H9003/H11036/H20849nR+nL−1 /H20850 −/H9003−2nF−/H9003/H11036/H20849ei/H9261¯/2nL+e−i/H9261¯/2nR/H20850 /H9003−2/H208491−nF/H20850+/H9003/H11036/H20851ei/H9261¯/2/H208491−nR/H20850+e−i/H9261¯/2/H208491−nL/H20850/H20852/H9003−/H208492nF−1 /H20850+/H9003/H11036/H20849nR+nL−1 /H20850/H20876. Then the determinant − Det /H20849dˆ 0−1−/H9018ˆK/H20850=/H92752+/H20849/H9003/H11036+/H9003−/H208502+/H9003/H110362/H20851nL/H208491−nR/H20850/H20849ei/H9261¯−1 /H20850+nR/H208491−nL/H20850/H20849e−i/H9261¯−1 /H20850/H20852 +2/H9003−/H9003/H11036/H20853/H20851nF/H208491−nR/H20850+nL/H208491−nF/H20850/H20852/H20849ei/H9261¯/2−1 /H20850+/H20851nF/H208491−nL/H20850+nR/H208491−nF/H20850/H20852/H20849e−i/H9261¯/2−1 /H20850/H20854. /H2084937/H20850 The GFs of interest are then given by Det /H20849dˆ 0−1−/H9018ˆK/H20850Dˆbb=/H20875−/H9275+i/H20851/H9003−/H208492nF−1 /H20850+/H9003/H11036/H20849nR+nL−1 /H20850/H20852 i/H20851/H9003−2nF+/H9003/H11036/H20849ei/H9261¯/2nL+e−i/H9261¯/2nR/H20850/H20852 −i/H20853/H9003−2/H208491−nF/H20850+/H9003/H11036/H20851ei/H9261¯/2/H208491−nR/H20850+e−i/H9261¯/2/H208491−nL/H20850/H20852/H20854/H9275+i/H20851/H9003−/H208492nF−1 /H20850+/H9003/H11036/H20849nR+nL−1 /H20850/H20852/H20876.TOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-9Inserting the calculated GFs into the fundamental relation /H2084935/H20850results in /H11509 /H11509/H9261−U/H20849/H9261/H11007/H20850=−i2/H9003/H11036/H20885d/H9275 2/H9266I/H20849/H9275/H20850 Det /H20849dˆ 0−1−/H9018ˆK/H20850/H2084938/H20850 with I/H20849/H9275/H20850=/H9003/H110362/H20851nL/H208491−nR/H20850ei/H9261¯−nR/H208491−nL/H20850e−i/H9261¯/H20852 +2/H9003−/H9003/H11036/H20851nF/H208491−nR/H20850ei/H9261¯/2−nF/H208491−nL/H20850e−i/H9261¯/2/H20852. To proceed, we split the /H9275integral in Eq. /H2084938/H20850into two parts for negative and positive energies and change /H9275→−/H9275in the second integral. In doing so observe that under this transfor-mation n F→1−nF,nR→1−nL, and nL→1−nR. Therefore the denominator stays invariant while the numerator changesas I/H20849− /H9275/H20850=/H9003/H110362/H20851nL/H208491−nR/H20850ei/H9261¯−nR/H208491−nL/H20850e−i/H9261¯/H20852 +2/H9003−/H9003/H11036/H20851nL/H208491−nF/H20850ei/H9261¯/2−nR/H208491−nF/H20850e−i/H9261¯/2/H20852. Equation /H2084938/H20850thus becomes /H11509 /H11509/H9261−U/H20849/H9261/H11007/H20850=−1 2/H20885 0/H11009d/H9275 2/H9266I/H20849/H9275/H20850+I/H20849−/H9275/H20850 Det /H20849dˆ 0−1−/H9018ˆK/H20850. Observe that, crucially,/H11509K /H11509/H9261−=i 2/H20851I/H20849/H9275/H20850+I/H20849−/H9275/H20850/H20852so that the/H9261in- tegration can be performed as before. The following exact formula for the statistics, valid at finite temperatures, followsimmediately: ln /H9273/H20849/H9261/H20850=T/H20885 0/H11009d/H9275 2/H9266ln/H208531+T2/H20849/H9275/H20850 /H11003/H20851nL/H208491−nR/H20850/H20849e2i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849e−2i/H9261−1 /H20850/H20852 +T1/H20849/H9275/H20850/H20851/H20851nF/H208491−nR/H20850+nL/H208491−nF/H20850/H20852/H20849ei/H9261−1 /H20850 +/H20851nF/H208491−nL/H20850+nR/H208491−nF/H20850/H20852/H20849e−i/H9261−1 /H20850/H20852/H20854, /H2084939/H20850 where the effective “transmission coefficients” /H20849two of them now /H20850are T2/H20849/H9275/H20850=/H9003/H110362 /H92752+/H20849/H9003−+/H9003/H11036/H208502,T1/H20849/H9275/H20850=2/H9003−/H9003/H11036 /H92752+/H20849/H9003−+/H9003/H11036/H208502. /H2084940/H20850 In the more general case of finite magnetic field /H9004and/H9003+the result /H2084939/H20850is exactly the same up to the modified transmis- sion coefficients /H20849derived in Appendix A /H20850, T2=/H9003/H110362/H20849/H92752+/H9003+2/H20850 /H20851/H92752−/H90042−/H9003+/H20849/H9003/H11036+/H9003−/H20850/H208522+/H92752/H20849/H9003++/H9003−+/H9003/H11036/H208502, T1=2/H9003/H11036/H9003−/H20849/H92752+/H9003+2/H20850+2/H90042/H9003/H11036/H9003+ /H20851/H92752−/H90042−/H9003+/H20849/H9003/H11036+/H9003−/H20850/H208522+/H92752/H20849/H9003++/H9003−+/H9003/H11036/H208502. /H2084941/H20850 In fact, since the refermionized Hamiltonian describes local scattering of noninteracting /H20849Majorana /H20850particles, the result /H2084939/H20850can as well be derived using the approach originallyconceived by Levitov and Lesovik for systems with known scattering matrix.28For the corresponding calculation see Appendix B. Using the properties nF/H208491−nR/H20850+nL/H208491−nF/H20850 =nL/H208491−nR/H20850/H208491+exp /H20851−V/T/H20852/H20850 and nF/H208491−nL/H20850+nR/H208491−nF/H20850 =nR/H208491−nL/H20850/H208491+exp /H20851V/T/H20852/H20850we can rewrite the result in the form ln/H9273/H20849/H9261/H20850=T/H20885 0/H11009d/H9275 2/H9266ln/H208531+nL/H208491−nR/H20850/H20851T2/H20849/H9275/H20850/H20849e2i/H9261−1 /H20850+T1/H20849/H9275/H20850 /H11003/H20849ei/H9261−1 /H20850/H208491+e−V/T/H20850/H20852+nR/H208491−nL/H20850/H20851T2/H20849/H9275/H20850/H20849e−2i/H9261−1 /H20850 +T1/H20849/H9275/H20850/H20849e−i/H9261−1 /H20850/H208491+eV/T/H20850/H20852/H20854. /H2084942/H20850 We first take a look onto the T=0 situation, when exp /H20849−/H20841V/H20841/T/H20850→0. In that case one can reduce the generating function to the Levitov-Lesovik formula /H2084910/H20850for a spinful system51 /H9273/H20849/H9261/H20850=/H208511+Te/H20849ei/H9261−1 /H20850/H20852N, /H2084943/H20850 where Te=/H20881T2/H208490/H20850=/H9003/H11036//H20849/H9003/H11036+/H9003−/H20850. Hence in the low tempera- ture limit we obtain the conventional binomial statistics for the charge transfer through the dot. Needless to say this is inaccordance with the binomial theorem stated in the previoussection. However, the reduction /H2084943/H20850is not possible for finite temperatures and voltages. To the best of our knowledge Eq./H2084939/H20850is the first exact result showing nontrivial statistics at finite energy scales. It can be interpreted in terms of twodistinct tunneling processes: /H20849i/H20850tunneling of single electrons and /H20849ii/H20850tunneling of electron pairs with opposite spins. As has already been realized in Ref. 8, at least in the regimeT,V/H11270/H9004 tunneling of single electrons is energetically very costly as it requires a spinflip. A simultaneous tunneling oftwo electrons, which is described by the terms with 2 /H9261and T 2/H20849/H9275/H20850, leaves the dot spin effectively untouched, making that kind of process the dominant transport channel. In zero field the finite voltage is known to act as effective magnetic field8 so that this tunneling mechanism is always present regardlessof the precise value of /H9004. In the low energy sector /H9275/H11270V,Tthe integral of Eq. /H2084942/H20850 can be performed explicitly in the spirit of Ref. 29, resultingin /H9273/H20849/H9261/H20850= exp/H20875TT 2h/H20849u2−v2/H20850/H20876, where v=V/Tand cosh /H20849u/H20850− cosh /H20849v/H20850=T1/H20851cos/H9261− 1 + cosh /H20849v+i/H9261/H20850− cosh v/H20852 +T2/H20851cosh /H20849v+i2/H9261/H20850− cosh v/H20852. In the limiting case V/H11271Twe recover the result /H2084943/H20850while for V/H11270Twe obtain /H9273/H20849/H9261/H20850= exp/H20873−TT 2h/H9261*2/H20874, where sin2/H20849/H9261*/2/H20850=4Tesin2/H20849/H9261/2/H20850/H208511−Tesin2/H20849/H9261/2/H20850/H20852. The full transport coefficient T0as calculated in Ref. 8 turns out to be acomposite one and it is recovered from T1,2through a very simple relation T0=T2+T1/2. We have evaluated the first andA. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-10the second cumulant of the Kondo FCS Eq. /H2084939/H20850which are the same as calculated by SH at all VandT.8We shall not reproduce these two cumulants here and concentrate insteadon new results. First we would like to analyze the equilibrium statistics at V=0. From Eq. /H2084939/H20850it is obvious that as n L=nRall odd order cumulants are identically zero. Then for the even order cu-mulants we obtain /H20855 /H9254q2/H20856=T/H20885 0/H11009d/H9275 2/H9266nF/H208491−nF/H208504/H20849T1+2T2/H20850, /H20855/H9254q4/H20856=T/H20885 0/H11009d/H9275 2/H9266/H208514/H20849T1+8T2/H20850nF/H208491−nF/H20850 −4 8 /H20849T1+2T2/H208502nF2/H208491−nF/H208502/H20852, /H20855/H9254q6/H20856=T/H20885 0/H11009d/H9275 2/H9266/H208514/H20849T1+3 2T2/H20850nF/H208491−nF/H20850 + 1920 /H20849T1+2T2/H208503nF3/H208491−nF/H208503 − 240 /H20849T1+2T2/H20850/H20849T1+8T2/H20850nF2/H208491−nF/H208502/H20852. As for finite f/H208490/H20850one obtains /H208480/H11009d/H9275f/H20849/H9275/H20850nFn/H208491−nF/H20850n /H11015anTf/H208490/H20850with a1=1/2, a2=1/12, a3=1/60, a4=1/280, etc., allequilibrium cumulants are linear in temperature in the low energy sector. The lowest order cumulant is then theconventional thermal Johnson-Nyquist noise S JN/H110154G0T0T, where G0is the conductance quantum and T0is the transmis- sion coefficient of the dot at /H9275=0. In the opposite limit of finite voltage and T=0 we obtain for the third cumulant /H20855/H9254q3/H20856=T/H20885 0Vd/H9275 2/H9266/H20851T1+8T2−3 /H20849T1+2T2/H20850/H20849T1+4T2/H20850 +2 /H20849T1+2T2/H208503/H20852. This simplifies further in zero field /H20855/H9254q3/H20856 =T 2/H9266/H208772/H9003/H11036tan−1/H20851V//H20849/H9003/H11036+/H9003−/H20850/H20852−2V/H9003/H110362 /H20851/H20849/H9003/H11036+/H9003−/H208502+V2/H208522 /H11003/H20851/H20849/H9003/H11036+/H9003−/H208502+2/H9003−/H20849/H9003/H11036+/H9003−/H20850+3V2/H20852/H20878, possessing the following limiting forms: /H20855/H9254q3/H20856V→0/H11015TG02/H9003/H11036/H9003−/H20849/H9003−−/H9003/H11036/H20850 /H20849/H9003/H11036+/H9003−/H208503V, /H20855/H9254q3/H20856V→/H11009/H11015T/H9266G0/H9003/H11036. /H2084944/H20850 At low voltages the cumulant is negative for /H9003−/H11021/H9003/H11036. Gen- erally, under these conditions the n-th cumulant appears to possess n−2 zeroes as a function of V, according to numer- ics. The saturation value in the limit V→/H11009is independent of the coupling in the spin-flavor channel because the fluctua-tions in the biased conducting charge-flavor channel aremuch more pronounced than those in the spin-flavor channel, which experiences only relatively weak equilibrium fluctua-tions. For the general situation of arbitrary parameters, the cu- mulants can be calculated numerically. The asymptotic valueof the third cumulant at high voltages, similarly to the find-ings of Ref. 26, does not depend on temperature and is givenby the result /H2084944/H20850, see Fig. 1 of Ref. 22. In the opposite limit of small V,/H20855 /H9254q3/H20856can be negative. Sufficiently large coupling /H9003−or magnetic field, see Fig. 1, suppress this effect though. According to the result of Ref. 15, as long as the distri- bution is binomial /H20855/H9254q3/H20856//H20855/H9254q/H20856=/H20849e*/H208502, where e*is the effec- tive charge of the current carriers. This quantity is to be preferred to the Schottky formula because of its weak tem-perature dependence. Indeed we find numerically that theratio /H20855 /H9254q3/H20856//H20855/H9254q/H20856in the present problem is weakly tempera- ture dependent /H20849it is flat and levels off to 1 /H20850in comparison to /H20855/H9254q2/H20856//H20855/H9254q/H20856. 2. Corrections around the Toulouse point Thus far we dealt with a system which finds itself at one special point in the parameter space, when v1=0 and Jz−=0. While the latter requirement is reasonable for realistic sys-tems, the former is quite artificial. It has been shown bymeans of RG transformation procedure that at least in equi-librium the operators, describing deviations from the Tou-louse point are irrelevant in the RG sense and do not influ-ence the physics in the low energy sector strongly. There is,however, no a priori reason why that should hold in a non- equilibrium situation. Therefore the full analysis of the FCSmust incorporate the investigation of the statistics beyond theToulouse restrictions. We first concentrate on the situation offinite v1. As was pointed out above, an analytic solution in this situation is not possible. The only option to progress isperturbation theory in v1. To access the generating function we still can use the fundamental relation /H2084935/H20850. As we have the complete knowl- FIG. 1. Zero temperature voltage dependence of the third cumu- lant for different magnetic field values and /H9003±//H9003/H11036=0.1. Inset: tem- perature evolution of the curve for /H9004//H9003/H11036=1.5 for T//H9003/H11036=0, 0.2, and 1.5 /H20849from bottom to top /H20850.TOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-11edge of all GFs with respect to H/H11032, see Ref. /H2084930/H20850, the simplest thing we can do is to calculate perturbative corrections toD bbin the second order in v1. They are given by D¯ˆ bb=Dˆbb+v12Dˆbb/H9018ˆbDˆbb=Dˆbb+/H9254Dˆbb. The correction to the adiabatic potential is then given by /H9254/H20873/H11509U /H11509/H9261−/H20874=i/H9003/H11036 2/H20885d/H9275 2/H9266/H20853/H9254Dbb−−/H20849nR−nL/H20850 +/H9254Dbb−+/H20851ei/H9261¯/2/H208491−nR/H20850−e−i/H9261¯/2/H208491−nL/H20850/H20852/H20854. The self-energy matrix components are defined as /H9018bij/H20849/H9275/H20850=/H20885d/H92801 2/H9266Daaij/H20849/H9275−/H92801/H20850/H20885d/H92802 2/H9266Gsij/H20849/H92802/H20850Gsji/H20849/H92801+/H92802/H20850, where Gs/H20849t,t/H11032/H20850=−i/H20855TC/H9274s/H20849t/H20850/H9274s†/H20849t/H11032/H20850/H20856/H9261is the GF of the spin sec- tor fermion which is free. Therefore at T=0 it is easily found to be Gs/H20849/H9275/H20850=/H20900−i 2sgn /H20849/H9275/H20850i/H9008/H20849−/H9275/H20850 −i/H9008/H20849/H9275/H20850−i 2sgn /H20849/H9275/H20850/H20901. In the/H9004=0 and/H9003+=0 case only Daa−−/H20849++/H20850=±d0=±1//H9275are non-zero, so that only /H9018b−−needs to be calculated, resulting in /H9018b−−/H20849/H9275/H20850=−/H9018b++/H20849/H9275/H20850=/H208492/H9266/H20850−2/H9275/H20849ln/H20841/H9275/H20841−1 /H20850. It is an odd function of /H9275and vanishes in the infrared limit as expected from RG arguments since it is generated by anirrelevant operator. The corrections to the impurity GFs arethen /H9254Dbb−−=v12/H9018b−−/H20849Dbb−−Dbb−−−Dbb−+Dbb+−/H20850, /H9254Dbb−+=v12/H9018b−−Dbb−+/H20849Dbb−−−Dbb++/H20850. Furthermore, Dbb−−Dbb−−−Dbb−+Dbb+− = Det−1/H20849dˆ 0−1−/H9018ˆK/H20850 +2/H9275/H20853/H9275+i/H20851/H9003−/H208492nF−1 /H20850+/H9003/H11036/H20849nR+nL−1 /H20850/H20852/H20854 Det2/H20849dˆ 0−1−/H9018ˆK/H20850, and is an even function of /H9275. After multiplication with the self-energy, which is an odd function, and after an integra-tion over all frequencies we immediately see, that the off-Toulouse correction to the first contribution in Eq. /H2084935/H20850is identically zero: D bb−+/H20849Dbb−−−Dbb++/H20850=−2 i/H9275/H9003−2nF+/H9003/H11036/H20849ei/H9261¯/2nL+e−i/H9261¯/2nR/H20850 Det2/H20849dˆ 0−1−/H9018ˆK/H20850. Therefore for the correction to the derivative of the adiabatic potential we obtain/H9254/H20873/H11509U /H11509/H9261−/H20874=−v12/H9003/H11036/H20885d/H9275 2/H9266/H9018b−− Det2/H20849dˆ 0−1−/H9018ˆK/H20850 /H11003/H9275/H208532/H9003−nF/H20851ei/H9261¯/2/H208491−nR/H20850−e−i/H9261¯/2/H208491−nL/H20850/H20852 +/H9003/H11036/H20851ei/H9261¯nL/H208491−nR/H20850−e−i/H9261¯nR/H208491−nL/H20850/H20852/H20854. Comparing this result with Eq. /H2084938/H20850we conclude that the effect of v1is the correction to the transmission coefficients T¯i=Ti+/H9254Tiwith /H9254Ti/H20849/H9275/H20850=/H20873v1 2/H9266/H208742/H92752 /H92752+/H20849/H9003−+/H9003/H11036/H208502/H20849ln/H20841/H9275/H20841−1 /H20850Ti/H20849/H9275/H20850, since schematically the structure of the correction is 2T2e2i/H9261+T1ei/H9261 1+T2/H20849e2i/H9261−1 /H20850+T1/H20849ei/H9261−1 /H20850 +/H92542T2e2i/H9261+T1ei/H9261 /H208511+T2/H20849e2i/H9261−1 /H20850+T1/H20849ei/H9261−1 /H20850/H208522, which can be seen as the lowest order expansion of 2T2/H208491+/H9254/H20850e2i/H9261+T1/H208491+/H9254/H20850ei/H9261 1+T2/H208491+/H9254/H20850/H20849e2i/H9261−1 /H20850+T1/H208491+/H9254/H20850/H20849ei/H9261−1 /H20850. This correction vanishes for /H9275→0, hence the trivialization /H2084943/H20850still holds for the transmission coefficients away from the Toulouse point. It is not clear whether this picture is valid for /H9003+,/H9004/HS110050. As in this situation the magnetic field couples the aandb Majorana fields, the Daacorrelation functions have to be cal- culated from the Dyson equation with respect to the self- energy/H9018aij=/H20849ij/H20850/H20851/H90042Dbbij/H20849/H9004=0/H20850+/H9003+g/H9257/H9257/H208490/H20850ij/H20852. As our ultimate goal is the behaviour of the generating function in the limiting case of small V, we set V=0. We return to the finite Vsitu- ation at the end of this section. The resulting /H9018bcan be best given in terms of the retarded component /H9018bR, /H9018b−−/H20849++/H20850=± R e/H9018bR−i/H208492nF−1 /H20850Im/H9018bR, /H9018b−+=i2nFIm/H9018bR, /H9018b+−=−i2/H208491−nF/H20850Im/H9018bR, /H2084945/H20850 with /H9018bR=i /H208492/H9266/H2085021 /H20881b2−4d/H20858 j=1,2/H20851/H9024j2+/H90042−/H20849/H9003−+/H9003/H11036/H208502/H20852 /H11003/H20853/H20849/H9024j−i/H9275/H20850/H20851ln/H20849/H9024j−i/H9275/H20850−1 /H20852−/H9024j/H20849ln/H9024j−1 /H20850/H20854, where b=/H20849/H9003−+/H9003/H11036/H208502+/H9003+2−2/H90042,d=/H20851/H90042+/H9003+/H20849/H9003−+/H9003/H11036/H20850/H208522, and /H90241,22=/H20849b±/H20881b2−4d/H20850/2. The expansion for small energies is different from that at /H9004=0, Re/H9018R/H11015−/H9275 /H208492/H9266/H208502/H20881b2−4d/H20858 j=1,2/H20851/H9024j2+/H90042−/H20849/H9003−+/H9003/H11036/H208502/H20852ln/H9024j,A. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-12Im/H9018R /H11015−i/H92752 /H208492/H9266/H2085022/H20881b2−4d/H20858 j=1,2/H20851/H9024j2+/H90042−/H20849/H9003−+/H9003/H11036/H208502/H20852ln/H9024j /H9024j. The correction to the time ordered part is /H9254Dbb−−=v12/H20849Dbb−−/H9018b−−Dbb−−+Dbb−+/H9018b++Dbb+−+Dbb−+/H9018b+−Dbb−− +Dbb−−/H9018b−+Dbb+−/H20850=v12/H20849Dbb−−Dbb−−−Dbb−+Dbb+−/H20850Re/H9018bR +iv12Im/H9018bRF1/H20849/H9275/H20850, with F1/H20849/H9275/H20850=− /H208492nF−1 /H20850/H20849Dbb−−Dbb−−+Dbb−+Dbb+−/H20850 −2 /H208491−nF/H20850Dbb−−Dbb−++2nFDbb−−Dbb+−. This function is an odd function of /H9275while Dbb−−Dbb−− −Dbb−+Dbb+−is even. Taking into account the symmetry proper- ties of the self-energy we conclude that the whole contribu-tion to the generating function stemming from /H9254Dbb−−van- ishes. The other component can be written down in a similarway, /H9254Dbb−+=v12/H20849Dbb−−/H9018b−−Dbb−++Dbb−+/H9018b++Dbb+++Dbb−+/H9018b+−Dbb−+ +Dbb−−/H9018b−+Dbb++/H20850=v12Dbb−+/H20849Dbb−−−Dbb++/H20850Re/H9018bR +iv12Im/H9018bRF2/H20849/H9275/H20850, where we have introduced F2/H20849/H9275/H20850=/H208491−2 nF/H20850Dbb−+/H20849Dbb−−+Dbb++/H20850 +2 /H208491−nF/H20850Dbb−+Dbb−+−2nFDbb−−Dbb++. /H2084946/H20850 The analysis of the contribution arising from Re /H9018bRcan be done in the same way as before as it has exactly the samestructure. The substitution /H20849A1/H20850still can be applied and one immediately recognises that it only leads to the renormaliza-tion of the transmission coefficients /H9254Ti/H20849/H9275/H20850=v12Re/H9018bR/H9275/H20849/H92752+/H9003+2−/H90042/H20850 /H20851/H92752−/H90042−/H9003+/H20849/H9003/H11036+/H9003−/H20850/H208522+/H92752/H20849/H9003−+/H9003++/H9003/H11036/H208502Ti/H20849/H9275/H20850. /H2084947/H20850 Re/H9018bRis itself linear in /H9275in the low energy sector, that is why the corrections to the transmission coefficients /H2084947/H20850vanish at low energies. Now we turn to the contribution of Im /H9018bR. The first term in Eq. /H2084946/H20850can be shown to produce renormalization of the transmission coefficient similar to Eq. /H2084947/H20850, /H9254Ti/H20849/H9275/H20850=v12Im/H9018bR/H9003−/H20851/H9003−/H20849/H92752+/H9003+2/H20850+/H90042/H9003+/H20852 /H20851/H92752−/H90042−/H9003+/H20849/H9003/H11036+/H9003−/H20850/H208522+/H92752/H20849/H9003−+/H9003++/H9003/H11036/H208502Ti/H20849/H9275/H20850. Although the two remaining terms of Eq. /H2084946/H20850cannot be reduced to renormalisation of the transmission coefficients ina simple way, their contribution to the derivative of the adia-batic potential can be evaluated directly, /H9254I/H20873/H11509U /H11509/H9261−/H20874=v2/H9003/H11036 2/H20885 0Vd/H9275 2/H9266Im/H9018bRei/H9261¯/2Det−2/H20849dˆ 0−1−/H9018ˆK/H20850 /H11003/H20875/H20873/H9275−/H90042/H9275 /H92752+/H9003+2/H208742 +/H20873/H9003−+/H90042/H9003+ /H92752+/H9003+2/H208742 +/H9003/H110362ei/H9261¯/H20876. Taking into account that the leading behaviour of the imagi- nary part of the self-energy is /H11011/H92752for small energies one immediately verifies, that the above correction is cubic in theapplied voltage and therefore leads to qualitatively the samepicture as the renormalisation of the transmission coeffi-cients. For that particular evaluation we used the equilibrium self-energy/H9018ˆ b. Nevertheless, after a lengthy but straightfor- ward calculation, we find that the same conclusion is stillvalid for the proper nonequilibrium one as the corresponding corrections to/H9018ˆ bis of exactly the same order in Vand/H9275. Thus at least in the low energy sector the predictions of Sec.III F 1 remain valid beyond the Toulouse point.3. Resonant level problem in Luttinger liquids We now briefly turn to the g=1/2 RL setup. This setup has caused much interest recently, see Ref. 23, and refer-ences therein. The Hamiltonian now is H=H 0+/H20849/H9253L/H9274Ld†+/H9253Rd/H9274R†+ H.c. /H20850+/H9004d†d+HC, where H0stands for two biased Luttinger liquids /H20849LLs /H20850,dis the electron operator on the dot, /H9253R/H20849L/H20850are the tunneling am- plitudes to R/H20849L/H20850electrode and HCis an electrostatic interac- tion /H20849see also Ref. 23 /H20850, HC=/H9261Cd†d/H20858 i/H9274i†/H208490/H20850/H9274i/H208490/H20850. The contacting electrodes are supposed to be one- dimensional half-infinite electron systems. We model themby chiral fermions living in an infinite system: the negativehalf-axis then describes the particles moving towards theboundary, while the positive half-axis carries electrons mov-ing away from the end of the system. In the bosonic repre-sentation H 0/H20851/H9274i/H20852are diagonal even in presence of interactions /H20849for a recent review see, e.g., Ref. 20; we set the renormal- ized Fermi velocity v=vF/g=1, the bare velocity being vF/H20850:TOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-13H0/H20851/H9274i/H20852=/H208494/H9266/H20850−1/H20885dx/H20851/H11509x/H9278i/H20849x/H20850/H208522. Here the phase fields /H9278i/H20849x/H20850describe the slow varying spatial component of the electron density /H20849plasmons /H20850, /H9274i†/H20849x/H20850/H9274i/H20849x/H20850=/H11509x/H9278i/H20849x/H20850/2/H9266/H20881g. The electron field operator at the boundary is given by52 /H9274i/H208490/H20850=ei/H9278i/H208490/H20850//H20881g//H208812/H9266a0, where a0is the lattice constant of the underlying lattice model. Here gis the conventional LL parameter /H20849coupling constant /H20850connected to the bare interaction strength Uviag =/H208491+U//H9266vF/H20850−1/2.20,53In the chiral formulation the bias volt- age amounts to a difference in the densities of the incoming particles in both channels far away from the constriction.54,55 The current is then proportional to the difference between thedensities of incoming and outgoing particles within eachchannel. Construction of the operator /H208492/H20850is unproblematic and leads to T /H9261=/H9253L/H20849d†/H9274Lei/H9261/4+/H9274L†de−i/H9261/4/H20850+/H9253R/H20849d†/H9274Re−i/H9261/4+/H9274R†dei/H9261/4/H20850, where, contrary to the Anderson impurity calculation, we choose to build in the counting field in a symmetric mannerfor the reasons which will become clear later. After theEmery-Kivelson rotation, refermionization to new fermions /H9274and after the introduction of the Majorana components as in Eq. /H2084929/H20850/H20849Refs. 21 and 24 /H20850we find T/H9261=/H20851ei/H9261/4/H20849/H9253Ld†/H9274+/H9253R/H9274d/H20850+e−i/H9261/4/H20849/H9253Ld†/H9274†+/H9253R/H9274†d/H20850/H20852 =−i/H9253+b/H20851cos /H20849/H9261/4/H20850/H9264− sin /H20849/H9261/4/H20850/H9257/H20852 +i/H9253−a/H20851sin/H20849/H9261/4/H20850/H9264− cos /H20849/H9261/4/H20850/H9257/H20852, /H2084948/H20850 where/H9253±=/H9253L±/H9253R. In case of the symmetric coupling /H9253−=0 the corresponding T/H9261has exactly the same shape as Eq. /H2084931/H20850. In fact we find the same set of equations as for the Kondodot, Eqs. /H2084930/H20850and /H2084931/H20850, but with/H9261→/H9261/2 and J /H11036=/H9253+,J± =0. Consequently, the FCS is given by a modification of the Levitov-Lesovik formula /H2084910/H20850: /H92731/2/H20849/H9261/H20850=/H92730/H20849/H9261;2V;/H20853T/H9004/H20849/H9275/H20850/H20854/H20850, /H2084949/H20850 with the effective transmission coefficient T/H9004/H20849/H9275/H20850=4/H92534/H92752/ /H208514/H92534/H92752+/H20849/H92752−/H90042/H208502/H20852of the RL setup in the symmetric case.23 All the cumulants are thus obtainable from those of the non- interacting statistics /H2084910/H20850. The/H9004=0 RL setup is equivalent to the model of direct tunneling between two g=2 LLs.24The latter model is con- nected by the strong to weak coupling /H208491/g→g/H20850duality ar- gument to the g=1/2 Kane and Fisher model,53,56which is, in turn, equivalent to the CB setup studied in Refs. 25 and 26/H20849for a more general case of arbitrary interaction strength see also Refs. 57 and 58 /H20850. Therefore their FCS must be related to our Eq. /H2084949/H20850at/H9004=0 by means of the transformation: T 0 →1−T0andV→V/2. Indeed after some algebraic manipu-lation with Eq. /H2084912/H20850of Ref. 26, for details see Appendix C, we find that the FCS for the CB setup can be rewritten as /H9273CB/H20849/H9261/H20850=/H92730/H20849−/H9261;V;/H208531−T0/H20849/H9275/H20850/H20854/H20850. For the asymmetric coupling /H9253−/HS110050 the problem cannot be mapped onto the Kondo dot any more. The correspondingcalculation is nevertheless straightforward and is presentedin Appendix D. There is no fundamental difference in theresult up to the more involved transmission coefficient,which has already been derived for the case of the nonlinearI-Vin Ref. 23. IV. CONCLUSIONS To conclude, we present a detailed study of the charge transfer statistics through the Anderson impurity model. Wefind an expression for the exact generating function in termsof the impurity self-energy calculated in the presence of themeasuring field /H9261: Eq. /H2084920/H20850. Based on this formula we con- clude that T=0 linear response statistics is universal and bi- nomial for the AIM and similar models: we call this factbinomial theorem. The only effect of correlations is to definean effective transmission coefficient. For the symmetricAIM, for example, there is a perfect transmission and nofluctuations of the current at all in this case. In the search for nontrivial interaction effects one has, therefore, to go to higher values of TandV. To this end we have calculated the exact FCS distribution function in theToulouse limit /H20849Kondo regime /H20850: it is given by Eq. /H2084939/H20850. This formula uncovers rather profound, if model dependent, con-sequences of correlations: there are two distinct tunnelingprocesses /H20849T 1andT2/H20850, that of single electrons and electron pairs with opposite spin. The latter process is, in fact, domi-nant in zero field. The structure of higher moments is alsodetermined by these two processes as discussed in detail inthe main text. At T=0 linear response all this rich physics is masked by Eq. /H2084939/H20850collapsing to the universal binomial dis- tribution. We checked this universality by extensively study-ing corrections to the distribution function due to departuresfrom the Toulouse limit. We close by outlining some possible directions for future developments. Formula /H2084920/H20850could be used to develop Fermi liquid theory for the noise and possibly higher moments.Perhaps more importantly, the ideas of this paper could beapplied to models with many conduction channels, whereone would expect some equivalent of the binomial theoremto hold, as seems to be compatible with recent experiments. 59 ACKNOWLEDGMENTS We wish to thank H. Saleur, H. Grabert, K. Schönham- mer, A. Tsvelik, A. Chitov, Y. Adamov, F. Siano, and R.Egger for many inspiring discussions. We would also like tothank W. Belzig for illuminating discussions and for pointingout to us the reduction: Eq. /H2084943/H20850. Part of this work was done during AOG’s visits to the Brookhaven National Laboratoryand to the University of Göttingen, the hospitality is kindlyacknowledged. The AK’s financial support has been pro-A. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-14vided by the EU RTN programme, by the DFG and by the Landesstiftung Baden-Württemberg. AK thanks the Alex-ander von Humboldt foundation for support. APPENDIX A The calculation of the impurity GF in finite magnetic field /H9004and J+/HS110050 is accomplished again by inversion of dˆ 0−1 −/H9018ˆ/H9004with /H9018ˆ/H9004=/H20875/H90042Daa−−+/H9018K−−−/H90042Daa−++/H9018K−+ −/H90042Daa+−+/H9018K+−/H90042Daa+++/H9018K++/H20876, where/H9018ˆKis given in Eq. /H2084936/H20850.Dˆaahas to be evaluated with respect to the Hamiltonian H+=H0/H20851/H9257f/H20852−iJ+a/H9257f. A relatively simple calculation yields Daa/H20849/H9275/H20850=1 /H92752+/H9003+2/H20875/H9275+i/H9003+/H208492nF−1 /H20850 i2/H9003+nF −i2/H9003+/H208491−nF/H20850−/H9275+i/H9003+/H208492nF−1 /H20850/H20876. From now on the calculation can be performed in exactly the same way as before. However, writing down explicitly the expression for /H9018ˆ/H9004one observes, that it can be constructed from the corresponding /H9018ˆKvia trivial substitution /H9275→/H9275−/H90042/H9275 /H92752+/H9003+2,/H9003−→/H9003−+/H90042/H9003+ /H92752+/H9003+2. /H20849A1 /H20850 This can be used to obtain the transmission coefficients /H2084941/H20850 from the ones given by Eq. /H2084940/H20850. APPENDIX B For systems with known scattering matrix s/H9251/H9252,mnbetween terminals/H9251,/H9252and channels mandn, there is a ready formula for the FCS generating function, derived in Ref. 28, ln/H9273/H20849/H9261/H20850=T 2/H9266/H20885d/H9275ln Det /H208511+fˆ/H20849/H9275/H20850/H20849sˆ†s˜−1 /H20850/H20852, /H20849B1/H20850 where s˜/H9251/H9252,mn=ei/H20849/H9261/H9251−/H9261/H9252/H20850s/H9251/H9252,mn,/H9261/H9251,/H9252being the fields counting the particles in the respective terminals. fˆ/H20849/H9275/H20850=/H9254mn/H9254/H9251/H9252f/H9251/H20849/H9275/H20850 is diagonal in both channel /H20849m,n/H20850and terminal /H20849/H9251,/H9252/H20850indices and describes the energy distribution function in the respec- tive terminal. In the simplest situation, when /H9004=0 and J+ =0, we have four terminals with one channel in each of them. The scattering part of the Hamiltonian is HI=J/H11036/H20849/H9274† −/H9274/H20850b+J−/H20849/H9274s†−/H9274s/H20850b, see Eq. /H2084928/H20850/H20849for simplicity we ignore the unimportant numerical prefactors /H20850. The equations of mo- tion /H20849EOMs /H20850for the participating operators read i/H11509t/H9274s=−i/H11509x/H9274s+J−b/H9254/H20849x/H20850,i/H11509t/H9274=−i/H11509x/H9274+J/H11036b/H9254/H20849x/H20850, i/H11509tb=J/H11036/H20849/H9274†−/H9274/H20850+J−/H20849/H9274s†−/H9274s/H20850. Integrating the first equation over time and then around the point x=0 we obtain i/H20851/H9274s/H208490+/H20850−/H9274s/H208490−/H20850/H20852=J−b. Acting with i/H11509tfrom the left and using the EOM for the b Majorana one obtains, −/H11509t/H20851/H9274s/H208490+/H20850−/H9274s/H208490−/H20850/H20852=J−J/H11036/H20849/H9274†−/H9274/H20850+J−2/H20849/H9274s†−/H9274s/H20850, −/H11509t/H20851/H9274/H208490+/H20850−/H9274/H208490−/H20850/H20852=J−J/H11036/H20849/H9274s†−/H9274s/H20850+J/H110362/H20849/H9274†−/H9274/H20850, where the last equation is obtained by symmetry. Now we employ the plain wave decomposition similar to that used inRefs. 23, 54, and 60 /H9274/H20849x/H20850=/H20885dk 2/H9266eik/H20849x−t/H20850/H20877akforx/H110210 bkforx/H110220/H20878, /H9274s/H20849x/H20850=/H20885dk 2/H9266eik/H20849x−t/H20850/H20877ckforx/H110210 dkforx/H110220/H20878. Since the dispersion relation of both fermion species is trivial,/H9275=k, we can use/H9275both for momentum and energy. Employing the regularisation scheme /H9274i=/H20851/H9274/H208490+/H20850+/H9274/H208490−/H20850/H20852/2 we obtain −i/H9275/H20849d/H9275−c/H9275/H20850=J/H11036J− 2/H20849a−/H9275†+b−/H9275†−a/H9275−b/H9275/H20850 +J−2 2/H20849c−/H9275†+d−/H9275†−c/H9275−d/H9275/H20850, −i/H9275/H20849b/H9275−a/H9275/H20850=J/H11036J− 2/H20849c−/H9275†+d−/H9275†−c/H9275−d/H9275/H20850 +J/H110362 2/H20849a−/H9275†+b−/H9275†−a/H9275−b/H9275/H20850. Comparing these relations with their adjunct at − /H9275we iden- tify that b−/H9275†=a−/H9275†−b/H9275+a/H9275, d−/H9275†=c−/H9275†−d/H9275+c/H9275, and that b/H9275−a/H9275=/H20849d/H9275−c/H9275/H20850J/H11036/J−. Using these expressions we can find both b/H9275andd/H9275as functions of a/H9275andc/H9275, e.g., b/H9275=1 /H9275+i/H20849/H9003/H11036+/H9003−/H20850 /H11003/H20851/H20849/H9275+i/H9003−/H20850a/H9275+i/H9003/H11036a−/H9275†−iJ/H11036J−c/H9275+iJ/H11036J−c−/H9275†/H20852. That leads to the following scattering matrix:TOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-15/H20898b/H9275b−/H9275† d/H9275 d−/H9275†/H20899=s/H20898a/H9275 a−/H9275† c/H9275 c−/H9275†/H20899=1 /H9275+i/H20849/H9003/H11036+/H9003−/H20850/H20898/H9275+i/H9003− i/H9003/H11036 −iJ/H11036J−iJ/H11036J− i/H9003/H11036/H9275+i/H9003−iJ/H11036J−−iJ/H11036J− −iJ/H11036J−iJ/H11036J−/H9275+i/H9003/H11036 i/H9003− iJ/H11036J−−iJ/H11036J− i/H9003−/H9275+i/H9003/H11036/H20899/H20898a/H9275 a−/H9275† c/H9275 c−/H9275†/H20899. The actual charge transport through the system is conveyed by the charge flavour channel, e.g., by scattering of /H9274fermi- ons across the constriction. The physical picture is similar tothat discussed in Ref. 23: the incoming particles—chiral fer-mions in terminal 1, which are described by a koperators and which have chemical potential /H92621=V—are transferred into all other terminals 2–4 /H20849bk,ckanddkoperators /H20850, which are unbiased/H92622,3,4=0, that is why we have to set fˆ =diag /H20849nL,nF,nF,nF/H20850. Then/H92611=/H9261counts particles which leave channel 1. However, the very same fermion reappears in the channel 2. Since 1 and 2 are physically one lead wehave/H9261 2=−/H9261. We are not interested in change of particle numbers in the other channels, that is why /H92613,4=0.61There- fore the matrix s˜is given by s˜=1 /H9275+i/H20849/H9003/H11036+/H9003−/H20850 /H11003/H20898/H9275+i/H9003− i/H9003/H11036e−i2/H9261−iJ/H11036J−e−i/H9261iJ/H11036J−e−i/H9261 i/H9003/H11036ei2/H9261/H9275+i/H9003− iJ/H11036J−ei/H9261−iJ/H11036J−ei/H9261 −iJ/H11036J−ei/H9261iJ/H11036J−e−i/H9261/H9275+i/H9003/H11036 i/H9003− iJ/H11036J−ei/H9261−iJ/H11036J−e−i/H9261i/H9003−/H9275+i/H9003/H11036/H20899. Plugging these relations into Eq. /H20849B1/H20850, folding the integration over energy to the domain /H208510,/H11009/H20850and using the properties nF/H20849−/H9275/H20850=1− nF/H20849/H9275/H20850andnL/H20849−/H9275/H20850=1− nR/H20849/H9275/H20850immediately leads then to the result /H2084939/H20850.APPENDIX C Here we establish the relation between our findings and the result of Kindermann and Trauzettel /H20849KT /H20850calculation.26 Let us consider Eq. /H2084912/H20850of Ref. 26, ln/H9273KT/H20849/H9261/H20850=T 4/H9266/H20849−iV/H9261−T/H92612/H20850+T/H20885 0/H11009d/H9275 2/H9266ln/H208531+T0/H20849/H9275/H20850 /H11003/H20851f+/H208491−f−/H20850/H20849ei/H9261−1 /H20850+f−/H208491−f+/H20850/H20849e−i/H9261−1 /H20850/H20852/H20854. The ffunctions are given by f±=nL/R /H208491−nL/R/H20850e±i/H9261+nL/R, where nR,L/H20849/H9275/H20850=nF/H20849/H9275±V/2/H20850. Obviously 1−f±=/H208491−nL/R/H20850e±i/H9261 /H208491−nL/R/H20850e±i/H9261+nL/R. Taking into account that T0/H20849/H9275/H20850=4/H92534//H20849/H92752+4/H92532/H20850, the identifi- cation with the KT’s impurity strength is 2 /H9253=TB. In order to proceed we define the object ln/H9273KT/H20849/H9261;/H9251/H20850=T 4/H9266/H20849−iV/H9261−T/H92612/H20850+T/H20885 0/H11009d/H9275 2/H9266ln/H208531+/H9251T0/H20849/H9275/H20850 /H11003/H20851f+/H208491−f−/H20850/H20849ei/H9261−1 /H20850+f−/H208491−f+/H20850/H20849e−i/H9261−1 /H20850/H20852/H20854, /H20849C1/H20850 where/H9251is a parameter. The derivative of Eq. /H20849C1/H20850with respect to this parameter /H11509ln/H9273KT/H20849/H9261;/H9251/H20850 /H11509/H9251=T/H20885 0/H11009d/H9275 2/H9266T0/H20849/H9275/H20850/H20851f+/H208491−f−/H20850/H20849ei/H9261−1 /H20850+f−/H208491−f+/H20850/H20849e−i/H9261−1 /H20850/H20852 1+/H9251T0/H20849/H9275/H20850/H20851f+/H208491−f−/H20850/H20849ei/H9261−1 /H20850+f−/H208491−f+/H20850/H20849e−i/H9261−1 /H20850/H20852. Substituting explicit expressions for the ffunctions into the long fraction gives −T0/H20849/H9275/H20850/H20851nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852 /H20851/H208491−nL/H20850ei/H9261+nL/H20852/H20851/H208491−nL/H20850e−i/H9261+nR/H20852−/H9251T0/H20849/H9275/H20850/H20851nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852. After simple algebra this simplifies as −T0/H20849/H9275/H20850/H20851nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852 1+ /H208511−/H9251T0/H20849/H9275/H20850/H20852/H20851nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852. Integrating with respect to /H9251therefore results in ln/H9273KT/H20849/H9261;/H9251/H20850=T/H20885 0/H11009d/H9275 2/H9266ln/H208531+ /H208511−/H9251T0/H20849/H9275/H20850/H20852/H20851nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852/H20854+C. To fix the constant C, evaluate the above integral at /H9251=0, see also Ref. 29.A. O. GOGOLIN AND A. KOMNIK PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-16T/H20885 0/H11009d/H9275 2/H9266ln/H208511+nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850+nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852=T 4/H9266/H20849−iV/H9261−T/H92612/H20850, so that C=0. The following identity is therefore established /H20849put/H9251=1/H20850: ln/H9273KT/H20849/H9261/H20850=T/H20885 0/H11009d/H9275 2/H9266ln/H208531+ /H208511−T0/H20849/H9275/H20850/H20852/H20851nL/H208491−nR/H20850/H20849e−i/H9261−1 /H20850 +nR/H208491−nL/H20850/H20849ei/H9261−1 /H20850/H20852/H20854 /H20849 C2/H20850 The two equations /H20849C1/H20850and /H20849C2/H20850define the same function, which means that the KT statistics, similar to the g=1/2 statistics, is reducible to the generic Levitov-Lesovik for-mula, its relation to the noninteracting statistics Eq. /H2084910/H20850be- ing /H9273KT/H20849/H9261/H20850=/H92730/H20849−/H9261;V;/H208531−T0/H20849/H9275/H20850/H20854/H20850. Finally the explicit relation between the KT statistics and our g=1/2 result is /H9273KT/H20851/H9261;V;/H20853T0/H20849/H9275/H20850/H20854/H20852=/H92731/2/H20851−/H9261;V/2;/H208531−T0/H20849/H9275/H20850/H20854/H20852, which is a direct consequence of the duality shown in Ref. 24. APPENDIX D The calculation starts as usual with the adiabatic potential /H20851see also Eq. /H2084932/H20850/H20852, /H11509 /H11509/H9261−U/H20849/H9261±/H20850=−1 4/H20885d/H9275 2/H9266/H20853/H9253+/H20851sin/H20849/H9261¯/4/H20850Gb/H9264−−+ cos /H20849/H9261¯/4/H20850Gb/H9257−−/H20852 +/H9253−/H20851cos /H20849/H9261¯/4/H20850Ga/H9264−−− sin /H20849/H9261¯/4/H20850Ga/H9257−−/H20852/H20854. /H20849C3/H20850 The most compact way to evaluate the inhomogeneous GFs entering this expression is through their reduction to GFsinvolving only the resonant level Majoranas aandb. This is accomplished by the following relations: G b/H9257−−=i/H9253+/H20851Dbb−−sin/H20849/H9261−/4/H20850g/H9257/H9257−−−Dbb−+sin/H20849/H9261+/4/H20850g/H9257/H9257+− −Dbb−−cos /H20849/H9261−/4/H20850g/H9264/H9257−−+Dbb−+cos /H20849/H9261+/4/H20850g/H9264/H9257+−/H20852 +i/H9253−/H20851Dba−−sin/H20849/H9261−/4/H20850g/H9264/H9257−−−Dba−+sin/H20849/H9261+/4/H20850g/H9264/H9257+− +Dba−−cos /H20849/H9261−/4/H20850g/H9257/H9257−−−Dba−+cos /H20849/H9261+/4/H20850g/H9257/H9257+−/H20852, Gb/H9264−−=i/H9253+/H20851Dbb−−sin/H20849/H9261−/4/H20850g/H9257/H9264−−−Dbb−+sin/H20849/H9261+/4/H20850g/H9257/H9264+− −Dbb−−cos /H20849/H9261−/4/H20850g/H9264/H9264−−+Dbb−+cos /H20849/H9261+/4/H20850g/H9264/H9264+−/H20852 +i/H9253−/H20851Dba−−sin/H20849/H9261−/4/H20850g/H9264/H9264−−−Dba−+sin/H20849/H9261+/4/H20850g/H9264/H9264+−+Dba−−cos /H20849/H9261−/4/H20850g/H9257/H9264−−−Dba−+cos /H20849/H9261+/4/H20850g/H9257/H9264+−/H20852, Ga/H9257−−=i/H9253+/H20851Dab−−sin/H20849/H9261−/4/H20850g/H9257/H9257−−−Dab−+sin/H20849/H9261+/4/H20850g/H9257/H9257+− −Dab−−cos /H20849/H9261−/4/H20850g/H9264/H9257−−+Dab−+cos /H20849/H9261+/4/H20850g/H9264/H9257+−/H20852 +i/H9253−/H20851Daa−−sin/H20849/H9261−/4/H20850g/H9264/H9257−−−Daa−+sin/H20849/H9261+/4/H20850g/H9264/H9257+− +Daa−−cos /H20849/H9261−/4/H20850g/H9257/H9257−−−Daa−+cos /H20849/H9261+/4/H20850g/H9257/H9257+−/H20852, Ga/H9264−−=i/H9253+/H20851Dab−−sin/H20849/H9261−/4/H20850g/H9257/H9264−−−Dab−+sin/H20849/H9261+/4/H20850g/H9257/H9264+− −Dab−−cos /H20849/H9261−/4/H20850g/H9264/H9264−−+Dab−+cos /H20849/H9261+/4/H20850g/H9264/H9264+−/H20852 +i/H9253−/H20851Daa−−sin/H20849/H9261−/4/H20850g/H9264/H9264−−−Daa−+sin/H20849/H9261+/4/H20850g/H9264/H9264+− +Daa−−cos /H20849/H9261−/4/H20850g/H9257/H9264−−−Daa−+cos /H20849/H9261+/4/H20850g/H9257/H9264+−/H20852. Inserting these results into Eq. /H20849C3/H20850leads to /H11509 /H11509/H9261−U/H20849/H9261±/H20850=−1 4/H20885d/H9275 2/H9266/H20853/H9253+2/H20851Dbb−−g/H9257/H9264− cos /H20849/H9261¯/4/H20850Dbb−+g/H9257/H9264+− + sin /H20849/H9261¯/4/H20850Dbb−+g/H9257/H9257+−/H20852+/H9253−2/H20851Daa−−g/H9257/H9264 − cos /H20849/H9261¯/4/H20850Daa−+g/H9257/H9264+−+ sin /H20849/H9261¯/4/H20850Daa−+g/H9257/H9257+−/H20852 +/H9253+/H9253−/H20851/H20849Dba−−−Dab−−/H20850g/H9257/H9257−−− sin /H20849/H9261¯/4/H20850/H20849Dba−+−Dab−+/H20850 /H11003g/H9257/H9264+−− cos /H20849/H9261¯/4/H20850/H20849Dba−+−Dab−+/H20850g/H9257/H9257+−/H20852/H20854. /H20849C4/H20850 Finally, the calculation of the composite 4 /H110034 matrix object Dˆ=/H20900Dbb−−Dbb−+Dba−−Dba−+ Dbb+−Dbb++Dba+−Dba++ Dab−−Dab−+Daa−−Daa−+ Dab+−Dab++Daa+−Daa++/H20901, can be done by calculation of /H20851/H20849Dˆ/H208490/H20850/H20850−1−/H9018ˆg/H20852−1, where Dˆ/H208490/H20850 =diag /H208491//H9275,−1//H9275,1//H9275,−1//H9275/H20850is the corresponding matrix in the absence of the tunnelling couplings and where the corre- sponding self-energy is given by /H9018ˆg=/H20875/H9018ˆbb/H9018ˆba /H9018ˆab/H9018ˆaa/H20876, and the components of this object are /H20849we set/H9003±=/H9253±2/2 and /H9003/H11036=/H9253−/H9253+/2/H20850, /H9018ˆbb=/H20875/H90042//H9275+i/H9003+/H20849nR+nL−1 /H20850 −i/H9003+/H20849nLei/H9261¯/4+nRe−i/H9261¯/4/H20850 i/H9003+/H20851/H208491−nR/H20850ei/H9261¯/4+/H208491−nL/H20850e−i/H9261¯/4/H20852−/H90042//H9275+i/H9003+/H20849nR+nL−1 /H20850/H20876,TOWARDS FULL COUNTING STATISTICS FOR THE ¼ PHYSICAL REVIEW B 73, 195301 /H208492006 /H20850 195301-17/H9018ˆba=/H9003/H11036/H20875nR−nL nLei/H9261¯/4−nRe−i/H9261¯/4 /H208491−nR/H20850ei/H9261¯/4−/H208491−nL/H20850e−i/H9261¯/4 nR−nL/H20876, /H9018ˆab=/H9003/H11036/H20875nL−nR −nLei/H9261¯/4+nRe−i/H9261¯/4 −/H208491−nR/H20850ei/H9261¯/4+/H208491−nL/H20850e−i/H9261¯/4 nL−nR/H20876, /H9018ˆaa=/H20875/H90042//H9275+i/H9003−/H20849nR+nL−1 /H20850 −i/H9003−/H20849nLei/H9261¯/4+nRe−i/H9261¯/4/H20850 i/H9003−/H20851/H208491−nR/H20850ei/H9261¯/4+/H208491−nL/H20850e−i/H9261¯/4/H20852−/H90042//H9275+i/H9003−/H20849nR+nL−1 /H20850/H20876. 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PhysRevB.76.195327.pdf

Saturation of spin-polarized current in nanometer scale aluminum grains Y . G. Wei, C. E. Malec, and D. Davidovi ć School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA /H20849Received 15 October 2007; published 27 November 2007 /H20850 We describe measurements of spin-polarized tunneling via discrete energy levels of single aluminum grains. In high resistance samples /H20849/H11011G/H9024/H20850, the spin-polarized tunneling current rapidly saturates as a function of the bias voltage. This indicates that the spin-polarized current is carried only via the ground state and the fewlowest in energy excited states of this grain. At the saturation voltage, the spin-relaxation rate T 1−1of the highest states excited by tunneling is comparable to the electron tunneling rate, T1−1/H110151.5/H11003106and 107s−1,i nt w o samples. The ratio of T1−1to the electron-phonon relaxation rate is in agreement with the Elliot-Yafet scaling, an evidence that spin relaxation in Al grains is governed by the spin-orbit interaction. DOI: 10.1103/PhysRevB.76.195327 PACS number /H20849s/H20850: 73.21.La, 72.25.Hg, 72.25.Rb, 73.23.Hk I. INTRODUCTION Electron tunneling through single nanometer scale metal- lic grains at low temperatures can display a discrete energylevel spectrum. 1Tunneling spectroscopy of the energy spec- tra has led to numerous discoveries, including Fermi-liquidcoupling constants between quasiparticles, 2spin-orbit interactions,3,4and superconducting correlations in zero- dimensional systems.5Some information regarding the spin of an electron occupying a discrete level can be obtainedusing spin-unpolarized tunneling, such as spin-multiplicityand electron gfactors. 1 In this paper, we report on spin-polarized tunneling via discrete energy levels of single aluminum grains. Spin-polarized electron transport permits studies of spin relaxationand spin dephasing. 6,7By comparison, spin-unpolarized spectroscopy is suitable for the studies of energy relaxationin the grains. 1,2Since spin-relaxation times are generally many orders of magnitude longer than energy relaxationtimes, spin-unpolarized spectroscopy is not an easy tool tostudy spin relaxation in the grains, and spin-polarized tunnel-ing is needed. We find that some electron spin-relaxationtimes in Al grains are exceptionally long compared to bulkAl with similar disorder, on the order of microseconds. Spin-polarized transport via metallic grains has recently generated a lot of theoretical interest. 8–12In addition, there is a major effort to study nanospintronics using carbon nano-tubes; see Ref. 13and references therein. Spin-coherent elec- tron tunneling via nanometer scale normal metallic grainshas been confirmed in arrays 14,15and in single grains.16How- ever, the electron spin-relaxation time T1in a metallic grain has not been reported yet. II. SAMPLE FABRICATION Our samples are prepared by electron beam lithography and shadow evaporation, similar to the technique describedpreviously. 3First, we define a resist bridge placed 250 nm above the Si wafer; this bridge acts as a mask. Next/H20851Fig. 1/H20849A/H20850/H20852, we deposit 11 nm Permalloy /H20849Py=Ni 0.8Fe0.2/H20850 onto oxidized silicon substrate at 4 /H1100310−7Torr base pres- sure, measured near the gate valve, along the direction indi-cated by the arrow. Then, we rotate the sample by 36° with-out breaking the vacuum and deposit 1.2 nm of Al 2O3by reactive evaporation of Al,3at a rate of 0.35 nm /s and at an oxygen pressure of 2.5 /H1100310−5Torr. Now, oxygen flow is shut down. When pressure decreases to the 10−7Torr range, we deposit a 0.6 nm thick film of Al, as shown in Fig. 1/H20849B/H20850. Al forms isolated grains with a typical diameter of 5 nm. Thegrains are displayed by the scanning electron microscope/H20849SEM /H20850image in Fig. 1/H20849D/H20850. Finally, we deposit another 1.2 nm layer of Al 2O3by the reactive evaporation and top it of an 11 nm thick film of Py /H20851Fig. 1/H20849C/H20850/H20852. We make many samples on the same silicon wafer, and vary the overlap from0 to 50 nm and select the devices with the highest resistance,as they have the smallest overlap. Figures 1/H20849E/H20850and 1/H20849F/H20850 show SEM images of a typical device. III. DISCRETE ENERGY LEVELS Transport properties of the samples at low temperatures were measured using an Ithaco current amplifier. Thesamples were cooled down to /H110150.035 K base temperature. The sample leads were cryogenically filtered to reduce theelectron temperature down to /H110150.1 K. The majority of samples /H20849/H1102280% /H20850exhibit Coulomb block- ade at low temperature. About 150 samples were measured at 4.2 K and 16 samples at 0.035 K. In this paper, we describetwo samples. The I-Vcurves of two samples are shown in Figs. 1/H20849G/H20850and1/H20849H/H20850. The tunneling current increases in dis- crete steps as a function of bias voltage, corresponding todiscrete electron-in-a-box energy levels of the grain. In sample 1, the average electron-in-a-box level spacing caused by electron geometric confinement is /H9254/H110150.8 meV, which corresponds to diameter D/H110156 nm assuming a spheri- cal Al grain. The average current step I¯/H110150.47 pA. We make a connection with the tunneling rates from the leads to thegrain and the measured current response. The tunnel junc-tions are highly asymmetric, and therefore, one of the tun-neling rates is much smaller than the other, and thus ratelimiting. Throughout this paper, we choose the rate limitingstep to be across the left junction, corresponding to the tun-neling rate /H9003 L. Therefore, our measured current corresponds to the average tunneling-in rate of /H9003¯L=I¯/2/H20841e/H20841/H110151.5/H11003106s−1. Similarly, in sample 2, /H9254/H110152.7 meV, D/H110154 nm, and /H9003¯L/H110159.6/H11003106s−1.PHYSICAL REVIEW B 76, 195327 /H208492007 /H20850 1098-0121/2007/76 /H2084919/H20850/195327 /H208496/H20850 ©2007 The American Physical Society 195327-1The spin-conserving energy relaxation in Al grains takes place by phonon emission with the relaxation rate,2 /H9270e-ph−1/H20849/H9275/H20850=/H208732 3EF/H208742/H92753/H9270e/H9254 2/H9267/H60365vS5, /H208491/H20850 where EF=11.7 eV is the Fermi energy, /H9275is the energy difference between the initial and the final states, /H9267=2.7 g /cm3is the ion-mass density, and vs=6420 m /si s the sound velocity. We obtain /H9270e-ph−1/H20849/H9254/H20850/H110151.6/H11003109and 4.1/H110031010s−1in samples 1 and 2, respectively. Sample 2 has significantly larger relaxation rate because of the larger levelspacing. Since the tunneling rates in our samples are/H1101110 6s−1, if the grain is excited by electron tunneling in and out, it will instantly relax to the lowest energy state acces-sible by spin-conserving transitions. As shown by Fig. 2, the energy levels exhibit Zeeman splitting as a function of an applied magnetic field. In sample1, the I-Vcurve probes the same energy spectrum at negative and positive bias voltages. This is evident from the equiva-lence of the magnetic field dependencies at negative andpositive biases. The lowest tunneling threshold is twofolddegenerate at zero magnetic field, showing that N 0, the num- ber of electrons on the grain before tunneling in, is even. Theconductance peaks are similar in magnitude at negative bias,because the first tunneling step, in which an electron tunnelsinto the grain through the higher resistance junction, is rate limiting. At positive bias, the first conductance peak is muchlarger than the subsequent conductance peaks, because thefirst tunneling step takes place via the lower resistance junc-tion, and the rates are limited by the electron discharge pro-cess across the high resistance junction. In sample 1, the first two peaks split corresponding to g factors: g=1.83±0.05 and 1.95±0.05. Slight reduction of the gfactors from sample 2 indicates spin-orbit interaction in Al. 1The avoided level crossings are clearly resolved in Fig. 2, near points /H20849−11.5 mV, 5 T /H20850and /H20849−13 mV, 11.5 T /H20850. The corresponding avoided crossings at positive bias are located near /H2084913.5 mV, 5 T /H20850and /H2084915.5 mV, 11.5 T /H20850, respectively. In the regime, where gfactors are slightly reduced, the spin- orbit scattering rate /H20849/H9270SO−1/H20850can be obtained from the avoided crossing energies /H9004SO/H110150.1 meV.17Theory predicts that /H9270SO/H11015/H6036/H9254//H9266/H9004SO2,17within a factor of 2. Thus, we obtain /H9270SO−1/H110155.5/H110031010s−1. By the Elliot-Yafet relation,18/H9270SO−1is re- lated to the elastic scattering rate /H9270e−1:/H9270SO−1=/H9251/H9270e−1. Assuming ballistic grain, /H9270e−1/H11015vF/D=3.4/H110031014s−1. We obtain /H9251/H110151.6/H1100310−4, in excellent agreement with /H9251/H1101510−4in Al thin films.19 IV. SPIN-POLARIZED TUNNELING Now, we discuss magnetoresistance from the spin- polarized tunneling. In the magnetic field range of ±50 mT, G) H) FIG. 1. /H20849A/H20850,/H20849B/H20850, and /H20849C/H20850: sample fabrication steps. /H20849D/H20850: image of Al grains. /H20849E/H20850and /H20849F/H20850: image of a typical sample. /H20849G/H20850and /H20849H/H20850:I-V curves at the base temperature.WEI, MALEC, AND DA VIDOVI Ć PHYSICAL REVIEW B 76, 195327 /H208492007 /H20850 195327-2approximately 90% of the samples do not display any of the tunneling magnetoresistance /H20849TMR /H20850effect. By contrast, we tested about ten tunneling junctions without the embeddedgrains and with similar resistance /H20849empty junctions /H20850at 4.2 K. All of the empty junctions exhibit a significant TMR in thisfield range, comparable to 10%. Approximately one-half of the empty junctions display a simple spin-valve effect. So theabsence of TMR for electron tunneling via grains shows that the spin-dephasing rate T 2−1in 90% of the samples must be much larger than the tunneling rate. Nevertheless, approximately 10% of the samples with em- bedded grains display significant TMR, so the dephasing must be weak, e.g., T2−1must be smaller than or comparable to the tunneling rate in these samples. Here, T2variation among different samples could be explained by magnetic de-fects, such as paramagnetic impurities from the Py layer.Paramagnetic impurities are common sources of dephasing. 20 The defects would be located on the grain surface, since bulkAl does not support paramagnetism. Since the number ofatoms on the surface is relatively small /H20849/H110111000 /H20850, we could occasionally obtain a sample free of impurities. More insight into the nature of T 2in this device will require a more in depth theoretical study. Majority of the samples with nonzero TMR show positive TMR near the Coulomb-blockade conduction threshold; onlyabout 30% of the samples show negative TMR. The sign ofTMR in quantum dots is determined by the interplay be-tween charging effects and spin accumulation. 15,21For any given sample, the data in this paper correspond to the voltagerange within the first step of the Coulomb staircase. In thisrange, the sign of TMR is found to be constant as expected. TMR in our devices usually does not display a simple spin-valve effect. We believe that this is because there arespin-dependent interactions inside the grain, which induce acomplicated TMR even when the magnetic transitions in thedrain and source leads are sharp as expected. For example, arotation of stray magnetic field acting on the grain will alterthe direction of the spin-quantization axis in the grain,thereby changing the conductance. 8A rotation or a switch ofa remote domain can change the tunneling current through the grain via the magnetic field generated by the domain.Similarly, the orientation of the nuclear spin in the grain canchange the quantization axes via the hyperfine interaction. We select only those samples that display a simple spin- valve TMR effect, which is shown in Figs. 3. Figure 3/H20849A/H20850is the TMR of sample 1 at a bias voltage corresponding to thesecond current plateau. TMR is barely resolved in this case,since the current changes by only about 40 fA. We do nothave good data to display TMR at the first current plateau.By comparison, Figs. 3/H20849B/H20850and 3/H20849C/H20850display TMR at bias voltage where the numbers of electron-in-a-box levels ener-getically available for tunneling in are approximately 19 and48, respectively. To facilitate comparisons, the current inter-vals on the vertical axes in Figs. 3/H20849A/H20850–3/H20849C/H20850and3/H20849D/H20850–3/H20849F/H20850 have equal lengths. The main observation in this paper is that /H9004I=I ↑↑−I↑↓is nearly constant with current above a certain current. There ishardly any increase in /H9004Ibetween Figs. 3/H20849B/H20850and3/H20849C/H20850and between Figs. 3/H20849E/H20850and3/H20849F/H20850. This behavior is shown in more detail in Fig. 4/H20849A/H20850and4/H20849B/H20850, which displays /H9004Iversus bias voltage. Here, /H9004Iversus negative bias voltage in sample 1 is fully saturated at the third current plateau; at the second cur-rent plateau, /H9004Iis already at one-half of the saturation value. Similarly, in sample 2, /H9004Ireaches saturation at the second current plateau. Our samples should be contrasted with ordi-nary ferromagnetic tunneling junctions, where /H9004Iis propor- tional to the current over a significantly wider range of biasvoltage. 22,23 V. INTERPRETATION OF THE RESULTS In Coulomb-blockade samples containing magnetic leads, the electrochemical potential difference between the islandand leads can jump when the magnetization in one of theleads changes direction. 10This can lead to a sudden shift in energy levels, producing a jump in current that is constant asa function of bias voltage. The shift in energy levels is seen (A) (B)FIG. 2. /H20849A/H20850and /H20849B/H20850: differential conductance peaks /H20849dark /H20850versus bias voltage and the applied magnetic field in sample 1 at the basetemperature.SATURATION OF SPIN-POLARIZED CURRENT IN … PHYSICAL REVIEW B 76, 195327 /H208492007 /H20850 195327-3as a discontinuity near zero magnetic field in Fig. 2and is /H110110.1 mV. To show that the electrochemical shift is not responsible for the saturation of the spin-polarized current with voltagein our sample, we performed other measurements by sweep-ing the magnetic field both on and between the current pla-teaus, coming up with similar values for the electrochemicalshift. The shift is lower than the average level spacings of 0.8and 2.7 meV for sample 1 and sample 2, respectively. There-fore, since we measured magnetoresistance in the middle ofthe current plateau, the threshold voltage shift should notaffect our measurements of the saturation in /H9004I. To explain I ↑↑−I↑↓=const, we must discuss the relative magnitudes of three rates: /H9270e-ph−1, the rate of energy relaxation from excited to lower energy states by spin-conserving pho-non emission; /H9003 L, the rate electrons tunnel into the grain; and T1−1, the rate of transitions between levels that result in an electron flipping in its spin orientation. Moreover, /H9270e-ph−1is obtained theoretically, the measured I-Vspectrum fixes the tunneling rate, and T1−1is obtained from the saturation in I↑↑−I↑↓with bias voltage. Finally, we must deduce the relative magnitude of T1−1. The rate of spin-flip transitions is expected to be significantly smaller than /H9270e-ph−1.18In this case, the ground state would not necessarily be accessible by energy relaxation. The graincould remain in an excited, spin-polarized state, as sketched in Fig. 4/H20849C/H20850. These spin-polarized excited states are respon- sible for spin accumulation in the antiparallel magnetic con-figuration of the leads. If the relaxation rates for the spin-flip transitions are much smaller than the tunneling rate, thenvarious spin-polarized states would have similar probabili-ties, which are determined by the tunneling rates. In the an-tiparallel configuration of the leads, the probabilities of theexcitations with spin up would be enhanced by 1+ P, and probabilities of the excitations with spin down would be sup-pressed by 1− P, where Pis the spin polarization in the leads. In the parallel configurations, the probabilities of theexcitations with spin up and spin down are the same. In thisregime, I ↑↑−I↑↓is proportional to the current, similar to the usual ferromagnetic tunneling junctions. It is reasonable to expect that the spin-flip rate T1−1/H20849/H9275/H20850 increases rapidly with energy difference /H9275between the initial and the final states.24IfT1−1/H20849/H9275/H20850exceeds the tunneling rate above some /H9275, then the excitations with energy /H11022/H9275will occur with a reduced probability, in the ensemble of statesgenerated by tunneling in and out. Thus, /H9004Iis limited by tunneling via the ground state and those low-lying spin- polarized states, where T 1−1/H20849/H9275/H20850/H11021/H9003 L. Here, /H9004Iversus bias voltage approaches saturation approximately when T1−1/H20849/H9275/H20850=/H9003L, where /H9275is the highest excitation energy in the-22.95-22.90-22.85 I (pA) 60x10-340 20 0 -20 -40 -60 B (mT)-9.00-8.95-8.90 I (pA)-0.95-0.90-0.85 I (pA)0.5 0.4 0.3I (pA) 2.2 2.1 2.0I (pA) 5.1 5.0 4.9I (pA) -40x10-3-20 0 20 40 B(mT)Vbias=-39.2mVVbias=-26.3mVVbias=-13.2mV (A) (B)(D) (C)(E) (F)Vbias=4.2mV Vbias=7.3mV Vbias=15mVSample 1 Sample 2 FIG. 3. /H20849A/H20850–/H20849F/H20850: spin-valve effect in current versus applied magnetic field in two samples at the base temperature. The current magnitude is reduced in the antiparallel state.WEI, MALEC, AND DA VIDOVI Ć PHYSICAL REVIEW B 76, 195327 /H208492007 /H20850 195327-4ensemble of spin-polarized states generated by tunneling in and out: /H9275/H11015/H9254I /H20841e/H20841/H9003L. This is how we determine the spin- relaxation time T1/H20849/H9275/H20850at an energy /H9275in a given sample. In sample 1, /H9004Iis at 50% of the saturation value at the second current plateau, and /H9004Iis saturated at the third cur- rent plateau. At the second current plateau, the spin-relaxation rate of the highest energy excited state generated by tunneling must be close to the tunneling rate. Since thespin relaxation is very rapid in configurations more than 3 /H9254 above the ground state, and N0is even as noted above, the grain spends most of the time among the five configurations shown in Fig. 4/H20849C/H20850:N0,N0+,N0−,N0++, and N0−−. The highest energy spin-polarized states are N0++and N0−−. Thus, T1−1/H208493/H9254/H20850/H11015/H9003L=1.5/H11003106s−1. In sample 2, this analysis leads toT1−1/H208492/H9254/H20850/H11015107s−1. Now we discuss the origin of spin relaxation and its rapid enhancement with the energy difference. Note that the rate ofspin-conserving transitions in Eq. /H208491/H20850increases as /H9275.3We suggest that the electron-phonon transition rates without andwith spin-flip scale by the Elliot-Yafet relation: T1−1/H20849/H9275/H20850 =/H9251/H11032/H9270e-ph−1/H20849/H9275/H20850. This scaling would certainly explain the rapid increase in spin-relaxation rate with excitation energy. In me- tallic films, it is well established that the Elliot-Yafet scalingapplies for both elastic and inelastic scattering processes,with /H9251/H11015/H9251/H11032.19 In sample 1, Eq. /H208491/H20850leads to /H9270e-ph−1/H208493/H9254/H20850/H110154/H110031010s−1. Since T1−1/H208493/H9254/H20850/H110151.5/H11003106s−1, we obtain /H9251/H11032/H110150.4/H1100310−4. Similarly, in sample 2, /H9270e-ph−1/H208492/H9254/H20850/H110153.3/H110031011s−1, and we ob- tain/H9251/H11032/H110150.3/H1100310−4. Here, /H9251/H11032agrees with /H9251/H110151.5/H1100310−4ob- tained earlier, within an order of magnitude. So the ratio of /H9270e-phand T1is in agreement with the Elliot-Yafet scaling. This is an evidence that the spin-flip transitions in Al grainsare driven by the spin-orbit interaction. By this relaxationmechanism, the spin of an electron on the grain is coupled tothe phonon continuum via the spin-orbit interaction. An elec-tron in an excited spin-polarized state relaxes by an emissionof a phonon, which has an angular momentum equal to thedifference between the initial and final electron spins. FIG. 4. /H20849A/H20850and /H20849B/H20850:/H9004I=/H20841I↑↑−I↑↓/H20841versus bias voltage in samples 1 and 2, respectively, at the base temperature. The numbers near the circles indicate how many doubly degenerate electron-in-a-box levels are available for tunneling in. /H20849C/H20850: possible spin-polarized electron configurations caused by electron tunneling in and out, before an electron tunnels in, at the second current plateau, for N0even.SATURATION OF SPIN-POLARIZED CURRENT IN … PHYSICAL REVIEW B 76, 195327 /H208492007 /H20850 195327-5VI. CONCLUSION In summary, we have observed spin-coherent sequential electron tunneling via discrete energy levels of single Algrains. Spin-polarized current saturates quickly as a functionof bias voltage, which demonstrates that only the groundstate and the few lowest excited states can carry spin-polarized current in these samples. Higher excited stateshave a relaxation time shorter than the tunneling time, andthey do not carry spin-polarized current. The spin-relaxationtimes of the low-lying excited states are T 1/H110150.7 and 0.1 /H9262s in two samples. Finally, the ratio of the electron spin-fliptransition rate and the electron-phonon relaxation rate is inquantitative agreement with the Elliot-Yafet scaling ratio, an evidence that the spin-relaxation transitions are driven by thespin-orbit interaction and phonons. ACKNOWLEDGMENTS This work was performed in part at the Georgia-Tech electron microscopy facility. We thank Matthias Braun andMarkus Kindermann for consultation. This research is sup-ported by the DOE under Grant No. DE-FG02-06ER46281and David and Lucile Packard Foundation under Grant No.2000-13874. 1D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. 74, 3241 /H208491995 /H20850. 2O. Agam, N. S. Wingreen, B. L. Altshuler, D. C. Ralph, and M. Tinkham, Phys. Rev. Lett. 78, 1956 /H208491997 /H20850. 3D. Davidovi ćand M. Tinkham, Phys. Rev. Lett. 83, 1644 /H208491999 /H20850. 4J. R. Petta and D. C. Ralph, Phys. Rev. Lett. 87, 266801 /H208492001 /H20850. 5C. T. Black, D. C. Ralph, and M. Tinkham, Phys. Rev. Lett. 76, 688 /H208491996 /H20850. 6M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 /H208491985 /H20850. 7F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Baselmans, and B. J. van Wees, Nature /H20849London /H20850416, 713 /H208492002 /H20850. 8M. Braun, J. Konig, and J. Martinek, Europhys. Lett. 72, 294 /H208492005 /H20850. 9I. Weymann and J. Barnas, Phys. Rev. B 73, 205309 /H208492006 /H20850. 10S. J. van der Molen, N. Tombros, and B. J. van Wees, Phys. Rev. B73, 220406 /H20849R/H20850/H208492006 /H20850. 11W. Wetzels, G. E. W. Bauer, and M. Grifoni, Phys. Rev. B 74, 224406 /H208492006 /H20850. 12A. Cottet and M. S. Choi, Phys. Rev. B 74, 235316 /H208492006 /H20850. 13A. Cottet, T. Kontos, S. Sahoo, H. T. Man, M. S. Choi, W. Belzig, C. Bruder, A. F. Morpurgo, and C. Schonenberger, Semicond.Sci. Technol. 21, S78 /H208492006 /H20850. 14L. Zhang, C. Wang, Y . Wei, X. Liu, and D. Davidovi ć, Phys. Rev. B72, 155445 /H208492005 /H20850. 15F. Ernult, K. Yakushiji, S. Mitani, and K. Takanashi, J. Phys.: Condens. Matter 19, 1652140 /H208492007 /H20850. 16A. Bernand-Mantel, P. Seneor, N. Lidgi, M. Munoz, V . Cros, S. Fusil, K. Bouzehouane, C. Deranlot, A. Vaures, F. Petroff et al. , Appl. Phys. Lett. 89, 062502 /H208492006 /H20850. 17S. Adam, M. L. Polianski, X. Waintal, and P. W. Brouwer, Phys. Rev. B 66, 195412 /H208492002 /H20850. 18Y . Yafet, Solid State Phys. 14,1/H208491963 /H20850. 19F. J. Jedema, M. S. Nijboer, A. T. Filip, and B. J. van Wees, Phys. Rev. B 67, 085319 /H208492003 /H20850. 20G. Bergman, Phys. Rep. 107,1/H208491984 /H20850. 21J. Barnas and A. Fert, Phys. Rev. Lett. 80, 1058 /H208491998 /H20850. 22J. S. Moodera, J. Nowak, and R. J. M. van de Veerdonk, Phys. Rev. Lett. 80, 2941 /H208491998 /H20850. 23S. Zhang, P. M. Levy, A. C. Marley, and S. S. P. Parkin, Phys. Rev. Lett. 79, 3744 /H208491997 /H20850. 24In bulk metals, the spin-orbit scattering rate increases rapidly with electron excitation energy.WEI, MALEC, AND DA VIDOVI Ć PHYSICAL REVIEW B 76, 195327 /H208492007 /H20850 195327-6

PhysRevB.97.195410.pdf

PHYSICAL REVIEW B 97, 195410 (2018) Superlattice-induced minigaps in graphene band structure due to underlying one-dimensional nanostructuration A. Celis,1M. N. Nair,2,1M. Sicot,3F. Nicolas,2S. Kubsky,2D. Malterre,3A. Taleb-Ibrahimi,2and A. Tejeda1,2,* 1Laboratoire de Physique des Solides, CNRS, Université Paris–Sud, Université Paris–Saclay, 91405 Orsay, France 2Synchrotron SOLEIL, Saint-Aubin, 91192 Gif-sur-Yvette, France 3Institut Jean Lamour, UMR 7198 CNRS, Université de Lorraine, 54506 Vandoeuvre lès Nancy, France (Received 22 December 2017; revised manuscript received 12 April 2018; published 8 May 2018) We have studied the influence of one-dimensional periodic nanostructured substrates on graphene band structure. One-monolayer-thick graphene is extremely sensitive to periodic terrace arrays, as demonstrated ontwo different nanostructured substrates, namely Ir(332) and multivicinal curved Pt(111). Photoemission showsthe presence of minigaps related to the spatial periodicity. The potential barrier strength of the one-dimensionalperiodic nanostructuration can be tailored with the step-edge type and the nature of the substrate. The minigapopening further demonstrates the presence of backward scattered electronic waves on the surface and the absenceof Klein tunneling on the substrate, probably due to the fast variation of the potential, of a spatial extent of theorder of the lattice parameter of graphene. DOI: 10.1103/PhysRevB.97.195410 I. INTRODUCTION Quantum confinement and structural superperiodicities are well-known procedures to tailor electronic properties in mas-sive fermion systems [ 1–3]. Superperiodic potentials of spatial period Lcan lead to the apparition of band replicas (umklapps), displaced in the reciprocal space by G=2π/L .A tt h ei n t e r - section between the original band and the replicas (i.e., at theBrillouin zone edge of the new periodicity), minigaps open updue to the backscattering of electrons at the potential barrier [ 4– 6]. In graphene, the situation can be different if Klein tunneling protects the backscattering [ 7]. In this situation, when the barrier is sharper than the electron wavelength, electrons do notbackscatter at normal incidence from the potential barrier, asexperimentally demonstrated [ 8,9]. This particularity of Dirac electrons has been studied in slowly varying superperiodicpotentials [ 10–12], where the spatial variation of the potential is much smaller than the intercarbon distance, in order toavoid intervalley scattering. In this situation, therefore, thechirality of graphene prevents the band-gap opening [ 10,11], although new Dirac points associated with the periodicity areobserved. The influence of electron backscattering in graphene un- der a superperiodic potential can be studied by graphenegrowth on nanostructured substrates. The ultimate monolayerthickness of graphene makes graphene extremely sensitiveto underlying or overlying nanostructuration. It has beenobserved that graphene feels a two-dimensional potential whenit is grown on noble metal [ 13,14], directly attached to the SiC (buffer layer) [ 15–17] or deposited on a BN substrate [10,18], because of the stacking-induced moiré pattern. One- dimensional structurations rely on corrugations [ 19], graphene suspension on nanotrenches [ 20], e-beam induced nanostruc- *antonio.tejeda@u-psud.frtures [ 21], and growth on different vicinal surfaces [ 22–25]. Scanning tunneling spectroscopy has allowed to observe insome cases the appearance of the superlattice Dirac points[18,19,22,26], observed as extra dips in the density of states. On two-dimensional moirés, minigaps have even been observeddirectly in the band structure [ 14,27]. On one-dimensional systems, and despite all the interest that these systems haveevoked [ 10–12,28–32], no gap has ever been observed in direct measurements of the band structure [ 23,24,33]. In this paper, we focus precisely on graphene grown on one-dimensional nanostructured vicinal surfaces. We successfullyidentify minigap openings at the crossing of the Dirac conesalong the superperiodicity direction, making explicit the partialconfinement of the electronic waves and the absence of Kleintunneling. We also quantify the strength of the potential barrierwhen graphene grows on different periodic nanostructures. II. EXPERIMENTAL DETAILS The experiments were carried out in three different ultrahigh-vacuum chambers equipped with low-energy elec-tron diffraction (LEED). We measured at CASSIOPEEbeamline and at the Surface Laboratory of SOLEIL forhigh-resolution angle-resolved photoemission spectroscopy(ARPES) with a spot size of the order of 50 microns, and room-temperature scanning tunneling microscopy (STM), respec-tively, and we performed low-temperature STM measurementsat the Institut Jean Lamour setup (77 K). Photoemissionexperiments were performed at a photon energy of 36 eV and77 K. The substrates were Ir(332) from Surface PreparationLaboratory and a curved multivicinal Pt(111) crystal fromBihurCrystal. The pristine Ir(332) surface is composed of 1.25nm width steps that extend along the [10 1] direction. The macroscopic normal of multivicinal Pt(111) ranges from 0◦on the (111) surface up to 16◦toward the [11 2] and [ 112] direc- tions. Ir(332) was prepared by Ar+sputtering at 1 keV kinetic 2469-9950/2018/97(19)/195410(6) 195410-1 ©2018 American Physical SocietyA. CELIS et al. PHYSICAL REVIEW B 97, 195410 (2018) FIG. 1. (a) Scheme of a curved multivicinal Pt(111) crystal. (b) Step-edge structure of a noble metal with step edges running along the [10 1] direction. Step edges have square (triangular) atomic dispo- sition in the A(B) step edges. (c) Three-dimensional representation of an STM image of graphene on the multivicinal Pt(111) crystal at a vicinal angle of ∼− 7d e g a n d 3 .5±0.5 nm width facets (100 mV , 2.8 nA). The original surface relaxes into (111) terraces ( T)a n d step bunching areas (SB). (d) STM image of graphene on Ir(332)(0.9 V , 0.5 nA). (e) Zoom on a T-SB boundary. The border width is of the order of the lattice parameter of graphene. Some hexagons are superimposed to indicate the continuity of graphene across theboundary. (f) Representation of the superperiodic structure (top) and energetic representation of the potential (bottom). Lis the width of theTand SB regions, bis the varying potential region a few ˚Aw i d e , andU 0is the potential barrier. energy followed by a short annealing at 920 K. Multivicinal Pt(111) crystal was prepared by four-step cycles consisting ofAr +sputtering at 1 keV , annealing at 970 K, Ar+sputtering at 1 keV , and annealing under oxygen atmosphere (1 ×10−7 mbar) for 10 min. A final flash at 1220 K ends the preparation. Graphene is grown by 10 min ethylene exposure to the substratekept at 1070 K. This method is self-limited to 1 ML, as thegrowth needs the catalytic substrate to dissociate ethylene[34–36]. Dirac cones of the differently rotated moirés can be easily studied due to the angular resolution of photoemission. III. RESULTS AND DISCUSSION Figure 1illustrates the one-dimensional nanostructured substrates for graphene growth. We have studied simultane-ously graphene on different superperiodic potentials by usinga multivicinal Pt(111) substrate [Figs. 1(a) and 1(c)] andIr(332) [Figs. 1(d) and1(e)]. The multivicinality of the curved substrate allows us to tune the spatial periodicity and to controlthe potential barrier depending on the facet width and thestep-edge type. Figure 1(b) shows a scheme of the step-edge atomic structure at both sides of the (111) direction. Asteps are characterized by a square disposition of edge atoms, whileB-step atoms are arranged in a triangular fashion. The different atomic coordination at these step edges varies its reactivity. When graphene is grown on the substrates, the original periodicity is modified due to faceting [ 33], so we have deter- mined the spatial periodicity after growth by LEED and STM.Figure 1(c) shows an STM image of the multivicinal Pt(111) at −7 ◦off normal, where a periodic arrangement of terraces ( T) and step-bunching areas (SB) of similar width (3 .5±0.5n m ) is shown. Graphene grows like a carpet covering the steps,as appreciated in the STM images on Figs. 1(d) and 1(e). A sharp boundary between terraces and the step-bunchingareas is appreciated. Graphene grown on these nanostructuredsubstrates feels a one-dimensional potential varying at theT-SB boundaries, as illustrated in Fig. 1(f)and demonstrated in the following. The spatial extension of this potential is muchsmaller than the smooth potential in moiré structures where the mismatch between the atoms in the stacked layers varies progressively. Any change in the graphene band structure induced by the periodic step potential should appear in photoemissionmeasurements. It is therefore crucial to identify the directionwhere the potential varies, which can be determined from theisoenergetic cuts of the band structure. Figure 2(a) shows an isoenergetic cut at −330 meV . Different rotated Brillouin zones (indicated by dotted lines) appear due to the rotational domainsof graphene. These domains correspond to graphene alignedwith the substrate ( R0 domain) or rotated by 25 ◦,3 0◦,o r3 5◦ (R25,R30, and R35 domains, respectively). These domains are explicit in the isoenergetic cut due to the spectral weightarcs at the Kpoints of the rotated Brillouin zones [Fig. 2(a)]. These arcs correspond to partially observed Dirac conesdue to the photoemission matrix elements. These cones aresensitive to the one-dimensional potential along the directionperpendicular to the /Gamma1K of Ir, so its effect is better observed on the R0 domain, which is far apart from spectral features from other rotated domains. Figures 2(b) and 2(c) show the band structure of this domain along /Gamma1K and perpendicular to it. The E(k) maps use a color scale proportional to the second derivative of the photoemission intensity. This representationallows us to observe faint spectral features even when bands arebroad because of the step-width distribution on the substrate. Figure 2(b) shows the dispersion along a direction per- pendicular to /Gamma1K through the Kpoint, i.e., perpendicular to the step edges. The low dispersing spectral features at theFermi level, ∼− 1 and ∼− 1.6 eV indicated by S1,S2, and I, correspond to substrate features [ 37,38]. Coexisting with them, the characteristic dispersion of graphene is observed. TheDirac point is above the Fermi level, as expected for grapheneon noble metal [ 14,38–41], so a band-gap opening there is unobservable by photoemission. Strikingly, the spectral weightdecreases below the Fermi level at two different reciprocalspace locations when the electronic structure is proven perpen-dicular to the step edges [arrows in Fig. 2(b)]. On the contrary, the graphene dispersion is unaffected parallel to the step edges 195410-2SUPERLATTICE-INDUCED MINIGAPS IN GRAPHENE … PHYSICAL REVIEW B 97, 195410 (2018) FIG. 2. Electronic structure of graphene on Ir(332). (a) Isoen- ergetic map of the band structure at −330 meV . The Brillouin zones associated with the different graphene rotations are shown by the dotted lines. “Ir” labels the substrate spectral features [ 38]. dI2/d2Emaps of the dispersion relations E(k) along the directions (b) perpendicular and (c) parallel to the steps [see the continuous lines in (a)]. The insets show the experimental geometry. S1,S2, and Iare substrate-related features. The graphene dispersion exhibits minigaps of∼400 meV ( Eg) perpendicularly to the steps, while no band-gap opening is appreciated in the parallel direction. [Fig. 2(c)], which indicates that the gap opening is related to the one-dimensional periodicity. The dispersion exhibitssome slope changes, as previously observed [ 42,43]. Moreover, theklocations where the intensity decreases are compatible with a translation of the Dirac cone by multiples of G= 2π/L , where Lis the one-dimensional structural periodicity. The one-dimensional surface potential has, therefore, opened FIG. 3. LEED pattern of Ir(332) at 148 eV showing the G1×1 vector of the Ir(111) substrate and the Gvector due to the facet peri- odicity. This Gvector allows us to determine the spatial periodicity. In this case it is L =1.8±0.2 nm along the [1 21] direction. minigaps in graphene band structure. The spatial periodicity Lcan be obtained from STM images and from LEED. In LEED patterns, Lis deduced from the distance between the (1×1) substrate spots and the replicas associated with the step-induced periodicity (Fig. 3). The minigap opening is not exclusive of Ir(332). We have also determined the electronic structure in three differentregions of a multivicinal Pt(111) single crystal [Figs. 4(a)– 4(c)]. Minigaps appear in a reciprocal space position π/L away from the Kpoint as shown in the second derivative images [Figs. 4(a) and4(b)] as well as in the original spectra (Fig. 5). Spatial periodicities of L=3.2 and 3.5 nm exhibit minigaps at different klocations [Figs. 4(a) and 4(b)]. The potential inducing the gap is different from the moiré potentialexisting in the nonvicinal surface [ 14], as the gap openings appear at different kin both cases (Fig. 6). Also, when the step-edge type varies from AtoB[panels (b) and (c), respectively] and therefore the graphene is anchored differentlyto the substrate, the amplitude of the minigap varies. We thusconclude that the minigaps are correlated to the periodicityof the one-dimensional nanostructured substrate and also to itsstrength. The potential strength can be estimated by comparingthe experimental data to the numerical solution of a DiracHamiltonian model [ 44–46]: cos(k xL)=cos(k1a)cos(k2b) +k2 y¯h2v2 f−(V−E)2 ¯h2v2 fk1k2sin(k1a)sin(k2b)(1) with k1=/parenleftBigg [V−E]2 ¯h2v2 f−k2 y/parenrightBigg1/2 (2) k2=/parenleftBigg [V−E]2−U2 0 ¯h2v2 f−k2 y/parenrightBigg1/2 (3) where vfis the Fermi velocity and Vdescribes the exper- imental doping of the system. The potential strength U0b 195410-3A. CELIS et al. PHYSICAL REVIEW B 97, 195410 (2018) FIG. 4. Electronic structure of graphene on multivicinal Pt(111). dI2/d2Emaps of the dispersion relations E(k) for different facet size and orientation angle with respect to the (111) direction: (a) 3 .2±0.1n ma n d −5.5◦,( b )3.5±0.1n ma n d −2◦,a n d( c )3 .5±0.1n ma n d +2◦.T h e Dirac Hamiltonian model is shown as continuous lines, corresponding to a periodicity and potential strength of (a) 3.2 nm and 3.35 eV ˚A, (b) 3.5 nm and 2.8 eV ˚A, and (c) 3.5 nm and 1.8 eV ˚A. The vertical dashed lines indicate the klocation of the band-gap opening ( k0). (d) Spectra of the original photoemission intensity around k0,i n t e g r a t i n gi na krange of 0.1 ˚A−1. Gaps are indicated by arrows. The Fermi level corresponds to 0 eV . depends on the energy barrier U0and its spatial extent b [Fig. 1(f)]. By adjusting our experimental results with this model, we obtain for Ir(332) minigaps of 390 meV and a potential strength of 4.4 eV ˚A. For the different regions of the Pt(111) multivicinal substrate, the minigaps are explained for potential strengths of 3.2 eV ˚A [panel (a)], 2.8 eV ˚A [panel (b)], and 1.8 eV ˚A [panel (c)]. The 2 π/L reciprocal vector determined by photoemission is in agreement with thestructural periodicities observed by LEED and STM, whichare 3.2±0.1, 3.5±0.1, and 3 .5±0.1 nm, respectively. Theseresults allow us to understand the observed minigaps promoted by the different step periodicities and different coupling tothe substrate (Table I). In facets on the same material and with the same step edge but with different widths, a strongerpotential barrier appears for smaller terrace widths, wherethe electronic confinement is enhanced. If we compare thegraphene anchoring to the substrate in surfaces with A- andB- type step edges, we observe a doubling of the potential barrierforAsteps, probably because of a higher graphene-substrate coupling. Finally, we observe that the Ir promotes higher FIG. 5. Energy distribution curves of photoemission intensity. (a) EDCs for graphene on Ir(332). Black markers show the graphene dispersion and green markers show the iridium states I,S1, and S2. The arrows indicate the minigaps. (b) EDCs for graphene on multivicinal Pt corresponding to Fig. 3(c). 195410-4SUPERLATTICE-INDUCED MINIGAPS IN GRAPHENE … PHYSICAL REVIEW B 97, 195410 (2018) FIG. 6. Comparison between gap openings due to moiré and a one-dimensional periodicity. Simulation of gap openings in the one-dimensional periodicity of Fig. 2(a) (blue) and on the moiré of a Ir(111) surface (red). The minigaps due to Ir(111) moiré do not correspond to the gaps by nanostructuration as they appear at different klocations. potential barriers because of the smaller graphene-substrate distance [ 47–49]. In all the previous situations, the physical origin of the gap is puzzling, since the Klein tunneling forbids the backscattering.However, the band gap can be understood by taking intoaccount that Klein tunneling applies only for normal incidenceto the potential barrier. As the system is a two-dimensionallayer feeling a one-dimensional potential, Klein tunneling isnot strict when electrons exhibit an out-of-normal incidence tothe potential barrier. An alternative or supplementary origin tothe band gap is the surface moiré. The moiré exists already onflat surfaces and promotes a band gap, which was attributed to amoiré-induced inequivalency between the AandBsublatticesTABLE I. Gap opening in the studied materials and vicinalities. Substrate θ◦Terrace width (nm) Step type Gap (eV) Pt multi. −5.5 3.2 A 0.36 Pt multi. −2 3.5 A 0.29 Pt multi. +2 3.5 B 0.20 Ir (332) +11 3.5 B 0.39 of graphene [ 14]. Vicinality may modify and amplify the moiré gap, possibly indicated by the presence of several Fouriercomponents in the potential, associated with the vicinality andthe moiré. The potential consists indeed of different Fouriercomponents, as E g∼E/prime g, whereas a single Fourier component potential should promote gaps of decreasing amplitude. IV . SUMMARY In conclusion, we have studied the effects of one- dimensional potentials on graphene by growing graphene onIr(332) and on multivicinal Pt(111) substrates, demonstratingthe band-gap tailoring in graphene band structure. Photoemis-sion shows the presence of minigaps related to the spatialperiodicity, whose strength can be varied with the step-edgetype, the nature of the substrate, and in a more general way asa function of the anchoring of the graphene to the substrate.The minigap opening further demonstrates the presence ofbackward scattered electronic waves on the surface and theabsence of Klein tunneling on the substrate, probably due tothe fast-varying potential, of a spatial extent of the order ofthe lattice parameter of graphene. These results show a wayto tailor graphene band structure in a solid-state system andpossibly in analogous graphene metamaterials. ACKNOWLEDGMENTS We acknowledge the financial support of the Agence Na- tionale de la Recherche (France) under contract CoRiGraph.The authors acknowledge L. Brey for useful discussions andthe support from F. Bertran and P. Le Fèvre and all theCASSIOPEE staff during ARPES beam time. [1] M. F. Crommie, C. P. Lutz, and D. M. 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PhysRevB.75.245206.pdf

Thermal evolution of defects in as-grown and electron-irradiated ZnO studied by positron annihilation Z. Q. Chen *and S. J. Wang Hubei Nuclear Solid Physics Key Laboratory, Department of Physics, Wuhan University, Wuhan 430072, People’ s Republic of China M. Maekawa, A. Kawasuso, and H. Naramoto Advanced Science Research Center, Japan Atomic Energy Agency, 1233 Watanuki, Takasaki, Gunma 370-1292, Japan X. L. Yuan and T. Sekiguchi Nanomaterials Laboratory, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan /H20849Received 31 January 2007; published 13 June 2007 /H20850 Vacancy-type defects in as-grown ZnO single crystals have been identified using positron annihilation spectroscopy. The grown-in defects are supposed to be zinc vacancy /H20849VZn/H20850-related defects, and can be easily removed by annealing above 600 °C. VZn-related defects are also introduced in ZnO when subjected to 3 MeV electron irradiation with a dose of 5.5 /H110031018cm−2. Most of these irradiation-induced VZnare annealed at temperatures below 200 °C through recombination with the close interstitials. However, after annealing ataround 400 °C, secondary defects are generated. A detailed analysis of the Doppler broadening measurementsindicates that the irradiation introduced defects and the annealing induced secondary defects belong to differentspecies. It is also found that positron trapping by these two defects has different temperature dependences. Theprobable candidates for the secondary defects are tentatively discussed in combination with Raman scatteringstudies. After annealing at 700 °C, all the vacancy defects are annealed out. Cathodoluminescence measure-ments show that V Znis not related to the visible emission at 2.3 eV in ZnO, but would rather act as nonradi- ative recombination centers. DOI: 10.1103/PhysRevB.75.245206 PACS number /H20849s/H20850: 61.72.Ji, 78.70.Bj, 71.55.Gs I. INTRODUCTION ZnO is a II-VI compound semiconductor with wide band gap of 3.4 eV and large exciton binding energy of 60 meV.These features enable its potential application in short wave-length light emitting devices. Due to the successful fabrica-tion of large area ZnO single crystals, 1there is a growing interest in the investigation of this materials in recentyears. 2,3Among these investigations, study of defects is one of the most important subjects because of the strong influ-ence of these defects on the electrical and optical properties.Due to deviation from the chemical stoichiometry, variousdefects are introduced inevitably after crystal growth. It isknown that the native defects, such as zinc interstitial /H20849Zn i/H20850 and oxygen vacancy /H20849VO/H20850, might be the reason for the intrin- sicn-type conduction in undoped ZnO.4,5These defects will also compensate acceptors and cause difficulty in producing p-type ZnO.6On the other hand, defects may also act as nonradiative recombination centers and degrade the lightemission efficiency. There have been several theoretical investigations on na- tive defects in ZnO, 6–9which calculated the formation en- ergy and ionization level of various defects. A variety ofexperiment methods have also been used to characterize thedefects in ZnO, such as electron paramagnetic resonance/H20849EPR/H20850, photoluminescence, cathodoluminescence /H20849CL/H20850, and deep level transient spectroscopy /H20849DLTS /H20850. These methods provided detailed information about defects from differentaspects. However, direct discrimination of defects in ZnO isalways a difficult problem because of the limitation of thesemethods. Until now, there are still some fundamental prob-lems in which no clear consensus seems to exist. For ex- ample, the EPR signal of g=1.96 was attributed to oxygen vacancy, 10but some others argued that the oxygen vacancy signal was at g=1.99.11The green luminescence at about 2.4 eV was correlated with oxygen vacancy by manyresearchers, 12–15but it was also attributed to other defects such as VZn,16,17Zni,18,19OZn,20,21and even Cu impurities.22 DLTS revealed a major defect level L2 at Ec−0.3 eV in the as-grown ZnO, but the assignment of this level is also ratherdifficult. 23 Positron annihilation spectroscopy /H20849PAS/H20850has been proven to be a powerful tool for the study of vacancy-type defects insemiconductors. 24When positrons are emitted into materials, they will first lose their energy /H20849thermalization /H20850in a few picoseconds, then diffuse inside the lattice. If there arevacancy-type defects within the range of positron diffusionlength, positron will be trapped by these vacancies. Due tothe reduced electron density and lower probability of posi-tron annihilating with high momentum core electrons, theannihilation characteristics of positrons at defects will be dif-ferent from the delocalized bulk state: the positron lifetimewill become longer and the Doppler broadening of annihila-tion radiation will be narrower. Therefore, by measuring suchpositron annihilation characteristics, we can get detailed in-formation about defects. Up to now, several works have beenconducted on the study of defects in ZnO by PAS. 25–33 Except for defects in the as-grown state, they can also be introduced by many other ways, such as irradiation or plasticdeformation. High energy electron irradiation is one of theeffective ways to introduce simple and number controllabledefects. This will help us study defects and their thermalPHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 1098-0121/2007/75 /H2084924/H20850/245206 /H208499/H20850 ©2007 The American Physical Society 245206-1stability in more detail. On the other hand, study of radiation-induced defects is also important for the space ap-plication of ZnO, because these defects play an importantrole in the degradation of devices used in the space environ-ment. A full understanding of the resistance to radiation isthus necessary before its space application. Nevertheless, thestudy of radiation effect in ZnO using PAS is still veryscarce 25,27,30,31and far to be comprehensive. In this paper, we studied the vacancy defects introduced by 3 MeV electronirradiation in ZnO single crystals and investigated their ther-mal recovery process using positron annihilation, Ramanscattering, and cathodoluminescence measurements. II. EXPERIMENT ZnO samples are hydrothermal grown single crystals pur- chased from the Scientific Production Company /H20849SPC Good- will/H20850. They are undoped ntype with /H208490001 /H20850orientation. Two series of sample are used in this work, which are purchasedat different times. The first series /H20849sample A /H20850was annealed from room temperature up to 1000 °C. No electron irradia-tion experiment was conducted for these samples. The sec-ond series is further divided into two categories according todifferent treatments. One sample /H20849sample B /H20850was annealed at 1000 °C before electron irradiation, and another sample/H20849sample C /H20850was directly irradiated with electrons without preannealing. Annealing was performed in N 2or O 2atmo- sphere for a period of 2 h. Electron irradiation was per-formed at room temperature using a dynamitron accelerator.The samples were irradiated on Zn face. The incident energyof electrons was 3 MeV and the average fluence rate was4.5/H1100310 13cm−2s−1. The total electron dose was 5.5 /H110031018cm−2. During electron irradiation, samples were cooled by a water cooling system, and the temperature waskept below 70 °C. The irradiated samples were annealedfrom 100 to 700 °C in nitrogen ambient. Each annealingtime was 30 min. Positron lifetime measurement was performed using a fast-fast coincidence system with time resolution of about210 ps. Conventional Doppler broadening spectra were mea-sured using a high purity /H20849HP/H20850-Ge detector with energy reso- lution of about 1.3 keV at 511 keV. The Doppler spectra arecharacterized by the SandWparameters, which are definedas the ratio of the central region /H20849511±0.85 keV /H20850and wing region /H20849511±3.4 to 511±6.8 keV /H20850to the total area of the 511 keV annihilation peak, respectively. In this work, the S andWparameters were normalized to the defect-free bulk value. Therefore, S/H110221o r W/H110211 means existence of vacancy defects. Coincidence Doppler broadening measurements were performed using two HP-Ge detectors. The 22Na posi- tron source was sandwiched between two pieces of identicalsample for measurements. Temperature dependent measure-ments were carried out by mounting the source-sample sand-wich to the cold finger of a He-cycling refrigerator. The tem-perature range is from 5 to 295 K. The positron sourceintensity was about 10 /H9262Ci for room temperature measure- ment and about 30 /H9262Ci for low temperature measurement, respectively. CL was measured at room temperature using a modified scanning electron microscope /H20849TOPCON DS-130 /H20850.34A monochromator with grating of 100 lines/mm /H20849Jobin Yvon HR320 /H20850and a charge-coupled device were used for the de- tection of spectra. The electron beam energy was 5 keV andbeam current was about 1 nA. The acquisition time for eachmeasurement was 5 s. Micro-Raman-scattering measure-ments were performed using the Nanofinder spectrometer.The 488.0 nm line of an Ar +-ion laser was used for excita- tion. The incident laser power was /H110111 mW and the measure- ment time for each spectrum was 60 s. All the above mea-surements were repeated at least one time to ensurereproducibility of the data. III. RESULTS AND DISCUSSION A. Grown-in defects in ZnO In this paper, we studied three ZnO samples which have different properties and are subjected to different treatments.The sample treatments and measured positron lifetime datafor the three samples are listed in Table I. The positron life- time spectra are analyzed by PATFIT program.35In all the three as-grown samples, decomposition of positron lifetimespectra shows only one lifetime component. In the firstsample /H20849sample A /H20850, the positron lifetime is about 189 ps, while in samples B and C, the positron has the same lifetimevalue, i.e., 183 ps. The only one lifetime component does notTABLE I. ZnO samples and the positron lifetime data measured before and after annealing or electron irradiation with dose of 5.5 /H110031018cm−2. ZnO Treatment/H92701 /H20849ps/H20850/H92702 /H20849ps/H20850I2 /H20849%/H20850/H9270av /H20849ps/H20850 Sample A As grown 189 0 189 1000 °C annealed 182 0 182 Sample B As grown 183 0 183 1000 °C annealed 183 0 183 Electron irradiated 152 230 77.3 212 Sample C As grown 183 0 183 Electron irradiated 157 231 71.9 210CHEN et al. PHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 245206-2mean that there are no defects that trap positrons. Obviously, a higher positron lifetime in sample A suggests the existenceof vacancy defects. Figure 1shows the variation of positron lifetime with an- nealing temperature in sample A. The positron lifetime hasno change up to annealing temperature of 400 °C. However,above 400 °C, it begins to decrease, drops to about 181 ps at600 °C, and then remains nearly constant up to 1000 °C.The Doppler broadening Sparameter shows similar change as shown in Fig. 1. Another as-grown sample A was annealed in O 2ambient at 900 °C, and the positron lifetime also de- creases to about 181 ps. For sample B, after annealing at1000 °C in N 2, the positron lifetime shows no change /H20849Table I/H20850. These results confirm that sample A contains vacancy- type defects that trap positrons, and they can be removedafter annealing above 600 °C. The value of 182±1 ps mightbe the positron bulk lifetime. Figure 2shows the positron lifetime and Sparameter as a function of temperature from 5 to 300 K measured forsample A before and after annealing. For the as-grownsample, the positron lifetime or Sparameter shows notable temperature dependence. The positron lifetime decreases toabout 180 ps at /H110115 K. However, after annealing at 900 °C, such temperature dependence is greatly reduced, and the pos-itron lifetime shows only about 2 ps decrease, which mightbe attributed to lattice thermal expansion effect. The temperature behavior exhibited in as-grown sample A is generally explained by the competitive positron trappingby negative ions at low temperatures. 36As the positron binding energy of these negative ions is rather small/H20849/H11021100 meV /H20850, they are called shallow positron traps. The positron annihilation characteristics at these centers are very close to the bulk state. With increasing temperature, posi-trons are detrapped from these centers and shift to vacancy-type defects. Therefore, the positron lifetime or Sparameter increases.However, another possible reason for such temperature behavior cannot be excluded, i.e., the change of the positrontrapping rate with temperature. By supposing that the ther-mally activated trapping of positrons is analog to the captureof free carriers through multiphonon emission, 37–39the posi- tron trapping rate will increase with temperature. Therefore,detailed fundamental studies are still needed to find out theunderlying mechanism for such temperature behavior. In any case, the temperature dependence of positron an- nihilation parameter /H20849lifetime or Sparameter /H20850appears only in the existence of positron trapping by vacancy defects. With-out vacancy trapping centers to compete with, the shallowtraps cannot be observed. Therefore, the disappearance oftemperature dependence after annealing sample A at 900 °Cconfirms that all the vacancies are annealed out, and positronbulk lifetime in the hydrothermally grown ZnO sample isabout 182±1 ps. The origin of the positron trapping centers in as-grown sample A might be V Zn-related defects, as VOis not an effec- tive positron trap.30In hydrothermally grown ZnO crystals, hydrogen is easily incorporated into the sample duringgrowth. These hydrogen atoms may fill the zinc vacancy siteand form complexes as suggested by Lavrov et al. 40The positron lifetime in VZnwill then be reduced due to the oc- cupation of hydrogen and become closer to the positron bulklifetime in ZnO. This may explain why the defect lifetimecomponent cannot be resolved from the lifetime spectrum.However, the hydrogen will become unstable at high tem-peratures and desorption of hydrogen occurs when annealingthe ZnO sample at around 500–700 °C. 41,42This will restore the open space of VZnand cause an increase of positron lifetime. However, in our ZnO sample, we observed a de-crease of positron lifetime after annealing. This reveals thatthe defects are not hydrogen-vacancy complexes. The con-centration of V Znmay be too low, so that the defect related positron lifetime cannot be resolved.FIG. 1. Average positron lifetime /H9270avandSparameter as a func- tion of annealing temperature measured for ZnO sample A. τ FIG. 2. Temperature dependences of the average positron life- time and Sparameter in ZnO sample A measured before and after annealing at 900 °C.THERMAL EVOLUTION OF DEFECTS IN AS-GROWN AND … PHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 245206-3B. Electron-irradiation-induced defects and their thermal evolution Samples B and C have the same positron lifetime of 183 ps in the as-grown state, which is close to the lifetimevalue of annealed sample A. The annealing of sample B doesnot cause any change of positron lifetime /H20849Table I/H20850. So, we believe that no or very few vacancy defects exist in these two samples which trap positrons. After electron irradiation witha dose of 5.5 /H1100310 18cm−2, the average positron lifetime in- creases up to about 212 and 210 ps for samples B and C,respectively. The Doppler broadening Sparameter also in- creases to about 1.016 and 1.012 /H20849Figs. 3and4/H20850. This means that vacancy-type defects are introduced by electron irradia-tion. After irradiation, we can decompose the lifetime spectrainto two components, and /H92702corresponds to positron lifetime at vacancy defects, which is about 230 ps. This value is inagreement with that reported by Tuomisto et al. 30As the ratio of /H92702//H9270bis around 1.26, this corresponds to positron lifetime at monovacancies. In ZnO, both VZnandVOmay be introduced after electron irradiation. However, as described above, VOmight be less possible to be positron trapping centers. Therefore, the va-cancies observed by positrons in the electron-irradiated ZnOare also V Zn-related defects. The positron trapping rate /H9260was calculated according to the two state trapping model: /H9260 =/H20849/H9270av−/H9270b/H20850//H20849/H92702−/H9270av/H20850//H9270b. According to the trapping model, there is a quantitative relationship between trapping rate and defect concentration: /H9260=/H9262Cd, where /H9262is the specific posi- tron trapping rate. Taking the value of /H9262=3/H110031015s−1for VZnas suggested by Tuomisto et al. ,30the electron- irradiation-induced vacancy concentration is about 2.6/H1100310 17and 2.1 /H110031017cm−3for samples B and C, respec- tively. This corresponds to a defect introduction rate of0.04–0.05 cm −1, in agreement with the result obtained by Tuomisto et al.31The two electron-irradiated samples are subjected to an- nealing up to 700 °C. The change of average positron life-time and Sparameter after annealing is shown in Figs. 3and 4. The annealing behavior is nearly the same for these two samples. A fast decrease of both positron lifetime and Spa- rameter is seen below 200 °C. After that, both of them beginto increase and reach a maximum value at 350–400 °C.Above 400 °C, the positron lifetime and Sparameter de- crease again and approach the bulk value at 700 °C. The decrease of positron lifetime and Sparameter below 200 °C is attributed to the recombination of V Zn-Zn iclose Frenkel pairs. Tomiyama et al.25observed similar annealing stage for the 28 MeV electron-irradiated ZnO, which wasaround 150–200 °C. This was in agreement with our result.They attributed this annealing stage to the recovery of irra-diation induced oxygen vacancies, but that is not likely, asoxygen vacancies cannot be detected by positrons. On theother hand, oxygen vacancy may remain stable up to400 °C. 43The same annealing stage at 50–150 °C was also observed by Brunner et al.27in the 2 MeV electron- irradiated ZnO, and it was attributed to the annealing of Znmonovacancies. As for the abnormal increase of the positron lifetime after annealing at 350–400 °C, there are two possible reasons.One reason might be the shift of the Fermi level to a highervalue after annealing of the irradiated sample; therefore, thevacancy defects become more negatively charged, whichleads to an increase of positron trapping rate. In order tocheck such possibility, we measured the electrical conductiv-ity of the irradiated sample after annealing by Hall effect.The results are listed in Table II. The resistivity of the sample before electron irradiation is about 250 /H9024cm, but it becomes semi-insulating after irradiation. Even after 400 °C anneal-ing, it still keeps high resistivity. Only after annealing at600 °C, the resistivity recovers to that of the unirradiatedτ FIG. 3. Average positron lifetime and Sparameter as a function of annealing temperature in electron-irradiated ZnO /H20849sample B /H20850 with dose of 5.5 /H110031018cm−2. The annealing time was 30 min. τ FIG. 4. Average positron lifetime and Sparameter as a function of annealing temperature in electron-irradiated ZnO /H20849sample C /H20850 with dose of 5.5 /H110031018cm−2. The annealing time was 30 min.CHEN et al. PHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 245206-4state, i.e., 275 /H9024cm. This means that the Fermi level does not move after annealing up to 400 °C, which is pinned at around the midgap position. Therefore, the increase of posi-tron lifetime is not due to the shift of the Fermi level. By excluding the possibility of the Fermi level movement, the second reason for the increase of positron lifetime afterannealing at 400 °C would be the introduction of some ad-ditional open volume defects. These defects might be formeddue to defect reaction at high temperatures. Nevertheless, wedo not observe much change of the defect lifetime /H92702after formation of the secondary defects. However, this does notmean that the secondary defect is still V Zn. After annealing at 400 °C, the defect concentration is largely reduced, so thedecomposition of positron lifetime spectrum is not reliable.Even if the positron lifetime for the secondary defect is dif-ferent from that of V Zn, we cannot get its accurate value. On the other hand, the secondary defects might be VZn-impurity complexes, which have positron lifetime close to that of VZn. Brunner et al.27also observed an increase of positron life- time and Sparameter after annealing the electron irradiated ZnO at around 300 °C, and they believe that it is due toagglomeration of nonannealed defects to more stable com-plexes, such as V Zn-donor complex.27 A detailed analysis of the correlation between SandW parameters from the Doppler broadening measurementsgives us more complementary information about the defectevolution that occurred during annealing. Figure 5shows the S-Wplot obtained from the annealing measurements for electron-irradiated samples B and C. In the annealing pro-cess, the S-Wdata are concentrated on two different lines, which correspond to two different types of defects, i.e., theelectron-irradiation-induced V Zn/H20849defect 1 /H20850and the annealing produced secondary defects /H20849defect 2 /H20850. It is interesting to note that in samples B and C, the S-Wdata are concentrated on the two same lines. Therefore, this confirms that anneal-ing at 350–400 °C produces additional vacancy defectswhich are different from V Znintroduced by irradiation. To further identify the difference between the irradiation- induced VZnand the thermally generated secondary defects, we also conducted the coincidence Doppler broadening mea-surements for electron-irradiated sample C. The peak tobackground ratio of the measured spectrum is higher than10 5. The ratio curves are shown in Fig. 6. The reference sample is unirradiated sample C. It is clear to see that theratio curves at the low momentum region for the asirradiatedand 400 °C annealed sample are nearly the same; thus, the Sparameter has nearly the same value as seen in Figs. 3and4. However, the high momentum region shows much differ-ence. After 400 °C annealing, the center of the valley in thehigh momentum region shifted from /H1101120/H1100310 −3m0cto lower than 10 /H1100310−3m0c, and becomes much smaller. This coincides with the higher Wparameter after annealing, indi- cating different defect species. After annealing above 400 °C, the positron lifetime and S parameter show a decrease again. It can be inferred that theremaining V Znare removed in this stage through migration into sinks. The secondary defects created at around 400 °Calso become unstable and are annealed out at 700 °C as bothpositron lifetime and Sparameter decrease to the bulk value. We measured temperature dependence of the positron life- time and Doppler broadening spectra for the irradiatedsample B before and after annealing. The result is shown inFig.7. For the asirradiated sample, the average positron life- time first remains nearly constant below 100 K, and thenTABLE II. Resistivity /H9267in the ZnO sample /H20849sample B /H20850before and after electron irradiation and annealing determined from theHall measurements. Sample treatment /H9267 /H20849/H9024cm/H20850 Type Unirradiated 247 n As irradiated SI200 °C annealed SI400 °C annealed 6.7 /H1100310 4SI 600 °C annealed 275 n FIG. 5. S-Wplot from the Doppler broadening measurement in electron-irradiated ZnO samples B and C. FIG. 6. Ratio curve of the Doppler broadening spectrum of electron-irradiated ZnO sample C before and after annealing mea-sured using a coincidence Doppler broadening spectrometer. Thereference sample is the unirradiated ZnO sample C.THERMAL EVOLUTION OF DEFECTS IN AS-GROWN AND … PHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 245206-5starts to increase with temperature. Afterward, it tends to be saturated above 250 K. The Sparameter shows similar tem- perature behavior. These results suggest that the positrontrapping by vacancy defects is temperature dependent, pos-sibly due to the trapping by shallow positron traps at lowtemperatures. The negatively charged antisites or interstitialslike O Znand O iwhich are produced by irradiation might act as such shallow traps. After annealing the irradiated sample at 400 °C, where we observe the production of secondary defects, the tempera-ture dependence of positron lifetime and Sparameter is, however, quite different from that of the asirradiated sample.There is an increase of positron lifetime at around100–180 K when the temperature decreases. The Sparam- eter also shows similar change at this temperature range.This means that positron trapping by the secondary defects isenhanced below 180 K, or the secondary defects undergo acharge state transition and become more negatively chargedat lower temperatures, which causes an increase of positronlifetime. Below 100 K, the negative ions begin to competewith deep traps, and positron lifetime shows a decrease. Werepeat the same temperature dependence measurement forthe irradiated sample C after annealing at 400 °C, and ob-served quite similar result, which is shown in Fig. 8. There- fore, from the temperature dependence of the positron anni-hilation parameters, it suggests again that the defects createdafter annealing at 400 °C are different from the irradiation-induced V Zn. After annealing the irradiated sample at 700 °C, both pos- itron lifetime and Sparameter show very slight increase or nearly no change with temperature /H20849Fig. 7/H20850. This confirms that all the vacancy defects have been removed by annealing. At present, we cannot clarify the species of the secondary defects. In order to understand more deeply the thermal evo-lution of the electron-irradiation-induced defects, we alsoperformed Raman scattering measurements. Figure 9shows the Raman spectra for the electron-irradiated ZnO /H20849sample B/H20850before and after annealing. The detailed assignment of the Raman peaks in ZnO can be found in many literature 44,45and also our previous paper.42The primary peak at 437 cm−1is the high frequency E2phonon mode, which represents the wurtzite structure of ZnO. The peak at 331 cm−1is due to the second order phonon. After electron irradiation, a broad peakat 575 cm −1appears. This broad peak is obviously inducedτ FIG. 7. Temperature dependences of the average positron life- time and Sparameter in electron-irradiated ZnO /H20849sample B /H20850before and after annealing. τ FIG. 8. Temperature dependences of the average positron life- time and Sparameter in electron-irradiated ZnO /H20849sample C /H20850before and after annealing. FIG. 9. Annealing effect on the Raman spectra measured for the electron-irradiated ZnO /H20849sample B /H20850.CHEN et al. PHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 245206-6by the defects produced by electron irradiation because of the relaxation of the Raman selection rules.44 After annealing up to 200 °C, the broad peak at 575 cm−1 shows very small decrease. This indicates that the defects arestable at least up to 200 °C. After 300 °C annealing, suchpeak becomes much weaker, and at 400 °C, it decreases tothe same level as that of the as-grown sample. Figure 10 shows the ratio of the integrated intensity of the broad peakat 575 cm −1to the sum of the 575 and 437 cm−1peaks. It is clear that the broad peak intensity begins to decrease at300 °C and reaches the value of the as-grown sample at400 °C. As the positron annihilation measurements showthat most of the V Zn-related defects disappear at temperatures below 200 °C, the broad Raman peak is apparently not re-lated to V Zn. This means that the defect is invisible to posi- trons. The broad peak at 575 cm−1has been suggested to be related to oxygen vacancies or zinc interstitials by manyresearchers, 46–48as they found that such broad peak will be enhanced in an oxygen deficient condition. The Raman measurements show that the 575 cm−1peak begins to recover above 200 °C and disappears at around400 °C. This is in good agreement with the result that theoxygen vacancies are annealed at 400 °C in electron-irradiated ZnO measured by Vlasenko and Watkins. 43There- fore, the possible candidate for this defect would be oxygenvacancy. The annealing behavior shows that V Obecomes mobile at 200–400 °C, which coincides with the tempera-ture for the production of secondary defects. Thus, we mayassume that while some of the V Odisappear through recom- bination with their interstitials or migration into sinks, manyof them might also take part in the following defect reac-tions: V O⇒VZn+Z n O, /H208491/H20850 VZn+VO⇒VZnVO. /H208492/H20850 Therefore, formation of VZn−Zn Ocomplexes or agglom- eration of VZninto divacancies is the possible candidate for the secondary defects, which result in the increase of posi-tron lifetime and Sparameter after annealing at 350–400 °C. However, further experiments and theoretical calculationsare still needed to confirm such speculation.C. Correlation between defects and luminescence centers The cathodoluminescence spectra measured for sample A in the as-grown state and after annealing at 900 °C areshown in Fig. 11. In the as-grown sample, there is only one weak peak at about 3.3 eV. This is the ultraviolet /H20849UV/H20850emis- sion due to the recombination of free excitons. The ratherweak UV emission indicates that a large number of nonradi-ative recombination centers exist in the as-grown ZnOsample. These nonradiative recombination centers are mostprobably grown-in defects, which suppress both UV anddeep level /H20849visible /H20850emission. After annealing sample A in N 2at 900 °C, the UV emis- sion peak is strongly enhanced. The peak height increases byalmost 20 times. The visible emission also appears, which islocated at around 2.3 eV. This emission obviously originatesfrom the deep level defects. When sample A is annealed inO 2ambient, both UV and visible emission also show in- crease. However, the UV emission is much lower comparedwith the N 2annealed one, while the visible emission is stron- ger. This indicates that annealing atmosphere has some influ-ence on the formation of deep level defects. Figure 12presents the CL spectra measured for ZnO sample C before and after electron irradiation and annealing.For the as-grown sample, the UV and visible emission inten-sities are higher than those of sample A. After irradiation, theUV emission peak is weakened, while the deep level emis-sion shows an increase. Annealing of the irradiated sampleup to 300 °C causes a gradual recovery of the light emission.However, after annealing at 400 °C, the visible emission at/H110112.3 eV shows abrupt increase. This reveals again that de- fect reaction really happened during annealing, which resultsFIG. 10. Ratio of the integrated 575 cm−1peak intensity to the sum of the 437 and 575 cm−1peaks as a function of annealing temperature for the electron-irradiated ZnO sample B. FIG. 11. Annealing effect on the cathodoluminescence spectra measured for the as-grown ZnO /H20849sample A /H20850.THERMAL EVOLUTION OF DEFECTS IN AS-GROWN AND … PHYSICAL REVIEW B 75, 245206 /H208492007 /H20850 245206-7in the formation of secondary vacancy defects at around 400 °C. The visible emission becomes even stronger afterannealing at 700–800 °C, and thereby the UV emission be-comes much weaker. Combining the positron annihilation and cathodolumines- cence results, we can find that there is no correlation betweenthe deep level emission and zinc vacancy. In the as-grownsample A, the deep level emission is too weak to be seen inthe spectrum. It becomes much stronger after annealing at900 °C. Contrarily, the positron measurements show the re-duction of V Znafter annealing. In the as-grown ZnO sample C, which has nearly no positron trapping by VZn, however, the deep level emission is stronger than that of sample A.Furthermore, in the electron-irradiated ZnO /H20849for example, sample C /H20850, the annealing process of deep level emission is also different from that of V Zn. Thus, VZnis not related to the deep level emission at 2.3 eV in ZnO. This defect wouldrather be responsible for the nonradiative recombination cen- ters. When these defects exist in ZnO, both the UV and vis-ible emission will be suppressed. Thus, we can explain theweak CL signal in as-grown sample A, the enhancement afterannealing, and also the stronger CL signal in as-grownsample C. In the electron-irradiated sample C, though theirradiation introduces nonradiative recombination centers,deep level emission centers are also introduced; thus, the UVemission is weakened, but the visible emission is enhanced. As for the candidates for the deep level emission centers, our results may provide some suggestions. When the as-grown ZnO sample was annealed in O 2ambient, the forma- tion of deep level centers is favored. This suggests that thedeep level defects might be associated with oxygen, for ex-ample, the antisite O Znas suggested by Lin et al. ,20or inter- stitial O i. These defects are easier to be formed during an- nealing in O 2atmosphere. Tuomisto et al. also found that the negative-ion-type defects such as O ior O Znhad ionization levels close to 2.3 eV, and thus are possibly involved in thegreen luminescence. 31 IV. CONCLUSION Defects in the as-grown and electron-irradiated ZnO crys- tals were studied by positron annihilation spectroscopy. Thegrown-in defects in ZnO seen by positron are V Zn-related defects and can be removed by annealing above 600 °C.Electron irradiation also introduces V Znin ZnO. Annealing below 200 °C causes partial recovery of these vacancies.However, after annealing at around 400 °C, additional va-cancy defects are produced. The S-Wcorrelation and coinci- dence Doppler broadening study reveals that these two va-cancy defects are not the same category. All the vacancydefects are annealed out at around 700 °C. The zinc vacancyin ZnO is found to be not related to the deep level emissionat 2.3 eV, but would rather be responsible for the nonradia-tive recombination centers. ACKNOWLEDGMENTS This work was partly supported by the Program for New Century Excellent Talents in University and the NationalNatural Science Foundation of China under Grant Nos.10475062 and 10075037. *chenzq@whu.edu.cn 1D. C. Look, D. C. Reynolds, J. R. Sizelove, R. L. Jones, C. W. Litton, G. Cantwell, and W. C. Harsch, Solid State Commun. 105, 399 /H208491998 /H20850. 2S. J. Pearton, D. P. Norton, K. Ip, Y. W. Heo, and T. Steiner, Prog. Mater. Sci. 90, 293 /H208492005 /H20850. 3U. 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PhysRevB.79.205209.pdf

Giant magnetoresistance of magnetic semiconductor heterojunctions N. Rangaraju,1Pengcheng Li,1and B. W. Wessels1,2,* 1Department of Materials Science and Engineering and Materials Research Center, Northwestern University, Evanston, Illinois 60208, USA 2Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60208, USA /H20849Received 22 July 2008; revised manuscript received 2 January 2009; published 18 May 2009 /H20850 The giant magnetoresistance characteristics of magnetic III–V semiconductor p-nheterojunctions are de- scribed. The origin of the extremely large positive magnetoresistance /H208492680% /H20850observed at room temperature and at a field of 18 T is attributed to efficient spin-polarized carrier transport. The magnetocurrent ratio of thejunction saturates with magnetic field. The field dependence of the magnetoresistance points to the existence ofa paramagnetic component, which determines the degree of spin polarization of the junction current. This workindicates that highly spin-polarized magnetic semiconductor heterojunction devices that operate at room tem-perature can be realized. DOI: 10.1103/PhysRevB.79.205209 PACS number /H20849s/H20850: 85.75. /H11002d, 73.40.Kp, 75.47. /H11002m, 75.50.Pp I. INTRODUCTION Semiconductor spintronics offer many unique capabilities that can potentially revolutionize electronics and comput-ing. 1,2A bipolar semiconductor spintronic junction has been proposed as a building block for various magnetoelectronicand magneto-optical device architectures via control over thespin degree of freedom of free carriers. 3Potential advantages include large magnetoresistance /H20849MR /H20850effects, control of magnetic properties by charge injection, and amplification.4 Spintronics can potentially lead to integration of both infor-mation processing and storage. Theoretical calculations ofthe transport properties of ferromagnetic p-nheterojunctions showed that the current through a device would increase in amagnetic field. 5The predicted increase is attributed to the presence of spin-polarized carriers due to the splitting in thevalence and conduction bands as result of giant Zeeman ef-fect. The achievement of magnetic semiconductor/semicon-ductor junctions, however, that operate at room temperaturewith a high degree of spin polarization remains elusive. While there has been extensive theoretical work on dilute magnetic semiconductor /H20849DMS /H20850heterojunctions much less is known about their experimental transport properties. 1,3–5A DMS n-n*-nheterojunction consisting of paramagnetic ZnBeMnSe, as the spin aligner, and the nonmagnetic semi-conductor ZnBeSe was fabricated and its junction character-istics measured. 6A large positive magnetoresistance of 25% at 1 T and 4 K was observed. The origin of this positivemagnetoresistance was attributed to the suppression of con-duction in one of the spin channels in the nonmagnetic semi-conductor. Spin-polarized electrons are injected from theDMS material and the resistance offered by the nonmagneticmaterial increases because only one spin channel is activeinstead of two. In addition, /H20849Ga,Mn /H20850As/GaAs Zener p-ndi- odes have been fabricated that have a large magnetoresis-tance with a relative change in current up to 8% /H20849Ref. 7/H20850and also have shown to emit circularly polarized light. 8,9These devices, however, are operated only at temperatures up to40 K. Our group has previously reported on the transport prop- erties of InMnAs/InAs p-nheterojunctions in fields as highas 9 T where InMnAs is the magnetic semiconductor. 10 InMnAs when grown by metalorganic vapor phase epitaxy has been shown to be ferromagnetic up to 330 K.11,12The high Curie temperature was attributed to Mn acceptor com-plexes. 13,14A giant positive magnetoresistance up to 1300% at room temperature was observed in these devices at a mag-netic field of 9 T. The magnetoresistance of these devicescould be attributed to scattering of free carriers due to aninhomogeneous distribution of magnetic Mn ions at or nearthe depletion region. 15A small, negative magnetoresistance of less than 1% is observed in the InMnAs thin films at lowtemperature and low magnetic fields related to reduced spinscattering. However, increasing the temperature leads to asmall positive magnetoresistance presumably related to a de-crease in mobility. 16This suggests that spin scattering may be also be the origin of the positive junction magnetoresis-tance. 16 Furthermore, magnetocapacitance measurements on these InMnAs/InAs junctions have recently been reported and in-dicate that spin transport is important. 17Magnetocapacitance as high as 7% was observed at 300 K that increased withmagnetic field. It was attributed to a giant Zeeman effect andto the presence of spin-polarized carriers. These spin-polarized carriers were due to splitting of the valence andconduction band in the magnetic semiconductor layer. 17 These measurements also indicated the presence of two mag-netic components, one ferromagnetic and another paramag-netic in the InMnAs layer. Previous superconducting quan-tum interference device /H20849SQUID /H20850measurements on InMnAs films also support the presence of two different magneticcomponents. 11Here we describe the magnetotransport prop- erties of the InMnAs/InAs heterojunctions in magnetic fieldsof up to 18 T over the temperature range of 80–300 K andpresent a physical model for the origin of the giant junctionmagnetoresistance. Taking into account the splitting of thebands due to the giant Zeeman effect we model the junctionelectrical characteristics and the effect of magnetic field. II. EXPERIMENT The InMnAs/InAs heterojunctions were fabricated by de- positing 100 nm of InMnAs on /H20849001 /H20850n-type InAs substratesPHYSICAL REVIEW B 79, 205209 /H208492009 /H20850 1098-0121/2009/79 /H2084920/H20850/205209 /H208495/H20850 ©2009 The American Physical Society 205209-1as previously described in Ref. 10. The carrier concentration in the n-type substrate was 2.6 /H110031016cm−3while the p-side carrier concentration was of the order of 1018cm−3. Photo- lithography was used to define circular Ti/Au contacts with a300 /H9262m diameter. Current-voltage characteristics were measured with the magnetic field applied parallel to the direction of currentflow through the mesa. Currents from −2 to 25 mA weresourced while the voltage was measured. These characteris-tics were measured at the MilliKelvin laboratory at the Na-tional High Magnetic Field Laboratory. A magnetic fieldfrom zero to 18 T was applied using a superconducting mag-net and the I-Vcharacteristics were measured at each field and at different temperatures. III. MODEL A modified diode equation /H20851Eq. /H208491/H20850/H20852was previously shown to describe the behavior of InMnAs/InAs hetero-diodes in a magnetic field. 10The current Iis given by I=I0exp/H20873qVA /H9257kBT/H20874exp/H20873−qIR 0 /H9257kBT/H20874exp/H20873−qIR /H20849H/H20850 /H9257kBT/H20874. /H208491/H20850 In Eq. /H208491/H20850,qis the charge of an electron, R0is the junction zero-field series resistance, R/H20849H/H20850is the magnetic-field- dependent series resistance, /H9257is the junction ideality factor, Tis the temperature, kBis Boltzmann’s constant, and VAis the applied voltage. For the InMnAs/InAs heterojunctions, ahighly conductive accumulation layer can potentially form atthen-type InAs interface /H20851Fig. 1/H20849a/H20850/H20852, and can influence transport. 18 The junction magnetoresistance is defined by10 MR /H20849%/H20850=/H20873dV/H20849H/H20850 dI−dV/H208490/H20850 dI/H20874/H11003100/H20882dV/H208490/H20850 dI, /H208492/H20850 where the derivative dV/H20849H/H20850/dIis calculated at a constant current for different applied magnetic fields. To explain the magnetic-field dependence of the magne- toresistance of a magnetic semiconductor/nonmagnetic semi-conductor junction a two channel conduction model is pro-posed. In this model the valence and conduction bands in the magnetic semiconductor layer are split due to the giant Zee-man effect, each forming two bands consisting of spin-upand spin-down carriers. 5As a result the holes in the valence band and electrons in the conduction band are spin polarized.The band structure of the p-type injection layer is shown in Fig.1/H20849b/H20850. For the present study, the Fermi level in the p-type layer is assumed to be at the top of zero-field valence band.As the applied magnetic field increases the magnetization ofthe spin injector layer increases. This leads to an increase inthe density of states of the spin-down hole band at the Fermilevel and as a consequence their number increases. As themagnetic field increases, the spin-down hole concentrationincreases while that of spin-up holes decreases leading to anincrease in spin polarization. Theory using Boltzmann statis-tics predicts that the spin-polarized current density for such adevice is proportional to exp /H20849 /H9264/kBT/H20850where /H9264depends on the amount of splitting in the band.4The parameter /H9264in turn depends on the magnetic field and the gfactor of the mag- netic semiconductor and is equal to g/H9262BH. The value of gis assumed to be constant. This assumption leads to the calcu-lation of an effective gover this range of magnetic fields. For the present study we also assume that in the junction thereare two spin channels that are in parallel that determine thetotal conductance. 6The conductance Gat a constant current for the heterojunction is thus described by the followingequation: G=1 2 cosh /H20849/H9264/kBT/H20850/H20853G1exp /H20849−/H9264/kBT/H20850+G2exp /H20849/H9264/kBT/H20850/H20854. /H208493/H20850 In Eq. /H208493/H20850,G1exp /H20849−/H9264/kBT/H20850andG2exp /H20849−/H9264/kBT/H20850are the con- ductances for holes with aligned and antialigned spins, re-spectively, the term 2 cosh /H20849 /H9264/kBT/H20850is a normalization factor where at zero field G=/H20849G1+G2/H20850/2. The terms exp /H20849−/H9264/kBT/H20850 and exp /H20849/H9264/kBT/H20850take into account the change in the popula- tion of the spin split bands with field. As the magnetic field isincreased, the number of carriers in the minority spin bandincreases while the number of carriers in the majority spinband decreases. For this model we assume a priori thatG 1 /H11022G2. This assumption is consistent with recent theoretical studies on magnetoresistance of dilute magnetic semiconduc-tors at high magnetic fields. 19In that study it was shown that a positive magnetoresistance results from scattering due tospatial disorder of magnetic ions and locally enhanced spinsplitting of the band. The mobility of carriers depends ontheir polarization. For G 1/H11022G2, a decrease in the total con- ductance with field /H20849positive magnetoresistance /H20850is predicted and is attributed to the exponential decrease in the conduc-tance of the majority spin channel with a concurrent increasein the conductance of the minority spin channel. Since bipolar devices exhibit highly nonlinear I-Vcharac- teristics, the magnetoresistance will depend on whether it ismeasured at a constant current or constant voltage. By ana-lyzing the various measures of magnetoresistance, we candifferentiate between the physical processes that lead to thegiant magnetoresistance effects. The relative change in thecurrent /H20849magnetocurrent ratio /H20850with magnetic field has alsoFIG. 1. /H20849a/H20850Band diagram for InMnAs/InAs p-nheterojunction. /H20849b/H20850Schematic of band structure for InMnAs which shows spin split- ting of the bands and the position of the Fermi level.RANGARAJU, LI, AND WESSELS PHYSICAL REVIEW B 79, 205209 /H208492009 /H20850 205209-2been previously used as a measure of magnetoresistance.7,20 The ratio /H20851MR I/H20849%/H20850/H20852at a constant voltage is defined as fol- lows: MR I/H20849%/H20850=/H20851I/H208490/H20850−I/H20849H/H20850/H20852/H11003100 /I/H208490/H20850, /H208494/H20850 where I/H20849H/H20850is the current measured at a field H. IV. RESULTS AND DISCUSSION The I-Vcharacteristics of the p-nInMnAs/InAs hetero- junctions are shown in Fig. 2for different magnetic fields and at a temperature of 300 K. The inset shows the I-V curves at zero and 18 T, respectively, with the currents plot-ted on the log scale to show the differences of the currents athigh fields. The forward bias junction I-Vcharacteristics are clearly exponential when there is no magnetic field applied,consistent with Eq. /H208491/H20850. 10An increase in the magnetic field, however, leads to flattening of the characteristics and an ap-parent loss of its exponential character. Indeed at high fieldstheI-Vbehavior appears to be nearly Ohmic. The junction I-Vcharacteristics and its magnetic-field dependence are de- scribed by Eq. /H208491/H20850up to and including fields of 18 T. The conductance of the junction as a function of magnetic field is shown in Fig. 3. The conductance for different cur- rents is calculated by determining tangents to the current-voltage curves at a constant current. We see that the conduc-tance initially sharply decreases with field, but at fieldshigher than 9 T the conductance approaches a constant value.The dependence of the conductance on current at a low mag-netic field results from the internal resistance of the diode,which is a function of bias. As the magnetic field is in-creased, the current-voltage behavior is nearly linear and thisleads to the convergence of the measured conductance fordifferent values of the currents. To explain the observed magnetoresistance a two channel spin transport model was applied. We fit the conductancedata at 300 K to the expression given by Eq. /H208493/H20850. Shown in the inset of Fig. 3is the conductance data for a junction current of 5 mA and the theoretical fit where G 1,G2, and g are the free parameters. We can see that there is good agree-ment between theory and experimental data. Values extractedfrom the fit are 0.335 and 0.013 S for G 1andG2, respec- tively, assuming an effective gfactor of 95. The large g factor is consistent with the presence of a giant Zeeman ef-fect. High gfactors have been reported previously in II–VI DMS materials. For comparison, the gvalue for CdMnSe is greater than 500. A large gfactor is expected to result in giant magnetoresistance effects in magnetic/nonmagneticsemiconductor heterojunctions. 21 The low conductance at high fields is attributed to the conduction mostly via the low mobility spin channel. Thedifference in the conductance for the two spin channels ispresumably a result of the spin polarity dependant mobilityof DMS materials. 19It is also noted that at low fields, the junction conductance model deviates from the experimentalresults for junction currents of 10 and 15 mA. This deviationis due to the current dependence of the junction resistance atzero magnetic field. The zero-field junction conductivity in-creases with current. The deviation may also result from thepresence of other competing conduction channels such astunneling. 22 The magnetoresistance of the device at a constant current of 15 mA at temperatures of 80, 190, and 300 K is shown inFig. 4. At 300 K, the low bias region of the magnetoresis- tance has a nonlinear dependence on field, while at magneticfields greater than 1.5 T, a linear magnetoresistance is ob-served. At an applied field of 18 T, the magnetoresistancedefined by Eq. /H208492/H20850has a value of 2680% while at lower temperatures of 190 and 80 K, the value of magnetoresis-tance decreases to 710% and 405%, respectively. At lowertemperatures, both the turn-on voltage for the junction andits internal resistance are much larger at zero field. As theparameter MR /H20849%/H20850is taken relative to the zero-field value, consequently the change is smaller at lower temperatures. The field dependence of the magnetocurrent ratio at a constant voltage given by Eq. /H208494/H20850is shown in Fig. 5.A ta/s52 /s48 /s51 /s48 /s50 /s48 /s49 /s48 /s48/s67 /s117 /s114 /s114 /s101 /s110 /s116 /s40 /s109 /s65 /s41 /s49 /s46 /s53 /s49 /s46 /s48 /s48 /s46 /s53 /s48 /s46 /s48 /s86 /s111 /s108 /s116 /s97 /s103 /s101 /s40 /s86 /s41/s48 /s46 /s48 /s48 /s48 /s49/s48 /s46 /s48 /s48 /s49/s48 /s46 /s48 /s49 /s49 /s46 /s53 /s49 /s46 /s48 /s48 /s46 /s53 /s48 /s46 /s48/s48 /s84 /s49 /s56 /s84 /s48 /s84/s57 /s84 /s49 /s56 /s84 FIG. 2. /H20849Color online /H20850Current-voltage curves at magnetic fields from 0 to 18 T at 300 K. The inset /H20849a semilog plot /H20850is shown to point to the large variation in the I-Vcharacteristics with field./s48 /s46 /s51 /s48 /s48 /s46 /s50 /s53 /s48 /s46 /s50 /s48 /s48 /s46 /s49 /s53 /s48 /s46 /s49 /s48 /s48 /s46 /s48 /s53/s67 /s111 /s110 /s100 /s117 /s99 /s116 /s97 /s110 /s99 /s101 /s40 /s83 /s41 /s49 /s53 /s49 /s48 /s53 /s48 /s65 /s112 /s112 /s108 /s105 /s101 /s100 /s70 /s105 /s101 /s108 /s100 /s40 /s84 /s41/s48 /s46 /s49 /s54 /s48 /s46 /s49 /s52 /s48 /s46 /s49 /s50 /s48 /s46 /s49 /s48 /s48 /s46 /s48 /s56 /s48 /s46 /s48 /s54 /s48 /s46 /s48 /s52 /s48 /s46 /s48 /s50/s67 /s111 /s110 /s100 /s117 /s99 /s116 /s97 /s110 /s99 /s101 /s40 /s83 /s41 /s49 /s53 /s49 /s48 /s53 /s48 /s65 /s112 /s112 /s108 /s105 /s101 /s100 /s70 /s105 /s101 /s108 /s100 /s40 /s84 /s41/s69 /s120 /s112 /s101 /s114 /s105 /s109 /s101 /s110 /s116 /s97 /s108 /s68 /s97 /s116 /s97 /s70 /s105 /s116/s51 /s48 /s48 /s75 /s40 /s49 /s53 /s109 /s65 /s41 /s51 /s48 /s48 /s75 /s40 /s49 /s48 /s109 /s65 /s41 /s51 /s48 /s48 /s75 /s40 /s53 /s109 /s65 /s41 FIG. 3. /H20849Color online /H20850Conductance of the junction vs magnetic field calculated for currents of 5, 10, and 15 mA. /H20849Inset: fit to conductance of junction vs applied field at 300 K. The current is5 mA. /H20850GIANT MAGNETORESISTANCE OF MAGNETIC … PHYSICAL REVIEW B 79, 205209 /H208492009 /H20850 205209-3bias of 0.13 V where this quantity is calculated, the current though the device is high at low magnetic fields and dropsrapidly as the magnetic field is increased. A similar trend isseen at other bias voltages. We observe that at magneticfields greater than 9 T, the magnetocurrent ratio starts tosaturate. To understand the origin of the magnetoresistance the relative change in the current with magnetic field given byEq. /H208494/H20850has also been measured and analyzed. Saturation of the magnetocurrent ratio with field /H20849Fig. 5/H20850is observed. Similar behavior was previously observed in a magneticsemiconductor heterojunction and was attributed to the satu-ration of the magnetization in the magnetic semiconductorlayer. 20The saturation is attributed to a spin polarization ap- proaching 100%. As can be seen a very high magnetic field is required /H20849at least 9 T in the present case /H20850to saturate the magnetocurrent ratio. The high magnetic field required for complete satura-tion indicates the InMnAs has a large paramagnetic compo-nent in its magnetization. The coexistence of both ferromag-netism and paramagnetism has been previously observed inthese materials. 11,17Furthermore if the ferromagnetism in the layer is carrier mediated, a reduction in this componentwould be expected in the junction space charge region due tolack of carriers. While the magnetocurrent ratio shown in Fig. 5was mea- sured at 0.13 V , increasing the bias also shows a similardependence of the magnetocurrent with magnetic field. Thisis expected due to the exponential behavior of the I-Vchar- acteristics observed with and without magnetic field. As tosaturation of the magnetocurrent ratio it should be noted thatit would be expected to saturate just above the coercive field if only a ferromagnetic component is present in the InMnAslayer. 20Saturation of paramagnetism, on the other hand re- quires very high magnetic fields. The magnetocurrent and itsfield dependence indicate that the spin polarization continuesto increase up until very high magnetic fields. V. CONCLUSION In summary, the magnetoresistance of InMnAs/InAs p-n heterojunctions was measured for magnetic fields of up to 18T. Giant magnetoresistance effects /H208492680% at 18 T /H20850are ob- served at 300 K. The positive giant magnetoresistance is at-tributed to conduction of spin-polarized carriers due to thepresence of spin-split bands. High magnetic fields are re-quired to saturate the junction magnetoresistance measuredat a constant voltage. These results indicate that paramagnet-ism of the magnetic semiconductor layer is responsible forthe spin-polarized carriers at high magnetic fields. This workindicates that highly spin-polarized, all-semiconductor junc-tion devices that operate at room temperature can be realized,potentially leading to a new class of spintronic devices forinformation storage and processing. ACKNOWLEDGMENTS We would like to thank Alexey V . Suslov at the National High Field Laboratory for assistance with the measurements.This work was supported by AFOSR under Grant No.FA9550-07-1-0381. Extensive use of MRSEC facilities un-der Grant No. DMR-0520513 is also acknowledged./s49 /s48 /s48 /s56 /s48 /s54 /s48 /s52 /s48 /s50 /s48 /s48/s77 /s82/s73/s40 /s37 /s41 /s49 /s53 /s49 /s48 /s53 /s48 /s65 /s112 /s112 /s108 /s105 /s101 /s100 /s70 /s105 /s101 /s108 /s100 /s40 /s84 /s41 FIG. 5. Magnetocurrent ratio /H20851MR I/H20849%/H20850/H20852vs magnetic field at 300 K and at a voltage of 0.13 V calculated using Eq. /H208494/H20850/H20849the line is a guide to the eyes /H20850./s50 /s53 /s48 /s48 /s50 /s48 /s48 /s48 /s49 /s53 /s48 /s48 /s49 /s48 /s48 /s48 /s53 /s48 /s48 /s48/s77 /s97 /s103 /s110 /s101 /s116 /s111 /s114 /s101 /s115 /s105 /s115 /s116 /s97 /s110 /s99 /s101 /s40 /s37 /s41 /s49 /s53 /s49 /s48 /s53 /s48 /s65 /s112 /s112 /s108 /s105 /s101 /s100 /s70 /s105 /s101 /s108 /s100 /s40 /s84 /s41/s51 /s48 /s48 /s75 /s56 /s48 /s75/s49 /s57 /s48 /s75 FIG. 4. /H20849Color online /H20850Magnetoresistance vs magnetic field at I=15 mA. The data are reported at T=300, 190, and 80 K.RANGARAJU, LI, AND WESSELS PHYSICAL REVIEW B 79, 205209 /H208492009 /H20850 205209-4*b-wessels@northwestern.edu 1H. Akinaga and H. Ohno, IEEE Trans. Nanotechnol. 1,1 9 /H208492002 /H20850. 2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y . Chtchelkanova, and D. M.Treger, Science 294, 1488 /H208492001 /H20850. 3I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 /H208492004 /H20850. 4I. Zutic, J. Fabian, and S. Das Sarma, Phys. Rev. Lett. 88, 066603 /H208492002 /H20850. 5N. Lebedeva and P. Kuivalainen, J. Appl. Phys. 93, 9845 /H208492003 /H20850. 6G. Schmidt, G. Richter, P. Grabs, C. Gould, D. Ferrand, and L. W. Molenkamp, Phys. Rev. Lett. 87, 227203 /H208492001 /H20850. 7H. Holmberg, N. Lebedeva, S. Novikov, J. Ikonen, P. Kuiva- lainen, M. Malfait, and V . V . Moshchalkov, Europhys. Lett. 71, 811 /H208492005 /H20850. 8Y . Ohno, D. K. Young, B. Beschoten, F. Matsukura, H. Ohno, and D. D. Awschalom, Nature /H20849London /H20850402, 790 /H208491999 /H20850. 9E. Johnston-Halperin, D. Lofgreen, R. K. Kawakami, D. K. Young, L. Coldren, A. C. Gossard, and D. D. Awschalom, Phys.Rev. B 65, 041306 /H20849R/H20850/H208492002 /H20850.10S. J. May and B. W. Wessels, Appl. Phys. Lett. 88, 072105 /H208492006 /H20850. 11A. J. Blattner and B. W. Wessels, J. Vac. Sci. Technol. B 20, 1582 /H208492002 /H20850. 12A. J. Blattner, P. L. Prabhumirashi, V . P. Dravid, and B. W. Wessels, J. Cryst. Growth 259,8 /H208492003 /H20850. 13P. T. Chiu, B. W. Wessels, D. J. Keavney, and J. W. Freeland, Appl. Phys. Lett. 86, 072505 /H208492005 /H20850. 14P. T. Chiu and B. W. Wessels, Phys. Rev. B 76, 165201 /H208492007 /H20850. 15S. J. May, Ph.D. thesis, Northwestern University, 2006. 16S. J. May, A. J. Blattner, and B. W. Wessels, Phys. Rev. B 70, 073303 /H208492004 /H20850. 17N. Rangaraju and B. W. Wessels, J. Vac. Sci. Technol. B 26, 1526 /H208492008 /H20850. 18C. Affentauschegg and H. H. Wieder, Semicond. Sci. Technol. 16, 708 /H208492001 /H20850. 19M. Foygel and A. G. Petukhov, Phys. Rev. B 76, 205202 /H208492007 /H20850. 20F. Tsui, L. Ma, and L. He, Appl. Phys. Lett. 83, 954 /H208492003 /H20850. 21T. Dietl, Handbook of Semiconductors /H20849North-Holland, Amster- dam, 1994 /H20850. 22W. Yang and K. Chang, Phys. Rev. B 72, 075303 /H208492005 /H20850.GIANT MAGNETORESISTANCE OF MAGNETIC … PHYSICAL REVIEW B 79, 205209 /H208492009 /H20850 205209-5

PhysRevB.88.245308.pdf

PHYSICAL REVIEW B 88, 245308 (2013) Experimental verification of the surface termination in the topological insulator TlBiSe 2using core-level photoelectron spectroscopy and scanning tunneling microscopy Kenta Kuroda,1,*Mao Ye,2,3Eike F. Schwier,2Munisa Nurmamat,1Kaito Shirai,1Masashi Nakatake,2Shigenori Ueda,4 Koji Miyamoto,2Taichi Okuda,2Hirofumi Namatame,2Masaki Taniguchi,1,2Yoshifumi Ueda,5and Akio Kimura1,† 1Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526, Japan 2Hiroshima Synchrotron Radiation Center, Hiroshima University, 2-313 Kagamiyama, Higashi-Hiroshima 739-0046, Japan 3State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China 4Synchrotron X-ray Station at SPring-8, National Institute for Materials Science, Hyogo 679-5148, Japan 5Kure National College of Technology, Agaminami 2-2-11, Kure 737-8506, Japan (Received 22 July 2013; revised manuscript received 10 October 2013; published 23 December 2013) The surface termination of the promising topological insulator TlBiSe 2has been studied by surface- and bulk-sensitive probes. Our scanning tunneling microscopy has unmasked for the first time the unusual surfacemorphology of TlBiSe 2obtained by cleaving, where islands are formed by residual atoms on the cleaved plane. The chemical condition of these islands was identified using core-level spectroscopy. We observed thalliumcore-level spectra that are strongly deformed by a surface component in sharp contrast to the other elements. Wepropose a simple explanation for this behavior by assuming that the sample cleaving breaks the bonding betweenthallium and selenium atoms, leaving the thallium layer partially covering the selenium layer. These findings willassist the interpretation of future experimental and theoretical studies of this surface. DOI: 10.1103/PhysRevB.88.245308 PACS number(s): 73 .20.−r, 79.60.Bm, 68 .37.Ef, 72.15.Lh I. INTRODUCTION Topological insulators are known as a new class of materials where the bulk is an insulator with a band inversion due to a strong spin-orbit coupling, which results in the emergenceof topological surface states (TSSs) forming Dirac-cone-likebands with high spin polarizations. 1,2Owing to the existence of their nontrivial surface state, the surfaces of topologicalinsulators provide a playground to realize intriguing proximityeffects such as the Majorana fermion at the interface with a superconductor 3and topological magnetoelectric effect with magnetic materials,4–6being of great significance for both the fundamental physics and their applications. So far, several binary chalcogenides, such as Bi 2Se3and Bi2Te3, are known as the prototypical topological insulators. They form a tetradymite crystal structure with a stacking ofquintuple layers (QLs) (X-Bi-X-Bi-X atomic layers, X: Seor Te) weakly coupled via van der Waals forces as shown inFig. 1(a). Due to the weak bonds between the QLs, the outer- most chalcogenide layer of a QL terminates the surface aftercleaving, and it is therefore thought that dangling bonds willnot emerge. The scenario that layered materials cleave alongtheir van der Waals gap is widely accepted. 7Actually, abinitio calculation8with the chalcogenide termination model has reproduced their characteristic electronic structures obtainedby angle-resolved photoemission spectroscopy (ARPES). 9,10 After finding the tetradymite-type topological insulators, TlBiSe 2has been discovered to be a topological insulator,11–13 which forms a three-dimensional crystal structure without van der Waals gaps, as shown in Fig. 1(b). The crystal is built up by stacking [-Tl-Se-Bi-Se-]nlayers along the caxis of its hexagonal unit cell. In this compound, the observed TSSfeatures an in-gap Dirac point, which is well isolated fromthe continuum states of the bulk. 12This may prove essential for the realization of an ambipolar gate-control in spin-currentdevices as well as for studying novel topological properties, )b( )a( Bi2Se 3 TlBiSe 2 Bi SeBiTl SeQL QL FIG. 1. (Color online) Crystal structures of (a) Bi 2Se3and (b) TlBiSe 2. (left) Part of the crystal structures where 10 layers are shown. (right) Side view from the [100] direction. The crystalstructure of Bi 2Se3and TlBiSe 2with the quintuple layers (QLs) stacking in the sequence Se-Bi-Se-Bi-Se and the atomic layer stacking in the sequence -Tl-Se-Bi-Se-Tl- without the van der Waals gap. where the Dirac point is required to be close to the Fermi energy.1,2,4–6In this respect, TlBiSe 2is known as one of the most promising topological insulators today. However, dueto the absence of any van der Waals gaps, and the overallcovalent and ionic natures of the interatomic bonds, it isunknown which layer terminates the surface after cleaving.A theoretical study predicted that trivial surface states shouldcoexist with the nontrivial TSS due to dangling bonds atthe cleaved surface, and even the TSS will be affected bythe surface termination. 14However, no trivial surface states have been observed, and the observed band structures ofthe TSS apparently are the same among previous ARPESmeasurements. 11–13This discrepancy indicates that the actual cleaved surface would be different from the assumed one inthe calculation, which is in sharp contrast to the situation forthe prototypical topological insulators. Therefore, studying 245308-1 1098-0121/2013/88(24)/245308(7) ©2013 American Physical SocietyKENTA KURODA et al. PHYSICAL REVIEW B 88, 245308 (2013) the surface termination is an important key to solving the discrepancy with a precise surface model in TlBiSe 2. In this paper we present an experimental approach to determine the surface termination of TlBiSe 2by using a combination of scanning tunneling microscopy (STM) andcore-level photoelectron spectroscopy (CL-PES). First, wepresent STM results revealing that residual islands are formedon the cleaved surface. This fact makes a case for a preferentialcleaving plane between the Tl and Se layers, leaving a Sesurface covered with the residual Tl atoms forming the islands.This particular situation is strongly supported by employingPES on the core levels of TlBiSe 2using different surface and bulk sensitive photon energies ( hν) ranging from the vacuum ultraviolet (VUV) to the hard x-ray (HAX) regime. II. EXPERIMENT Single crystalline samples of TlBiSe 2were grown by the Bridgman method using high purity elements (Bi, Se:99.999%, Tl: 99.99%). The materials were heated in anevacuated quartz ampule above the melting point around800 ◦C and kept at the constant temperature for two days. It was then cooled down to 100◦C over a period of 20 days. The STM experiment was conducted at 78 K in an ultrahigh vacuum with a base pressure better than 1 ×10−8Pa using a low temperature scanning tunneling microscope (OmicronNano Technology). STM images were acquired in the constant-current mode with a bias voltage ( V s) applied to the sample. The samples were cleaved in situ at room temperature as well as low temperature below 100 K and then transferred to themeasurement chamber. Surface-sensitive PES in the VUV regime (VUV-PES) was conducted at BL-7 of the Hiroshima Synchrotron Ra-diation Center with a hemispherical photoelectron analyzer(VG-SCIENTA SES 2002) at 80 K in ultra-high vacuumconditions better than 1 ×10 −8Pa. The excitation energies ranged from 24 to 250 eV . The total energy resolution forthe VUV-PES measurement was obtained by Fermi edgefitting of polycrystalline Au and set to be approximatelyE//Delta1E =1000. The samples were cleaved in situ at 80 K. In order to reduce photoelectron diffraction effect the VUV-PESspectra were recorded at normal emission in angle integratedmode, covering an angular window of ±7 ◦along the ¯/Gamma1-¯M direction. Bulk-sensitive PES with hard x-ray (HAX-PES) was measured at BL15XU of SPring-815at room temperature. An excitation energy of hν=5.95 keV was used, which results in an inelastic mean free-path (IMFP) for the photoelectrons aslarge as 50 ˚A. This corresponds to a probing depth of up to 30 atomic layers for TlBiSe 2. The total energy resolution for the HAX-PES measurement was set as ∼250 meV . The samples were cleaved in air and immediately installed into the vacuumchamber. III. RESULTS AND DISCUSSION First, the cleaved surface of TlBiSe 2has been directly examined by STM. Figure 2(a) shows a typical large area image acquired at a sample bias voltage of Vs=− 0.8V .A couple of step edges are clearly seen in the cleaved plane,whose height profile along A–B is shown in Fig. 2(b).T h e Height (A)Vs=-0.8 V; I t= 0.20 nA V s=-1.8 V; I t= 0.15 nA B A)c( )a( (d) (b)hgiH hgiH woL woL 4 2 0 00 1.5 1.0 3.0 3.5 2.0 0.551015 80 120 Distance (nm)Height (nm) Area (arb. units)1.1 A~A 04 0B FIG. 2. (Color online) (a) Large scale STM image of TlBiSe 2; (b) height profile of the steps on the cleaved surface of TlBiSe 2along the A–B line in (a); (c) small area of the cleaved surface, showing atomically resolved structure of the residual clusters on the cleavedsurface; (d) histogram of the surface shown in (c). observed terraces have a typical width larger than 20 nm. It can also be seen that their step height is around 8 ˚A, which is comparable to the five-layer thickness (-Tl-Se-Bi-Se-Tl-).The uniform step height strongly indicates that identicalatomic layers terminate the different terraces at the cleavedsurface. In Fig. 2(c), we show a small-area STM image on a single terrace with atomic resolution. It can be clearly seenthat the terrace is made up by residual islands on top of thesurface layer. It should be noted that one can see the closepacked lattice structure within the islands, which correspondsto the TlBiSe 2(001) surface. These features indicate that the atoms forming the islands remain on top of the surface layerafter the sample is cleaved. Note that the residual islands havebeen observed also for the surface cleaved at low temperaturebelow 100 K (not shown), which means that the formation ofthe islands does not depend on the cleaving temperature. It iscertainly surprising that a single TSS has been identified byARPES measurement 11–13regardless of the existence of the islands found by the STM. This unambiguously represents thetopologically nontrivial character of this surface state. To qualify the island coverage of the surface, we show that the height distribution of the topographic STM image ina histogram in Fig. 2(d), where two peaks can be identified. Note that the height difference between two peaks representsthe average heights of the island in the topographic image. Inaddition the area of the each peak represents the area of islands(right peak) and the rest of the area (left peak), respectively.The height of the islands is found to be 1.1 ˚A, which is smaller than the interlayer distance along the caxis∼1.8˚A. 16This difference may indicate the altered local density of states ofthe islands including an apparent change in height and aninward relaxation of the islands toward the bulk. In both cases,it can be safely concluded that the islands are formed by asingle monolayer of atoms. An analysis of the area of thesurface and island peaks shows that almost half of the surfaceis covered by islands. One can notice that bright spots arevisible on the terraces [arrows in Fig. 2(a)]. By their height, we consider them to be of different surface terminations than 245308-2EXPERIMENTAL VERIFICATION OF THE SURFACE ... PHYSICAL REVIEW B 88, 245308 (2013) )b( )a( TlSe BiSe TlSe Tl layerTl islands Bi layer Se layerSe layerAB FIG. 3. (Color online) (a) Schematic image for the cleaving of TlBiSe 2from the [100] direction. Dashed lines with marked A (B) shows the torn bonding with the lower (upper) Se layer. (b) Proposedcleaved surface model. the islands. We note, however, that the occupancy of such difference termination on the terrace is small compared to theislands. Therefore, we can safely disregard their influence onthe surface properties. In the crystal structure of TlBiSe 2shown in Fig. 1(b),t h e Tl layers are sandwiched by Se layers. By taking into accountthe previous abinitio study, which has shown that the bonding strength between Tl and Se layers is weaker than that forthe others, 14it can be expected that cleaving would happen between these layers. Note that there are two possibilities forthe surface termination as shown in Fig. 3(a): the breaking of the bond with the lower Se layer [dashed line A in Fig. 3(a)] and the upper layer (dashed line B). In the former case, theSe layer would terminate the surface, whereas the Tl layerwill remain on top of the Se layer in the latter case. Sincethese two possibilities can be considered to be equally likely,we propose a model of the cleaved TlBiSe 2surface as seen in Fig. 3(b). The islands on the cleaved surface would then consist of the residual Tl atoms, covering half the Se layer.Note that the Tl-Se swap model where the surface Tl layer isinterchanged with the Se layer, i.e., Se-Tl-Bi-Se-Tl-, has beenproposed as another possible way to terminate the surface, butthe Tl termination was predicted to be energetically preferredto the swap model. 14 Commonly, a variation of atomic configurations leads to a modification of chemical bonding, which has been widelystudied by CL-PES in clusters and step edges at surfaces. 17,18 Considering the proposed model as shown in Fig. 3(b), one can expect that the chemical condition of Tl atoms formingthe islands and buried in the bulk are different. In this respect,studying the chemical character with CL-PES is useful toexamine our model. Therefore, we focus on the core-levelspectra obtained by the surface-sensitive VUV-PES and thebulk-sensitive HAX-PES techniques. Figure 4(a) shows VUV- PES (top) and HAX-PES (bottom) results in a wide bindingenergy ( E B) range. Three core levels are identified as Se 3 d, Bi 5d, and Tl 5 d. It becomes evident that VUV-PES results for Bi and Se core levels are overall similar to those measured byHAX-PES. Note that the both binding energies for VUV-PESresults are apparently corresponding to that for HAX-PESresults, which indicates that a band bending effect is consideredto be negligible in this material. By contrast, a distinctdifference in the spectrum is visible in the Tl 5 dcore levels; HAX-PES result shows a single spin-orbit doublet, whereasan additional peak appears for each spin-orbit component in Intensity (arb. units) Intensity (arb. units)(a) (b)50 40 30 20 10 EFSe3dBi5d Tl5d VUVPES (h = 79 eV)ν HAXPES (h = 5.95 keV)ν56 54 52 56 54 52 Tl5d VUVPES h= 79eVν h= 5.95 keVνexperiment HAXPES 18 16 14 12 10 Binding energy (eV)total fit fit TlB fit TlSTlSTlB FIG. 4. (Color online) Results of the photoelectron spectroscopy with vacuum ultraviolet radiation (VUV-PES) and hard x-ray (HAX- PES) (a) in the wide energy region and (b) Tl 5 dcore-level energy region. VUV-PES (HAX-PES) result is denoted by the upper (lower) line. The solid lines in (b) indicate (black) total fitting result, (red) the Tl 5 dpeaks denoted with TlBas the main line, and (blue) TlS as the satellite line. The observed Tl 5 dspectra are fitted with V oigt functions and a Shirley-type background. The fitting parameters are listed in Table I. the VUV-PES data. A similar spectral feature of the Tl 5 dcore level has already been reported for another thallium-basedtopological insulator TlBiTe 2with VUV radiation;13however, a detailed explanation of this feature was not provided. Themagnified Tl 5 dcore-level spectra are shown in Fig. 4(b), where one can clearly see two components (named Tl Band TlS) in the VUV-PES data (red), whereas HAX-PES data (blue) show only a single component of Tl 5 dthat energetically coincides with the component of TlBin VUV-PES data. Considering the surface sensitivity of VUV-PES experimentsas well as the bulk sensitivity of HAX-PES experiments, theTl Sline is considered as the satellite, which is possibly linked to the formation of the islands at the surface as seen in STM,and the Tl Bemission is determined to be emitted from the bulk as the main line. Then these components as seen in theVUV-PES spectra are decomposed by a fitting procedure usingfour V oigt functions and a Shirley-type background. The fittingresults are indicated by the solid lines in Fig. 4(b). To obtain the best fit, the Gaussian width of all V oigt profiles had to be setlarger than the experimental energy resolution, probably dueto lattice vibration. 19The fitting parameters are tabulated in Table I.20We find that the Lorentzian and Gaussian widths for TlSpeaks are slightly larger than those for the TlBpeaks. The size of the splitting between TlBand TlScomponents in the VUV-PES spectrum is estimated to be /Delta1E=420±20 meV . 245308-3KENTA KURODA et al. PHYSICAL REVIEW B 88, 245308 (2013) TABLE I. Fitting parameters for the components of the Tl 5 d core level in Fig. 3(b).T lBand TlSshow the components of the Tl 5 d state located at lower and higher EB, respectively. All listed energy units are eV . The Lorentzian and Gaussian widths refer to the full width at half maximum. /Delta1Eindicates the size of splitting for the two components. TlBTlS VUV-PES ( hν=79 eV) EBof 5d5/2 12.81 ±0.01 13.23 ±0.01 Spin-orbit splitting 2.22 ±0.02 2.22 ±0.02 Lorentzian width 0.16 ±0.01 0.20 ±0.01 Gaussian width 0.19 ±0.02 0.27 ±0.02 /Delta1E 0.42±0.02 HAX-PES ( hν=5.95 keV) EBof 5d3/2 12.82 ±0.03 – Spin-orbit splitting 2.23 ±0.05 – Lorentzian width 0.16 ±0.01 – Gaussian width 0.30 ±0.02 – To further confirm the surface and bulk origin of the two components observed in the VUV-PES measurement, we haveexamined the Tl 5 dcore-level spectra using various excitation energies. The measurement mirrors similar experiments usedto separate surface and bulk components at conventionalsemiconductor surfaces. 21,22In these reports the kinetic energy (Ekin) of photoelectrons was tuned to a low energy below 10 eV , where the IMFP will rapidly increase as shown in the universalcurve of the IMFP. 23This can be expected to result in amixture between surface and bulk sensitivity. To continuously change the surface sensitivity, we use the tunable photonsource provided by the synchrotron radiation in the hνrange of 24–190 eV . The observed spectra are summarized in Fig. 5(a) together with their fitting results. The Lorentzian widths shownin Table Iare used for the fitting functions. The total spectra are found to strongly depend on hν.A thν=27 eV , the main Tl 5 d core-level emission (Tl B) is stronger than the emission from the satellite (TlS). With increasing hνto 46 eV , the intensity of TlSbecomes stronger than the main line. Further increase of the excitation energy toward hν=190 eV again reduces the weight of TlSclose to its low hνvalue. Figures 5(b) and 5(c) show a magnified view of the Tl 5 dcore-level spectra normalized by the intensity of the TlSpeak in the energy window of Tl 5 d3/2and Tl 5 d5/2components, respectively. For both peaks in the doublet, a similar hνdependence is observed, except for the fact that the intensities of TlBfor the 5 d3/2and 5d5/2peaks show their minimal at different photon energies, hν=49 and 52 eV , respectively. This corresponds roughly to the spin-orbit splitting of the doublet and demonstrates that thechanges in the intensity of the spectra can be better explainedwith a dependence on E kinrather than hν. In the following, we will analyze the intensity ratio between the main line and itssatellite by defining the normalized intensity ratio: r j=IB j IS j+IB j, (1) where IB jis the fitted intensity of the TlBline and IS j the corresponding intensity from the TlSline of the 5 dj, respectively ( j=5/2o r3 /2). The Ekindependence of r5/2 (b) (d)(c)Photoelectron intensity (arb. units) Photoelectron intensity (arb. units)fit h= 27 eVν 190 eV58 eV46 eV33 eV 16 14 12 Binding energy (eV)Binding energy (eV)15.6 15.2 14.8 13.6 13.2 12.8 12.4Tl5d3/2 Tl5d5/2 46 eV 46 eV 55 eV55 eV 52 eV 49 eV43 eV 43 eV 49 eV 52 eVIntensity ratio 0.8 0.61.0 0.2 00.4 Kinetic energy of photoelectrons (eV)10 1007 8 9 20 30 40 60 80 200Tl 5d5/2 Tl 5d3/231eV~420 meV 124 126 122 120 118 116 Binding energy (eV)123 122250 eV 180 eVTl5d core-levels (a) h= 180 eVν 190 eV 250 eV 5.95 keVEk~ 57 eV 67 eV 127 eV 5.8 keVTl4f core-levels (e) total fit ~410 meVTlSTlB TlSTlB FIG. 5. (Color online) (a) Core-level spectra for Tl 5 dwith vacuum ultraviolet (VUV) radiation ranging from 27 to 190 eV . The corresponding Ekinof the photoelectrons from 5 d5/2are shown. The red (blue) solid line shows the TlB(TlS) lines decomposed with fitting, and the Lorentzian widths are fixed at the same value listed in Table I. (b) and (c) Intensity evolution of the TlBfor Tl 5 d3/2and Tl 5 d5/2with respect to that of TlS peaks, respectively. (d) Intensity ratios between TlBand TlSfor each doublet as a function of Ekinof photoelectrons. The intensity ratios are defined with the equation as described in the main text. (e) Core-level spectra for Tl 4 fwith VUV radiations ranging from 180 to 250 eV as well as hard x-ray. 245308-4EXPERIMENTAL VERIFICATION OF THE SURFACE ... PHYSICAL REVIEW B 88, 245308 (2013) Ek~ 18 eV3d3/23d5/2 5d3/25d5/2Photoelectron intensity (arb. units)h = 76 eVν 90 eV32 eV 160 eV 192 eV31 eV61 eV 250 eV190 eV 5.95 keVSe3d core-levels ) b( )a( (c)Bi2Se3 Bi2Se3 (d)Bi5d core-levels Binding energy (eV) Binding energy (eV)56 54 52 30 28 26 245.95 keVVek 29.5 Vek 09.5h = 46 eVν h = 61 eVν h = 90 eVνEk~ 16 eV Ek~ 31 eVEk~ 32 eVBiOx FIG. 6. (Color online) Core-level spectra for (a) Se 3 dand (b) Bi 5 dwith vacuum ultraviolet radiations and hard x-ray. The corresponding Ekinof the photoelectron from 3 d5/2and 5d5/2are denoted in the figure. Solid lines show fitting results with parameters listed in Table II. We found the tails in Bi 5 dspectra taken with hard x-ray (bottom) at higher EB, which is attributed to the oxidized Bi (BiO x) (shaded area). (c) and (d) Se 3 dand Bi 5 dcore-level spectra for Bi 2Se3obtained by the selected photon energy to obtain the most surface sensitivity. andr3/2is summarized in Fig. 5(d). As suggested from the preceding qualitative analysis in Fig. 5(b) and5(c), it is found that the Ekindependency of both Tl 5 ddoublets are indeed identical. The intensities of the TlBlines increase toward lower and higher Ekinfrom their minimum near Ekin=31 eV , which is in excellent agreement with the universal curve ofphotoelectron IMFP. 23This result can further confirm the bulk origin of TlBobserved in our HAX-PES experiment and is consistent with our HAX-PES result that TlSand TlBstates originate from the surface and bulk, respectively. To analyze the relation between Tl emission from the surface and the bulk, we focus on the core-level shift (CLS)between the Tl Sand TlBstates. The size of the CLS generally depends on several parameters, such as the bonding distance,the valency of the atom, and the local chemical surroundings. 24 The Tl 5 dcore level is normally thought to be a semicore state because it is located energetically close to the valence band(E B=12–15 eV), and thus it is highly sensitive to the physical and chemical environments compared to other deeper lyingcore levels. Therefore, in order to reduce the possibility thatshifts are induced from beyond the nearest neighbor latticesite, it is important to compare the CLS of the Tl 5 dline with other Tl core levels that are more strongly screened. Todo so, we chose to investigate the Tl 4 fcore level, which is located at higher E B(∼118 eV) than that of the Se 3 d,TABLE II. Fitting parameters of Bi 5 dand Se 3 dspectra in TlBiSe 2and Bi 2Se3. The values of HAX-PES are shown in parentheses. The Lorentzian width refer to the full width at half maximum.25All listed energy units are in eV . TlBiSe 2 Bi2Se3 Bi 5dcore level EBof 5d5/2 25.06±0.01 24.83 ±0.01 (24.98 ±0.03) Spin-orbit split 3.04 ±0.02 3.05 ±0.02 (3.03±0.05) Lorentzian width of 5 d5/2 0.32±0.02 0.28 ±0.02 (0.20±0.02) Lorentzian width of 5 d3/2 0.38±0.02 0.34 ±0.02 (0.24±0.02) Se 3dcore level EBof 3d5/2 53.41±0.01 53.26 ±0.01 (53.50 ±0.03) Spin-orbit split 0.84 ±0.02 0.86 ±0.02 (0.85±0.05) Lorentzian width of 3 d5/2 0.20±0.01 0.16 ±0.01 (0.20±0.01) Lorentzian width of 3 d3/2 0.20±0.01 0.16 ±0.01 (0.20±0.01) Bi 5d, and Tl 5 dcore levels. Figure 5(e) shows the spectral features for Tl 4 fcore levels excited with several different hνand fitted with V oigt profiles. Even without resorting to an analysis of the fitting results, one can already see that thespectral weight on the higher E Bside increases if the Ekinof photoelectrons is close to the highly surface sensitive values(see the inset). Accordingly, the spectral weight at lower E B increases at more bulk-sensitive Ekin. This behavior is verified in more detail by the fitting functions. Similar to Tl 5 d,w ea r e able to identify a TlBmain line by comparing the VUV-PES data with the spectrum taken with HAX-PES, which againshows a single component. The Tl Band TlSlines of the Tl 4fpeak show comparable Ekindependence of the Tl 5 dcore levels. The size of the CLS between these lines is estimated tobe 410 ±20 meV , which is well comparable to that of the Tl 5 d level. This finding again support our notion that the chemicalstate of Tl is strongly deformed by the chemical environment atthe surface. In order to get a proof that no similar CLS exists for the Se or Bi atoms, we focus on the shallow core levels fromthese elements next. Figures 6(a) and 6(b) summarize the Se 3dand Bi 5 dcore-level spectra, respectively. The fitting parameters are listed in Table IIand compared with those for Bi 2Se3as a reference (as discussed later). We find that no additional features, such as shoulders or peak shifts, arepresent in the VUV-PES spectra. Both the Se 3 dand the Bi 5 d core-level spectra measured by VUV-PES can be reproducedby a single V oigt profiles with the same parameters listedin Table II. 25Tuning hνto obtain a highly surface sensitive condition ( Ekin∼31 eV) does not change the spectral shape, which is in a strong contrast to the Tl core levels. We noticethat the peak positions slightly shift in different directionsfor the Se 3 dand Bi 5 dpeaks if the HAX-PES results are compared to those from the VUV-PES measurements. TheBi 5dspectrum measured by HAX-PES is located at lower 245308-5KENTA KURODA et al. PHYSICAL REVIEW B 88, 245308 (2013) EBwith respect to that acquired by the VUV-PES, while the Se 3dshifts to higher EBwith a comparable energy shift of∼80 meV . For Se 3 dcore level, the CLS can be partially attributed to a recoil effect in HAX-PES measurement,26which is expected to induce an energy shift of ∼40 meV to higher EB, and a single V oigt profile with the same Lorentzian width canreproduce the HAX-PES spectrum. However, the Lorentzianwidth of the HAX-PES peak for Bi 5 dis smaller than that measured by VUV-PES. This result probably indicates that allobserved VUV-PES spectra of Bi 5 dinclude an unresolved surface component. In order to understand the possible contribution of dangling bonds on the Bi 5 dand Se 3 dcore levels, we investigate the core levels of Bi 2Se3in which dangling bonds are believed to be absent. Figures 6(c) and6(d) show the observed Se 3 d and Bi 5 dspectra in Bi 2Se3at highly surface sensitive Ekin. We find that both core levels are located at lower EBwith respect to those of TlBiSe 2, indicating the different bonding conditions as well as a spontaneous carrier doping effect dueto the presence of defects in the bulk for Bi 2Se3. We note that the spectral shape in Bi 2Se3is quantitatively close to that in TlBiSe 2for both core levels, and these features are independent of the Ekin(not shown). Thus, the contribution of the dangling bonds is found to be weak for Bi and Se corelevels in TlBiSe 2, and we consider that an unresolved surface component in TlBiSe 2may be attributed to a surface relaxation effect which is predicted to decay slowly into the bulk.14A previous CL-PES measurement on Bi 2Se3with Fe deposition has demonstrated that the spectral feature of Se 3 dcore level are stable against the chemical environment.27Hence, we do not deny the existence of the Se atoms affected by the differentchemical surroundings in TlBiSe 2. The proposed model for the surface termination as shown in Fig. 3(b) is therefore supported by our core-level PES where only the Tl core levels are strongly deformed by thesurface component. Since a flat surface is generally assumedfor the previous calculation, 14the fact that residual islands exist on the surface provides us with a reason why thetrivial surface states which are present in the calculated bandstructure are absent in the ARPES measurements. 11–13In the following, we want to give the possible explanationsusing our model. Firstly, due to the small domain size ofthe islands, the dispersing features of the dangling bondstates would be absent, which results from the fact that thedangling bond states are localized on the islands as wellas in the Se terminated surface area. The localized statesmay be energetically located above the Fermi energy orweakly contributing to the intensity of photoelectrons andthus be absent from the ARPES measurements. Secondly, asmentioned in the STM analysis that the island height is smallerthan the lattice constant in the bulk, the island deformation maylead to a saturation of the dangling bonds. Finally, the strongionic nature of the interlayer bonding between Tl and Se canlead to an absence of the dangling bond states. Our VUV-PESfor Tl 5 dcore level shows that the surface component is locatedat∼420 meV higher E Bthan the bulk one. If we assume that the stoichiometry of TlBiSe 2formally assigns the Tl atoms a1+oxidation state, the Bi atoms a 3 +state, and the Se atoms a 2 −state, the energy difference can be expected to be originating from a higher oxidation state more than 1 +of the partially remaining Tl atoms on the surface. According tothe coverage of the Tl islands estimated to be half of thesurface, the valence number of the rest of the Tl ions can beassigned nearly 1.5 +states to keep neutrality in the whole of the crystal. The speculation of the higher oxidation state of therest of the Tl ions actually has good agreement with the shrink-ing of the interlayer distance in STM analysis because theionic radius of the residual Tl ions would constrict, resultingfrom the higher oxidation. For this reason, the considerationof the ionic type of bonding can give an interpretation ofour experimental results and furthermore provide a reasonableexplanation for the absence of the dangling bond states. Thepresented surface model may motivate future experiments onthis surface. Possible experiments include the study of surfacedeposition, which has been widely used on the surface ofBi 2Se3,27–31as well as theoretical studies of the surface of TlBiSe 2including the island structure. IV . CONCLUSION In conclusion, we have experimentally approached the open question of the surface termination problem of TlBiSe 2 utilizing STM and CL-PES. The STM results have shownthat islands are formed on the cleaved surface. The CL-PESmeasurements with tunable surface sensitivity have revealedthat the spectral features of Tl core levels are strongly deformedby the surface component in contrast to the other elements.From these findings, we proposed a model for the surfacetermination that includes Tl islands covering half of thesurface, which is terminated by a Se layer. We demonstratedthat the residual Tl island model is the most likely explanationfor the pronounced CLS found in Tl core levels. The model wasalso able to account for the absence of the trivial surface statein ARPES measurements. It will motivate further experimentaland theoretical studies of the surface properties. ACKNOWLEDGMENTS We thank K. Mimura and H. Sato for fruitful discussions. The STM and VUV-PES experiments were performed atHiSOR, Hiroshima University, with the approval of theProposal Assessing Committee of HSRC (Proposal Nos.12-B-12, 12-B-46, and 13-B-5). The HAXPES experiment wasperformed at BL15XU of SPring-8 with the approval of theNIMS Beamline Station (Proposal No. 2012B4908). This workwas partly supported by KAKENHI (23340105), Grant-in-Aidfor Scientific Research (B) of Japan Society for the Promotionof Science. K.K. acknowledges support from the Japan Societyfor the Promotion of Science for Young Scientists. *kuroken224@hiroshima-u.ac.jp †akiok@hiroshima-u.ac.jp1M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 045302 (2010). 2X. L. Qi and S. C. Zhang, Rev. Mod. Phys 83, 1057 (2011). 245308-6EXPERIMENTAL VERIFICATION OF THE SURFACE ... PHYSICAL REVIEW B 88, 245308 (2013) 3L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). 4X. L. Qi, T. L. Hughes, and S. C. Zhang, Phys. Rev. B 78, 195424 (2008). 5R. Li, J. Wang, X. L. Qi, and S. C. Zhang, Nature Phys. 6, 284 (2010). 6K. Nomura and N. Nagaosa, Phys. Rev. Lett. 106, 166802 (2011). 7L. Despont, F. Clerc, M. G. Garnier, H. Berger, L. Forro, andP. 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PhysRevB.86.155410.pdf

PHYSICAL REVIEW B 86, 155410 (2012) Electronic charge redistribution in LaAlO 3(001) thin films deposited at SrTiO 3(001) substrate: First-principles analysis and the role of stoichiometry Alexandre Sorokine,1,2,*Dmitry Bocharov,1,3Sergei Piskunov,2,3and Vyacheslavs Kashcheyevs1,3 1Faculty of Physics and Mathematics, University of Latvia, 8 Zellu Str., Riga LV-1002, Latvia 2Institute for Solid State Physics, University of Latvia, 8 Kengaraga Str., Riga LV-1063, Latvia 3Faculty of Computing, University of Latvia, 19 Raina blvd., Riga LV-1586, Latvia (Received 18 May 2012; published 9 October 2012) We present a comprehensive first-principles study of the electronic charge redistribution in atomically sharp LaAlO 3/SrTiO 3(001) heterointerfaces of both nandptypes allowing for nonstoichiometric composition. Using two different computational methods within the framework of the density functional theory (linear combinationof atomic orbitals and plane waves) we demonstrate that conducting properties of LaAlO 3/SrTiO 3(001) heterointerfaces strongly depend on termination of LaAlO 3(001) surface. We argue that both the “polar catastrophe” and the polar distortion scenarios may be realized depending on the interface stoichiometry.Our calculations predict that heterointerfaces with a nonstoichiometric film—either LaO-terminated ntype or AlO 2-terminated ptype—may exhibit the conductivity of norptype, respectively, independently of LaAlO 3(001) film thickness. DOI: 10.1103/PhysRevB.86.155410 PACS number(s): 68 .35.Ct, 68.35.Md, 73 .20.At I. INTRODUCTION The discovery of conducting interfaces between two ini- tially insulating materials—TiO 2-terminated (001) surface of SrTiO 3(STO) substrate and LaAlO 3(LAO) thin film deposited on top of it1—has attracted strong scientific interest during the past few years.2–6The high application potential of LAO/STO heterointerfaces has been demonstrated, e.g., by fabrication of highly voltage-tunable oxide diodes7that utilize the advantage of the electric-field controlled interfacialmetal-insulator transition of LAO/STO. 8,9 Conductivity of an atomically flat interface in the limit of large film thickness can be understood1from electrostatic considerations within the “polar catastrophe” picture. From theperspective of formal charges, the atomic planes in the [001]direction [which we refer to as monolayers (ML)] are neutral for STO (SrO 0and TiO0 2) but charged for LAO (LaO+and AlO− 2). Transition from STO to LAO can be ptype (from SrO0 to AlO− 2)o rntype (from TiO0 2to LaO+). The corresponding jump of the surface charge at the interface would createan electric field inside LAO increasing linearly with thedistance from the interface—a “polar catastrophe.” This bulkpolarization of the LAO film can be compensated (thus averting the “catastrophe”) if 0 .5eper unit cell area is transferred from the LAO film surface onto the interface, resulting in a maximalsheet carrier density of n=0.5/a 2=3.3×1014cm−2(here a=3.90˚A is the lattice constant of STO assuming epitaxial matching of the LAO film). This estimate is immune to dielec-tric relaxation and bond covalency/charge smearing effects 6 and, thus, provides a useful reference in the thick film limit. For sufficiently thin films, however, the polar catastrophe may be tolerated10and a metal-insulator transition occurs11 as a function of the number of epitaxial monolayers of LAOdeposited. From an electrostatic perspective, in a sufficientlythin film the internal field does not develop a potentialdifference large enough to overcome the dielectric gap. The accumulation of oppositely charged monolayers leads to progressive band bending until the critical thickness (5 u.c.or 10 monolayers for n-type structures, according to experimental 11and theoretical12–14evidence) is reached be- yond which the chemical forces are overcome and the chargeredistribution occurs. This mechanism is known as “polardistortion.” 4,10As far as is known, p-type interfaces do not exhibit this mechanism, as covalent forces overwhelmelectrostatic ones. Similar electostatic arguments may be applied to nonsto- ichiometric LAO/STO structures, i.e., the ones with an oddnumber of LAO monolayers. The LAO films in these structurespossess one extra electron (or a hole for p-type interfaces) per unit cell area compared to the parent bulk material, thus, theyshould be conducting (with n=1/a 2=6.6×1014cm−2) irrespective of the thickness or the presence of STO substrate.However, as seen from the stoichiometric example, for thinfilms, the competition between the semicovalent bonds andlong-range electrostatics is very sensitive to the the number ofmonolayers deposited. The nature of the conducting layer innonstoichiometric LAO/STO interfaces is the main subject ofourab initio investigation. Recent experimental reports indicate that La /Al ratio in nonstoichiometric LAO films may be controlled duringepitaxial growth. 13,15Atomically sharp interfaces are produced by molecular beam epitaxy (MBE) in which thermal energiesof evaporated incident ions are low, about 0.1 eV , thus, MBEavoids intermixing of cations at the interface. 5,11However, the vaporization process used to facilitate transfer throughthe vapor LAO phase does not guarantee preservation of thetarget stoichiometry, 5which makes room for a possibility to control the film growth monolayer by monolayer. We notethat for different preparation methods of LAO/STO interfaces,e.g., pulsed laser deposition (PLD), other mechanisms maygive rise to conductivity. One of proposed mechanisms isformation of the high density of oxygen vacancies, whichare generated in the STO substrate while depositing LAOthin film and can be responsible for increase of sheet carriersdensity up to 5 ×10 17cm−2for PLD-grown n-LAO/STO interfaces if the sample is not annealed.1,16The insulating 155410-1 1098-0121/2012/86(15)/155410(10) ©2012 American Physical SocietySOROKINE, BOCHAROV , PISKUNOV , AND KASHCHEYEVS PHYSICAL REVIEW B 86, 155410 (2012) behavior of p-LAO/STO has been also ascribed to that the holes can be trapped by two electrons located at the oxygenvacancies created in the STO substrate. 17 Yet another scenario for LAO/STO interface conductivity that may take place in PLD-prepared structures is basedon the suggestion that the La/Sr cation intermixing due toion bombardment effect (inherent in PLD and postgrowthtreatment) may lead to the formation of one or two layers ofmetallic La 1−xSrxTiO 3.5,18,19The thermodynamical stability for intermixed configurations has been recently reported.5,20 In this paper, we aim to construct a clear picture of charge-density redistribution both in stoichiometric and non-stoichiometric interfaces of either type and LAO film thicknessfrom 1 to 11 monolayers (0.5–5.5 u.c.). The ab initio calcu- lation methods employed are based on the density functionaltheory (DFT) using a hybrid exchange-correlation functional.We contrast stoichiometric/nonstoichiometric and p-type/n- type structures utilizing identical methods and computationparameters. The B3PW functional 21used in the CRYSTAL code22with atomic basis set (BS) contains a “hybrid” of the DFT exchange and correlation functionals with exactnonlocal Hartree-Fock (HF) exchange. For comparison, theselected set of interface configurations has been also modeledusing the Perdew-Wang generalized gradient approximation(PW91-GGA) density functional 23,24as implemented in the periodic plane-wave (PW) code V ASP .25 We find that covalent effects in nonstoichiometric films are less pronounced than in stoichiometric ones and the structuresare metallic in accordance with formal charges considerations.As Ti–O bond strength exceeds Sr–O bond strength by ca.120 kJ mol −1(obtained considering formation enthalpies for respective oxides), in p-type IFs (where SrO monolayer is at the IF) we can expect covalent forces to be stronger thaninn-type IFs. This, in its turn, leads to an approximately uniform free charge distribution through the film, whereaswhen covalent forces are weaker—as in n-type IFs—the free charge is forced to the edges of the LAO film (the surface andthe IF), resulting in a bilayered electron gas structure. Experimental works show that stoichiometric p-type inter- faces exhibit no measurable conductivity, 11but annealed stoi- chiometric n-type interfaces with LAO film thickness /greaterorequalslant5u . c . have free electron density in the range 1–3 ×1013cm−2.11 Similar densities (2–7 ×1013cm−2) are obtained from first- principles calculations12–14(cf. 3.3×1014cm−2predicted from electrostatic considerations). The paper is structured as follows. Section IIdescribes the computational details of our calculations. The main part of thepaper is formed by Sec. III. In Sec. III A we give an estimate of the thermodynamic stability and discuss the electronicstructure of ideal LaO- and AlO 2-terminated LAO(001) sur- faces. Section III B presents electronic charge distributions forn-LAO/STO and p-LAO/STO heterointerfaces and discusses their relation to the experimental and computational dataavailable in the literature. Our conclusions are summarizedin Sec. IV. II. COMPUTATIONAL DETAILS In this study LAO/STO heterointerfaces are modeled by means of two different methods: (i) linear combination ofatomic orbitals (LCAO) within the framework of a hybrid density functional approach and (ii) PW calculations using theGGA density functional. To perform hybrid LCAO calculations, we used the periodic CRYSTAL code,22which employs Gaussian-type functions centered on atomic nuclei as the BSs for expansion of thecrystalline orbitals. The BSs used in this study were taken fromthe following sources: For Sr, Ti, and O in the form of 311d1G,411d311dG, and 8-411d1G, respectively, from Ref. 26;f o rA l in the form of 8-621d1G from Ref. 27; and for La in the form of 311-31d3f1 from CRYSTAL ’s homepage22(f-type polarization Gaussian function with the exponent α=0.475 has been added according to prescription given in Ref. 28). For Al and O, all electrons are explicitly included. The inner core electronsof Sr and Ti are described by small-core Hay-Wadt effectivepseudopotentials, 29while the nonrelativistic pseudopotential of Dolg et al.30was adopted for La. We employ the hybrid B3PW exchange-correlation functional21which accurately reproduces the basic bulk and surface properties of a number of ABO 3perovskite materials.26,31–33The cutoff threshold parameters of CRYSTAL for Coulomb and exchange integrals evaluation (ITOL1– ITOL5) have been set to 7, 8, 7, 7, and 14, respectively.Calculations were considered as converged only when the totalenergy obtained in the self-consistency procedure differed byless than 10 −7a.u. in two successive cycles. Effective charges on atoms as well as net bond populations have been calculatedaccording to the Mulliken population analysis. 34–37 As the second method, the periodic total-energy code V ASP25based on the use of a PW BS was applied. The cut-off energy was chosen to be 520 eV . The nonlocal GGAexchange-correlation functional Perdew-Wang-91 (PW91)was employed. 23,24Scalar relativistic projector augmented wave (PAW) pseudopotentials in our calculations contain11 valence electrons (5 s 25p65d16s2) for La, 3 electrons (3s23p1) for Al, 10 electrons (4 s24p65s2) for Sr, 12 electrons (3s23p63d24s2) for Ti, and 6 electrons (2 s22p4)f o rO , respectively. Bader topological analysis38was adopted to obtain net charges on atoms in V ASP calculations. In both V ASP and CRYSTAL calculations the reciprocal space integration was performed by sampling the Brillouinzone with the 8 ×8×1 Pack–Monkhorst mesh 39for all surface structures under consideration. For bulk computationswe applied sampling with the 8 ×8×8 Pack-Monkhorst mesh. Such samplings provide balanced summation in directand reciprocal lattices. Taking into account that STO substrate at room temperature possesses perfect cubic structure, in our study we treat bothLAO and STO in their high symmetry Pm¯3mcubic phase. In fact, the bulk crystal structure of LAO, having space groupR¯3c(rhombohedral) with a 0=5.364 ˚A and c0=13.108 ˚A at room temperature,40can be represented by a pseudocubic unit cell with a0=3.790 ˚A. At 821 K the structure of LAO transforms to become cubic with a0=3.811 ˚A.40 Though the heterointerface assumes the transition between two intrinsically different crystal symmetries, Pm¯3mthe substrate andR¯3cin the film, whereby thin films are expected to adapt to the substrate.44 Table Ilists main bulk properties for both crystals. We note that the band gaps obtained by means of hybrid B3PW 155410-2ELECTRONIC CHARGE REDISTRIBUTION IN LaAlO ... PHYSICAL REVIEW B 86, 155410 (2012) TABLE I. Calculated equilibrium lattice constants ( a0in˚A), atomic net charges ( Qatomin e), cation–O bond populations ( PA/B−O in milli e), and band gaps ( δin eV) of bulk LAO and STO in their high-symmetry Pm¯3mcubic phase. Shown are data obtained by means of both hybrid B3PW and standard GGA PW91 functionals.Negative bond population means atomic repulsion. Last two columns contain available experimental results for comparison. LAO LAO STO STO LAO STO (B3PW) (PW91) (B3PW) (PW91) (Expt.) (Expt.) a0 3.802 3.808 3.910 3.918 3.811a3.905b QLa/Sr 2.43 2.14 1.87 1.60 – – QAl/Ti 2.07 3.00 2.35 2.10 – – QO −1.50 −1.78 −1.41 −1.23 – – PLa/Sr−O 4– −10 – – – PAl/Ti−O 152 – 88 – – – δ 5.51 3.18 3.64 1.77 5.6c3.25d aReference 40. bReference 41. cReference 42. dReference 43. computation scheme are in better agreement with experimen- tally observed results. Therefore, in this paper, we mainlydiscuss the results obtained by means of B3PW while resultsobtained using PW91 functional are published for comparativepurposes in order to make our study consistent with earlier ab initio calculations performed basically on LDA- or GGA-DFT ground. Surface structures were modeled using a single slab model for LCAO calculations and a multislab model with vacuumgap of 20 ˚A for PW calculation. To compensate the dipole moment arises at charged surfaces, our slabs are symmetricallyterminated. STO substrate contains 11 alternating (SrO) 0and (TiO 2)0atomic monolayers, while from 1 to 11 alternating (LaO)+and (AlO 2)−atomic monolayers were used for LAO film of the LAO/STO interface. Coordinates of all atoms inthe LAO/STO heterointerfaces were allowed to relax. Dueto symmetry constrains atomic displacements were allowedonly along the zaxis. Taking into account that the mismatch of∼2.5% between LAO and STO lattice constants arises during LAO epitaxial growth, in our modeling we have allowedrelaxation of their joint lattice constant to minimize the straineffect. III. RESULTS AND DISCUSSION A. LAO(001) surfaces Before general discussion of LAO/STO interfaces studied here, in this subsection we provide a comprehensive descrip-tion of electronic and thermodynamic properties of both LaO-and AlO 2-terminated pristine LAO(001) thin films. 1. Electronic properties Pristine LAO(001) thin films were modeled using a symmetrical nine-monolayer slab model. Considering formalionic charges, LAO(001) has alternating (LaO) +and (AlO 2)− surface monolayers and can be either a LaO- or AlO 2- terminated surface. Both LaO and AlO 2terminations areTABLE II. Calculated deviations in surface monolayer net charge (/Delta1Q ine), and deviations of cation–O bond populations ( /Delta1P A/B−O in milli e) in the corresponding atomic monolayer relative to the bulk values (see Table I). Shown are data obtained by means of a hybrid B3PW exchange-correlation functional. Surface monolayers are numbered beginning from the center of the slab (0 means the central monolayer of the symmetrical slab unit cell). LaO terminated AlO 2terminated No. ML /Delta1Q /Delta1P A/B−O ML /Delta1Q /Delta1P A/B−O 4L a O −0.32 10 AlO 2 0.46 100 3A l O 2−0.02 −16 LaO −0.02 −4 2L a O −0.09 0 AlO 2 0.02 −10 1A l O 2 0.00 −2L a O −0.02 −4 0L a O −0.05 −2A l O 2 0.02 −10 studied. The La /Al excess ratio is 1.25 and 0.8 for LaO- and AlO 2-terminated LAO(001) films, respectively. Monolayers in LAO(001) possess a net charge, and the repeat slab unitcell has a nonzero dipole moment, and, therefore, LAO(001)is type III polar surface according to Tasker’s classification. 45 This means that perfect and unreconstructed (1 ×1) LAO(001) surfaces considered here can be stabilized by transferring ofa half an electron (or hole) from the surface to the slab bodythat normally results in atomic and electronic reconfigurationat the surface. In Table IIwe list the changes in surface (LaO) +and (AlO 2)−monolayer net charges with respect to their bulk values (see Table I). Due to the partly covalent nature of La–O and Al–O bonds (positive PA/B−Oin Table I), net charges of La, Al, and O deviate from their formal ionic values of +3, +3, and −2, respectively. The La–O hybridization between La 5dand O 2 pstates lead to atomic charges of 2 .43e,2.07e, and −1.50efor La, Al, and O, respectively. As a result, LaO and AlO 2monolayers possess a bulk monolayer charge of ±0.93e instead of a formal ionic ±1echarge. According to Table II, the surface monolayer of LaO-terminated LAO(001) attracts0.32 electrons, while other monolayers of the slab get the restof 0.14 electrons to compensate the surface polarity. On thecontrary, the surface monolayer of AlO 2-terminated LAO(001) solely receives 0.46 holes. Covalency of the surface La–Obond is only slightly increased (bond population increasedonly by 10 milli e), while the calculated covalency of the surface Al–O bond is practically 2 times larger than in thebulk, which, to some extent, may compensate relatively modestsurface relaxation of AlO 2-terminated LAO(001) with respect to LaO-terminated one. Figure 1shows the density of states (DOS) projected onto all orbitals of La, Al, and O atoms of LAO bulk and bothLaO- and AlO 2-terminated LAO(001) surfaces as well. In case of LAO bulk [Fig. 1(c)] the top of valence band is formed by O 2 porbitals, while the bottom of conduction band is formed mainly by La 5 dstates. La–O hybridization is well pronounced. Calculated band gap of 5.51 eV is in excellentagreement with its experimental value of 5.6 eV . 42In case of LaO-terminated surface [Fig. 1(a)] gained excess of electrons shifts the Fermi level up to unoccupied level that gives raise toelectron conductivity. In its turn the AlO 2-terminated surface [Fig. 1(b)] experiences the lack of electrons that shifts Fermi 155410-3SOROKINE, BOCHAROV , PISKUNOV , AND KASHCHEYEVS PHYSICAL REVIEW B 86, 155410 (2012) FIG. 1. (Color online) Projected density of states as calculated by means of B3PW hybrid exchange-correlation functional: (a) LaO- terminated LAO(001), (b) AlO 2-terminated LAO(001), and (c) LAO bulk. TVB stands for the top of valence band. level down to valence band and, thus, reveals the existence of hole conductivity. 2. Thermodynamic stability The thermodynamic formalism adopted in the current study to estimate the stability of both LaO- and AlO 2-terminated LAO(001) surfaces has been thoroughly described in Refs. 46 and47(see also references therein). The stable crystalline surface has to be in equilibrium with both LAO bulk andsurrounding oxygen atmosphere, assuming that an exchangeof atoms between the surface and environment is allowed.Therefore, the most stable surface has the lowest Gibbs freesurface energy, defined as /Omega1 t(T,p)=1 2A/bracketleftbig Eslab t−NAlELAO bulk−(NLa−NAl)/Delta1μ La −(NO−3NAl)/Delta1μ O(T,p)/bracketrightbig , (1) where tindicates the surface terminations, Athe unit cell surface area, Nithe number of atoms of type iin the slab unit cell, Eslab tis the total energy of a slab with tsurface terminations and ELAO bulkis the LAO total energy averaged per five-atom perovskite unit cell. /Delta1μi=μi−Ei bulk,(i=La,Al) are deviations of chemical potentials for metal atoms fromtheir energy in the bulk metals. For the oxygen atom, such adeviation is considered with respect to the energy of an oxygenatom in the ground triplet state of an O 2molecule /Delta1μ O= μO−1 2EO2. Because the pVterm (Vis unit cell volume) and the differences in the vibrational Gibbs free energy betweenthe bulk solid and a corresponding slab is negligibly small, 48TABLE III. Formation energies per formula unit used in analysis of surface stability. Experimental values are taken from Ref. 49. Material Ef(eV) Expt. Ef(eV) La2O3 −17.52 −18.64 Al2O3 −16.68 −17.37 LaAlO 3 −17.68 we omit these two contributions. This permits replacing the Gibbs free energies in Eq. (1)and in the following formulas with the total energies obtained from ab initio calculations. In order to avoid the precipitation of relevant metals and oxides at LAO surface, as well as to prevent metal atoms toleave the sample, the following conditions must be satisfied: 0>/Delta1 μ La,0>/Delta1 μ Al, (2) Ef LaAlO 3−Ef Al2O3<2/Delta1μ La+3/Delta1μ O<Ef La2O3, (3) where Ef nis the formation energies of material nlisted in Table III. We evaluate the oxygen chemical potential /Delta1μ O(p,T)a s a function of partial gas pressure and temperature using thestandard experimental thermodynamical tables 49as it was done in Refs. 47and 48./Delta1μ O(T,p O2)i st h ev a r i a t i o no f oxygen chemical potential due to temperature and pressure ofthe surrounding oxygen atmosphere. In addition to the experi-mental variation, it contains a correction term δμ 0 O=0.03 eV , which compensates the difference between the experimentallydetermined variation of the oxygen chemical potential and thereference state in current theoretical calculations (see Refs. 50 and51for a thorough discussion). B a s e do nE q s . (1)–(3), the thermodynamic stability diagram is plotted in Fig. 2, showing the regions of stability of pristine LAO(001) surfaces with respect to precipitation of La 2O3and Al2O3oxides. Figure 3shows the thermodynamic stability diagram along the lines corresponding to precipitation ofLa 2O3and Al 2O3oxides as a function of /Delta1μ Orelated to the temperature scale at an oxygen pressure typical duringLAO/STO synthesis ( P=10 −6mbar). To make such a diagram possible, according to the prescription given inRef. 52, we replaced /Delta1μ Laby /Delta1μ La=1 2/parenleftbig Ef La2O3−3/Delta1μ O/parenrightbig , (4) which corresponds to precipitation of La 2O3( l i n e s3i nF i g . 3), and by /Delta1μ La=Ef LaAlO 3−1 2/parenleftbig Ef Al2O3−3 2/Delta1μ O/parenrightbig , (5) which corresponds to precipitation of Al 2O3(lines 4 in Fig. 3). Formation energies for oxides are taken from Table III. From the calculated thermodynamic stability diagrams we can predict that at ultrahigh vacuum (UHV) conditions typicalduring PLD synthesis of LAO/STO interfaces and low temper-atures ( T< 550 K), the most stable is the AlO 2-terminated sur- face, while at elevated temperatures ( T> 1100 K) stabilizes the LaO-terminated surface. Between these temperatures, bothsurface terminations may coexist. Further lowering of oxygen 155410-4ELECTRONIC CHARGE REDISTRIBUTION IN LaAlO ... PHYSICAL REVIEW B 86, 155410 (2012)ΔμO (eV) AlO2 - term. LaO - term.ΔμLa (eV) T (1000 K) T (1000 K)p = 0.2p 0 ΔμLa (eV) ΔμO (eV)13 4 2103 104 10-4 10-6 10-4BA FIG. 2. (Color online) Thermodynamic stability diagram as a function of O and La chemical potentials built for both LaO- and AlO 2- terminated LAO(001) surfaces. Diagram contains precipitation conditions for both La and Mn metals, as well as for their trivalent oxides(La 2O3and Al 2O3). The stable region is shown as a shaded area between the La 2O3and Al 2O3precipitation lines. The numbers from 1 to 4 in the circles indicate segregation lines for precipitation of /p65Al,/p66La,/p67La2O3,/p68Al2O3. The right side shows a family of oxygen chemical potentials under different conditions. The label mindicates the O 2gas partial pressure: 10mmbar. Red (gray) line corresponds to oxygen partial pressure p=0.2p0as in the ambient atmosphere. Point A denotes room temperature and ambient oxygen pressure, and point B denotes typical temperature and pressure during LAO/STO(001) synthesis. pressure shifts down these demarcated temperatures. Thus, our prediction is in good qualitative agreement with time-of-flightscattering and recoiling spectrometry (TOF-SARS), atomicforce microscopy (AFM), and a photoelectron spectroscopy(PES) study performed by Rabalais and coworkers. 53,54They found that at temperatures under 423 K, the surface isexclusively terminated by an Al-O layer, while at temperaturesabove 523 K the surface is exclusively terminated by aLa-O layer. Between 423 K and 523 K surface stoichiometry 4C D 3 FIG. 3. (Color online) The thermodynamic stability diagram calculated along the La 2O3and Al 2O3precipitation lines (numbers 3 and 4 in the circles, respectively) with /Delta1μ Ladefined according to Eqs. (4)and(5). The dependence on the oxygen chemical potential is converted to the appropriate temperature scale at an oxygen pressure typical during LAO/STO(001) synthesis ( P=10−6mbar). The interval between points C and D correspond to temperature range where both LaO- and AlO 2-terminated LAO(001) surfaces are stable and may coexist.changed from AlO xto LaO xand, thus, mixed terminations were proposed. Moreover, this change was found to befully reversible. Rabalais and coworkers suggested that thesurface termination change was caused by the formation ofsurface oxygen vacancies at high temperature, which drivesthe migration of the La atom to the surface and the Al atominto the bulk. A more recent experimental study based onx-ray crystal truncation rod (CRT) analysis 55demonstrates that LAO(001) possesses Al-terminated structure at both room andhigh (670 K) temperatures with no evidence for the reversalof surface termination or for the formation of surface oxygenvacancy. Authors of Ref. 55explain the observation of La-rich termination in ion-scattering experiments 53,54by the effect of increasing access to the lanthanum atom because of consid-erable surface oxygen relaxation that leads to a significantenhancement of the lanthanum atom signature. On the otherhand, Marx and coworkers have observed the La-terminatedLAO(001) with stoichiometry of (VLa 4O5)−0.5, where V is the lanthanum cation vacancy, i.e., each surface La is coordinate to four surface oxygens and four oxygens in the subsurfacelayer. 56Therefore, one may conclude that the experimental analyses have been performed at various conditions and reporteither LaO- and AlO 2-terminated LAO(001) or a mixture of them, so it is not clear if surfaces reached thermodynamicequilibrium. Ab initio thermodynamical stability diagrams previously calculated for LAO(001) show that the LaO-terminated surfaceis more stable with respect to the AlO 2-terminated one57 and that the LaO-terminated surface containing an oxygenvacancy is more stable than oxygen-deficient AlO 2-terminated LAO(001).58Mixed surfaces with LaO and AlO 2terminations were not predicted. In fact, our thermodynamic analysis doesnot support this prediction. From our point of view, the mainreason for such a discrepancy may be the different computa-tional approach, DFT within the local density approximation,used by the authors of Refs. 57and58. 155410-5SOROKINE, BOCHAROV , PISKUNOV , AND KASHCHEYEVS PHYSICAL REVIEW B 86, 155410 (2012) B. LAO/STO heterointerfaces 1. Charge redistribution and electronic properties Calculations of electronic properties of the LAO/STO(001) heterointerfaces were carried out using the symmetricallyterminated slab model. The STO(001) substrate consisted of11 atomic monolayers and could be terminated with either(TiO 2) monolayer in n-type heterostructures or with the (SrO) monolayer in p-type heterostructures. Monolayer-by- monolayer epitaxial growth then was modeled, adding a pairof respective monolayers of LAO(001) symmetrically to bothsides of a substrate slab until deposited LAO(001) thin filmreached thickness of up to 11 monolayers. In such way, weconstruct 22 heterostructures of both types and of differentLAO film thicknesses to model. Note that a 11-monolayer-thick substrate and a 20- ˚A-thick vacuum gap used for V ASP GGA calculations is enough to avoid an undesirable interaction of neighboring surfaces/interfaces and allows us to reach theequilibrium charge density redistribution in heterointerfacesunder study. Due to the restrictions by imposed symmetry, inour calculations atomic positions of all the heterointerfacesunder study were relaxed along the zaxis. If we consider atomic displacements, we can see that cations and anions in LAO monolayers have considerably differentdisplacements, thus, the electric dipole moment appears andaccumulates within the thin film. Stoichiometric heterointer-faces have greater displacement differences between anionsand cations than nonstoichiometric ones in LAO monolayers,while the situation is diametrically opposite for the STOmonolayers. As we shall see further, the dipole moment createsan electric field, and its potential strongly correlates withthe distortion of the band edges (so-called polar distortion),which then gives rise to the conductivity in stoichiometricLAO/STO(001) heterointerfaces of ntype. To predict the charge redistribution in heterointerfaces, we calculated the changes of net atomic Mulliken charges incomparison with the bulk phase of the LAO and STO parentmaterials. These charge deviations are shown in Figs. 4(a)–4(d) for LAO/STO(001) heterointerfaces of nandptypes. From these, one can clearly see that deviation of charges arerelatively small in the inner monolayers of the LAO filminn-type LAO/STO(001), not exceeding 0 .03e, whereas the same layers in the p-type LAO/STO(001) show quite large charge deviations ±(0.35–0.40)efrom the parent bulk, and these are negative for AlO 2monolayers and positive for LaO monolayers. In both n- andp-type interfaces, charges on the substrate monolayers did not vary substantially. For stoichiometricn-type and nonstoichiometric p-type interfaces, these are about ±0.04efor TiO 2and SrO, respectively. On the other hand, stoichiometric p-type interfaces show a small positive deviation of TiO 2monolayer charges (ca. 0 .01e) and about 10 times bigger negative charge deviation for SrO monolayers.Charge shifts in the substrates of stoichiometric n-type structures are all negative, and SrO shifts (ca. 0 .04e)a r e smaller than TiO 2shifts of ca. 0 .06e. Most significant deviations in atomic charges of n-type structures are located in the top-most monolayer— +0.2efor stoichiometric structures and −0.25efor nonstoichiometric ones—due to the surface effects and, thus, compensate the (a) (c) (b) (d) FIG. 4. (Color online) Calculated deviations of Mulliken effective charge densities ( /Delta1PQ)i nA Oa n dB O 2monolayers of [(a) and (b)] n-LAO/STO(001) and [(c) and (d)] p-LAO/STO(001) heterostruc- tures with respect to charge densities in AO and BO 2monolayers of STO and LAO bulk, correspondingly. Calculations are performed using a B3PW hybrid exchange-correlation functional. The xaxis shows the atomic monolayers from which atoms are originated. STOand LAO monolayers are numbered starting from the center of slab (0 means the central monolayer of the symmetrical slab unit cell). Monolayers (planes) are numbered separately for STO(001) substrateand for LAO(001) nanofilm. Panels (a) and (c) show a charge-density deviation for N LAO=10, while panels (b) and (d) for NLAO=11. “polar catastrophe” as proposed from a pure ionic model.59 Inp-type structures charge shifts in the surface layers are less pronounced than in the inner layers of the film and are+0.05eand−0.27efor LaO- and AlO 2-terminated structures, respectively. Here charge redistribution only in the thickeststructures investigated is shown. Respective graphs for thinnerstructures can be found in Refs. 60and61. Another way to look at the problem of charge redistribution is to calculate what happens with the electronic charge densityin the heterostructures compared to the isolated LAO andSTO slab parts. Charge-density redistribution is defined asthe electronic density in the heterointerface minus the sum ofelectron densities in separately isolated STO(001) substrateand LAO(001) thin film slabs and is depicted in Fig. 5for both n- andp-type LAO/STO(001) interfaces. These plots show us that the most significant distortions occur at the interface due to the compensation of the surfaceeffects of the slabs. They also show that the electronicstructure of the substrate of nonstoichiometric heterostructuresis distorted stronger than that of stoichiometric ones. Thesituation in the thin films is opposite. This fact correlates withthe argument in the section on atomic structure. 155410-6ELECTRONIC CHARGE REDISTRIBUTION IN LaAlO ... PHYSICAL REVIEW B 86, 155410 (2012) FIG. 5. (Color online) Difference electron charge-density maps calculated for [(a)–(d)] n-LAO/STO(001) and [(e)–(h)] p-LAO/STO(001) heterostructures: [(a) and (e)] (110) cross section for NLAO=10, [(b) and (f)] (100) cross section for NLAO=10, [(c) and (g)] (110) cross section for NLAO=11, and [(d) and (h)] (100) cross section for NLAO=11. Red (dark gray), blue (light gray), and gray isolines describe positive, negative, and zero values of the difference charge density, respectively. Isodensity curves are drawn from −0.025 to +0.025 e˚A−3 with an increment of 0.0005 e˚A−3. Right-side bar shows the atomic monolayers from which atoms are originated. Calculations are performed using a B3PW hybrid exchange-correlation functional. STO and LAO monolayers are numbered beginning from the center of the slab (0 means the central monolayer of the symmetrical slab unit cell). Monolayers (planes) are numbered separately for STO(001) substrate and forLAO(001) nanofilm. More illustrative property to consider is the polarization of all four of stoichiometric and nonstoichiometric n- and p-type heterointerfaces, which was already briefly introduced. It allows us to explain certain phenomena, such as the polardistortion, as well as to provide a mechanism for a partialcompensation of the “polar catastrophe.” Let us assume that each one of considered interfaces possesses no net charge, thus, it can be divided into multipleneutral slabs normal to z, in which net charge is also zero and average polarization of such slabs can be calculated. Thecharge-density function that should be used in the calculationsis estimated as if the charge of each atom Ais uniformly distributed over the plane z=z A, reducing the task to one dimension. Thus, the projection of polarization vector on z axis can be calculated as ¯Pi=/summationtext AzAQA /Delta1z, (6)where QAis the charge on atom Aand/Delta1zis the thickness of the neutral allocated slab, to which the atom Abelongs and summation is performed over all the atoms in the i-th neutral slab. In order to divide the interface in neutral slabs, itsometimes is necessary to split one monolayer’s charge: Onepart of it compensates the remaining charge of the previousslab and the remainder goes to the next one. The resulting polarization function ¯P(z) is averaged using the moving average function, and the results for n- and p-type interfaces are shown together with the energies of band boundaries E TVBandEBCBand the potential due to intrinsic electrostatic field Vin Fig. 6. Here one can see that LAO films of stoichiometric interfaces are strongly polarized, givingrise to the polar distortion of band edges. On the other hand,there is rather weak LAO polarization in the nonstoichiometricinterfaces, meaning a weak polar distortion as is observed.The substrate is polarized more in the nonstoichiometric case,which corresponds to Figs. 5(c) and 5(d) and Figs. 5(g) 155410-7SOROKINE, BOCHAROV , PISKUNOV , AND KASHCHEYEVS PHYSICAL REVIEW B 86, 155410 (2012) a 0 5 10 15 20 25 300.60.40.20.00.20.40.6c 0 5 10 15 20 25 300.60.40.20.00.20.40.6 b 0 5 10 15 20 25 300.60.40.20.00.20.40.6 Åd 0 5 10 15 20 25 300.60.40.20.00.20.40.6 au 102V TVB10 eV BCB 10 eV Å FIG. 6. (Color online) Polarization [as calculated using Eq. (6)], band edges and electrostatic potential of [(a) and (b)] n- LAO/STO(001) and [(c) and (d)] p-LAO/STO(001) heterostructures with [(a) and (c)] NLAO=10 and [(b) and (d)] NLAO=11 LAO monolayers. Zero at the energy scale corresponds to the Fermi level. Distances are measured from the central monolayer of the symmetrical slab unit cell. TVB stands for the top of valence bandand BCB stands for the bottom of the conduction band. and 5(h). AlO 2-terminated structures possess substantial po- larization in the top-most monolayer. The top-most layer’spolarization of LaO-terminated structures, on the other hand,is negligible. The interface monolayers of n-type structures are substantially polarized. Electronic properties in a more experimentally measurable way can be represented as band gaps for insulating structuresor as the concentration of charge carriers for conductors. Thesedata, obtained with CRYSTAL and V ASP, are represented in Tables IVandVforn- and p-type structures, respectively. First, one can see that all the nonstoichiometric interfacesare conducting and the free charge concentration is roughlyequal within a type and does not depend on the LAO filmthickness. p-Type structures possess greater carrier density thann-type structures, though experiments never showed conductive behavior in the former. For stoichiometric structures insulating behavior is the default one. The thickness of the band gap decreases with thethickness of the LAO film both for n- andp-type structures. This eventually leads to the closing of the gap for the n-type interfaces with N LAO/greaterorequalslant10 monolayers, which is in a good accordance with experimental works.11The gap-diminishing tendency is less pronounced for the p-type structures and, thus, they are not found conducting at any thickness withinthis study. The results obtained with V ASP are given for qualitative comparison. They showed out to be in accordance with CRYSTAL results, but, due to the specifics of the nonhybridTABLE IV . Band gaps ( δin eV) or sheet carrier density ( nsin 1014cm−2)o fn-LAO/STO heterointerfaces as calculated by means of hybrid B3PW and PW91 exchange-correlation functionals. NLAO tot stands for the total number of LAO(001) monolayers deposited atop STO(001) substrate. B3PW ( CRYSTAL )P W 9 1 ( VA S P ) NLAO tot Term. ML δn s δn s 1 LaO – 6.04 – 2A l O 2 3.65 – 1.41 – 3 LaO – 6.07 – 4A l O 2 2.91 – 1.03 – 5 LaO – 5.91 – 6A l O 2 1.96 – 0.40 – 7 LaO – 6.20 –8A l O 2 1.07 – 0.03 – 9 LaO – 6.27 – 10 AlO 2 – 1.56 – 0.16 11 LaO – 6.13 – 0.54 functional, band gaps and free charge concentrations are far too small. Taking into account that the largest difference betweenthat calculated using the CRYSTAL code and the experimentally observed band gap of bulk materials is 0.39 eV (see Table I), we note that our CRYSTAL calculations give plausible results compared to experimental data. The total band gap described above gives us some valuable data on the conducting-insulating behavior of the interfaces of different types. Nevertheless, it does not give us muchinformation about the origin of conductivity. Thus, it is moreworthy to look at the positions of the band edges in energyscale separately for each monolayer. Such a decompositionis depicted in aforementioned Figs. 6(a) and 6(b) and then Figs. 6(c) and 6(d) for n- andp-type structures, respectively. From these plots one can see that band edges for stoichiometricinterfaces are distorted; in addition, such a distortion leads ton-type conductivity in n-type structures that are thick enough and might hypothetically lead to the p-type conductivity in thicker p-type structures than investigated. Nonstoichiometric interfaces show little or no polar distortion, but it is notnecessary for the appearance of the conductivity, because such TABLE V . The same as in Table IVbut for p-LAO/STO(001) heterostructures. B3PW ( CRYSTAL )P W 9 1 ( VA S P ) NLAO tot Term. ML δn s δn s 1A l O 2 – 6.65 – 2 LaO 4.00 – 1.60 – 3A l O 2 – 7.27 – 4 LaO 4.05 – 1.69 – 5A l O 2 – 9.08 – 6 LaO 4.05 – 1.51 – 7A l O 2 – 7.90 – 8 LaO 3.80 – 0.48 –9A l O 2 – 6.97 – 10 LaO 2.92 – 0.25 – 11 AlO 2 – 10.2 – 0.12 155410-8ELECTRONIC CHARGE REDISTRIBUTION IN LaAlO ... PHYSICAL REVIEW B 86, 155410 (2012) structures contain nonstoichiometric LAO films, which are al- ready conducting on their own. Our prediction on conductivityof nonstoichiometric LaO-terminated n-type LAO/STO(001) interfaces is in agreement with a recent theoretical studyperformed by Pavlenko and Kopp (see Ref. 62) in which they show that the LaO-terminated n-type LAO/STO(001) interface is metallic. IV . SUMMARY AND CONCLUDING REMARKS We have performed large-scale first-principles calculations on a number of both stoichiometric and nonstoichiometricLAO/STO(001) heterostructures. Two different ab initio ap- proaches have been applied: LCAO with hybrid B3PW andPW with PW91 exchange-correlation functionals within DFT.Consistently within both approaches, we predict that thereexists a distortion in energies of band edges for stoichiometricstructures which eventually leads to the appearance of theconductivity at a critical thickness in n-type interfaces or to the reduction of the band gap for p-type interfaces. Nonstoichiometric interfaces were found to be conductingindependently of the LAO film thickness and possessing littleor no distortion of band edges. The conductivity appears dueto the nonstoichiometry of the thin film which is a conductoron its own, as we demonstrate by a separate analysis of anisolated film. The degree of distortion of the band edges agree well with the estimates of the internal electric field generated bychanges in the atomic charges and the geometric relaxationof the atomic structure. We confirm these factors as the onesresponsible for the rise of conductivity in stoichiometric n-type heterostructures. Calculated concentration of the free chargein the interfaces roughly agrees with the experimental data,being somewhat underestimated. For nonstoichiometric n-type interfaces, the electron gas structure is monolayered with uniform distribution over boththe film and the substrate, while for p-type interfaces it is bilayered with one part of free charge carriers located on3dorbitals of Ti at the IF, while the other is located on La orbitals at the surface. The total calculated n≈6×10 14cm−2 well agrees with that predicted from electrostatic assumptions nES=1/a2=6.6×1014cm−2. Of that, the IF gas layer getsnIF≈1.3×1014cm−2and the surface gas layer gets nS≈4.7×1014cm−2. Thermodynamic analysis that we have performed for the pristine LAO(001) surface reveals that both its LaO andAlO 2terminations may coexist at temperatures above 550 K. If the LAO/STO(001) heterointerface is covered by a LaOmonolayer, charge compensation mechanism of depositedpolar nonstoichiometric LAO film leads to the tendency ofTi 3+formation at the interface (see Fig. 4). To some extend it may explain the unexpected observation of Ti3+photoemission spectroscopy peak from n-type LAO/STO interfaces grown at 873 K.63 In general, we conclude that one should not disregard the stoichiometry aspect when considering ways to makethe LAO/STO interfaces conducting as nonstoichiometricinterfaces possess unique quasi-2D electron gas structure thatgives an overall 2 times greater free charge carrier density incomparison with stoichiometric interfaces. For stoichiometricn-type structures, the interplay of covalent and electrostatic forces leads to a metal-insulator transition at critical filmthickness but, for nonstoichiometric structures, it lead to theformation of a bilayered ( n-type IFs) or monolayered ( p-type IFs) quasi-2D electron gas. 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PhysRevB.67.155418.pdf

Phonons in a nanoparticle mechanically coupled to a substrate Kelly R. Patton and Michael R. Geller Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602-2451 ~Received 19 February 2002; revised manuscript received 5 February 2003; published 30 April 2003 ! The discrete nature of the vibrational modes of an isolated nanometer-scale solid dramatically modifies its low-energy electron and phonon dynamics from that of a bulk crystal. However, nanocrystals are usuallycoupled—even if only weakly so—to an environment consisting of other nanocrystals, a support matrix, or asolid substrate, and this environmental interaction will modify the vibrational properties at low frequencies. Inthispaperweinvestigatethemodificationofthevibrationalmodesofasphericalinsulatingnanoparticlecausedby a weak mechanical coupling to a semi-infinite substrate. The phonons of the bulk substrate act as a bath ofharmonic oscillators, and the coupling to this reservoir shifts and broadens the nanoparticle’s modes. Thevibrational density of states in the nanoparticle is obtained by solving the Dyson equation for the phononpropagator, and we show that environmental interaction is especially important at low frequencies.As a probeof the modified phonon spectrum, we consider nonradiative energy relaxation of a localized electronic impuritystate in the nanoparticle, for which good agreement with a recent experiment is found. DOI: 10.1103/PhysRevB.67.155418 PACS number ~s!: 63.22. 1m, 78.67.Bf I. INTRODUCTION There is currently great interest in properties of nanometer-scale mechanical systems, such as cantilevers,nanoparticles, and resonators. 1Because of the extremely small size and volume-to-surface ratio of these systems, in-teractions with their surroundings can dramatically alter theirproperties. In particular, it is well known that the vibrationalspectrum of an isolated nanometer-scale crystal, being dis-crete, is qualitatively different than that of the same bulksolid, leading to important changes in any property depen-dent on the phonon density of states ~DOS!. The differences between the vibrational DOS in a nanoparticle and a bulksolid are most evident at low frequencies: A spherical nano-particle with diameter dand characteristic bulk sound veloc- ity vcannot support a mode with frequency less than about pv/d. Thus an acoustic energy ‘‘gap’’ in the low-energy phonon spectrum is present in contrast with that of the bulk,which has a continuous spectrum down to zero energy. How-ever, mechanical interaction with the environment willmodify the vibrational modes. In an interesting experiment byYang and co-workers, 2the phonon DOS deep inside this gap was measured in insulating Y2O3nanoparticles. The experiment used nanoparticles whose sizes ranged from 7 to 23 nm in diameter and wasperformed by measuring the nonradiative lifetimes of an ex- cited electronic state of a Eu 31dopant. The lowest vibra- tional mode, referred to as the Lamb mode, for a nanopar-ticle with the mean size 3of 13 nm has a frequency of approximately 9 cm21.A t3c m21the DOS measured was more than 100 times smaller than that of bulk Y 2O3at 3 cm21. In this paper we propose and investigate a mechanism that could be responsible for the observed broadening of a nano-particle’s phonon modes. Several broadening mechanismscould be responsible for the observed effect. For example,anharmonicity leads to broadening and therefore to a low-energy DOS, but anharmonicity is ineffective at low energyand was found to be too small to account for theexperiment. 4Another possibility could be adsorbed ‘‘dirt’’on the outside of the nanoparticle. This might lower the qualityfactorQof the nanoparticle, regarding it as a resonator, re- flecting a broadening of the vibrational modes. A third, andin our opinion more likely mechanism, follows from the re-alization that these nanoparticles are not isolated, but instead were prepared in a powered form. Thus each nanoparticle isin weak contact with a cluster of other nanoparticles. Be-cause the cluster is relatively large, including at least severalhundred nanoparticles, each nanoparticle is mechanicallycoupled to a reservoir that has a continuous vibrational DOSat low energy. This interaction broadens the modes and al-lows phonons in the nanoparticle to escape and be absorbedinto the cluster.We will investigate the effect this mechanicalenvironmental interaction has on the nanoparticle’s phononspectrum. Because we are only interested in determining the correct origin of the observed broadening, 5and do not hope to be able to exactly reproduce the experimental results of Ref. 2,we propose the following simplified model: The cluster ofnanoparticles is replaced by a semi-infinite elastic substrate,and one nanoparticle is placed in weak mechanical contactwith it. The weak contact is imagined to be a few atomicbonds or small neck of elastic material, which we model bya harmonic spring. For simplicity, we take the substrate andthe nanoparticle to be made out of the same isotropic elasticmaterial. Because we are interested in the low-energy re-gime, continuum elasticity theory will be used to describethe dynamics of the nanoparticle and substrate. After defin-ing and analyzing our simplified model, in Secs. II–V, weexplain in Sec. VI how the model can be adapted to addressthe experiment of Ref. 2, and good agreement is obtained. The simple model we study is related to, but different than, models used to study energy relaxation by moleculesadsorbed on surfaces. 6However, in surface science the inter- est is usually in the relaxation of rigid translational motion,rotational motion, or simple internal vibrations of adsorbates.In contrast, we investigate the broadening of complex inter-nal vibrational modes of much larger objects ~which arePHYSICAL REVIEW B 67, 155418 ~2003! 0163-1829/2003/67 ~15!/155418 ~10!/$20.00 ©2003 The American Physical Society 67155418-1crystalline !. Our work also has much in common with that of Gurevich and Schober,7where many of the same consider- ations and modeling were used to study the Lamb-mode de-cay rate of nanoparticles caused by both anharmonicity andcoupling to an enviroment of other nanoparticles. II. NANOPARTICLE AND SUBSTRATE MODEL As discussed in the introduction, the model we study is that of a single nanoparticle in weak mechanical contact witha semi-infinite substrate. Linear elasticity theory will be usedto describe the phonons of this system. We assume the nano-particle and substrate to be made of an isotropic nonpolardielectric. Because we take the nanoparticle and substrate tobe made of the same material, we will use the same density rand Lame´coefficients landmfor both. The Lagrangian for the entire system is L5E Vd3r@1 2r~]tu!221 2luii22muij2#, ~1! whereu(r,t) is the displacement field, and uij[~]iuj1]jui!/2 ~2! is the strain tensor. Vis the combined volume of the nano- particle, substrate, and connecting material, as shown in Fig.1. Because the Lagrangian density is local, the integration volume in Eq. ~1!can be split into three independent parts: the nanoparticle, the substrate, and the connecting region. In the limit of weak coupling ~diameter d cof connecting region much smaller than d), the surface area on the nanoparticle and substrate over which the actual boundary conditions dif-fer from stress-free conditions are negligible, and the Hamil-tonian can be written as H5H nano1Hsub1dH, ~3! whereHnanois the Hamiltonian for an isolated nanoparticle ~with stress-free boundaries !,Hsubis that for an isolated sub- strate, and dHis the interaction between the two. The con- necting region is taken to be a few atomic bonds or smallneck of material, as shown in Fig. 1, and is discussed furtherbelow. Our analysis will require the vibrational normal modes and spectra of the isolated nanoparticle and semi-infinitesubstrate, calculated with stress-free boundary conditions.The long-wavelength modes of interest here may be obtainedfrom elasticity theory, to which we now turn. A. Isolated nanoparticle Here we derive the normal modes of an isolated elastic sphere. The method we shall use is different than ~but equivalent to !that used in the classic paper by Lamb,8but is better suited for our purposes. The equation of motion givenby Eq. ~1!is ]t2u2vl2~u!1vt233u50, ~4!where vl[A(l12m)/ris the bulk longitudinal sound veloc- ity and vt[Am/ris the transverse velocity. To solve Eq. ~4! the displacement field can be decomposed into longitudinaland transverse parts, u5u l1ut, ~5! where 3ul50 ~6! and ut50. ~7! With harmonic time dependence, the equation of motion ~4! then separates into two vector Helmholtz equations for thelongitudinal and transverse parts, ~„21p2!ul50,p[v/vl ~8! and ~„21q2!ut50,q[v/vt. ~9! FIG. 1. ~a!Model of nanoparticle, substrate, and connecting region.dis the diameter of the nanoparticle. ~b!Expanded view of cylindrical connecting region with dimensions dcandlc. In our numerical study we assume d510 nm,lc52.5 nm, and dc 50.5 nm.KELLY R. PATTON AND MICHAEL R. GELLER PHYSICAL REVIEW B 67, 155418 ~2003! 155418-2The longitudinal equation ~8!can be solved by introducing a scalar potential ul5f(p), ~10! where f(p)is a solution of the scalar Helmholtz equation („21p2)f(p)50. The transverse equation ~9!has two lin- early independent solutions, ut5MandN, where M5f(q)3r ~11! and N51 q3M. ~12! Here f(q)is a solution of ( „21q2)f(q)50. The prefactor 1/qis included for dimensional convenience. The scalar Helmholtz equations are separable in spherical coordinatesand the solutions can be written as flm~r![jl~kr!Ylm~u,w!,k5p,q ~13! where jl~x![Ap 2xJl11 2~x! ~14! is a spherical Bessel function of the first kind ~regular at origin !and Ylm~u,w![~21!mA2l11 4p~l2m!! ~l1m!!Plm~cosu!eimw. ~15! Here Plm~x![~12x2!m 2]m ]xmPl~x!, ~16! where the Pl(x) are Legendre polynomials. Now we use the flmto construct three linearly indepen- dent solutions of Eq. ~4!, Llm[1 pflm~pr!, ~17! Mlm[flm~qr!3r, ~18! Nlm[1 q3Mlm. ~19! The general solution is a linear combination of Llm,Mlm, andNlm, u~r!5( lm@almLlm1blmMlm1clmNlm#. ~20! Although they are linearly independent, the vector fields Llm,Mlm, andNlmare not orthogonal in space. However, they can be rewritten in terms of orthogonal vector spherical harmonics Plm,Blm, andClm, defined as Plm~V![Ylm~V!er, ~21!Blm~V![1 Al~l11!S]uYlm~V!eu1imYlm~V! sinuewD, ~22! Clm~V![1 Al~l11!SimYlm~V! sinueu2]uYlm~V!ewD, ~23! with the following properties: EdVXlm*Xl8m85dll8dmm8~24! forXPB,C,Pand EdVXlm*Xl8m8850 ~25! forXÞX8. Expressed in terms of orthogonal vector spheri- cal harmonics, Llm,Mlm, andNlmare given by Llm5jl8~pr!Plm~V!1Al~l11! prjl~pr!Blm~V!,~26! Mlm5Al~l11!jl~qr!Clm~V!, ~27! and Nlm5l~l11! qrjl~qr!Plm~V!1Al~l11! qr 3@jl~qr!1qrjl8~qr!#Blm~V!, ~28! where prime denotes differentiation with respect to the argu- ment. Next we impose stress-free boundary conditions sijnj50 ~29! at the surface r5Rof the nanoparticle. Here nis an outward pointing normal vector and sijis the strain tensor. In an isotropic elastic continuum, sij5l~u!dij12muij. ~30! In spherical coordinates ~29!implies srr5sur5swr50. ~31! The three conditions ~31!require that l~u!12murr50, ~32! uur50, ~33! and uwr50. ~34! In terms of the displacement field, urr5]rur, ~35!PHONONS IN A NANOPARTICLE MECHANICALLY. . . PHYSICAL REVIEW B 67, 155418 ~2003! 155418-3uur51 2S]ruu21 ruu11 r]uurD, ~36! uwr51 2S1 rsinu]wur1]ruw21 ruwD. ~37! The boundary-condition equations ~31!then become alm@2lpjl~pR!Ylm12mpjl9~pR!Ylm# 1clm2ml~l11!EYlm50, ~38! alm2D]uYlm1blmimEYlmcscu1clmF]uYlm50,~39! and alm2imDYlmcscu1blmE]uYlm1clmimFYlmcscu50, ~40!where D[jl8~pR! R2jl~pR! pR2, ~41! E[jl8~qR! R2jl~qR! qR2, ~42! F[qjl9~qR!1l~l11! qR2jl~qR!22jl~qR! qR2. ~43! Finally, we rewrite Eqs. ~38!–~40!in matrix form as S2lpjl~pR!Ylm12mpjl9~pR!Ylm 02 ml~l11!EYlm 2D]uYlm imEYlmcscuF]uYlm 2imDYlmcscu E]uYlmimFYlmcscuDSalm blm clmD50. ~44! For a nontrivial solution of Eqs. ~38!–~40!to exist, the de- terminant of the above matrix must vanish. Taking the deter-minant and simplifying we obtain @2lpjl~pR!12mpjl9~pR!#EF24ml~l11!E2D50. ~45! This implies that either E5jl8~qR! R2jl~qR! qR250 ~46! or @2lpjl~pR!12mpjl9~pR!#F24ml~l11!ED50. ~47! If condition ~46!is met, then this imposes certain con- straints on alm,blm, andclm, which require alm5clm50. This can easily be seen in the matrix of Eq. ~44!by setting E50. If Eq. ~47!is met,blmhas to be zero. In conclusion, we have two branches of vibrational modes: The branch in which Eq. ~46!is satisfied, u~r!5blmnMlmn~r!, ~48! is referred to as the torsional branch, where nspecifies the radial quantum number @nth solution of Eq. ~46!#. These modes have no radial component @see Eq. ~27!#and arenot broadened by the mechanism considered in this paper. Theother branch is found when Eq. ~47!is satisfied, u ~r!5almnLlmn~r!1clmnNlmn~r!, ~49! which is called the spheroidal branch.We define the Lamb mode to be the vibrational eigenfunc- tion associated with the lowest frequency solution of Eq.~45!. For the cases considered in this paper, the Lamb mode is a fivefold degenerate l52 torsional mode with angular frequency vL5j0vt R, ~50! where, according to Eq. ~46!,j0’2.50 is the smallest posi- tive solution of j28~j!21 jj2~j!50. ~51! Thus the Lamb frequency is simply vL’1.593pvt d. ~52! We emphasize, however, that the Lamb mode has no radial displacement component and is not broadened in our model. To quantize the vibrational modes we write the displace- ment field as9 unano~r!5( JA\ 2rvJ@aJCJ~r!1aJ†CJ*~r!#,~53! where J5@So rT ,n,l,m# ~54! is a label uniquely specifying a nanoparticle eigenmode. The first entry S or T specifies whether the mode is in the sphe-KELLY R. PATTON AND MICHAEL R. GELLER PHYSICAL REVIEW B 67, 155418 ~2003! 155418-4roidal or torsional branch, respectively. nis the radial quan- tum number and landmare the usual angular-momentum quantum numbers. aanda†are phonon annihilation and cre- ation operators which satisfy the Bose commutation relation @aJ,aJ8†#5dJJ8. ~55! The CJare vibrational eigenvectors normalized such that E VuCJ~r!u2d3r51, ~56! whereVis the volume of the nanoparticle. Assuming ~with- out proof !that the modes CJform a complete set, ( JCJi*~r!CJj~r8!5dijd~r2r8!, ~57! it can easily be shown that usatisfies the correct equal-time canonical commutation relation with p[r]tu, namely @ui~r!,pj~r8!#5i\dijd~r2r8!. ~58! B. Isolated substrate The vibrational modes for a semi-infinite isotropic elastic substrate, with a free surface at the xyplane and extending to infinity in the negative zdirection, were quantized previously by Ezawa.10Therefore the details will be left out here. The displacement field can be written as9 usub~r!5( IA\ 2rvI@bIfI~r!1bI†fI*~r!#, ~59! wherebandb†are the annihilation and creation operators for the substrate phonons. The index I, like the index Jfor the nanoparticle, uniquely specifies a phonon mode for the substrate. fIare eigenfunctions of Eq. ~4!subject to stress- free boundary conditions at the z50 plane. In what follows we will need the spectral density of the isolated substrate, which is defined as Nsub~r,v![21 pImDsubzz~r,r,v!, ~60! whereDsubij(r,r8,v) is the Fourier transform of the retarded phonon Green’s function Dsubij~r,r8,t![2iu~t!^@usubi~r,t!,usubj~r8,0!#&~61! of the substrate. The spectral density at the free surface of silicon, regarding it as an isotropic elastic continuum, wascalculated in Appendix B of Ref. 11. There we obtained N sub~v!5CSiv,CSi’1.4310246cm2s2.~62! It turns out that Eq. ~62!is quite close to that resulting from the simpler Debye model for three-dimensional bulk Si. C. Nanoparticle-substrate interaction As discussed above, the connection between the nanopar- ticle and substrate is assumed to be weak, allowing perturba-tion theory to be applied. The connection consists of a small cylindrical neck of elastic material, with diameter dcand lengthlc. In addition, we assume that dc!lc, in which case the dominant interaction between the nanoparticle and thesubstrate is mediated by the longitudinal compression/extension of this cylinder. Thus the interaction between thenanoparticle and substrate is that of a harmonic spring dH51 2K:@unanoz~r0!2usubz~r0!#2:, ~63! whereKis an effective spring constant, and unanozandusubz are thezcomponents of the displacement field of the nano- particle and substrate at the point of contact r0. We take the zdirection to be along the upward pointing normal to the substrate surface. The Hamiltonians we will introduce below @see Eq. ~75!#forHnanoandHsubare normal-ordered; it is therefore necessary to normal order dHas well. This opera- tion is denoted by the colons in Eq. ~63!. The force constant Kgiven by K5pdc2 4lcY, ~64! whereY’1.331012dyn cm-2is the Young modulus for Si. Assuming neck dimensions of lc52.5 nm and dc50.5 nm, we obtain K51.03104erg cm-2. ~65! Of course, Eq. ~65!is just an estimate of the actual interac- tion strength and should not be taken seriously beyond theorder-of-magnitude level. III. GOLDEN-RULE LIFETIMES The relaxation rate or inverse lifetime of the perturbed eigenmodes of the nanoparticle can be calculated using Fer- mi’s golden rule ~setting \51), tJ2152p( fu^fudHui&u2d~vi2vf!, ~66! where the initial and final states are ui&5aJ†u0&anduf&5bI†u0&. ~67! Using Eqs. ~53!,~59!, and ~63!, leads to9 tJ215pK2 2r2uCJz~r0!u2 vJ( IufIz~r0!u2 vJd~vJ2vI!.~68! Noting that ( IufIz~r0!u2d~vJ2vI!52rvNsub~v!, ~69! we obtain ~reinstating factors of \) tJ215pK2 \rNsub~vJ! vJuCJz~r0!u2. ~70!PHONONS IN A NANOPARTICLE MECHANICALLY. . . PHYSICAL REVIEW B 67, 155418 ~2003! 155418-5UsingK51.03104erg cm22andr52 . 3gc m23, which are appropriate ~see above !for our model, the resulting re- laxation rates and quality factors are given in Table I forsome low-lying modes. The quality factor Qis defined here as the lifetime tdivided by the period T, Q[t T5\v 2pg, ~71! where g[\t21is an energy width. The values of the Qfactors we obtain for the low lying modes are incredibly large, reflecting the fact that the reser-voir~substrate !is extremely ineffective at absorbing energy at these low frequencies. As we will discuss below in Sec.VI, the lifetimes ~andQfactors !for the model considered here cannot be directly compared with the experiment ofRef. 2 without accounting for the difference in sound speedsbetween a solid Si substrate and a weakly bound nanoparticlecluster ~as well as some other less important modifications !. There we shall show that the coefficient Cin Eq. ~62!should be enhanced by a factor of about 10 3before making such a comparison, which decreases the Qfactors by this same fac- tor. However, the Qfactors corrected in this way are still huge, and the good agreement with the observed low-frequency DOS ~see below !suggests that the Qfactors of the nanoparticles studied experimentally in Ref. 2 are also verylarge. In Ref. 12 we used the golden-rule result ~70!to estimate the phonon DOS at low energies. This is achieved by replac-ing, in accordance with Fermi’s golden rule, each discretemode in the isolated nanoparticle by a Lorentzian with awidth given by Eq. ~70!.~More precisely, this amounts to approximating the energy-dependent phonon self-energy for each mode Jwith its value at v5vJ, a procedure often called the quasiparticle-pole approximation. !However, this procedure is unreliable at low energies because the actualline shapes of the broadened modes are non-Lorentzian in the tails. Nevertheless, we obtained a DOS at 3 cm 21that was only 20 times smaller than that observed.13 IV. MANY-BODY THEORY OF THE DOS A. Local DOS To leading order in the electron-phonon interaction strength, the electronic population relaxation rate due to pho-non emission ~for example, as measured in Ref. 2 !is given by Fermi’s golden rule, 5which states that the rate ~for a deformation potential electron-phonon interaction !is propor-tional to the square of the electron-phonon coupling strength times the phonon DOS. In a translationally invariant systemthe DOS does not have any position dependence, but in ananoparticle one must distinguish between the globalDOS ~the DOS relevant for thermodynamics !and the local eigenfunction-weighted DOS, which is the one that deter-mines phonon emission rate. We will call this position- dependent DOS the local DOS and denote it by g(r, v). The precise definition of g(r,v) will be given below in Eq. ~77!. From a theoretical point of view, the quantity describing the local vibrational dynamics in the nanoparticle is the ~re- tarded !phonon Green’s function DRij~r,r8,t![2iu~t!^@ui~r,t!,uj~r8,0!#&H, ~72! where ^&H[Tr~e2bH! Tre2bH, ~73! with the Hamiltonian given by H5H01dH. ~74! HereH0is the Hamiltonian of the isolated nanoparticle and substrate, H05( JvJaJ†aJ1( IvIbI†bI, ~75! and, as mentioned in Sec. II, dHis a harmonic spring poten- tial given in Eq. ~63!. In this section the phonon Green’s function Dalways re- fers to the nanoparticle, and the label ‘‘nano’’ will be sup-pressed. The imaginary part of the Fourier transform of D Rij(r,r,t) defines the nanoparticle’s phonon spectral density Nij~r,v![21 pImDRij~r,r,v!. ~76! For an electron system ~or any system of particles !, the spec- tral density in Eq. ~76!is precisely the local DOS. However, because the elasticity theory equation of motion ~4!is second order in time, the spectral density and DOS ~both local and global !differ by a factor of 2 rv. In addition, the vibrational spectral density ~76!is a tensor, whereas the phonon emis- sion rate probes some coupling-constant-weighted sum oftensor elements. Because we are ascribing the observed re-duction in phonon emission ~in going from bulk to nanopar- ticle!to a reduction in the local DOS, our results are not sensitive to the precise way in which a scalar quantity isconstructed from the tensor, as long as the same measure isused in both the nanoparticle and bulk. It will be most con-venient to investigate the trace of the local DOS tensor.Therefore the quantity we calculate in this paper is g ~r,v![2rv( i513 Nii~r,v!, ~77! which we shall refer to as the local DOS. g(r,v) character- izes the number of states per unit energy per unit volumeTABLE I. Afew representative relaxation rates and quality fac- tors. Only the m50 spheroidal modes are broadened by the mecha- nism considered in this paper. (S,l,m,n) v~rad s21) t21(s21)Qfactor (S,2,0,1) 3.1 310122.5310242.231015 (S,1,0,1) 3.4 310121.0310287.831019 (S,0,0,1) 1.0 310133.6310244.931015 (S,0,0,2) 1.6 310131.1310232.531015KELLY R. PATTON AND MICHAEL R. GELLER PHYSICAL REVIEW B 67, 155418 ~2003! 155418-6near position r. In a bulk material with Debye spectrum, Eq. ~77!reduces at low frequency to g~r,v!5v2 2p2S1 vl312 vt3D, ~78! independent of r. Equation ~78!is the well-known Debye formula for the vibrational DOS of a three-dimensional bulkcrystal. The local DOS g(r, v) controls the phonon emission rate for an impurity atom sitting at position r. Although the im- purity locations in a real nanoparticle are assumed to be ran-dom, dopants near the surface are known to be opticallyinactive. Hence most optical experiments ~including that of Ref. 2 !do not probe impurities near the nanoparticle’s sur- face. Therefore we introduce a particular volume-averagedDOS g ¯~b,v![E r<bd3rg~r,v! 4 3pb3, ~79! which characterizes the average g(r,v) within a sphere of radiusb. In the limit b!R, in which case the local DOS is averaged over the full nanoparticle volume, we obtain theglobal ~or thermodynamic !DOS, which for an isolatednano- particle would be simply g¯~R,v!51 V( Jd~v2vJ!. ~80! Physically, we expect bto be somewhere between R/2 andR. B. Perturbative calculation of the local DOS The retarded Green’s function ~72!for the nanoparticle can be obtained by calculating the Euclidean time-ordered~or imaginary time !Green’s function defined by D ij~r,r8,t!52^Tui~r,t!uj~r8,0!&H. ~81! In the interaction representation, Dij~r,r8,t!52^Tui~r,t!uj~r8,0!e2*0bdH(t8)dt8&H0 ^e2*0bdH(t8)dt8&H0, ~82! where the expectation values are with respect to H0.B y expanding the exponentials to leading order in the perturba-tion~63!and Fourier transforming, Eq. ~82!can be written as D ij~r,r8,v!5D0ij~r,r8,v!1( klED0ik~r,r9,v! 3Pkl~r9,r-,v!D0lj~r-,r8,v!d3r9d3r-, ~83! whereD0ij~r,r8,v!5( JCJi~r!CJj*~r8! 2rvJF1 iv2vJ21 iv1vJG ~84! is the free propagator and Pijis the leading-order phonon self-energy, given at zero temperature by9 Pij~r,r8,v!5K2 2r( IufIz~r0!u2 vIF1 iv2vI21 iv1vIG 3dizdjzd~r2r0!d~r82r0!. ~85! ThefI(r) are substrate eigenfunctions discussed in Sec. II, andr0is the point at which the nanoparticle is connected to the substrate. The self-energy ~85!is correct away from the centers of the vibrational modes, but ignores small elastic scattering contributions that shift the modes without broad-ening them. Retarded quantities are obtained by analytically continuing i v!v1i01. To calculate the local DOS ~77!we need to solve the Dyson equation for the nanoparticle Green’s function, writ-ten symbolically as D5D 01D0PD. ~86! The solution to Eq. ~86!can be obtained by introducing ma- trix representations for D,D0, and P, in which case D5~D0212P!21. ~87! The matrix representation we use is defined by O~J,J8![( ijE Vd3rd3r8CJi*~r!Oij~r,r8!CJ8j~r8!, ~88! whereO5D,D0,o rP. In Eq. ~88!the integration is over the volume Vof the nanoparticle, and the CJ(r) are the nanoparticle eigenfunctions. The inverse transformation is Oij~r,r8!5( JJ8CJi~r!O~J,J8!CJ8j*~r8!. ~89! A nanoparticle with a diameter of 10 nm has approximately 8000 atoms in it. Thus, there are roughly 24000 acousticvibrational modes. By knowing the total number of modes, aDebye energy can be defined: The Debye energy is the en-ergy at which there are 24000 elasticity-theory modes that liebelow in energy. For our nanoparticle, the Debye energy is about 320 cm 21. The Debye energy cutoff truncates the Hil- bert space, which leads to finite-size matrices.14This enables every mode Jof the nanoparticle to be included in the cal- culation of the Green’s function ~87!. V. RESULTS In this section we present our results for the phonon DOS in a 10-nm Si nanoparticle, obtained by solving the Dysonequation ~86!for the phonon Green’s function, as explained above. As we have discussed, the DOS g(r, v), defined in Eq.~77!, is a local quantity that varies with position within the nanoparticle, and, as mentioned in the previous section,PHONONS IN A NANOPARTICLE MECHANICALLY. . . PHYSICAL REVIEW B 67, 155418 ~2003! 155418-7the quantity we are interested in is g¯(b,v), which is g(r,v) averaged over a sphere of radius bpositioned at the center of the nanoparticle. Because we have found no significant de- pendence of g¯(b,v)o nb, for the physically relevant values ofbbetweenR/2 andR, we plot simply g¯(R,v). As stated above in Sec. IV A, g¯(R,v) is just the globalphonon DOS in the nanoparticle. For simplicity we assume both the nanoparticle and the substrate to be made of Si; this allows us to use the surfacespectral density ~62!calculated in Appendix B of Ref. 11, where Si is treated as an isotropic elastic continuum withlongitudinal and transverse sound velocities vl58.53105cm s21, vt55.93105cm s21, ~90! and mass density r52 . 3gc m23. In the final section of this paper, where we compare our results to the experiment ofRef. 2, we will introduce an important correction to accountfor the differences between a solid Si substrate and a nano-particle cluster. In Fig. 2 the global DOS g ¯(R,v) of a 10-nm diameter nanoparticle is given up to 100 cm21. The ~unbroadened ! Lamb mode has a frequency @see Eq. ~52!#of 15.7 cm21, but is obscured by the nearby ( l52,m50,n51) spheroidal mode at 16.5 cm21. The modes above 100 cm21were in- cluded in the calculation, but the long-wavelength approxi-mation of elasticity theory becomes invalid at high energy.Thus only the lower part of the spectrum is shown. Figure 3 shows the low-energy phonon DOS up to about 23 cm 21.A n expanded view of the low-energy DOS is given in Fig. 4. The phonon DOS at 3 cm21is approximately 4.5 31010 states per wave number per cm3.VI. COMPARISON WITH EXPERIMENT In this section we compare our results to the experiment of Ref. 2, where the one-phonon emission rate ~and therefore the phonon DOS at 3 cm21) in a cluster of Y 2O3nanopar- ticles was observed to be 8.2 31023times that in bulk Y 2O3. In particular, the excited5D1(II) state of Eu31in the nano- particles had a phonon-emission lifetime of 27 ms, compared with a bulk value of 221 ns. In order to make a comparisonof our results to that of the experiment, two modifications ofour calculation have to be performed. ~i!In our model, the cluster of nanoparticles has been replaced by a solid substrate. However, the spectral density~62!of the substrate, which at long wavelengths is deter- mined by the sound speeds and mass density of Si, is muchsmaller than that of the nanoparticle cluster. Treating thelong-wavelength modes of the cluster with elasticity theory~or, even simpler, approximating the random cluster by an ordered cubic lattice !, shows that the spectral density ~62! FIG. 2. The phonon DOS, given in states per wave number per cm3, of a 10-nm Si nanoparticle, weakly coupled to a semi-infinite Si substrate. FIG. 3. Vibrational DOS at low energies. FIG. 4. Expanded view of the low-energy DOS. Note that the DOS vanishes at zero energy, as expected.KELLY R. PATTON AND MICHAEL R. GELLER PHYSICAL REVIEW B 67, 155418 ~2003! 155418-8should be replaced by ~the subscript ‘‘cl’’referring to cluster ! Ncl~v!5Cclv, ~91! where15 Ccl5vSi3rSi vcl3rclCSi. ~92! Here vclis a characteristic sound speed in the cluster and rcl is its mass density. The 1/ vcl3dependence in Eq. ~92!comes from the well-known velocity dependence of the Debye DOS, and the 1/ rclfactor comes from the definition of spec- tral density @see the discussion following Eq. ~76!#. Approxi- mating the cluster by an ordered cubic array with lattice con-stantd~the nanoparticle diameter !yields vcl’AK Md ~93! and rcl’p 6rSi, ~94! whereKis the effective spring constant connecting the nano- particles, given in Eq. ~65!, and M54 3prSiSd 2D3 ~95! is the mass of one nanoparticle. Using d513 nm ~the mean nanoparticle diameter in Ref. 2 !, we obtain an enhancement factor of Ccl CSi59.83102. ~96! This factor increases the nanoparticle DOS ~at frequencies below the Lamb mode !by nearly three orders of magnitude. We emphasize that this correction originates from the in-creased spectral density of the environment—the nanopar-ticle cluster—compared with that of a solid substrate. There are several other marginally important corrections, most of which will be ignored, and one that will be includedfor completeness. ~ii!In the model analyzed above, the nanoparticle was connected to its surroundings by only a single contact point,whereas a nanoparticle in a cluster most likely has more thanone connection. As the number of contacts increases, thissimply scales the DOS ~away from the peaks !linearly with the number of contact points. Conservatively, we expect thatthe multiple contact points present in the real system willincrease the decay rate of the nanoparticles’ modes, andhence the phonon DOS well below the Lamb mode, by afactor of 2. The following additional corrections have also been con- sidered and were found not to be significant ~and are not included in our final results !: ~iii!The actual experiment of Ref. 2 was done on an en- semble of nanoparticles with mean diameter 313 nm andstandard deviation of 5 nm. To understand the effects of this size distribution, we have calculated the DOS at 3 cm21 averaged over a Gaussian distribution of diameters centered at 10 nm. Even for very wide distributions ~standard devia- tions up to 8 nm !, the ensemble averaged DOS at 3 cm21is increased by no more than a factor of 2.16In addition, the correction for re-centering the size distribution from 10 nmto the experimentally observed 13-nm average size leads tocorrections only of order unity. ~iv!Our calculations were done for a nanoparticle and substrate made of Si, while the experiment was done on Y 2O3nanoparticles, which, of course, have different mass density and sound velocities.The differences in sound speedsand mass only shift the modes of the nanoparticle by a smallamount. 17This has the same effect as a small change in the diameter of the nanoparticles, which we have found to benegligible. As for the substrate ~or more precisely the re- placement of the substrate with a cubic lattice of nanopar-ticles !, the change in mass density does effect the spectral function ~62!by changing the velocity ~93!and mass density ~94!, but this change is only of order unity. ~v!The nanoparticles of Ref. 2 were immersed in He, either liquid ~forT,4.21 K) or gas ( T.4.21 K). However, the results were found not to change through the liquid-gastransition, presumably because of the large sound-speed mis- match between superfluid He and Y 2O3. Therefore we have ignored the presence of He in our theory. ~vi!The experiment of Ref. 2 was done at temperatures between 1.5 and 10 K ~excluding the interval 2.17–4.21 K !, whereas our calculations assume zero temperature. The ef-fect of finite temperature is to stimulate phonon emissioninto the bath ~substrate or cluster !. However, this is not im- portant until the Bose distribution function of the bath at the Lamb mode frequency ~approximately 9 cm 21) becomes of order unity, which does not occur until the temperature ex-ceeds about 13 K. Taking into account the first two modifications, and ignor- ing the others, we obtain a 3-cm 21DOS given by g¯~R,3cm21!54.531010states wave number cm3398032 58.831013states wave number cm3. ~97! The DOS of bulk Si is given by ( le2 2p2\3vl353.931015E2states wave number cm3,~98! whereEis the energy in wave numbers. Thus the theoretical ratio of the DOS of the nanoparticle to that of the bulk ma- terial at 3 cm21is 2.5 31023. As stated above, the experi- mental ratio of nanoparticle to bulk DOS was found to be approximately 8.2 31023. The agreement between our theory and the experiment of Ref. 2 is excellent consideringthe simplicity and robustness of our model.We conclude thatthe low-energy phonon DOS observed in Ref. 2 is consistentwith our enviromental broadening mechanism.PHONONS IN A NANOPARTICLE MECHANICALLY. . . PHYSICAL REVIEW B 67, 155418 ~2003! 155418-9VII. CONCLUSIONS Motivated by an interesting experiment2measuring the low-energy phonon DOS in a insulating nanoparticle, wehave thoroughly investigated a simplified model of a singlenanoparticle weakly coupled to its environment, a semi-infinite substrate. The environmental interactions were foundto significantly affect the DOS at energies below the Lambmode. Additionally, we have used the results of our model to predict the effect of environmental interaction in a cluster ofnanoparticles like that studied in Ref. 2. Although it is nec-essary to estimate the value of several quantities appearing inthe model, we believe that we can do this accurately enoughto obtain a final result that is correct at the order-of-magnitude level, with no free parameters. Because our re- sults for the 3-cm 21DOS is only about three times smaller than that observed in Ref. 2, our broadening mechanism andthe resulting phonon spectrum is clearly consistent with thatexperiment. Our theory makes several predictions that could be inves- tigated in future experiments:The vibrational DOS as a func-tion of energy at energies below the Lamb mode could bemeasured by observing the one-phonon emission rate from amagnetic impurity, whose energy splitting can be changedaway from 3 cm 21by varying an external magnetic field. Also, the predicted sensitivity of the low-energy DOS to the sound velocity ~and hence spectral density !of the environ- ment could be probed by modifying the mechanical proper-ties of the nanoparticle cluster, say, by applying externalpressure and measuring the cluster sound velocity along withthe phonon emission rate. The lifetime of the broadened vi-brational modes could be directly measured as well ~by neu- tron scattering, for example !, but splitting of the ~degenerate ! vibrational modes caused by deviations from spherical sym-metry would then be important. We hope that detailed inves-tigations such as these will be carried out in the future. ACKNOWLEDGMENTS This work was supported by the National Science Foun- dation under CAREER Grant No. DMR-0093217, and by theResearch Corporation. It is a pleasure to thank Michael Dun-can, David Leitner, Steve Lewis, Vadim Markel, RichardMeltzer, Mark Stockman, and Ho-Soon Yang for useful dis-cussions, and Patrick Sprinkle for help with the numerics.We would like to thank Dan Murray for his correspondenceand for pointing out an error in a previous version of thiswork. The authors are especially grateful to Bill Dennis forhis encouragement and help with every aspect of this work. 1A. N. Cleland, Foundations of Nanomechanics ~Springer-Verlag, Berlin, 2002 !. 2H. S. Yang, S. P. Feofilov, D. K. Williams, J. C. Milora, B. M. Tissue, R. S. Meltzer, and W. M. Dennis, Physica B 263, 476 ~1999!. 3The average nanoparticle size was stated incorrectly in Ref. 2. The correct value is 13 nm. See H. S. Yang, K. S. Hong, S. P.Feofilov, B. M. Tissue, R. S. Meltzer, and W. M. Dennis, J.Lumin.83, 139 ~1999!. 4V. A. Markel and M. R. Geller, J. Phys.: Condens. Matter 12, 7569 ~2000!. 5The experiment of Ref. 2 did not directlyprobe the phonon spec- trum of the nanoparticle. However, both the observed exponen-tial decay of the excited electronic state and our estimates of theelectron-phonon interaction strength suggest that the nanopar-ticle is in the relaxational regime where the electron’s lifetime isdetermined by the phonon DOS through Fermi’s golden rule.See M. R. Geller, W. M. Dennis, V.A. Markel, K. R. Patton, D. T. Simon, and H. S.Yang, Physica B 316, 430 ~2002!for a brief discussion of this question. 6A. Zangwill, Physics at Surfaces ~Cambridge University, New York, 1988 !. 7V. L. Gurevich and H. R. Schober, Phys. Rev. B 57,1 12 9 5 ~1998!. 8H. Lamb, Proc. London Math. Soc. 13, 189 ~1882!. 9Throughout this work we use Ito label the vibrational modes ofthe substrate and Jfor the nanoparticle. 10H. Ezawa, Ann. Phys. ~N.Y.!67, 438 ~1971!. 11K. R. Patton and M. R. Geller, Phys. Rev. B 64, 155320 ~2001!. 12K. R. Patton and M. R. Geller, J. Lumin. 94, 747 ~2001!. The vibrational eigenmodes in this preliminary report have frequen-cies slightly different than the present work because of a numeri-cal error brought to our attention by Dan Murray. 13This agreement is fortuitous, however, because we did not ac- count for the modified sound velocity of the bath. After thismodification is made, the golden-rule estimate for the low-frequency DOS is too large ~by a factor of 50 at 3 cm 21), as expected. 14Each such double-precision complex matrix requires about 9 GB of memory. 15For Si we use the branch-averaged sound velocity vSi[S1 3( l1 vl3D21/3 56.43105cms21. 16We note, however, that when the Lamb mode frequency becomes resonant with the electronic two-level system, the electronic re-laxation rate is not simply related to the phonon DOS. 17For example, the Lamb mode frequency for a 13-nm Y2O3nano- particle is about 9 cm21, and for one made of Si is 12 cm21. These estimates follow from Eq. ~52!using the Y2O3transverse sound speed of vt54.33105cm s21as measured by C. Proust, Y. Vaills, and L. E. Husson, Solid State Commun. 93, 729 ~1995!.KELLY R. PATTON AND MICHAEL R. GELLER PHYSICAL REVIEW B 67, 155418 ~2003! 155418-10

PhysRevB.72.245419.pdf

Electronic structure of boron nitride cone-shaped nanostructures Chunyi Zhi, *Yoshio Bando, Chengchun Tang, and Dmitri Golberg Advanced Materials Laboratory, National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan /H20849Received 7 September 2005; published 19 December 2005 /H20850 Boron nitride nanohorns /H20849BNNHs /H20850were synthesized in a large-scale. Detailed transmission electron micros- copy investigations reveal that a BNNH consists of boron nitride cone-shaped nanostructures /H20849BNCNSs /H20850with different disclination angles. Systematic studies of electronic structure of a BNCNS with different disclinationangles and different defects were performed by first-principles calculations. The results imply that the elec-tronic structure of a BNCNS weakly depends on the disclination angles; by contrast, it is strongly influencedby the types of line-bond defects /H20849B-B or N-N bonds /H20850. Importantly, our calculations show that a BNNH can be the junction made of semiconducting nanostructures with various energy gaps. DOI: 10.1103/PhysRevB.72.245419 PACS number /H20849s/H20850: 73.61.Wp, 81.05.Tp I. INTRODUCTION Carbon-based nanostructures, such as fullerene clusters, nanotubes, and cone-shaped particles have attracted signifi-cant attention due to a variety of novel physical properties.Boron nitride /H20849BN/H20850and carbon can be found in nature in a similar layered structure. However, the BN system has sev-eral unmatched properties when compared with carbon. For example, opposed to carbon nanotubes /H20849CNT /H20850, the electronic properties of a boron nitride nanotube /H20849BNNT /H20850have been predicted to be independent of various morphological and/orgeometrical factors. 1In addition, BN nanomaterials have ex- cellent mechanical properties, a high resistance to oxidation,and chemical stability, 2,3which makes them highly valuable in electronic devices as far as usage at elevated temperaturesor in hazardous environments are concerned. Recently, anelectric polarization induced by broken symmetry along theBNNT axis has been predicted in a theoretical study, 4which may stimulate the BNNT potential applications in a widerange of technological fields. Following the successful synthesis of BNNTs, 5,6the search for related BN nanostructures has been initiated.7–9A BN cone-shaped nanostructure /H20849BNCNS /H20850is of special inter- est since its properties are only related to a disclination angle/H20849disclination angle is defined as the angle of the sector re- moved from a flat sheet to form a cone /H20850and defects. 10,11 Cone-shaped carbon nanostructures, such as nanocones and nanohorns, have been successfully synthesized and a numberof theoretical and experimental works have been carried outto shed light on their electronic and mechanicalproperties. 12–17However, the analogous works in the BN sys- tem are still lacking. It is worth noting that BN conical nano-tubes have been found in a material containing large quanti-ties of standard BNNTs. 18These conical nanotubes have a solid core and are made of stacked BN conical structures orformed by a continuous BN sheet wrapping in a helical fash-ion. However, this novel nanostructure was only a by-product of the BNNT synthesis, and its yield was scarce.Recently, using chemical vapor deposition with boron andmetal oxide as the reactant /H20849BOCVD /H20850, BN nanohorns /H20849BN- NHs/H20850were synthesized with high yield. 19Using selected area electron diffraction patterning /H20849SAED /H20850, it was found that the horns are made of nanostructures with several disclinationangles. This indicates that BNNHs may have interesting electronic properties. However, the systematic investigationsof the electronic structure of BNCNS are not available in theliterature. Contradicting results have been obtained throughtheoretical calculations using different methods. The energygap of a B 31N31cone was calculated to be 0.8 eV by mo- lecular orbital calculations using the DV- X/H9251method,20while a much wider band gap of /H110113.0 eV was shown by density function theory calculations.21Moreover, the regarded works have presented the electronic structures of BNCNSs withspecial disclinations, such as 240°, 20–26whereas a wide vari- ety of disclination angles was observed in our experiments. In this research, BNNHs were fabricated on a large scale by the BOCVD method. Detailed transmission electron mi-croscopy /H20849TEM /H20850observation revealed that BNNHs were made of BNCNSs with different disclination angles. Theelectronic structure of BNCNSs with single and multiple dis-clinations was systematically investigated by first-principlescalculations. In addition, the variation of electronic statescaused by the appearance of B-B and N-N bonds in aBNCNS was also analyzed. II. EXPERIMENT BNNHs were synthesized through the BOCVD method, the detailed procedure was described elsewhere.19Figure 1/H20849a/H20850is the typical scanning electron microscopy /H20849SEM /H20850im- age of a product, which contains numerous cone-shapednanostructures. The diameters of their bottom parts are typi-cally 100–200 nm, and the lengths vary from500 nm to 1 /H9262m. Figure 1 /H20849b/H20850is the TEM image of a single BNNH with SAED patterns depicted in the insets. Typically,a single disclination is indicated by the two BN 002reflections separated by an angle, as shown in inset A, which is causedby the scattering from the two opposite sides of the horn’swall parallel to an incident electron beam. The BN 002reflec- tions become arcs when the BNNH diffracts as a whole, asshown in inset B, which indicates that several disclinationangles coexist within a single BNNH. High-resolution TEM /H20849HRTEM /H20850was used to thoroughly investigate the disclination angles appearing in the BNNHs.The sharp tip parts were selected as the objects since a singlePHYSICAL REVIEW B 72, 245419 /H208492005 /H20850 1098-0121/2005/72 /H2084924/H20850/245419 /H208495/H20850/$23.00 ©2005 The American Physical Society 245419-1BNNH is too big for HRTEM and the disclination angle at a tip part may be straightforwardly determined. Figure 2 /H20849a/H20850 shows the cone-shaped tip part of a BNNH. It is clear thatthere is a bend at the site marked with an asterisk, whichindicates the change in the disclination angle. Detailed mea-surements reveal that the apex angles vary from /H1101138.6° to 20.1°. This corresponds to the disclination angle variationsfrom 240° to 300°. Similarly, in Fig. 2 /H20849b/H20850, it is observed that the apex angle varies from 81.3° to 39.8°, that implies thatthe disclination angle varies from 120° to 240°. Taking into account numerous previous attempts to synthesize and char-acterize atomic junctions formed within nanostructures ofdifferent compositions and morphologies, the atomic andelectronic structures of the present BNNHs are highly valu-able for further studies. III. ATOMIC STRUCTURES OF BNCNS A conical BN nanostructure can be geometrically con- structed by rolling a BN sheet sector. For carbon, in order toensure continuity of the hexagonal lattice along the entire surface of a cone and to minimize energy, the disclinationsmust be multiples of 60° /H20849disclination angle D /H9258=60°, 120°, 180°, 240°, and 300° /H20850, which correspond to the apex angles of/H9258apex=112.9°, 83.6°, 60°, 38.9°, and 19.2°, respectively.10 For BN, a conical nanostructure is formed from the 120°multiple disclinations /H20849120° and 240° /H20850. 27The 120° multiple disclinations BNCNSs are usually modeled as those havingfour-member atomic rings at the apex, 27–29as shown in Figs. 3/H20849a/H20850and 3 /H20849b/H20850, which have been also adopted in the present calculations. Recently, Bourgeois et al.30have shown the existence of BNCNS with a disclination angle of 300°, which was alsoconfirmed in our experiments, as shown in Fig. 2 /H20849a/H20850. One possibility for the formation of BNCNS with non-120° mul-tiples disclination angles is that there is line quadrangle-octagon defects in it, as highlighted in Fig. 3 /H20849c/H20850. Another possibility is the existence of antiphase boundaries, whichcontain a line defect of non-BN bonds. 31The line defect of non-BN bonds can consist of the B-B bonds or N-N bonds,and can be a series of parallel B-B or N-N bonds /H20849mol B-B or mol N-N, for simplicity /H20850or zig-zag type, as illustrated in Figs. 3 /H20849d/H20850and 3 /H20849e/H20850. It is believed that similarly to carbon nanocones, the disclination angles of BNCNSs can be mul-tiples of 60°, since the disclination angle of 300° was indeedobserved in the experiments. For a BNCNS with a non-120°disclination angle, theoretical calculations indicate that thestructures with an antiphase boundary can be more stable FIG. 1. /H20849a/H20850SEM image of BNNHs. /H20849b/H20850TEM image of a BNNH with SAED patterns shown in the insets. Inset A is a SAED patterntaken from a very small area within the BNNH, while inset B dis-plays a pattern recorded from a BNNH as a whole. FIG. 2. HRTEM image of the tip parts of BNNHs. /H20849a/H20850The 240° and 300° disclination angles and /H20849b/H20850120° and 240° disclination angles coexist in individual BNNHs, respectively.ZHI et al. PHYSICAL REVIEW B 72, 245419 /H208492005 /H20850 245419-2than those without it.31Even so, in our calculations, the elec- tronic structures of both kinds of BNCNSs were analyzed inorder to exploit the influence of non-BN bonds on the elec-tronic structure. IV. ELECTRONIC STRUCTURE OF BNCNS The electronic structure of BNCNSs was investigated by the first-principles calculations based on density functionaltheory /H20849DFT /H20850within the local density approximation /H20849LDA /H20850. 32Periodic boundary conditions and a plane wave expansion were adopted for the wave functions. StandardTroullier-Martins pseudopotentials 33were used with a plan wave cutoff of 300 eV, while a further increase in the cutoffshows little influence on the results. This method has alreadybeen successfully applied to study carbon and BNnanostructures. 28,34–36A 9 Å wide vacuum region was in- cluded in the supercell to avoid interactions with the mol-ecule periodic images. The geometric structures of BNCNSswith different disclinations were allowed to relax completelyuntil the residual forces were less than 0.05 eV/ Å. 37–39 All dangling bonds were saturated by H atoms in the calculations. A BNCNS with a 240° disclination was used to initially check the accuracy of calculations affected by the differentnumbers of atoms in the supercell. The density of states/H20849DOS /H20850was calculated for the B 23N23,B 46N46, and B 137N137 structures. The results show that the electronic states of B23N23are slightly less compared to those of B 46N46and B137N137, while DOS of B 46N46and B 137N137are quite simi- lar. The energy gaps presented by the DOS are almost iden-tical for all three cases. The variations in electronic states areattributed to the “tip effect,” which is induced by the numberratio of atoms on the tips to the entire BNCNS. Taking into account that the interlayer distance in hexagonal BN is0.33 nm, the strong interactions between atoms in the tip partare expected. This may induce the change of local electronicstates. To ensure the precision of the present calculations,enough atoms should be included into a supercell in order toavoid the “tip effect.” Thus for every structure, more than the100 atoms within a supercell were adopted for the DOScalculations. 35 Figure 4 shows DOS of BNCNS with 60° multiples as disclinations. Only B-N bonds are permitted in all the struc-tures studied here, which means for BNCNSs with 60°, 180°,and 300° disclinations, the line quadrangle-octagon defectswere solely adopted to avoid the appearance of B-B or N-Nbonds. It can be seen that the influence of disclination type isweak, although some minor variations can be noticed. Forinstance, for BN nanocones with 60°, 120°, and 180° discli-nations, the energy gap is /H110113.6 eV, while it becomes /H110113.1 eV for the BN nanocones with the 240° and 300° dis- clinations. It is known that a cone is the intermediate struc-ture between a sheet and a tube, and only curvature makesthe difference. Strong hybridization effects may occur beingcaused by the tube curvature. This may strongly modify theband structure in line with the band-folding analysis. Forexample, for a CNT, 36a/H9266*-/H9268*hybridized state significantly reduces the gap predicted by the band-folding analysis. ForBNNTs, the hybridization effects are less important than fora CNT, since a BNNT is always a wide band gap semicon-ductor /H20849even at a large curvature /H20850. However, the present variations in energy gap of BNCNS are thought to be in fact FIG. 3. /H20849Color /H20850/H20849a/H20850and/H20849b/H20850the models of BNCNSs with the 240° and 120° disclination angles, the four-member rings are present attheir apexes. /H20849c/H20850,/H20849d/H20850, and /H20849e/H20850show the 300° disclination BNCNS with the quadrangle-octagon defects, the mol-type bond defects,and zig-zag-type bond defects, respectively. FIG. 4. /H20849Color /H20850Calculated DOS of BNCNSs with different dis- clination angles.ELECTRONIC STRUCTURE OF BORON NITRIDE CONE- … PHYSICAL REVIEW B 72, 245419 /H208492005 /H20850 245419-3related to curvature. The electronic structure of BNCNSs with 60°, 120°, and 180° disclinations is more similar to aBN sheet, whereas the other two kinds of BNCNSs resemblevery thin BNNT. 20Even so, the influence of curvature is limited because a varying curvature is present in a givencone; which is quite different from the case of nanotubes. Noobvious electronic state variations were observed, albeit vari-ous quadrangle-octagon defects exist in the BNCNSs with60°, 180°, and 300° disclinations. This is different fromCNTs, in which the local density of states /H20849LDOS /H20850sensi- tively depends on the number and relative positions of thepentagons constituting a conical tip. 16 To investigate the influence of different defects on the electronic structure, DOS was consecutively calculated forBNCNSs /H2084960°, 180°, and 300° disclinations /H20850with the quadrangle-octagon defects, mol B-B bonds and mol N-Nbonds, as shown in Fig. 5. The general trend is that no matterwhether the B-B or N-N bonds are introduced; the moreunoccupied states appear at the bottom of the conductionband, which makes the energy gap smaller. The presence ofN-N bonds decreases the number of electronic states near theFermi level. These results imply that the type of chemicalbonds in the BNCNS is more important than the specifictopology for the determination of band-edge states. 22The variations of electronic states induced by different chemicalbonds indicate that scanning tunneling spectroscopy /H20849STS/H20850can be a right way to determine the detailed chemical bond information in the BNCNSs, since the current observed in aSTS experiment is proportional to the charge density associ-ated with the states near the Fermi level. Based on the above-reported results, it is clear that when the disclination is significantly changed or unfavorable bonds/H20849B-B or N-N /H20850are introduced, the electronic structure of BNCNS is modified. Since the junctions consisting ofBNCNS with different disclinations were observed in thepresent experiments, the corresponding electronic states ofthis structure were further investigated. The LDOS at differ-ent parts along the axis of a 240°–300° junction is plotted inthe left-hand part of Fig. 6. The line mol B-B bond defects,other than quadrangle-octagon defects, were adopted in thestructure in its 300° part of the junction, since the BN coneswith an antiphase boundary were predicted to be more stablethan those without it. 31As expected, the main feature is that the LDOS in the 240° disclination section presents a/H110113.1 eV energy gap, while the LDOS in the 300° disclina- tion section reveals a small /H110111.2 eV gap. Similar results were obtained for a 300°–180° junction, as shown in theright-hand part of Fig. 6. These results indicate that a BNNHmay be a junction consisting of semiconductors with differ-ent energy gaps. Although the nanotubes can form variousheterojunctions, the structure presented here is particularlyinteresting. In fact, the electronic structure of a cone-shapedstructure can be solely determined by a topological defectlocated at its apex and a finite number of various defects. Bycontrast, in nanotubes, the number of chiralities can be infi-nite within a given multiwalled nanotube, which makes theprecise property control difficult. V. CONCLUSIONS BNNHs were fabricated by the BOCVD method with a high yield. Detailed TEM investigations reveal that a singleBNNH consists of BNCNSs with different disclinations. Theelectronic structure of BNCNS was studied systematically bythe first-principle calculations. It is found that an energy gap FIG. 5. Calculated DOS of non-120° disclination BNCNSs with different defects. /H20849a/H20850,/H20849b/H20850, and /H20849c/H20850correspond to a quadrangle- octagon defect, mol N-N bond, and mol B-B bond defects, respec-tively. FIG. 6. LDOS at different parts along the axis of 240°–300° and 300°–180° junctions.ZHI et al. PHYSICAL REVIEW B 72, 245419 /H208492005 /H20850 245419-4of BNCNS depends weakly on the disclination type, how- ever, the B-B or N-N bonds appearing within the BNCNSsstrongly affect the band-edge states. The present results im-ply that the type of chemical bonds in BNCNSs is moreimportant than nanostructure specific topology for the result-ant energy gap and band-edge state characteristics. The cal-culations also indicate that BNNHs can be a junction madeof semiconductors with different energy gaps, which mayexhibit interesting transport and luminescence properties. ACKNOWLEDGMENT The authors thank Dr. M. Mitome, Dr. K. Kurashima, Dr. L. Bourgeois, and Dr. F. F. Xu for their cooperation and kindhelp. *Electronic address: zhi.chunyi@nims.go.jp 1X. Blase, A. Rubio, S. G. Louie, and M. L. Cohen, Europhys. Lett. 28, 335 /H208491994 /H20850. 2A. P. Suryavanshi, M. Yu, J. Wen, C. Tang, and Y . 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PhysRevB.70.113105.pdf

Absence of the d-density-wave state from the two-dimensional Hubbard model A. Macridin,1M. Jarrell,1and Th. Maier2 1University of Cincinnati, Cincinnati, Ohio 45221, USA 2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Received 16 March 2004; revised manuscript received 25 May 2004; published 24 September 2004 ) Using the dynamical cluster approximation (DCA )we calculate the alternating circulating-current suscepti- bility and investigate the transition to the d-density wave (DDW )order in the two-dimensional Hubbard model. The 2 32 cluster used in the DCA calculation is the smallest that can capture d-wave order; therefore, due to the mean-field character of our calculation, we expect to overestimate the DDW transition temperatures.Despite this, we found no transition to the DDW state. On the other hand, DCA captures well the pseudogapfeatures, showing that the pseudogap is not caused by the DDW order. In the pseudogap region the DDWsusceptibility is enhanced, as predicted by the slave boson SUs2dtheory, but it still is much smaller than the d-wave pairing susceptibility. DOI: 10.1103/PhysRevB.70.113105 PACS number (s): 71.10.Hf, 74.20.Mn, 71.10.Fd, 74.25.Sv INTRODUCTION The high Tccuprates display a variety of unusual proper- ties, which remain unexplained by conventional theories.The most intriguing physics occurs at small doping, in theproximity of antiferromagnetism and superconductivity, andis characterized by non-conventional behavior of many ob-servables including the spin susceptibility, optical conductiv-ity, specific heat and transport properties. Many of these un-usual properties are associated with the presence of apseudogap in the one- and two-particle spectra. Photoemis-sion spectra show that, at small doping and above T c, the states around the sp,0dpoint in the Brillouin zone are gapped and Fermi segments appear around sp/2,p/2d, sug- gesting that the symmetry of the pseudogap in the hole- doped cuprates is consistent with the d-wave symmetry of the superconducting gap.1 Based on the cuprate phenomenology, Chakravarty et al.2 proposed that the pseudogap results from the competition between two ordering processes. One is d-wave supercon- ductivity (DSC )and the other is a state characterized by long-range order of alternating orbital currents.The latter is astaggered-flux state which breaks the translational and thetime-reversal symmetry and represents in fact a charge den-sity with d-wave symmetry, 3i.e., it is a d-density wave (DDW )state. In this scenario the system evolves continu- ously from the DDW state to the DSC state with decreasingtemperature or increasing doping, and the two states coexistup to optimal doping. The experimentally observed one-particle spectra in the pseudogap region can be well under-stood on the basis of the DDW state, 4which makes it a very appealing candidate for the origin of the pseudogap physics.Other properties of the cuprates, such as the resonant peak inthe superconducting state and the doping dependence of su-perfluid density seem also to be well captured by this model. 5 Recently it was proposed the interplay of the DDW and theinterplanar tunneling of Cooper pairs to be responsible fortheT cdependence on the number of CuO 2layers which char- acterizes different materials.6In principle, the presence of the DDW state has subtle experimental consequences, such asthe formation of a magnetic moment associated with the or- bital currents.7The interpretation of the experimental data in this respect however is still controversial. Whereas the theory of the DDW is phenomenological, slave boson theory holds the promise of a microscopic basis which may explicitly consider static or fluctuating DDW or-der, as well as d-wave pairing. These are uncontrolled theo- ries for the t−Jmodel, which is equivalent to the strong coupling limit of the Hubbard Hamiltonian. They are influ-enced by the idea of Anderson et al. 8of resonance valence bond state. The charge and the spin degrees of freedom areseparated by introducing auxiliary slave bosons. The result-ing mean-field theories explicitly decouple the fermion hop-ping along the bonds, the fermion pairing and the bosonicfield, and produce phase diagrams similar to the experimen-tal one. We briefly discuss the main results of slave boson theo- ries. The t−JHamiltonian has a local SUs2dsymmetry at half filling. 9As a result of this symmetry the p-flux state (a staggered-flux state with the flux per plaquette equal to p) and thed-wave paring state are degenerate in the undoped model. Doping breaks the symmetry to Us1dand thed-wave state becomes energetically favored.10Thed-wave state is characterized by a finite value of the fermion pairing opera-tor and a real superconductor emerges below the condensa-tion temperature of the bosons.At low doping s dł0.05dand forT.0 thed-wave pairing becomes unstable toward the p-flux or staggered-flux state of spinons.11This is the stan- dard picture of the Us1dslave boson mean field theory of the t−Jmodel. However, the inclusion of other terms in the mean-field decoupling, such as the holon’s flux,12results in the existence of a DDW state at finite doping and above Tc, but excludes the coexistence of the two states. One of the drawbacks of the Us1dtheory is that its solu- tion is not stable against the fluctuations of the gauge field.13 Fluctuations are especially important at small doping, where the energy difference between states connected via an SUs2d transformation is very small (since they are degenerate at zero doping ). Therefore all these states have an important contribution in the determination of the free energy.PHYSICAL REVIEW B 70, 113105 (2004 ) 1098-0121/2004/70 (11)/113105 (4)/$22.50 ©2004 The American Physical Society 70113105-1Leeet al.developed a slave boson mean field theory which is SUs2dsymmetric at finite doping.14The price paid is that one must deal with two slave bosonic fields and three constraints. The advantage of this approach is that the SUs2d mean-field solution is likely superior at small doping, since it accounts better for the fluctuations between different low en-ergySUs2dconnected states. Their solution for the pseudogap region is a staggered-flux of fermions which is gauge equivalent with the d-wave pairing of fermions. How- ever, the fermion staggered-flux state is not the same as thestaggered-flux state of electrons (or the DDW state ), and neither breaks time-reversal nor translation symmetry.There-fore, inSUs2dtheory, the pseudogap is not a broken symme- try state with long-range order as it is DDW, but is rather is characterized by strong spatial and dynamic fluctuations be-tweend-wave,s-flux and other SUs2drelated states. The goal of this paper is to investigate the interplay be- tween DDW and DSC order in the two-dimensional (2D) Hubbard model and to establish whether the pseudogap iscaused by the DDW order. Using the dynamical cluster ap-proximation (DCA ) 15,16we calculate the response functions associated with these two types of order. The DCA system-atically adds non-local corrections to the dynamical meanfield approximation (DMFA ) 17,18by mapping the lattice onto a finite-size periodic cluster. The DCA mapping from thelattice to the cluster is accomplished by coarse-graining allthe internal propagators in irreducible Feynman graphs inreciprocal space. Correlations at short length scales, withinthe cluster, are treated explicitly with a quantum MonteCarlo (QMC )simulation, while those at longer length scale are treated at the mean field level. Due to the residual meanfield character of our approximation, we expect our calcula-tion tooverestimate the transition temperatures of both the DDW and DSC critical temperatures . Generally, we expect to see the the most pronounced mean-field behavior from the smallest cluster that can reflectthe broken symmetry. A similar situation occurs in DMFAsimulations of the antiferromagnetic phase of the Hubbardmodel, where Néel order is possible since the impurity spinand the mean-field host may have opposite spin orientations.Since non-local fluctuations are suppressed, the DMFAover-estimates the Néel transition temperature. In the present case,theN c=4(i.e., 2 32)is the smallest possible cluster allowing ford-wave pairing or a circulating current. Orbital antiferro- magnetism is possible since the moment in the cluster andthe host can have opposite orientations. Since fluctuations onlonger length scales are suppressed, we would expect tooverestimate both the d-wave superconducting and the d-density wave transition temperatures. FORMALISM We present DCA calculations for the conventional 2D Hubbard model describing the dynamics of electrons on asquare lattice. The model H=−to ki,jl,sci,s†cj,s+Uo ini#ni", s1d is characterized by a hopping integral tbetween nearest neighbor sites and a Coulomb repulsion Utwo electrons feelwhen residing on the same site. As the energy scale we set t=0.25 eV so that the bandwidth W=8t=2 eV, and study the intermediate coupling regime U=W. We study the dynamics on short length-scales by setting the cluster size to Nc=4. This cluster size is large enough to capture the qualitativelow-energy physics of the cuprate superconductors. 19,20The corresponding phase diagram resembles the generic phasediagram of cuprates, 20displaying regions characterized by antiferromagnetism, d-wave superconductivity, Fermi liquid and pseudogap regimes, in qualitative agreement with ex-perimental results. In this paper we calculate the static (i.e., v=0)suscepti- bilities which correspond to the circulating current (cc)op- erator, W=io k,sgskdck+Q,s†ck,s, s2d and, respectively, to the d-wave pairing operator, P=o k,sgskdck#c−k", s3d wheregskd=cos skxd−cos skydis thed-wave symmetry factor. RESULTS The pseudogap temperature, T* is determined from the maximum in the uniform magnetic susceptibility (see the inset in Fig. 1 )when accompanied by a suppression of spec- tral weight in the DOS. We show this in Fig. 1 where thetotal and the K-dependent DOS, below T*, at d=0.05 doping is plotted. The DCA on an Nc=4 cluster implies a coarse graining of the Brillouin zone in four cells around K=s0,0d, s0,pd,sp,0dand sp,pdand theK-dependent DOS corre- sponds to the average over all kbelonging to a coarse- grained cell of the single particle spectra Ask,vd. This poor resolution in the reciprocal space allows one to study only the gross features in the single-particle spectra. Despite this,it can be seen from Fig. 1 that the pseudogap in the total FIG. 1. The one particle total and K-dependent DOS at d =0.05 doping. Inset:The uniform magnetic susceptibility, xspin, ver- susT. The maximum defines the pseudogap temperature T*.BRIEF REPORTS PHYSICAL REVIEW B 70, 113105 (2004 ) 113105-2DOS is a result of the suppression of spectral weight in the cell at s0,pd. Therefore, we believe that our calculations capture well the experimentally observed features of the pseudogap. However, our calculations show that these features are not a consequence of the DDW state. In Fig. 2 (a)we plot both thed-wave pairing susceptibility and the cc-susceptibility versus temperature, at d=0.05 doping. The pairing suscepti- bility diverges at Tc, indicating a d-wave superconducting instability. The cc-susceptibility does not diverge, indicatingthe absence of a possible transition to the DDW state. At large temperatures the d-wave pairing and the cc- susceptibilities are degenerate, and they both increase withdecreasing temperature. In the pseudogap region (left side of dotted line )thed-wave pair field susceptibility is much larger than the cc-susceptibility. Close to T c, the cc- susceptibility saturates and starts even decreasing with de-creasing temperature. The fact that in the pseudogap regionbothd-wave and the cc-susceptibilities are enhanced shows that fluctuations between these states are significant, as pre-dicted by the SUs2dtheory. 14 In Fig. 2 (b)we show the d-wave and the cc- susceptibilities at d=0.25 doping. No pseudogap20is ob- served at this doping. We notice that, starting well above Tc, the cc-susceptibility decreases with decreasing temperature.This behavior is different from the one observed at smalldoping where the cc-susceptibility increases with decreasingTup toT c. We therefore conclude that in the overdoped region the fluctuations between DSC and DDW above Tcare much less important.Calculations (not shown here )with other values of the parameters (different values of U), or with the inclusion of next-nearest-neighbor hopping corresponding to both elec-tron and hole doping, exhibit similar results. A divergentDDW susceptibility is never found. DISCUSSIONS AND CONCLUSIONS Other calculations of the Hubbard model are in agreement with ours. For example, the renormalization group studies21 found no divergence of the cc-susceptibility. A mean-fieldtreatment of an extended Hubbard model 22showed that an additional correlated hopping term was necessary to stabilizethe DDW state, and a strong-coupling approach 23studying the equation of motion of the Hubbard operators found that aweak nearest-neighbor attraction stabilizes the DDW order.All these calculations 21–23employ similar approximations which are static, neglecting the dynamical dependence of theself-energy. Unlike our method, their self-energy is fullyk-dependent. Our method correctly treats the dynamical cor- relations but reduces the k-dependence of the self-energy to only a few points in the Brillouin zone.The complementarityof these approaches to ours, together with the consistency ofthe results provides compelling evidence for their validity. In this paper we presented a DCA calculation of the two- dimensional Hubbard model, focusing on the competitionbetween DDW and DSC orders. We showed previously thatthe DCA calculation captures the generic features of thepseudogap region as seen in the photoemission and magneticmeasurements. Nevertheless, as we show here, these proper-ties are not a consequence of the existence of DDW state.The DCAshould overestimate any DDW transition tempera-ture but, despite this, we found no transition to such state.Showing the presence of the pseudogap and the absence ofthe DDW order, our results point against the DDW hypoth-esis of the pseudogap. We also found that both the cc-susceptibility and the d-wave pairing susceptibility are enhanced in the pseudogap region, indicating that the fluctuations between these states issignificant. This is not true in the overdoped region, wherewe found that the cc-susceptibility is suppressed above T c. ACKNOWLEDGMENTS We acknowledge useful conversations with G. Baskaran, C. Honerkamp, P. Lee, and W. Putikka. This research wassupported by the NSF grant DMR-0312680. Part of this re-search was performed by T.M. as Eugene P. Wigner Fellowand staff member at the Oak Ridge National Laboratory,managed by UT-Battelle, LLC, for the U.S. Department ofEnergy under Contract DE-AC05-00OR22725. 1B. O. Wells, Z-X. Shen, A. Matsuura, D. M. King, M. A. Kastner, M. Greven, and R. J. Birgeneau, Phys. Rev. Lett. 74, 964 (1995 ); A. Damascelli, Zahid Hussain, and Zhi-Xun Shen, Rev. Mod. Phys. 75, 473 (2003 )and references therein.2S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys. Rev. B63, 094503 (2001 ). 3C. Nayak, Phys. Rev. B 62, 4880 (2000 ). 4S. Chakravarty, C. Nayak, and S. Tewari, Phys. Rev. B 68, FIG. 2. The d-wave pairing, xDSC(circles ), and the circulating current, xcc(squares ), susceptibilities versus temperature, at (a)5% and (b)25% doping. The dotted vertical line in (a)is at the pseudogap temperature T*.BRIEF REPORTS PHYSICAL REVIEW B 70, 113105 (2004 ) 113105-3100504 (2003 ). 5S. Tewari, H.-Y. Kee, C. Nayak, and S. Chakravarty, Phys. Rev. B 64, 224516 (2001 ). 6S. Chakravarty, H.-Y. Kee, and K. Volker, Nature (London )428, 53(2004 ). 7S. Chakravarty, H.-Y. Kee, and C. Nayak, Int. J. Mod. Phys. B 15, 2901 (2001 ). 8G. Baskaran, Z. Zou, and P. W. Anderson, Solid State Commun. 63, 973 (1987 ); P. W. Anderson, Science 235, 1196 (1987 ). 9I. Affleck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745(1998 ). 10G. Kotliar and J. Liu, Phys. Rev. B 38, 5142 (1988 ). 11M. U. Ubbens and P. A. Lee, Phys. Rev. B 46, 8434 (1992 ). 12K. Hamada and D. Yoshioka, Phys. Rev. B 67, 184503 (2003 ). 13M. U. Ubbens and P. A. Lee, Phys. Rev. B 49, 6853 (1994 ). 14X.-G. Wen and P. A. Lee, Phys. Rev. Lett. 76, 503 (1995 );P .A . Lee, N. Nagaosa, T.-K. Ng, and X.-G. Wen, Phys. Rev. B 57,6003 (1998 ). 15M. H. Hettler, A. N. Tahvildar-Zadeh, M. Jarrell, T. Pruschke, and H. R. Krishnamurthy, Phys. Rev. B 58, R7475 (1998 ). 16M. H. Hettler, M. Mukherjee, M. Jarrell, and H. R. Krishnamurthy, Phys. Rev. B 61, 12 739 (2000 ). 17T. Pruschke, M. Jarrell, and J. Freericks, Adv. Phys. 44, 187 (1995 ). 18A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys.68,1 3 (1996 ). 19T. Maier, M. Jarrell, T. Pruschke, and J. Keller, Phys. Rev. Lett. 85, 1524 (2000 ). 20M. Jarrell, T. Maier, M. H. Hettler, and A. Tahvildarzadeh, Europhys. Lett. 56, 563 (2001 ). 21C. Honerkamp, M. Salmhofer, and T. M. Rice, Eur. Phys. J. B 27, 127(2002 ). 22C. Nayak and E. Pivovarov, Phys. Rev. B 66, 064508 (2002 ). 23T. D. Stanescu and P. Phillips, Phys. Rev. B 64, 220509 (2001 ).BRIEF REPORTS PHYSICAL REVIEW B 70, 113105 (2004 ) 113105-4

PhysRevB.102.115109.pdf

PHYSICAL REVIEW B 102, 115109 (2020) Universal Lindblad equation for open quantum systems Frederik Nathan and Mark S. Rudner Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark (Received 21 April 2020; revised 5 August 2020; accepted 9 August 2020; published 3 September 2020) We develop a Markovian master equation in the Lindblad form that enables the efficient study of a wide range of open quantum many-body systems that would be inaccessible with existing methods. The validity of the masterequation is based entirely on properties of the bath and the system-bath coupling, without any requirements onthe level structure within the system itself. The master equation is derived using a Markov approximation that isdistinct from that used in earlier approaches. We provide a rigorous bound for the error induced by this Markovapproximation; the error is controlled by a dimensionless combination of intrinsic correlation and relaxationtimescales of the bath. Our master equation is accurate on the same level of approximation as the Bloch-Redfieldequation. In contrast to the Bloch-Redfield approach, our approach ensures preservation of the positivity ofthe density matrix. As a result, our method is robust, and can be solved efficiently using stochastic evolutionof pure states (rather than density matrices). We discuss how our method can be applied to static or drivenquantum many-body systems, and illustrate its power through numerical simulation of a spin chain that wouldbe challenging to treat by existing methods. DOI: 10.1103/PhysRevB.102.115109 The theoretical description of a quantum system interact- ing with an environment is an important problem of bothfundamental and practical interest. The problem arises in adiverse array of settings, from chemistry to atomic, molecular,and optical physics, as well as condensed matter physics,high-energy physics, and quantum information processing[1–8]. Due to the importance and long history of the problem, there exists a wide range of well-established approaches fordescribing the dynamics of open quantum systems, see, e.g.,Refs. [ 9–17]. The Nakajima-Zwanzig (NZ) approach [ 11,12] provides a systematic framework for describing the evolution of openquantum systems. Although formally exact in its most generalform, in practice there are many challenges associated withapplication of the NZ equation, even in approximate form. Forexample, the Bloch-Redfield (BR) equation, which emergesas a lowest-order approximation to the time-convolutionlessNZ equation, is not guaranteed to preserve positivity of thedensity matrix of the system and may therefore yield un-physical solutions for long time evolution, with negative ordiverging probabilities. Moreover, solving these (NZ or BR)equations requires working with the density matrix of thesystem, whose dimension is the square of that of the system’sHilbert space. This requirement may make their numericalsolution prohibitively expensive, even for moderately sizedquantum systems [ 9]. For Markovian systems where the correlation (or “mem- ory”) time of the bath is sufficiently short, Lindblad-formmaster equations provide an alternative to the NZ approach[9]. The Lindblad form is the most general form of a time- local evolution equation that is guaranteed to preserve thetrace and positivity of the density matrix [ 18,19]. Importantly, the Lindblad form also admits efficient numerical solution viastochastic evolution of pure states [9,20–22], thus avoiding the computational cost of working with density matrices.However, derivations of Lindbladian master equations, suchas the quantum optical master equation [ 15], typically require stringent conditions on the level spacing of the system itself,thus limiting their applicability to specific classes of systems.In particular, the quantum optical master equation relies onthe rotating wave approximation (RWA), and hence is onlyvalid when the level broadening arising from bath-inducedtransitions is small compared with the smallest level spacingin the system. While this condition is well satisfied in manyimportant cases, for example in atomic physics and quantumoptics, many types of systems (including many-body systemswith dense spectra) and physical phenomena (such as Fanoresonances) cannot be described through this approach. Our motivation in the present work is based on the fol- lowing notion: when the correlation time of the bath ismuch shorter than a characteristic timescale of system-bathinteractions, we heuristically expect that the evolution of thesystem should be generated by a Markovian master equation.Hence, Markovianity should be a property of environmentalone, independent of details of the system itself. Noting thata Markovian master equation for the density matrix must bein the Lindblad form, we thus seek to systematically derive aLindbladian master equation without reference to any detailsof the system other than the operator(s) through which itcouples to its environment. The main result of this paper is the derivation of a “universal Lindblad equation” (ULE) that can be applied toanyopen quantum system whose bath satisfies a particular Markovianity condition that is defined in terms of the bathspectral function and the system-bath coupling strength. Inparticular, the derivation of the ULE does not rely on the 2469-9950/2020/102(11)/115109(24) 115109-1 ©2020 American Physical SocietyFREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) rotating wave approximation or any other assumption about the energy level spacings of the system. We provide explicitexpressions for the jump operators, and discuss their eval-uation for static, Floquet, and arbitrarily driven many-bodysystems. Importantly, the number of jump operators is equalto the number of independent terms (referred to as quantumnoise channels) that couple the system and bath, independentof the details of the system. As a result, for many cases, theULE features only one or a few jump operators. The jumpoperators are straightforward to compute, either through exactdiagonalization, or controlled expansions. The principle underlying our derivation of the ULE is that there is no unique way of implementing a Markov approxi-mation in the evolution of the density matrix. Instead, thereexists a continuous family of distinct approximations thatresult in Markovian dynamics of the system, all with errorbounds of the same order in a dimensionless Markovianityparameter (see below). One particular choice out of thisfamily of comparable Markov approximations leads to theBloch-Redfield equation. In this paper we employ a differentMarkov approximation from within this family which directlyleads to a Lindblad-form master equation without any furtherassumptions about the nature of the system. We provide rigorous bounds on the relative error induced by making the Markovian approximation that results in theULE. The error is controlled by a dimensionless “Marko-vianity” parameter, defined from a combination of correlationand relaxation timescales that we identify from the bath andits coupling to the system. We show that this error is of thesame order as that incurred in deriving the Bloch-Redfieldequation. Unlike the BR equation, however, the ULE pre-serves the physicality (i.e., positivity and normalization) of thedensity matrix. Hence it is intrinsically robust and amenableto solution using efficient stochastic methods [ 9,20–22]. The universal Lindblad equation that we present here can be used for a wide range of physical situations. In particular, itcan be used to efficiently simulate the dynamics of open andnoisy quantum many-body systems (i.e., systems with largeHilbert space dimension and small level spacing). In addition,it provides a straightforward, general approach for describingthe dynamics of driven systems coupled to Markovian baths. A master equation of the same form as we derive here was previously employed with phenomenological justificationin Ref. [ 23]. More recently, a similar master equation was also heuristically obtained in Ref. [ 24]. Here we provide a systematic, rigorous derivation of the universal Lindbladequation, and in particular show that it captures the dynamicsof the system at the same level of error as the Bloch-Redfieldequation. (Some of our arguments appeared in a preliminary,heuristic derivation in the Ph.D. thesis of one of the presentauthors [ 25].) The ansatz in Ref. [ 23] applies to systems cou- pled to independent bath observables, with static or weaklytime-dependent Hamiltonians, such that the jump operatorscan be computed within a quasistatic approximation for thesystem Hamiltonian. Our approach covers systems with ar-bitrary time dependence and system-bath couplings, and inparticular applies beyond the regime where the quasistaticapproximation is valid. Recently, another group of authors has also obtained a Lindblad-form master equation for open quantum systemswhose validity is independent of the details of the system [ 17]. The master equation of Ref. [ 17] is distinct from the ULE that we obtain, and was derived using a time–coarse-grainingapproach that is of a fundamentally different nature from theMarkov approximation that we employ here. Interestingly, theerror bounds obtained by the authors of Ref. [ 17] were defined in terms of a closely related Markovianity parameter to theone we identify here (see Appendix B). The ULE we derive is thus valid on an equivalent level of approximation as themaster equation of Ref. [ 17]. The simultaneous validity of these two distinct master equations reflects the nonuniquenessof the Markov approximation discussed above. The rest of this paper is organized as follows. In the main text we discuss the essential ideas of our work, whilewe provide technical details and derivations in several Ap-pendixes. In Sec. Iwe provide a summary of our main results. In Sec. IIwe formally introduce the general model of open quantum systems that we study, review existing approaches toanalyzing the dynamics of this class of systems, and presentimportant auxiliary results that are used to derive the ULE. InSec. IIIwe derive the ULE, allowing for multiple baths and arbitrarily time-dependent system Hamiltonians. In Sec. IV we discuss how to calculate and implement the jump operatorsof the ULE for a range of relevant special cases, includingsystems with time-independent Hamiltonians, periodicallydriven systems, and systems where exact diagonalization ofthe Hamiltonian is not feasible. In Sec. Vwe demonstrate our approach via numerical simulations of a spin chain coupledto two baths at different temperatures. We conclude with adiscussion in Sec. VI. I. SUMMARY OF RESULTS In this section we summarize the main ideas and results of this paper. We investigate the dynamics of a quantumsystem Sconnected to an external environment (bath) B. For simplicity, in this section we illustrate our results for thecase where the system’s Hamiltonian H Sis time independent, and the system and bath are connected through a single termH int=√γXBin the combined system-environment Hamil- tonian. Here Xand Bare observables of the system and bath, respectively, and the energy γdenotes the system-bath coupling strength, normalized such that Xhas unit spectral norm [ 26,27]. The results we present below for this system are derived and discussed in detail in Secs. IIandIII, where we also extend our results to general system-bath couplingsand time-dependent system Hamiltonians H S(t). In Sec. IIIwe seek conditions on the bath under which the time evolution of the reduced density matrix of the system ρ takes the Lindblad form [ 18,19,28]: ∂tρ=−i[HS+/Lambda1,ρ]−1 2{L†L,ρ}+LρL†. (1) In the above, Lis known as the jump operator, and determines the dissipative component of the system’s evolution, while theHermitian Lamb shift /Lambda1accounts for the renormalization of the Hamiltonian due to the system-bath coupling. Note thatwe set ¯ h=1 throughout. Importantly, the conditions for the Lindblad-form master equation that we identify in Sec. IIIare formulated purely in terms of properties of the bath and the system-bath coupling 115109-2UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) strength. This situation stands in contrast to the quantum optical master equation, which is also a Lindblad-form masterequation but is only valid under additional stringent require-ments on the level spacing of the system itself [ 9]. In a wide range of situations, all information of the bath Brequired to determine the evolution of ρis contained in the bath spectral function J(ω)[9] (see Sec. II Afor definition). The conditions we obtain in Sec. IIIfor the Lindblad-form master equation are also expressed in terms of this function:from J(ω) we identify an energy scale /Gamma1and timescale τ whose dimensionless product /Gamma1τserves as a measure of Markovianity. As the main result of this paper, we showthat, when /Gamma1τ/lessmuch1, the time evolution of ρis accurately described by a (Markovian) master equation in the Lindbladform [Eq. ( 1)], with the single jump operator L=/summationdisplay mn/radicalbig 2πγJ(En−Em)Xmn|m/angbracketright/angbracketleftn|. (2) Here{|n/angbracketright}and{En}denote the eigenstates and energies of the system Hamiltonian HS(not including the Lamb shift), respectively, while Xmn≡/angbracketleftm|X|n/angbracketright.T h eL a m bs h i f t /Lambda1is pro- portional to γand is defined from Xand the bath spectral function in Eq. ( 34) below. Stated more precisely, as we show in Sec. III, the time derivative of ρis given by Eqs. ( 1) and ( 2), up to a correction of order /Gamma12τ. In comparison, the magnitude of the right-hand side of Eq. ( 1) is typically well estimated by/Gamma1(hence the correction is smaller by a factor /Gamma1τ). Due to its system-independent applicability, we refer to the masterequation in Eqs. ( 1) and ( 2), along with its generalizations in Sec. IV,a st h e universal Lindblad equation . The two quantities /Gamma1andτthat determine the accuracy of the universal Lindblad equation [Eqs. ( 1) and ( 2)] are associated with the bath spectral function J(ω) and system- bath coupling γ. Specifically, /Gamma1andτare derived from a related function g(t) that we call the “jump correlator.” The jump correlator is defined via its Fourier transform g(ω)a s the square root of the spectral function: J(ω)=2π[g(ω)] 2.I n time domain this gives g(t)=1√ 2π/integraldisplay∞ −∞dω/radicalbig J(ω)e−iωt. (3) From this jump correlator, /Gamma1andτare given by /Gamma1=4γ/bracketleftbigg/integraldisplay∞ −∞dt|g(t)|/bracketrightbigg2 ,τ=/integraltext∞ −∞dt|g(t)t|/integraltext∞ −∞dt|g(t)|,(4) where γdenotes the system-bath coupling strength. As we explain in Sec. II B, the timescale τcan be seen as a measure of the characteristic correlation time of the bath observableB, while /Gamma1sets an upper bound for the rate of bath-induced evolution of the system, independent of any approximation.In the limit /Gamma1τ/lessmuch1, where the ULE is valid, the correlations of the bath decay rapidly on the characteristic timescaleof system-bath interactions /Gamma1 −1. In this case, the standard heuristic arguments behind the Markov-Born approximationsuggest that the dynamics of the system should be effectivelyMarkovian [ 9]. The results we obtain here hence put this intuition on rigorous footing, independent of properties of thesystem itself.We note that the bath-induced terms in Eq. ( 1) scale linearly with the system-bath coupling strength γ. In contrast, the correction to Eqs. ( 1) and ( 2) we identified above is of order/Gamma1 2τ, and thus scales as γ2. Hence, when the coupling- independent quantities /Gamma1/γ andτare finite, a small enough value of γcan in principle always be found such that the system’s dynamics are Markovian and well described by theuniversal Lindblad equation in Eq. ( 1). In this way, the condi- tion/Gamma1τ/lessmuch1 gives a well-defined notion of the weak-coupling limit. We demonstrate in Sec. II A 2 that the Bloch-Redfield equation is also valid up to a correction of order /Gamma1 2τ.I nt h i s sense, the ULE is valid on an equivalent level of approxima-tion as the Bloch-Redfield equation. The universal Lindblad equation in Eqs. ( 1) and ( 2)i s consistent with existing results for open quantum systems [ 9]. In particular, the ULE naturally reduces to the quantum opticalmaster equation in the limit where the latter is valid, namelywhen the rate of system-bath interactions /Gamma1is much smaller than any energy level spacing of the Hamiltonian H S(i.e., when the rotating wave approximation is valid). However, incontrast to the quantum optical master equation, the derivationof the ULE does not rely on the rotating wave approximation(or any other assumptions about the energy levels of thesystem); hence it can also be applied beyond the regime wherethe quantum optical master equation is valid. In addition tobeing consistent with the quantum optical master equation asexplained above, Eqs. ( 1) and ( 2) reproduce Fermi’s golden rule: Fermi’s golden rule states that the transition rate be-tween two energy levels of the Hamiltonian, mand n,i s given by /Gamma1 n→m=2π|Xmn|2J(En−Em). This result follows from Eqs. ( 1) and ( 2) by identifying /Gamma1n→m=/angbracketleftm|∂tρ(t)|m/angbracketright|t=0 when taking ρ(0)=|n/angbracketright/angbracketleftn|. We note that the expression for the jump operator Lin Eq. ( 2) was previously hypothesized in Ref. [ 23]. Reference [23] showed that the master equation in Eqs. ( 1) and ( 2) was consistent with Fermi’s golden rule, and reproducedthe quantum optical master equation in the regime γ→0 where the latter is valid. Based on these results and numericaldemonstrations, Ref. [ 23] conjectured that the master equation in Eqs. ( 1) and ( 2) could accurately describe the evolution of open quantum systems. In this work, by rigorous derivationwe recover the hypothesis of Ref. [ 23], and identify the pre- cise conditions under which the universal Lindblad equation[Eqs. ( 1) and ( 2)] holds. Crucially, the conditions we identify rely solely on the properties of the bath and system-bathcoupling, and hold well beyond the regime where the quan-tum optical master equation is valid. In addition, our resultsgeneralize the hypothesized master equation from Ref. [ 23]t o arbitrary system-bath couplings and time-dependent Hamilto-nians. II. OPEN SYSTEM DYNAMICS: FORMULATION AND CHARACTERISTIC TIMESCALES We now set out to derive the universal Lindblad equation by rigorous means, for general open quantum systems. Asa first step, in this section we define the model we studyand review standard theory for open quantum systems. Wemoreover present two auxiliary results that play an important 115109-3FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) role for the derivation of the ULE: we establish a rigorous upper bound for the correction to the Bloch-Redfield equation(Sec. II A 2 , see also Ref. [ 17]), and obtain an upper bound for the rate of bath-induced quantum evolution (a so-called“quantum speed limit”) [Eq. ( 13)]. These results, which may also be of interest on their own, are derived in Appendix A. The concepts and basic assumptions described in this sectionwill form the foundation for the derivation of the ULE in thenext section. The system we consider in this paper consists of a quantum (sub)system Swhich is connected to an external system, referred to as the bath B. The subsystem Sm a yb ea n y t h i n g from a two-level spin to a many-body system, while the bathBis typically a large system with a dense energy spectrum, such as a phononic or electromagnetic environment, or thefermionic modes in an electronic lead. The bath Bcan also consist of several “subbaths” with distinct physical origins andproperties. Without loss of generality, the Hamiltonian Hof the full system SB(including the bath) takes the form H=H S+HB+Hint, (5) where HSandHBare the Hamiltonians of the subsystem and bath, respectively, while Hintcontains all terms in the Hamiltonian that couple the two. In the following we allowH Sto depend on time, while we assume HBandHintto be time independent. It is useful to decompose Hintas follows: Hint=√γ/summationdisplay αXαBα, (6) where, for each α,Xαis a dimensionless Hermitian oper- ator on the subsystem S,Bαis a Hermitian operator act- ing on the bath B, with units of [Energy]1/2, and the en- ergyγparametrizes the system-bath coupling strength (see Ref. [ 27]). We normalize γandBαsuch that Xαhas unit spectral norm for each α[26]. While γcan still be absorbed into the operators {Bα}, and thus in principle remains arbitrary, we include it in Eq. ( 6) to highlight the scaling of various quantities with respect to the system-bath coupling in thediscussion below. We note that the decomposition above isalways possible with a sufficiently high, but finite, number ofterms in the sum N. We refer to each such term as a (quantum) noise channel in the following. For simplicity, in the remainder of this section, and in the derivation of the ULE in Sec. III A, we consider the case where the sum in Eq. ( 6) consists of a single term, and refer to the system and bath operators as XandB, respectively. In Sec. III C we generalize our results to the case where the sum in Eq. ( 6) contains multiple terms. To describe the dynamics of observables in the system S, it is sufficient to know the evolution of the reduced densitymatrix of S, ρ(t)≡Tr B[ρSB(t)]. (7) Here Tr Btraces out all the degrees of freedom in B, and ρSB(t) denotes the density matrix of the combined system SB. Crucially, it is possible to obtain an equation of motion forρ(t) which depends only on HS,X, and the statistical properties of the bath. Such an equation of motion is knownas a master equation. There exists several approximationschemes for obtaining master equations for ρ(see, for exam-ple, Refs. [ 9–15]). While useful in their respective regimes of applicability, each of these methods has its limitationson which cases they may be applied (either due to physicallimitations on the regime of applicability, or practical issuesassociated with numerical implementation). The goal of ourpaper is to derive a Markovian master equation that can beapplied to a wider range of cases, which unifies and extendssome of these previous approaches. A. Born-Markov approximation Before deriving the universal Lindblad equation, we review one of the existing approaches to obtaining a master equationforρ, namely the Born-Markov approximation. This standard approach leads to a master equation for ρknown as the Bloch- Redfield (BR) equation. The concepts introduced here will beused in the derivation of the Universal Lindblad equation inSec. III. 1. Derivation of Bloch-Redfield equation To derive the BR equation, we assume that the bath was in a steady state at some arbitrary time t0in the remote past. Specifically, we assume that ρSB(t0)=ρS(t0)⊗ρB, where ρBdescribes a steady state of the bath: [ HB,ρB]=0. The bath state ρBcan for example describe a thermal equilibrium state with a specific temperature and chemical potential. If B consists of several subbaths, ρBcan also be a direct product of thermal states out of equilibrium with each other. Due toits macroscopic size, the state of the bath remains practicallyunaffected by the system Sat later times, except for short- lived fluctuations arising from the system’s evolution in therecent past. Without loss of generality, we may assume thateach bath operator B αhas vanishing expectation value in the bath state ρB:T r B(BαρB)=0, since nonzero expectation values can be eliminated by appropriate redefinition of HSand Bαin Eqs. ( 5) and ( 6). We note that, due to the finite memory and relaxation times of the bath and of the system, respectively, the evolution ofρ(t) should be independent of the details of the initialization in the remote past. Supporting this, in Appendix A6we show that the evolution of the system is independent of the details ofthe state ρ S(t0) and the exact value of t0, when t0is sufficiently far in the past. The BR equation is most easily derived in the interaction picture. We transform the problem to the interaction pictureby applying a rotating frame transformation generated bythe Hamiltonian H S(t)+HB. After this transformation, the Hamiltonian of the combined system SBin the interaction picture is given by ˜H(t)=√γ˜X(t)˜B(t). (8) Here ˜X(t)≡U†(t)XU(t), and ˜B(t)≡eiHBtBe−iHBt,where U(t)≡Te−i/integraltextt 0dt/primeHS(t/prime)is the time-evolution operator of the subsystem Srelative to an arbitrary origin of time, and Tis the time-ordering operation. We let ˜ ρ(t) denote the reduced density matrix of Sin the interaction picture. Specifically, ˜ρ(t)≡TrB[˜ρSB(t)], where ˜ ρSB(t) denotes the state of the combined system SBwhen time evolved with ˜H(t)f r o mt h e state ˜ρSB(0)=ρSB(0). From ˜ ρ(t), one can straightforwardly 115109-4UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) obtain the time evolution of the system in the Schrödinger picture through the relation ρ(t)=U(t)˜ρ(t)U†(t). After transforming to the interaction picture, the sys- tem’s dynamics occur on a timescale which is set by thesystem-bath coupling γ. When this coupling is sufficiently weak, ˜ ρ(t) can be assumed static on the intrinsic correla- tion timescale of the bath (see Sec. I). This so-called weak- coupling limit forms the basis for the derivation of theBR equation, using the Born-Markov approximation [ 9]. To employ the Born-Markov approximation, we integrate thevon Neumann equation ∂ t˜ρSB(t)=−i[˜H(t),˜ρSB(t)] once, obtaining ∂t˜ρSB(t)=−/integraltextt t0dt/prime[˜H(t),[˜H(t/prime),˜ρSB(t/prime)]]. (Here we exploited the fact that Tr B[˜B(t0)ρSB]=TrB(BρSB) van- ishes by assumption as described above, to eliminate theterm arising from the boundary term of the integration). TheBorn approximation amounts to setting ˜ ρ SB(t/prime)≈˜ρ(t/prime)⊗ ρBinside the integral. The next step is to take the par- tial trace over the bath, to obtain an equation of motionfor the reduced density matrix of the system ˜ ρ(t). Using the fact that ˜X(t) acts only on the system, while ˜B(t) acts only on the bath, we obtain ∂ t˜ρ(t)≈−γ/integraltextt t0dt/primeJ(t− t/prime)[˜X(t),˜X(t/prime)˜ρ(t/prime)]+H.c., where we introduced the (two- point) bath correlation function J(t−t/prime)≡TrB(˜B(t)˜B(t/prime)ρB). (9) Finally, the Markov approximation is implemented by as- suming that ˜ ρ(t/prime) is stationary over the characteristic decay time of the bath correlation function J(t) (see below for discussion). By making the replacement ˜ ρ(t/prime)≈˜ρ(t)i n s i d e the integral over the history of the system ( t/prime), and taking the limit t0→− ∞ , we obtain [ 9] ∂t˜ρ(t)=DR(t)[˜ρ(t)]+ξ(t), (10) where DR(t)[ρ]≡−γ/integraldisplayt −∞dt/primeJ(t−t/prime)[˜X(t),˜X(t/prime)ρ]+H.c.,(11) andξ(t) denotes the correction arising from the Born and Markov approximations above. The Bloch-Redfield equationis obtained by assuming the error induced by the Born-Markov approximation ξ(t) to be negligible in Eq. ( 10). In Appendix Awe derive an upper bound for this correction, thus obtaining rigorous conditions for the validity of the BRequation. See Sec. II A 2 for a further discussion. Note that the last approximation in the above derivation resulted in an equation of motion for ˜ ρ(t) which is Markovian : in Eq. ( 10) the time derivative ∂ t˜ρ(t) depends only on the value of ˜ ρ(t) at the the same time t. As we demonstrate in Sec. III(see Sec. III B for discussion), the Born-Markov approximation above is not the only way of approximating ∂t˜ρ by a Markovian master equation in the weak-coupling limit. InSec. IIIwe will develop a different Markovian approximation for∂ t˜ρwhich is valid under the same conditions as the standard Markov-Born approximation above, but, unlike theformer, leads to a master equation in the Lindblad form. The bath correlation function in Eq. ( 9), or equivalently its Fourier transform J(ω)≡ 1 2π/integraltext∞ −∞dt J(t)eiωt, known as the bath spectral function, plays a crucial role for describing thedynamics of the system S: in the BR equation [Eq. ( 10)],J(t) contains all information of the bath required to determine theevolution of ˜ ρ. Importantly, even without the Born-Markov approximation, a wide class of baths (so-called Gaussianbaths) are fully characterized by the two-point correlationfunction J(t). This situation for instance arises if the bath consists of a large collection of decoupled subsystems, such ascontinua of independent fermionic or bosonic modes. Whilewe note that our approach below can also be applied tocases where higher-order bath correlations are relevant, in thispaper, we assume for simplicity that the bath is Gaussian. The spectral function J(ω), which is real and non-negative, can in many cases be computed or phenomenologically as-sumed [ 9] [see for example Sec. V, where we calculate J(ω) for a bath of bosonic modes]. While in the above ˜B(t) emerged as a time-evolved observable in a quantum mechanical bath,the results in this paper also apply to the case where ˜B(t)i s a classical noise signal, and the bath trace is replaced by thestatistical average over noise realizations. In this case, J(ω)i s symmetric in ωand gives the spectral density of the classical noise signal. 2. Correction to Bloch-Redfield equation As an important secondary result, in this paper we derive a rigorous upper bound for the correction to the Bloch-Redfieldequation ξ(t), which is independent of the details of the system Hamiltonian H S. This error bound is used in the next section, where we derive the ULE. To derive the error bound, we assume that the bath is Gaus- sian, and that the bath and system were decoupled at somepoint in the remote past (see beginning of this subsection).Using these assumptions, in Appendix Awe systematically expand the time derivative of ˜ ρin powers of the dimensionless “Markovianity parameter” /Gamma1τ, where the bath timescales /Gamma1 −1andτwere defined from the bath spectral function and the system-bath coupling strength γin Eqs. ( 3) and ( 4)( i n Sec. II B, we further discuss the physical meaning of these timescales). As we show in Appendix A, truncation of the expansion of ∂t˜ρto leading order in /Gamma1τyields DR[˜ρ(t)]. This truncation is thus is equivalent to making the Bornand Markov approximations, while the correction ξ(t) corre- sponds to the sum of all subleading terms in the expansion. InAppendix Awe obtain a bound for this subleading correction: /bardblξ(t)/bardbl/lessorequalslant/Gamma1 2τ, (12) where (here and in the following) /bardbl·/bardbl refers to the spectral norm (see Ref. [ 26]). Note that consistent correction bounds for the BR equation were recently obtained elsewhere [ 17,29]. Our derivation of Eq. ( 12) holds in the general case where the system and bath are connected through multiple noise channels, with thegeneralized definitions /Gamma1andτgiven in Eq. ( 26) below (see Ref. [ 9] or Appendix Afor the multichannel generalization of the BR equation). In Eq. ( 13) below, we also show that the spectral norm of∂ t˜ρon the left-hand side of Eq. ( 10) is bounded by /Gamma1/2. Comparing this bound with Eq. ( 12) above, we conclude that/Gamma1τ/lessmuch1 is a necessary condition for the Born-Markov approximation to be justified by our arguments. 115109-5FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) B. Characteristic timescales of the bath Above we found that the validity of the Born-Markov ap- proximation is determined from the characteristic timescales/Gamma1 −1andτ, which are intrinsic to the bath (and its coupling to the system). In Sec. IIIwe show that /Gamma1andτalso determine the accuracy of the universal Lindblad equation. Here webriefly discuss the nature of these quantities. As a demonstra-tion, we moreover explicitly calculate the jump correlator g(t) and the correlation time τfor the case of an Ohmic bath. To highlight the physical meaning of the timescale /Gamma1 −1,i n Appendix A3we show that /Gamma1provides a strict bound on the rate of change of ˜ ρ, /bardbl∂t˜ρ/bardbl/lessorequalslant/Gamma1/2. (13) Note that Eq. ( 13) is exact, and is derived without any ap- proximations, other than the assumption of a Gaussian bath. Inthis way, the rate /Gamma1/2 can be seen as a “quantum speed limit” for dissipative quantum evolution [ 30,31]. Heuristically, /Gamma1 −1 thus characterizes the (shortest) typical interval between real or virtual system-bath interaction events, such as, e.g., photonemission or absorption. Note that the inequality in Eq. ( 13) extends to the case of multiple noise channels, with /Gamma1as defined in Eq. ( 26) below (see Appendix A3). The timescale τcaptures the characteristic decay time of correlations in the bath. To see this, note from Eq. ( 4) that τgives the mean value of |t|associated with the normalized distribution |g(t)|/C, where C≡/integraltext ∞ −∞dt|g(t)|. The existence of a finite value of τrequires that g(t) effectively decays faster thanCτ/t2for|t|/greatermuchτ[32]. Noting from Eq. ( 3) that the bath correlation function J(t) is given by the convolution of the jump correlator with itself, J(t−t/prime)=/integraltext∞ −∞dsg(t−s)g(s− t/prime), the bath correlation function hence must also decay on a timescale of magnitude τ. The conditions described above hold under the assumption that τtakes a finite value; a diver- gent value of τindicates that long-term memory is present in the bath; in this case, the system cannot be well described bya Markovian master equation. To illustrate the above relationship between J(t),g(t), and the correlation time τ, we explicitly compute J(t),g(t), andτ, for an Ohmic bath. The Ohmic bath consists of a continuum ofbosonic modes with Hamiltonian H B=/integraltext∞ 0dωωb†(ω)b(ω), where b(ω) denotes the annihilation operator of modes with frequency ω, satisfying [ b(ω),b†(ω/prime)]=δ(ω−ω/prime) andδ(ω) denotes the Dirac delta function. In the framework of Eqs. ( 5) and ( 6), the bath operator Bis given by the bosonic field op- erator/integraltext∞ 0dω√S(ω)[b(ω)+b†(ω)], where S(ω) denotes the effective spectral density of the bath, including the frequencydependence of the system-bath coupling. The class of modelsabove is commonly used in the literature [ 9,10], and can for example describe a phononic or electromagnetic environmentof an electronic system. We consider the Ohmic spectral density S(ω)= ωe −ω2/2/Lambda12/ω0, with an ultraviolet energy cutoff set by the scale /Lambda1. The energy scale ω0is introduced to keep S(ω) dimensionless. Assuming the bath is in equilibrium at temperature T, a straightforward calculation [ 9,10] yields the bath spectral function J(ω)=1 ω0ωe−ω2 2/Lambda12 1−e−ω/T, (14)FIG. 1. Absolute value of the jump correlator g(t) (solid) and bath correlation function J(t) (black dashed line) for the Ohmic bath studied in Sec. II B. Red line indicates the fit used to obtain the exponential decay constant for g(see Sec. II B). where we work in units where kB=1. Using Eq. ( 3), we numerically compute the jump correlator g(t) from the spectral function in Eq. ( 14), for the case where /Lambda1=50T. By explicit computation [see Eq. ( 4)], we find for this case τ≈0.007T−1.I nF i g . 1we plot |g(t)|on a logarithmic scale (solid line), along with the bath correlationfunction |J(t)|(dashed line). As Fig. 1shows, both J(t) and g(t) decay exponentially, at approximately the same rate, after a sharp initial drop at short times. We confirm numerically(data not shown here) that the short-time peak arises from high-energy modes in the bath, and is controlled by the cutoff /Lambda1. By linear regression (red dashed line in Fig. 1) we find the slope of log g(t) outside this initial decrease to be given by approximately 0 .023T −1. The difference between the ex- ponential decay constant from τis caused by the short-time peak of g(t), and is thus controlled by /Lambda1. III. UNIVERSAL LINDBLAD EQUATION While useful, the standard Born-Markov approximation discussed in Sec. II Ahas some shortcomings. In particular, the Bloch-Redfield equation in Eq. ( 10) is not in the Lind- blad form. As a result, as was explained in the introduc-tion, integration of the BR equation may yield negative ordiverging probabilities, and can be impractical to implementnumerically even for moderately sized quantum systems. Inthis section we derive a master equation for ˜ ρwhich is valid under the same conditions as the BR equation, but will bein the Lindblad form and thus free of the limitations above.Specifically, our new master equation is accurate up to acorrection of the same magnitude as the correction bound/Gamma1 2τwe identified for the BR equation in Sec. II A 2 .T h e new master equation, which we term the universal Lindbladequation (ULE), constitutes the main result of our paper.Crucially, the ULE does not require any special conditions onthe system to be valid; rather, its validity relies solely on theproperties of the bath itself (along with its coupling to the system). To make the physical basis for the ULE most transparent, in Sec. III A we derive the ULE on an intuitive level of argumentation, focusing on the case where the system andbath are coupled through a single noise channel. In AppendixCwe provide a rigorous derivation of these results that also holds for general system-bath couplings. In Sec. III B we 115109-6UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) comment on the principle underlying our derivation, i.e., the existence of distinct but equivalently valid Markov approx-imations. The results for general system-bath couplings aregiven in Sec. III C, and expressed in the Schrödinger picture in Sec. III D. A. Single noise channel In this subsection we heuristically derive the universal Lindblad equation for the case of a single quantum noisechannel. As a first step in our derivation, we identify analternative form of Markov approximation, which is valid atthe same level of approximation as the standard Born-Markovapproximation (i.e., up to a correction of order /Gamma1 2τ). Sub- sequently, we demonstrate that this Markov approximationresults in a master equation in the Lindblad form (in contrast,the standard Born-Markov approximation does not lead to aLindblad-form master equation). The starting point for our derivation is the BR equation [Eq. ( 10)], whose error bounds we obtained in Sec. II A 2 above. Below we apply additional manipulations to the BRequation, which induce errors of the same magnitude asthose inherent in the Born-Markov approximation used inderiving Eq. ( 10). These additional steps hence lead to a master equation that is different from the BR equation, butis valid on the same level of approximation. Unlike the BRequation, our master equation is crucially in the Lindbladform. Our modification procedure is equivalent to employing adistinct Markovian approximation from the standard Markovapproximation (reviewed in Sec. II A) that is used to obtain the BR equation [note that our derivation still makes use ofthe “conventional” Born approximation, as described in theparagraph above Eq. ( 9)]. In Sec. III B below, we discuss the diversity of possible Markov approximations in more detail. As the first step of our derivation, we decompose the bath correlation function J(t−t /prime)[ E q .( 9)] as a convolution using “jump correlator” g(t) defined in Eq. ( 3):J(t−t/prime)=/integraltext∞ −∞dsg(t−s)g(s−t/prime). Using this decomposition, the BR equation [Eq. ( 10)] can be (exactly) rewritten as ∂t˜ρ(t)≈/integraldisplay∞ −∞dt/prime/integraldisplay∞ −∞dsF(t,s,t/prime)[˜ρ(t)], (15) where F(t,s,t/prime)[˜ρ] =γθ(t−t/prime)g(t−s)g(s−t/prime)[˜X(t),˜ρ˜X(t/prime)]+H.c. (16) The approximate equality in Eqs. ( 10) and ( 15) captures the correction to the BR equation ξ(t), whose bound (with respect to the spectral norm) we identified in Sec. II A 2 . For brevity we do not include this correction in the derivation below. NotethatF(t,s,t /prime) is a linear operator acting on system operators. Next, we integrate Eq. ( 15) with respect to tto compute the change of ˜ ρover a finite time interval from t1tot2, that we will choose much longer than τ: ˜ρ(t2)−˜ρ(t1)≈/integraldisplayt2 t1dt/integraldisplay∞ −∞dt/prime/integraldisplay∞ −∞dsF(t,s,t/prime)[˜ρ(t)]. (17) We now argue that the weak-coupling limit /Gamma1τ/lessmuch1 allows us to apply two approximations to the right-hand side above,which yield a new expression that is valid on an equivalent level of approximation as the standard Bloch-Redfield equa-t i o ni nE q .( 10). To make the first approximation, we note that, in the limit/Gamma1τ/lessmuch1, the condition /bardbl∂ t˜ρ/bardbl/lessorequalslant/Gamma1/2i nE q .( 13) ensures that ˜ρ(t)=˜ρ(s)+O(/Gamma1τ)f o r|t−s|/lessorsimilarτ. Additionally, since g(t) decays on the timescale τ(see Sec. II B),F(t,s,t/prime)i s suppressed when the difference between any two of its timearguments is much larger than τ[see Eq. ( 16)] [in particular, note that g(t−s)g(s−t /prime), and thus F(t,s,t/prime), must be small when t−t/prime/greatermuchτ]. These two results suggest that we may replace ˜ ρ(t)b y˜ρ(s) on the right-hand side of Eq. ( 17) when /Gamma1τ/lessmuch1. In Appendix Cwe implement this substitution in a systematic way, and prove that replacing ˜ ρ(t)b y˜ρ(s)i n Eqs. ( 15) and ( 17) results in a correction to ∂t˜ρof order /Gamma12τ. To make our second approximation, we again use the fact thatF(t,s,t/prime) decays when the difference between any of its time arguments exceeds τ. Thus, since τ/lessmucht2−t1by assumption, most of the contribution to the integral in Eq. ( 17) comes from the region where all three integration variables t,s,t/primeare located in the interval [ t1,t2]. As a result, the right- hand side of Eq. ( 17) is approximately unaffected if we change the integration domain from −∞<(s,t/prime)<∞,t1/lessorequalslantt/lessorequalslantt2 to the domain −∞<(t,t/prime)<∞,t1/lessorequalslants/lessorequalslantt2. Indeed, in Appendix Cwe show that this change of integration domain results in a correction to ˜ ρwhich is bounded by /Gamma1τ. After making the two approximations described above [i.e., setting ρ(t)≈ρ(s)i nE q .( 17), and subsequently changing the domain of integration], we obtain ˜ρ(t2)−˜ρ(t1)≈/integraldisplayt2 t1ds/integraldisplay∞ −∞dt/integraldisplay∞ −∞dt/primeF(t,s,t/prime)[˜ρ(s)]. (18) By taking the derivative with respect to t2, and renaming the variables of integration, we obtain the (time-local) masterequation ∂ t˜ρ(t)≈L(t)[˜ρ(t)],L(t)=/integraldisplay∞ −∞ds/integraldisplay∞ −∞ds/primeF(s,t,s/prime). (19) In Appendix Cwe put the above line of arguments on rig- orous footing: we identify a slightly modified density matrixρ /prime(t) whose norm distance to ˜ ρ(t) remains bounded by /Gamma1τat all times. Assuming that the bath is Gaussian, and that the bathand system were decoupled at some point in the remote past,we show that ρ /primeevolves according to the master equation ∂tρ/prime(t)=L(t)[ρ/prime(t)]+ξ/prime(t), (20) where L(t) is defined by Eqs. ( 16) and ( 19), and /bardblξ/prime(t)/bardbl/lessorequalslant 2/Gamma12τfor all times t. In the Markovian limit /Gamma1τ/lessmuch1,ρ/prime(t) is nearly identical to ˜ ρ(t), and the evolution of the system is thus well described by ρ/prime(t). The same condition /Gamma1τ/lessmuch1 is already required for the BR equation to be valid by ourarguments (see Sec. II A 2 ) and hence does not impose additional constraints on the system. Consistent with Eq. ( 13), we show in Appendix Cthat/bardblL(t)[ρ]/bardbl/lessorequalslant/Gamma1/2. Hence, by the same arguments as in Sec. II A 2 ,/Gamma1τ/lessmuch1 is also a necessary condition for error ξ /primeabove to be negligible. As a final step, we verify that the master equation in Eq. ( 19) is in the Lindblad form. By decomposing the step 115109-7FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) function θ(t−t/prime)i nE q .( 16) into its symmetric and antisym- metric components, θ(t)=1 2+1 2sgn(t), we find through a straightforward computation (see Appendix C3for details) that L(t)[˜ρ]=−i[˜/Lambda1(t),˜ρ]−1 2{˜L†(t)˜L(t),˜ρ}+˜L(t)˜ρ˜L†(t), (21) where the jump operator ˜L(t)i sg i v e nb y ˜L(t)=√γ/integraldisplay∞ −∞dsg(t−s)˜X(s) (22) and ˜/Lambda1(t)=γ 2i/integraldisplay∞ −∞dsds/prime˜X(s)g(s−t)g(t−s/prime)˜X(s/prime)s g n( s−s/prime). (23) T h eL a m bs h i f t ˜/Lambda1(t) is by construction Hermitian, due to the symmetry of the jump correlator g(t)=g∗(−t), which results from its definition in Eq. ( 3). Comparing with the BR equation [Eq. ( 10)], we see that the universal Lindblad equation [Eqs. ( 20)–(23) with ξ/prime(t) neglected] yields an expression for ∂tρ/prime(t) which is accurate up to a correction bounded by the same value as the correctionfor the BR equation (up to a factor of 2). In this sense, theULE and the BR equation are valid on an equivalent level ofapproximation [ 33]. To summarize this section, we showed that, in the weak- coupling limit /Gamma1τ/lessmuch1, the interaction picture density matrix of the subsystem S,˜ρ(t) evolves according to the Lindblad- form master equation in Eqs. ( 19)–(21). At each time t, the error in ∂ t˜ρis of the same magnitude as that of the Bloch-Redfield equation. Thus the ULE is valid over thesame regimes as the previously developed Markovian masterequations, while offering important gains in usability andapplicability. B. Equivalence of Markov approximations Above we derived a time-local master equation for ˜ ρ(t) that is distinct from the Bloch-Redfield equation, but is validon an equivalent level of approximation. As we explain here,the existence of distinct but equivalently valid time-localmaster equations reflects the existence of a class of distinctbut equivalently valid Markov approximations. We refer totwo approximations as being “equivalently valid” if they areboth valid up to an error of the same order in the Markovianityparameter /Gamma1τ. We demonstrate the existence of equivalently valid Markov approximations by means of a simple example. Considerthe master equation for ˜ ρ(t) that results from the Born ap- proximation [see text above Eq. ( 9)]:∂ t˜ρ(t)=−/integraltextt t0dt/primeJ(t− t/prime)[X(t),[X(t/prime),˜ρ(t/prime)]]+H.c. As explained in Sec. II A,t h e standard Markov approximation amounts to approximating ˜ρ(t/prime)≈ρ(t) in this expression. This approximation is justified when the bath correlation time τis much shorter than the char- acteristic timescale of system-bath interactions /Gamma1−1.B yt h e same arguments, however, instead of setting ˜ ρ(t/prime)≈˜ρ(t), we just as well could have approximated ˜ ρ(t/prime) by any weighted average of ˜ ρ(s) within a window of times snear s=t/prime,a s long as the width of the time window is much smaller than/Gamma1 −1. These different choices of weight functions result in distinct, but equivalently valid, time-local master equations.The infinite family of suitable weight functions can thus be seen as generating a class of distinct Markov approximations. As we show in Appendix C, there are also other classes of equivalent Markov approximations of more subtle origin thanthe simple example above. These other classes of equivalentapproximations can be identified using similar approaches asabove. The approximations in Eqs. ( 15)–(19) constitute such an alternative Markov approximation. A rigorous definitionand discussion of this approximation is given in Appendix C. C. General system-bath couplings In Sec. III A we derived the universal Lindblad equation for the case where the system-bath coupling Hintin Eq. ( 6) holds a single noise channel. In this subsection we extendour results to the most general case of system-bath couplings,namely the case where H intcontains an arbitrarily high (but finite) number of noise channels N:Hint=√γ/summationtextN α=1XαBα. Here, for each α,Xα, and Bα, are observables of the system Sand bath B, respectively. These are normalized such that /bardblXα/bardbl=1, while the bath operators {Bα}may have different scales of magnitude. For general system-bath coupling, the ULE can be derived through straightforward generalization of the single-channelcase in Sec. III A. Because of this, the derivation of the ULE in Appendix Cthat we quoted in Sec. III A considers the case of multiple noise channels. Here we present the results fromAppendix C. As for the single-channel case, the validity of the ULE is determined solely by properties of the bath correlation (or,equivalently, spectral) functions. In the case where the systemand bath are connected through Nquantum noise channels, the bath correlation function introduced in Eq. ( 9) takes values as an N×Nmatrix J(t) with matrix elements J αβ(t−s)≡TrB[˜Bα(t)˜Bβ(s)ρB]. (24) Here ˜Bα(t)≡eiHBtBαe−iHBtdenotes the interaction picture version of the bath operator Bα, and the indices αandβ label the noise channels, taking values 1 ,..., N.U s i n gt h e definition above, one can verify that the bath spectral func-tion J(ω)≡ 1 2π/integraltext∞ −∞dtJ(t)eiωtforms a positive-semidefinite matrix for all ω. The fact that Jis positive semidefinite generalizes the single-channel result that the scalar-valuedbath spectral function J(ω) is non-negative (see Sec. II A). To establish the conditions under which the ULE holds, we generalize the jump correlator g(t)f r o mE q .( 3)t ot h e multiple-channel case. Using the fact that J(ω)i sp o s i t i v e semidefinite, we define the matrix-valued jump correlator g(t) as follows: g(t)=/integraldisplay ∞ −∞dωg(ω)e−iωt,g(ω)=/radicalbig J(ω)/2π. (25) Here the square root in the second equation denotes the ma- trix square root; i.e., Jαβ(ω)=1 2π/summationtext λgαλ(ω)gλβ(ω),where {gαβ(ω)}denote the matrix elements of g(ω). Since J(ω) is positive semidefinite, the Fourier transform of the jumpcorrelator g(ω) is itself a well-defined positive-semidefinite matrix for all values of ω. From the multichannel jump-correlator g(t), we define the quantities /Gamma1andτfrom the mult-channel jump correlator g(t) 115109-8UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) as follows: /Gamma1=4γ/bracketleftbigg/integraldisplay∞ −∞dt/bardblg(t)/bardbl2,1/bracketrightbigg2 ,τ=/integraltext∞ −∞dt/bardblg(t)t/bardbl2,1/integraltext∞ −∞dt/bardblg(t)/bardbl2,1.(26) Here for any matrix Mwith elements Mαλ,/bardblM/bardbl2,1≡/summationtext λ(/summationtext α|Mαλ|2)1/2. The matrix norm /bardbl·/bardbl 2,1is also knownas the L2,1matrix norm [ 34], and is identical to the trace norm for diagonal matrices. As we require, for the special case of asingle noise channel ( N=1), the definitions of /Gamma1andτabove are identical to the definitions in Eq. ( 4). In Appendix Cwe show that when /Gamma1τ/lessmuch1 [with /Gamma1and τas defined in Eq. ( 26)], the system’s dynamics are effec- tively Markovian, and ˜ ρevolves according to the following Lindblad-form master equation: ∂t˜ρ(t)=−i[˜/Lambda1(t),˜ρ(t)]+N/summationdisplay λ=1/parenleftbigg ˜Lλ(t)˜ρ(t)˜L† λ(t)−1 2{˜L† λ(t)˜Lλ(t),˜ρ(t)}/parenrightbigg +ξ/prime(t), (27) where /bardblξ/prime(t)/bardbl/lessorequalslant2/Gamma12τ, and ˜Lλ(t) is the jump operator asso- ciated with the emergent noise channel λ(which may involve operators from multiple baths). Explicitly, ˜Lλ(t) is given by ˜Lλ(t)=√γ/summationdisplay α/integraldisplay∞ −∞dsgλα(t−s)˜Xα(s), (28) while the multiple-channel Lamb shift ˜/Lambda1(t) is given by ˜/Lambda1(t)=γ 2i/integraldisplay∞ −∞ds/integraldisplay∞ −∞ds/prime/summationdisplay αβ˜Xα(s)˜Xβ(s/prime)φαβ(s−t,s/prime−t). (29) Here{φαβ(t,s)}denote the matrix elements of the N×N matrix φ(t,s)≡g(t)g(−s)sgn( t−s).The above expressions for˜Lλ(t) and ˜/Lambda1(t) simplify further in the case of independent noise channels [i.e., when the bath spectral function J(ω)i s diagonal], since g(t) in this case is diagonal. Analogous to the single-channel case, Eqs. ( 27)–(29) hold up to a correction of order 2 /Gamma12τfor a density matrix ρ/prime whose norm distance to ˜ ρremains bounded by /Gamma1τat all times. Hence, we expect Eq. ( 27) to accurately describe the evolution of ˜ρin the weak-coupling limit /Gamma1τ/lessmuch1 (see discussion in Sec. III A). D. Schrödinger picture We conclude this section by expressing the universal Lindblad equation [Eqs. ( 27)–(29)] in the Schrödinger pic- ture. Using the transformation between the interaction andSchrödinger pictures below Eq. ( 8), we obtain the master equation for the reduced density matrix of the system in theSchrödinger picture ρ: ∂ tρ(t)=−i[HS(t)+/Lambda1(t),ρ(t)]+N/summationdisplay λ=1Dλ[ρ(t),t],(30) where we suppressed the correction of order /Gamma12τfrom Eq. ( 27). In the above, Dλdenotes the dissipator associated with emergent noise channel λ, and is given by Dλ[ρ,t]≡Lλ(t)ρL† λ(t)−1 2{L† λ(t)Lλ(t),ρ}. (31) Here the Schrödinger picture jump operators and Lamb shift are given by Lλ(t)=U(t)˜Lλ(t)U†(t) and /Lambda1(t)= U(t)˜/Lambda1(t)U†(t), where U(t) denotes the unitary evolutionoperator of the system S, while ˜/Lambda1(t) and ˜Lλ(t) were given in Sec.III C above. By direct computation, we find, in particular, Lλ(t)=√γ/summationdisplay α/integraldisplay∞ −∞dsgλα(t−s)U(t,s)XαU†(t,s),(32) where U(t,s)≡Te−i/integraltextt sdt/primeHS(t/prime)denotes the time-evolution operator of the system from time sto time t, defined such that U(s,t)=U†(t,s). An analogous expression can be obtained for/Lambda1(t). As we show in Sec. IV A , the jump operators {Lλ(t)} and Lamb shift /Lambda1(t) above are time independent when the system Hamiltonian HS(t) is time independent. IV . PRACTICAL IMPLEMENTATION In this section we discuss how to implement the univer- sal Lindblad equation [Eq. ( 32)] in practice. We separate our discussion into three often-arising cases: in Sec. IV A we consider systems with time-independent Hamiltonians, inSec. IV B we consider systems with time-dependent Hamil- tonians, and in Sec. IV C we demonstrate how the ULE can be implemented in cases where exact diagonalization of thesystem Hamiltonian is not feasible (such as, e.g., quantummany-body systems). A. Systems with time-independent Hamiltonians We first consider the case of time-independent Hamiltoni- ans. We moreover assume that the system’s Hamiltonian canbe efficiently diagonalized, either analytically or numerically.In this case the jump operators and Lamb shift can be easilycomputed from the Hamiltonian’s eigenstates and energies. InSec.IV C we discuss an efficient approximate implementation for cases where exact diagonalization is not practically possi-ble. When the system Shas a time-independent Hamiltonian H S, the time-evolution operator of the system is given by U(t,s)=/summationtext n|n/angbracketright/angbracketleftn|e−iEn(t−s)where {|n/angbracketright}and{En}denote the eigenstates and energy spectrum of HS. Inserting this result into Eq. ( 32), we obtain the following simple expression for the system’s jump operators: Lλ=2π√γ/summationdisplay m,n,αgλα(En−Em)X(α) mn|m/angbracketright/angbracketleftn|, (33) where X(α) mn≡/angbracketleftm|Xα|n/angbracketright. The result above holds for an arbi- trary number of quantum noise channels, and was quoted in 115109-9FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) Sec.Ifor the single-channel case. Note that the jump operators are time independent, as required by the time-translationsymmetry in this case of the problem. The Lamb shift /Lambda1can be expressed in similar terms as above, and also inherits the time independence of H S:i n Appendix Dwe find /Lambda1=/summationdisplay l,m,nfαβ(Em−El,En−El)X(α) mlX(β) ln|m/angbracketright/angbracketleftn|. (34) Here the functions {fαβ(E1,E2)}denote the elements of the matrix f(E1,E2)≡2πγP/integraltext∞ −∞dωω−1g(ω−E1)g(ω+ E2), with P/integraltext denoting the Cauchy principal value integral. We confirm that the master equation above reproduces previous results for open quantum systems in the limit of largelevel spacing and weak γ(i.e., in the regime where the quan- tum optical master equation is valid) [ 9]: in this limit, standard arguments [ 9] show that the rotating wave approximation can be applied to Eq. ( 31). Using the jump operator in Eq. ( 33), this approximation reduces Eq. ( 30) to the quantum optical master equation. Similarly, to first order in γ,t h eL a m bs h i f t /Lambda1renormalizes each energy level of the system E nby the amount δEn=/angbracketleftn|/Lambda1|n/angbracketright.U s i n g g(ω)2=2πJ(ω)i nE q .( 34), one can verify that δEnis identical to previous expressions for the Lamb-shift renormalization of energy levels in the small- γ limit [ 9]. B. Systems with time-dependent Hamiltonians We now consider the situation where the system’s Hamilto- nian HS(t) varies with time. While in this case one can always obtain the jump operators from Eq. ( 32), here we obtain more convenient expressions in two important situations of wideapplicability. The first case we consider arises when the time dependence ofH Sis slow on the bath correlation timescale τ. In this case, HS(t) may be assumed constant in Eq. ( 32), and the jump operators {Lλ(t)}and Lamb shift /Lambda1(t) can be calculated from the energies and eigenstates of the instantaneous HamiltonianH S(t). Specifically, as we show in Appendix E, calculating the jump operator from the instantaneous Hamiltonian as aboveyields a correction of order up to√ /Gamma1τ2/bardbl∂tHS/bardbl(recall that Lλ(t) has units of [Energy]1/2). See Appendix Efor further details. The above result confirms that Lλ(t) may be calculated from the eigenstates and energies of the instantaneous Hamil-tonian when its time-derivative ∂ tHSis sufficiently small compared to 1 /τ2. The approach in Appendix Ecan also be used to identify similar corrections for the Lamb shift. The second situation where the universal Lindblad equa- tion simplifies is the special case of periodically driven sys-tems, where H S(t)=HS(t+T) for some driving period T. In this case, Lλ(t) and /Lambda1(t) can be exactly computed from the time-periodic Floquet states |φn(t)/angbracketright=|φn(t+T)/angbracketrightand quasienergies of the system [ 35]εn: for periodically driven systems, the evolution operator of the system is given by U(t,s)=/summationtext n|φn(t)/angbracketright/angbracketleftφn(s)|e−iεn(t−s). Using this in Eq. ( 32), one can verify by straightforward computation that Lλ(t)=/summationdisplay m,n∞/summationdisplay z=−∞L(λ) mn;z|φm(t)/angbracketright/angbracketleftφn(t)|e−i/Omega1zt,where L(λ) mn;z≡/summationdisplay α/integraldisplayT 0dt T/angbracketleftφm(t)|Xα|φn(t)/angbracketrightei/Omega1ztgαλ/parenleftbig εz nm/parenrightbig .(35) Here/Omega1≡2π/T, while εz nm≡εn−εm+z/Omega1. Note that the jump operators inherit the time periodicity of the Hamiltonian,as required by discrete time-translation symmetry: L λ(t)= Lλ(t+T). The Lamb shift /Lambda1(t) has a similar expression in terms of the Floquet states and also satisfies /Lambda1(t+T)=/Lambda1(t). Interestingly, when the time dependence of HS(t)i ss l o w compared to the bath correlation time τ(which may be very short), the results above show that the jump operators inEq. ( 35) are equivalent to the jump operators generated from the instantaneous eigenstate basis of the Hamiltonian H S(t) through Eq. ( 33). The above form of the jump operators was used by the one of the authors to numerically simulate thedynamics of a driven-dissipative quantum cavity in Ref. [ 36]. The results above reproduce the generalization of the quan- tum optical master equation to periodically driven systems,for example derived in Refs. [ 37–39]. In these works, a Lindblad-form master equation is obtained for the systemusing a rotating wave approximation (RWA) which assumesthe relaxation rate ( /Gamma1) much smaller than the smallest possible level spacing in the system’s quasienergy spectrum δε min= min m/negationslash=n(εn−εm+z/Omega1). The approaches we present above do not rely on such a rotating wave approximation, and henceare valid for a wider class of systems. We note that bothapproaches presented here are equivalent to the above RWAmaster equations in the limit /Gamma1/lessmuchδεwhere the latter are valid. In this limit, one can verify that the steady state of thesystem is diagonal in the Floquet state basis. C. Obtaining jump operators without diagonalization We finally show how the universal Lindblad equation can be implemented in cases where diagonalization of the systemHamiltonian H Sis not feasible, such as for large quantum many-body systems. In this case, the jump operators andLamb shift of the ULE cannot be obtained from the eigenstatedecompositions presented above. Instead, as we show here,these operators can be easily obtained through a systematic,convergent expansion of Eq. ( 32)i np o w e r so f τ/τ X, where τ is the bath correlation time, and τXdenotes the characteristic timescale for the dynamics of the system observables {Xα} (see below for definition). For simplicity we consider here the time-independent single-channel case. The approach we present below gener-alizes straightforwardly to time-dependent Hamiltonians andmultiple noise channels. We moreover focus on computingthe jump operator L(we suppress λsince we consider the single-channel case); the Lamb shift can be obtained througha similar approach. To compute the jump operator L, we note that, for a time-independent Hamiltonian H S,U(t,s)=e−i(t−s)HS. Next, we note that eiHStXe−iHSt=/summationtext∞ n=0(it)n(adHS)n[X]/n! where ad HSdenotes the commutation operation by HS, i.e., adHS[O]=[HS,O]. Using these results in Eq. ( 32), we obtain L=√γ∞/summationdisplay n=0cn(adHS)n[X],cn≡in n!/integraldisplay∞ −∞dt g(t)tn.(36) 115109-10UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) FIG. 2. Simulation of the open Heisenberg spin chain model in Sec. V(see main text for further details): (a) Schematic depiction of the system. (b) Average zmagnetization in the chain, as a function of time, for the cases where the chain is connected to bath 2 (red), to bath 1 (blue), and to both baths (purple). Gray lines: Expectation values of the zmagnetization in the Gibbs states at the temperatures of baths 1 (lower) and 2 (upper). Shaded areas surrounding curves (only visible for purple curve) indicates the uncertainty due to the finite number ofsampling states. (c) Average zmagnetization in the chain /angbracketleftS z n/angbracketrightfor the final duration 150 η−1of the simulation, as a function of site index n [using same coloring scheme as in (b)]. Error bars indicate the uncertainty due to the finite number of sampling states in the simulation. Crucially, the coefficients {cn}are system independent, and can be easily computed from the jump correlator. The convergence of the series in Eq. ( 36) can be ensured by introducing a temporal cutoff τmax, such that g(t)i ss e t to zero for |t|/greaterorequalslantτmax. The error in Lresulting from this approximation is bounded by 2/integraltext∞ τmaxdt|g(t)|[see Eq. ( 32)], and can thus be made arbitrarily small by choosing τmax sufficiently large [ 40]. In particular, we expect the error to be negligible when τmax/greatermuchτ. With the temporal cutoff im- posed, cn∼τn max/n!f o rl a r g e n. As a result, the series in Eq. ( 36) converges at order τmax/τX, where τXdenotes the typical timescale associated with the dynamics of X, such that /bardbl(adHS)n[X]/bardbl∼1/τn X. The expansion of the jump operator simplifies further when the Hamiltonian HSis composed of an easily diagonalizable term H0(such as a quadratic term) and a weak nonintegrable perturbation V(such as an interaction term): HS=H0+V. By transforming to the interaction picture with respect to H0, we find e−iHSt=e−iH0tU/prime(t), where U/prime(t)≡Te−i/integraltextt 0dt/prime˜V(t/prime), and ˜V(t)=eiH0tVe−iH0t. Using this result in Eq. ( 32), and ex- panding U/prime(t)X[U/prime(t)]†in powers of ˜V(t)a sa b o v eE q .( 36), we obtain a series expansion, where the nth order term is bounded by ( τmax/˜τX)n/n!, where ˜ τXdenotes the timescale for the dynamics of Xinduced by the perturbation ˜V, such that /bardbladn ˜VX/bardbl/lessorsimilar˜τ−n X. As a result, only terms up to order τmax/˜τX contribute in the expansion of the jump operator. Importantly, ˜τXis inversely proportional to the strength of the perturbation V. Thus, in the limit of weak perturbations, where τmax/lessmuch˜τX, the expansion above can be truncated at order zero. In thiscase, we may thus ignore the nonintegrable perturbation inthe calculation of the jump operator, and obtain Lfrom the spectrum and eigenstates of the integrable Hamiltonian H 0. In the case where Vis small, but not completely negligible, the jump operator may still be efficiently approximated byincluding the first few terms of the expansion discussed above. V . NUMERICAL DEMONSTRATION: HEISENBERG SPIN CHAIN In this section we demonstrate how the universal Lindblad equation can be used in a numerical simulation. We considera ferromagnetic spin-1 /2 Heisenberg chain coupled to twoOhmic baths that are out of equilibrium with each other, as schematically depicted in Fig. 2(a). By numerically solving the ULE, we obtain the nontrivial steady states and transportproperties of the spin chain, along with its transient relaxationdynamics. The system we consider cannot be easily simulated by cur- rent master equation techniques, and hence our demonstrationhighlights the utility of the ULE. For instance, the quantumoptical master equation can only be employed when thesystem’s relaxation rate is small compared to the level spacingof the system Hamiltonian. For the spin chain we consider,this level spacing is exponentially suppressed in the numberof spins N, and hence, even for moderately sized chains, the quantum optical master equation only works for extremelyweak system-bath couplings. In contrast, the validity of theULE is independent of the level spacing in the system. Thusthe ULE is valid for system-bath couplings many orders ofmagnitude larger than allowed by the quantum optical masterequation. Another common master equation approach, the Bloch Redfield equation, is also ill-suited for the spin chain weconsider: the BR equation is often not stable, and may yieldunphysical results, as discussed in the beginning of Sec. III. In contrast, the ULE is in the Lindblad form, and thusinherently robust. Even without instabilities, integration ofthe BR equation is numerically expensive, since it requiresevolving the D×Ddensity matrix of the system ρ, where D=2 Nis the Hilbert space dimension of the system. On the other hand, Lindblad-form master equations can be integratedwith the stochastic-Schrödinger equation, which only requiresevolving a D-component state vector. This significantly re- duces the computational cost, with the relative gain scalingexponentially with the size of the system. The spin chain Hamiltonian is given by H S=−BzN/summationdisplay n=1Sz n−ηN−1/summationdisplay n=1Sn·Sn+1, (37) where Bzdenotes the strength of a uniform Zeeman field, ηis the nearest-neighbor coupling strength, and Sn=(Sx n,Sy n,Sz n), where Sμ ndenotes the spin- μoperator on site nin the chain. For the simulations below, we take N=12 sites. The system is connected to two baths, B1andB2, via spins S1and SNat the opposite ends of the chain, as schematically depicted 115109-11FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) in Fig. 2(a). For demonstration we couple the baths to the spins through their xcomponents, Sx 1andSx N, with coupling strengths γ1andγ2. The baths B1andB2are modeled as Ohmic baths in thermal equilibrium, with spectral functionsgiven in Eq. ( 14). For the simulations below we take the baths to have the same values of the cutoff /Lambda1andω 0, but distinct temperatures, T1andT2. The system-bath coupling of the system is hence given by Hint=√γ1Sx 1B/prime 1+√γ2Sx NB/prime 2, (38) where, for α=1,2,B/prime αis a bosonic field operator in the Ohmic bath Bαwith spectral function Jα(ω) given by Eq. ( 14) with T=Tα. To obtain the master equation for the system, we cast the above system-bath coupling into the form given in Eq. ( 6). Within this framework, the bath consists of two noise channelswith X 1=Sx 1andX2=Sx N. The corresponding bath operators are given by Bα=√˜γαB/prime αforα=1,2, where ˜ γα≡γα/γ denotes the relative system-bath coupling strength, and γ denotes the redundant energy scale we introduced in Eq. ( 6)t o parametrize the overall system-bath coupling [see discussionbelow Eq. ( 6)]. Straightforward calculations [ 9] show that the elements of the 2 ×2 spectral function matrix J(ω)a r eg i v e n byJ αβ(ω)=δαβ˜γαJα(ω), where δαβdenotes the Kronecker symbol and Jα(ω) denotes the spectral function of bath opera- torB/prime α(see above). Note that the coefficients ˜ γαappear in the spectral function due the parametrization of Hintin Eq. ( 6), in which a single common coupling scale γis factored out of the coupling Hamiltonian. A. Relaxation to thermal steady state We first seek to verify that, when connected only to bath 2, the system relaxes to a steady state in thermal equilibriumwith the bath, as we expect from basic thermodynamics.In our simulation, we therefore set γ 1=0,γ2=0.02η.T h e remaining parameters are set to Bz=8η,/Lambda1=100η,ω0=2η (for both baths), while T2=20η, and T1=2η. All parameters except for γ1andγ2are given by the same values throughout this section. As a first step, we compute the characteristic bath timescales /Gamma1−1andτ, which define the regime of applicability of the universal Lindblad equation. Using Eq. ( 26), we find /Gamma1=0.41ηandτ=0.00013 /η, resulting in /Gamma1τ=0.00053. Thus, following the discussion in Sec. III, we expect the ULE to be valid. In particular, the ULE correctly describesthe rate of change of the system’s density matrix ∂ tρup to a correction bounded by 2 /Gamma12τ=0.0043η. This error bound is several orders of magnitude smaller than the other energyscales of the model, and we thus expect the ULE to faithfullycapture the system’s evolution and steady states [ 33]. To solve the ULE, we computed the system’s jump opera- tors{L λ}by exact diagonalization of HS,u s i n gE q .( 33). Note that Eq. ( 36) can be used if diagonalization is not feasible. We excluded the Lamb shift from the simulation, since thisterm only weakly perturbs H S; thus we do not expect it to affect the system’s dynamics significantly [ 9]. In contrast, the jump operators, no matter how weak, break the unitarity oftime evolution, and hence cannot be neglected in the masterequation. We initialized the system in the state with all spinsaligned against the uniform field B z, and integrated the ULEnumerically using the stochastic Schrödinger equation, with an ensemble of 100 states [ 20–22]. In Fig. 2(b) we plot the expectation value of the average zmagnetization in the chain M=1 N/summationtextN n=1Sz nas a function of time (red line). The uncertainty of the expectation value/angbracketleftM/angbracketright≡Tr[ρ(t)M] arising from the finite number of ensemble states is smaller than the thickness of the line. As Fig. 2(b) shows, /angbracketleftM/angbracketrightreaches a stationary value after a transient re- laxation period of approximate duration 50 η −1. The steady- state value of /angbracketleftM/angbracketrightis identical to the expectation value of Min a Gibbs state at temperature T2(upper gray line), up to the accuracy of the simulation. A similar result arises inthe case where the chain is connected only to bath B 1:γ1= 0.1ηandγ2=0 [blue curve in Fig. 2(b)]. Thus we confirm that the universal Lindblad equation reproduces the expectedequilibrium steady states, further supporting its validity. B. Nonequilibrium steady state with two baths We now consider the case where the spin chain is simulta- neously connected to both baths, B1andB2, with γ1=0.1η andγ2=0.02η. In this case, due to the temperature difference between the baths, we expect the system to reach a nonequilib-rium steady state characterized by nonzero transport of energyand magnetization between the baths. With the parametersabove, the characteristic timescales as defined in Eqs. ( 4) evaluate to /Gamma1≈3.6ηandτ≈0.0032η −1. Thus, /Gamma1τ≈0.011 and 2/Gamma12τ≈0.079η, indicating that the universal Lindblad equation should accurately describe the system’s dynamics. In Fig. 2(b)we plot the the magnetization in the chain /angbracketleftM/angbracketright as a function of time (purple), obtained with the universalLindblad equation. Similar to the two equilibrium cases, themagnetization settles to a steady-state value after a transientrelaxation period of duration ∼50η −1. However, the relaxed system is not in a Gibbs state, but rather a more complicatednonequilibrium steady state: to demonstrate this, in Fig. 2(c) we show the site-resolved magnetization /angbracketleftS z n(t)/angbracketright, averaged over a time window of length 150 η−1at the end of the simulation. As Fig. 2(c) clearly shows, the local magnetiza- tion of the system is not uniform, but gradually increasesfrom the left to the right end of the chain, indicative of anonequilibrium steady state. In contrast, for the two caseswhere only a single bath is connected to the chain (red andblue), the local magnetization is uniform throughout the chain,consistent with a thermal Gibbs state at temperatures T 2and T1of the connected baths [horizontal gray lines in Fig. 2(c)]. The skewed magnetization profile in the nonequilibrium caseabove reflects a nonzero transport of heat and magnetization(magnons) between the two baths through the chain. By directcomputation (see Appendix Ffor details), we compute the average rate of heat transfer ¯I Eand magnetization transfer ¯IM from bath 2 to bath 1 over a time window of duration 75 η−1 at the end of the simulation, finding ¯IE=2.1η2±0.2η2and ¯IM=0.33η±0.02η. VI. DISCUSSION In this paper we derived a Lindblad-form master equation for open quantum many-body systems: the universal Lind-blad equation (ULE). We identified rigorous upper bounds 115109-12UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) for the correction to the ULE, expressed in terms of the intrinsic timescales of the bath and the system-bath coupling.Crucially, the correction bounds we obtained for the ULEare independent of the details of the system, and are of thesame magnitude as the error bounds we obtained for theBloch-Redfield (BR) equation, which is not in the Lindbladform. In this sense, the ULE is valid on an equivalent level ofapproximation as the BR equation. The universal Lindblad equation opens up new possibilities for systematically studying a wide class of open quantumsystems. These classes of systems include quantum many-body systems, and general driven quantum systems with denseenergy spectra, for which the stringent conditions of thequantum optical master equation are not met. In addition, theULE can be implemented with lower computational cost andgreater stability than the BR equation, since by construction itpreserves the positivity and trace of the reduced density matrixof the system. We have demonstrated the utility of the ULE in numerical simulations of an open Heisenberg spin chain, where we usedit to extract the transport characteristic of the system’s steadystate in a nonequilibrium setting. We expect the ULE can beused to easily infer other nonequilibrium characteristics ofthe chain, such as, e.g., the correlations of magnetization orheat current fluctuations, without adding any additional costin the simulation. In addition to the spin chain model weconsidered here for demonstration, the universal Lindblad wasrecently used by one of the authors to simulate the dynamicsof a periodically driven cavity-spin system in Ref. [ 36], and by our collaborators to study readout of topological qubitsin Ref. [ 41]. The universal Lindblad equation was also im- plemented in numerical simulations in Ref. [ 23], in order to support the hypothesized master equation there (see Sec. I). The principle underlying our derivation of the ULE is that there does not exist a unique Markov approximationin the Markovian regime /Gamma1τ/lessmuch1. Rather there exists an infinite family of Markov approximations yielding distincttime-local master equations for the system that each are validon an equivalent level of approximation. From this family ofequivalent master equations, we identified a master equationin the Lindblad form, the ULE. An interesting avenue of future studies is the mathematical exploration of this equivalence class of Markov approxima-tions; in particular, it will be interesting to investigate whetherthe above freedom of choice can be exploited further, to obtainmaster equations that are even more efficient or accurate,or perhaps explicitly respect desired symmetries or conser-vation laws. Another relevant question along this directionof research is whether higher-order bath correlations andnon-Markovian corrections can also be incorporated in theframework we develop here, and yield efficient and accuratemaster equations for the system. As stimulus for another direction of future work, we spec- ulate that the correction bounds we obtained can be improvedfurther. In particular, while we do not show it here, for Ohmicbaths, the energy scale /Gamma1scales linearly with the high-energy cutoff of the bath (see Sec. II B). However, this divergence arises from ultrashort (i.e., effectively time-local) correlationsand reflects a divergent renormalization of the Hamiltonianthrough the Lamb shift. Hence, adding a correction to thebare system Hamiltonian to compensate the divergent terms, we speculate that much better bounds can be obtained for thecorrection the ULE. Often, such a correcting counterterm isphysically well motivated. We believe further analysis of theproblem using this principle can lead to significant improve-ment of the error bounds for the ULE. As important secondary results, in this work, we obtained rigorous error bounds for the Bloch-Redfield equation, and es-tablished a “quantum speed limit” for the rate of bath-inducedevolution of open quantum systems. These results may alsobe relevant for future work. The results were establishedusing a perturbative approach in Appendix A, in which the time derivative of the reduced density matrix of the systemis systematically expanded in orders of the dimensionlessnumber /Gamma1τ. We speculate that this approach may be used in the future to obtain master equations that are valid at higherorders in /Gamma1τ. In summary, we have rigorously derived a Lindblad-form master equation for open quantum systems that offers severaladvantages over previously existing methods. We expect thatthe efficiency, wide applicability, and simplicity of our methodopens up new possibilities for future studies of open quantumsystems. ACKNOWLEDGMENTS We thank Ivar Martin, Gil Refael, Karsten Flensberg, Martin Leijnse, Morten I. K. Munk, Gediminas Kirsanskas,Evgeny Mozgunov, Tatsuhiko Ikeda, and Archak Purkayasthafor helpful comments and useful discussions. F.N. and M.R.gratefully acknowledge the support of Villum Foundation,the European Research Council (ERC) under the EuropeanUnion Horizon 2020 Research and Innovation Programme(Grant Agreement No. 678862), and CRC 183 of the DeutscheForschungsgemeinschaft. APPENDIX A: CORRECTION TO THE BLOCH-REDFIELD EQUATION Here we derive the rigorous upper bounds for the correc- tion to the Bloch-Redfield (BR) equation that were quotedin Sec. II A 2 in the main text. We derive the bounds for the general case where multiple noise channels connect thesystem and the bath. As a part of our derivation, in AppendixA3below we establish the upper bound for the rate of bath- induced evolution in the system /bardbl∂ t˜ρ/bardblthat we quoted in Sec. II Bof the main text. In this Appendix we work exclusively in the interaction picture (see Appendix II A). To avoid cumbersome notation, we therefore use different notation here than in the maintext, and neglect the ˜ ·accent on all interaction picture op- erators. Thus, throughout this Appendix, ρ(t),ρ SB(t),H(t), Xα(t), and Bα(t) denote the interaction picture operators ˜ ρ(t), ˜ρSB(t),˜H(t),˜Xα(t), and ˜Bα(t) from the main text, respec- tively. 1. Superoperator formalism Our derivation of error bounds for the BR equation exploits the fact that linear operators on a Hilbert space (such as den-sity matrices) can themselves be seen as vectors, or “kets.” To 115109-13FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) make this vector nature of operators explicit, in the following we use double brackets |·/angbracketright /angbracketrightto indicate operators acting on the Hilbert spaces HS,HB,andHSBof the system S, bath B, or the combined system SB. In this way, any operator which is denoted by Oin standard notation is denoted by the ket |O/angbracketright/angbracketrightin the derivation below. The vector space of operator kets that act on Hilbert space Hi(with i={S,B,SB}) defines an operator Hilbert space H2 i, defined with the inner product /angbracketleft/angbracketleftR|S/angbracketright/angbracketright ≡ Tr(R†S). This notation is commonly used in the literature, see, e.g., Ref. [ 43] for a recent example. Note that the operator space of the combined system SB,H2 SB, inherits the tensor product structure of the standard Hilbert space ofSB,H SB:H2 SB∼=H2 S⊗H2 B, where H2 SandH2 Bdenote the operator spaces of the system Sand bath B, respectively. In the superoperator notation above, the von-Neumann equation for the density matrix of the combined system (in theinteraction picture) ∂ tρSB(t)=−i[H(t),ρSB(t)] translates to a linear Schrödinger-type equation: ∂t|ρSB(t)/angbracketright/angbracketright = − iˆH(t)|ρSB(t)/angbracketright/angbracketright, (A1) where ˆH(t) denotes the commutator with H(t):ˆH(t)|O/angbracketright/angbracketright = |[H(t),O]/angbracketright/angbracketright. Note that ˆH(t) acts linearly on |ρSB/angbracketright/angbracketright, and hence it can be represented as a matrix acting on the operatorspaceH 2 SB. Below we furthermore show that ˆH(t)i sH e r - mitian, and hence can be seen as a “Hamiltonian” acting onH 2 SB. We refer to ˆH(t), and other linear transformations on operator kets, as superoperators. To make notation unambigu-ous, in the following we use the “hat” accent (ˆ) to indicatesuperoperators. A useful class of superoperators which we employ ex- tensively in the following is left and right multiplicationby some given operator: for any operator |A/angbracketright/angbracketrightinH 2 i(for i={S,B,SB}), we define the left- and right-multiplication superoperators ˆAland ˆAras ˆAl|O/angbracketright/angbracketright = | AO/angbracketright/angbracketright,ˆAr|O/angbracketright/angbracketright = |OA/angbracketright/angbracketright. (A2) From the above definition one can verify that the superoper- ator ˆH(t)i nE q .( A1) is given by ˆHl(t)−ˆHr(t), where, for any time-dependent operator A(t), and for m={l,r},w eu s e ˆAm(t) as shorthand for /hatwidestA(t)mto avoid cumbersome notation. The right- and left-multiplication superoperators have a fewuseful properties that we use below: first, we note that, byassociativity, ˆA lˆBl=(/hatwiderAB)l, while ˆArˆBr=(/hatwiderBA)r. Moreover, the Hermitian conjugate of the superoperator ˆAm[i.e., ( ˆAm)†] is given byˆ(A†)m. This follows from Eq. ( A2), along with the definition of the inner product /angbracketleft /angbracketleft·|·/angbracketright /angbracketright :/angbracketleft/angbracketleftO1|Al|O2/angbracketright/angbracketright = /angbracketleft/angbracketleftO1|AO2/angbracketright/angbracketright = /angbracketleft/angbracketleft A†O1|O2/angbracketright/angbracketright. For this reason, in the following, we let ˆAm†simply refer to ( ˆAm)†=(ˆA†)m. From these re- sults, it follows that ˆH(t) is Hermitian: ˆH(t)=ˆH†(t), as we claimed above. In deriving the error bounds for the Bloch-Redfield equa- tion, we will make use of the norms of superoperators. Wedefine the norm of the superoperator ˆAacting on H 2 ias /bardblˆA/bardbl≡ sup |O/angbracketright/angbracketright∈H2 i/bardblˆA|O/angbracketright/angbracketright/bardbl /bardbl|O/angbracketright/angbracketright/bardbl, (A3) where, here and in the following, /bardbl|O/angbracketright/angbracketright/bardbldenotes the spectral norm of the operator |O/angbracketright/angbracketright. Note that /bardbl|O/angbracketright/angbracketright/bardblisnotidentical to√/angbracketleft/angbracketleftO|O/angbracketright/angbracketright; rather,√/angbracketleft/angbracketleftO|O/angbracketright/angbracketrightgives the Frobenius norm of |O/angbracketright/angbracketright.From the definition above, it follows that the superoperator norm is submultiplicative: /bardblˆAˆB/bardbl/lessorequalslant/bardblˆA/bardbl/bardblˆB/bardbl. Moreover, using the submultiplicativity of the spectral norm along with thedefinitions in Eq. ( A2), we conclude that, for any operator |O/angbracketright/angbracketright and for m={l,r},/bardblˆO m/bardbl=/bardbl | O/angbracketright/angbracketright/bardbl. Using the superoperator notation above, we now consider the evolution of the reduced density matrix of the system inthe interaction picture |ρ(t)/angbracketright/angbracketright. Recalling that the superoperator ˆH(t)i nE q .( A1) is Hermitian, the “Schrödinger equation” for the density matrix of the combined system, Eq. ( A1), has the well-known solution |ρ SB(t)/angbracketright/angbracketright = ˆU(t,s)|ρSB(s)/angbracketright/angbracketright, (A4) where ˆU(t,s) denotes the unitary evolution superoperator of the combined system, given by ˆU(t,s)=Te−i/integraltextt sdt/primeˆH(t/prime). By taking the time derivative, one can verify that ˆU(t,s)= ˆUl SB(t,s)ˆUr† SB(t,s), where USB(t,s)=Te−i/integraltextt sdt/primeH(t/prime)denotes the (ordinary) time-evolution operator of the combined sys-tem. Thus |ρ SB(t)/angbracketright/angbracketright = | USB(t,s)ρSB(s)U† SB(t,s)/angbracketright/angbracketright.U s i n g the properties of the superoperator norm below Eq. ( A3), we conclude that that /bardblˆU(t,s)/bardbl=1. In the superoperator notation, the partial trace Tr Bover the bath degrees of freedom can be expressed as the dual vector(bra) of the bath identity operator |I B/angbracketright/angbracketright. Here, as for ordinary bra-ket notation, /angbracketleft/angbracketleftYB|is understood as the linear mapping H2 SB→H2 S, such that, for |MSB/angbracketright/angbracketright =/summationtext a,bMab|aS/angbracketright/angbracketright|bB/angbracketright/angbracketright, /angbracketleft/angbracketleftYB|MSB/angbracketright/angbracketright =/summationtext a,bMab|aS/angbracketright/angbracketright/angbracketleft/angbracketleftYB|bB/angbracketright/angbracketright, where aand blabel orthonormal bases for the operator spaces on SandB,r e - spectively. (Recall that the operator space H2 SBinherits the tensor product structure of HSB.) As a result, we may write the reduced density matrix of the system Sas|ρ(t)/angbracketright/angbracketright = /angbracketleft/angbracketleftIB|ρSB(t)/angbracketright/angbracketright. Inserting the above result into Eq. ( A4), and using our assumption that |ρSB(t0)/angbracketright/angbracketright = |ρ0/angbracketright/angbracketright|ρB/angbracketright/angbracketrightfor some time t0in the remote past (see Sec. IIin the main text), we find |ρ(t)/angbracketright/angbracketright = /angbracketleft/angbracketleft IB|ˆU(t,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright. (A5) To obtain a master equation for |ρ(t)/angbracketright/angbracketright, we explicitly take the time derivative in Eq. ( A5), obtaining ∂t|ρ(t)/angbracketright/angbracketright = − i/angbracketleft/angbracketleftIB|ˆH(t)ˆU(t,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright. (A6) Using the decomposition H(t)=√γ/summationtext αXα(t)Bα(t)[ E q .( 6) in the main text, translated to the interaction picture], we find ˆH(t)=√γ/summationtext m,ανmˆXm α(t)ˆBm α(t), where m={l,r}, withνl= 1 andνr=−1. Recalling that (for each α)Xα(t) acts trivially on the bath degrees of freedom, we obtain ∂t|ρ(t)/angbracketright/angbracketright = − i√γ/summationdisplay m,ανmˆXm α(t)/angbracketleft/angbracketleftIB|ˆBm α(t)ˆU(t,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright. (A7) 2. Statistical properties of the bath To obtain a convenient expression for the bath expectation value/angbracketleft/angbracketleftIB|ˆBm α(t)ˆU(t,t0)|ρB/angbracketright/angbracketrightin Eq. ( A7), we make use of our assumption that the bath is Gaussian. For simplicity, in thissection we assume that all bath operators are bosonic. Similarconsiderations can be applied for fermionic bath operators. For a Gaussian bath, the expectation value of any bath operator can be computed from the two-point correlation 115109-14UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) function using Wick’s theorem. In the superoperator notation we use, with ˆBj≡ˆBmj αj(tj) (where mj={l,r}, while αjrefers to the noise channel index), Wick’s theorem applied to aproduct of kbath operators takes the form /angbracketleft/angbracketleftˆB 1···ˆBk/angbracketright/angbracketright =k/summationdisplay j=2/angbracketleft/angbracketleftˆB1ˆBj/angbracketright/angbracketright/angbracketleft/angbracketleftˆA2,j−1ˆAj+1,k/angbracketright/angbracketright, (A8) where ˆAi,j=/producttextj n=iˆBnforj/greaterorequalslanti,ˆAi,j=1f o r j<i, and we introduced the shorthand /angbracketleft/angbracketleftˆO/angbracketright/angbracketright ≡ /angbracketleft/angbracketleft IB|ˆO|ρB/angbracketright/angbracketrightto simplify nota- tion. Wick’s theorem for superoperators, as stated in Eq. ( A8), can be proven by direct computation using the definitions ofthe superoperators {ˆB j}, along with Wick’s theorem for the (nonsuper) operators {Bαj(tj)}[9,10]. By iteration of Wick’s theorem [Eq. ( A8)], it is straightfor- ward to show that the expectation value /angbracketleft/angbracketleft· /angbracketright/angbracketrightof any polynomial functional of the bath superoperators {ˆBα(t)}can be expressed fully in terms of the (two-point) bath superoperator correlationfunctions J mn αβ(t−t/prime)≡/angbracketleftbig/angbracketleftbigˆBm α(t)ˆBn β(t/prime)/angbracketrightbig/angbracketrightbig . (A9) The bath superoperator correlation functions hold the same information as the ordinary bath correlation function J(t−s) [see Eq. ( 24) in the main text]: letting Jmn(t) denote the matrix with elements {Jmn αβ(t)}, and using the cyclic property of thetrace, one can verify that, for m={l,r},Jml(t)=J(t), while Jmr(t)=J†(t). Importantly, the unitary evolution superoperator of the combined system SB,ˆU(t,s), is analytic, and hence can be expanded as a polynomial of the bath superoperators{ˆB m α(t)}. By using this expansion along with Wick’s theorem [Eq. ( A8)], one can then verify that /angbracketleftbig/angbracketleftbigˆBm α(t)ˆU(t,s)/angbracketrightbig/angbracketrightbig =/integraldisplay∞ −∞dt/prime/summationdisplay β,nJmn αβ(t−t/prime)/angbracketleftBigg/angbracketleftBigg/angbracketleftBigg δˆU(t,s) δˆBn β(t/prime)/angbracketrightBigg/angbracketrightBigg/angbracketrightBigg , (A10) where δ/δˆBn β(t/prime) denotes the functional derivative with re- spect to ˆBn β(t/prime). Specifically, δˆBm α(t)/δˆBn β(t/prime)=δαβδmnδ(t− t/prime), where δijdenotes the Kronecker symbol, and δ(t)i st h e Dirac delta function. Using the Trotter decomposition of ˆU(t,s) along with ˆH(t)=−i√γ/summationtext m,ανmˆXm α(t)ˆBm α(t), one can verify that, for t/prime in the interval between sandt, δˆU(t,s) δˆBn β(t/prime)=−i√γνnU(t,t/prime)ˆXn β(t/prime)ˆU(t/prime,s), (A11) while δˆU(t,s)/δˆBn β(t/prime)=0 when t/primeis outside the interval between sandt. Inserting Eqs. ( A10) and ( A11) into Eq. ( A7) gives ∂t|ρ(t)/angbracketright/angbracketright = − γ/summationdisplay m,n;α,βνmνnˆXm α(t)/integraldisplayt t0dsJmn αβ(t−s)/angbracketleft/angbracketleftIB|ˆU(t,s)ˆXn β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright |ρ0/angbracketright/angbracketright. (A12) Equation ( A12) is a crucial result, and forms the basis for the derivation below. Importantly, the result is exact forGaussian baths, and does not rely on any other approximationsor assumptions. Equation ( A12) can be generalized to non- Gaussian baths by expanding the left-hand side of Eq. ( A10) in terms of the (nonvanishing) higher-order correlation func-tions of the bath. While such an extension to non-Gaussianbaths is in principle straightforward, in this Appendix werestrict ourselves for simplicity to the case of Gaussianbaths. 3. Upper bound for rate of bath-induced evolution While Eq. ( A12) looks somewhat complicated, we may already use it in its present form to infer important facts aboutthe evolution of the system. Specifically, in this subsection,using Eq. ( A12), we identify an upper limit for the rate of bath-induced evolution in the system /bardbl∂ t|ρ/angbracketright/angbracketright/bardbl. This result was quoted in Sec. II Bof the main text [recall that |ρ(t)/angbracketright/angbracketright in this Appendix is identical to ˜ ρ(t) in the main text]. The arguments and concepts we use here will also be used in thefollowing subsections, when we derive error bounds for theBloch-Redfield equation. To derive an upper bound for /bardbl∂ t|ρ/angbracketright/angbracketright/bardbl, we take the (spec- tral) norm on both sides in Eq. ( A12). Using the triangle in- equality along with /bardblˆXm α(t)|O/angbracketright/angbracketright/bardbl/lessorequalslant/bardbl|O/angbracketright/angbracketright/bardbl(this follows from the properties of the superoperator norm listed in AppendixA1and the fact that the operators X αare assumed to have unitspectral norm), we thereby obtain /bardbl∂t|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslantγ/summationdisplay m,n;α,β/integraldisplayt t0ds/vextendsingle/vextendsingleJmn αβ(t−s)/vextendsingle/vextendsinglekn β(t,s),(A13) where kn β(t,s)≡/bardbl /angbracketleft /angbracketleft IB|ˆU(t,s)ˆXn β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright/bardbl.W e now prove that kn β(t,s)/lessorequalslant1. To establish this bound, it is simplest to consider the cases n=landn=rseparately. Specifically, below we prove that kl β(t,s)/lessorequalslant1. The proof for kr β(t,s)/lessorequalslant1 proceeds along the same lines. To establish that kl β(t,s)/lessorequalslant1, we write kl β(t,s)=/bardbl |Q/angbracketright/angbracketright/bardbl, where |Q/angbracketright/angbracketright ≡ /angbracketleft/angbracketleft IB|ˆU(t,s)ˆXl β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright (A14) represents an operator on the system S. We now note that ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright = |ρSB(s)/angbracketright/angbracketright, since |ρSB(t0)/angbracketright/angbracketright = |ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright. Thus|Q/angbracketright/angbracketright = /angbracketleft/angbracketleft IB|ˆU(t,s)ˆXl β(s)|ρSB(s)/angbracketright/angbracketright. We now convert the above expression for |Q/angbracketright/angbracketrightinto standard (nonsuperoperator) notation for the corresponding operator Q that acts on system S: Q=TrB[USB(t,s)Xβ(s)ρSB(s)U† SB(t,s)], (A15) where USB(t,s)=Te−i/integraltextt sdt/primeH(t/prime)denotes the standard (i.e, nonsuper) unitary evolution operator of the combined systemin the interaction picture [see discussion below Eq. ( A4)]. We recall that the spectral norm of Q, also denoted /bardbl|Q/angbracketright/angbracketright/bardbl,i sg i v e n by the maximal value of |/angbracketleftφ|Q|ψ/angbracketright|for any two normalized states|ψ/angbracketright,|φ/angbracketrightin the system Hilbert space H S. To bound this 115109-15FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) number, we exploit the cyclic property of the trace along with Eq. ( A15) to write /angbracketleftφ|Q|ψ/angbracketright=TrSB[CρSB(s)], (A16) where C≡U† SB(t,s)(|ψ/angbracketright/angbracketleftφ|⊗IB)USB(t,s)Xβ(s), with IB denoting the identify operator on the bath Hilbert space HB. Next, we use the spectral decomposition of ρSB(s), ρSB(s)=/summationtext i|ni/angbracketright/angbracketleftni|pi, where {|ni/angbracketright}form an orthonormal ba- sis for the Hilbert space of the combined system HSB, and the eigenvalues {pi}are non-negative and have unit sum. Inserting this into Eq. ( A16), we find /angbracketleftφ|Q|ψ/angbracketright=/summationtext i/angbracketleftni|C|ni/angbracketrightpi.Using the triangle inequality, along with |/angbracketleftni|C|ni/angbracketright|/lessorequalslant/bardblC/bardbl, where /bardbl·/bardbl denotes the spectral norm, we find |/angbracketleftφ|Q|ψ/angbracketright|/lessorequalslant/bardblC/bardbl, (A17) where we also exploited the non-negativity and unit sum of the eigenvalues {pi}. Using the submultiplicativity of the spectral norm and the fact that /bardbl|ψ/angbracketright/angbracketleftφ|⊗IB/bardbl/lessorequalslant1 when |ψ/angbracketright and|φ/angbracketrightare normalized, one can verify that /bardblC/bardbl/lessorequalslant1. Thus, for any normalized states |ψ/angbracketrightand|φ/angbracketright,|/angbracketleftψ|Q|φ/angbracketright|/lessorequalslant1. We thus conclude that /bardbl|Q/angbracketright/angbracketright/bardbl = kl β(t,s) must be smaller than or equal to 1. The same line of arguments shows that kr β(t,s)/lessorequalslant1. Recalling that kn β(t,s) by construction cannot be negative, we thus conclude 0/lessorequalslantkn β(t,s)/lessorequalslant1. (A18) We now use Eq. ( A18)i nE q .( A13) to obtain /bardbl∂t|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslantγ/summationdisplay m,n;α,β/integraldisplayt t0ds/vextendsingle/vextendsingleJmn αβ(t−s)/vextendsingle/vextendsingle. (A19) Evaluating the sum, using the results below Eq. ( A9), we obtain /bardbl∂t|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant4γ/integraldisplayt t0ds/bardblJ(t−s)/bardbl1, (A20) where /bardbl·/bardbl 1denotes the entrywise matrix 1-norm, such that for any matrix Mwith elements {Mαβ},/bardblM/bardbl1≡/summationtext αβ|Mαβ|.To obtain Eq. ( A20), we used the relation /bardblM/bardbl1=/bardbl M†/bardbl1, which follows from the definition above. Extending the lower limit of integration in Eq. ( A20)t o −∞ and changing integration variables, we finally obtain /bardbl∂t|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma10,/Gamma1 0≡4γ/integraldisplay∞ 0dt/bardblJ(t)/bardbl1. (A21) Thus, the energy scale /Gamma10sets an upper bound for the rate of bath-induced evolution in the system, /bardbl|∂tρ(t)/angbracketright/angbracketright/bardbl.T h i s timescale was also identified in Ref. [ 17] (see main text and Appendix Bfor further discussion). In Appendix Bwe further show that /Gamma10/lessorequalslant/Gamma1/2, where /Gamma1was given in Eq. ( 26)i nt h e main text [see Eq. ( 4) for the special case of a single noise channel]. Thus, recalling that |ρ(t)/angbracketright/angbracketrightcorresponds to ˜ ρ(t)i n the main text, we have shown that /bardbl∂t˜ρ(t)/bardbl/lessorequalslant/Gamma1/2. (A22) This was the result quoted in Sec. II Bof the main text. 4. Error induced by the Born approximation Until now, our derivation has been exact, with our only assumptions being that the bath is Gaussian, and that thesystem and bath were decoupled at some point t 0in the remote past (see Sec. II A in the main text). At this point, exact manipulations cannot take us further, and we thus need tomake our first approximation: the Born approximation. To make the Born approximation, we integrate the equation of motion for the evolution superoperator of the combinedsystem ∂ tˆU(t,s)=−iˆH(t)ˆU(t,s): ˆU(t,s)=1−i/integraldisplayt sdt/primeˆH(t/prime)ˆU(t/prime,s). (A23) By directly substituting this expression in for the factor of ˆU(t,s)i nE q .( A12), we obtain ∂t|ρ(t)/angbracketright/angbracketright = − γ/summationdisplay m,n;α,βνmνn/integraldisplayt t0dsˆXm α(t)Jmn αβ(t−s)/parenleftbigg ˆXn β(s)|ρ(s)/angbracketright/angbracketright −i/integraldisplayt sdt/prime/angbracketleft/angbracketleftIB|ˆH(t/prime)ˆU(t/prime,s)ˆXn β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright/parenrightbigg .(A24) For the first term in the parentheses above we used that ˆXn β(t) acts trivially on the bath, along with ˆU(s,t0)|ρ0/angbracketright/angbracketright|ρB/angbracketright/angbracketright = |ρSB(s)/angbracketright/angbracketrightand/angbracketleft/angbracketleftIB|ρSB(s)/angbracketright/angbracketright = |ρ(s)/angbracketright/angbracketright, such that /angbracketleft/angbracketleftIB|ˆXn β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright = ˆXn β(s)|ρ(s)/angbracketright/angbracketright. Next, we separate the two terms in the parentheses in Eq. ( A24). Referring to the second term in the resulting expression as |ξB(t)/angbracketright/angbracketright(we discuss this term in further detail below), we find ∂t|ρ(t)/angbracketright/angbracketright = − γ/integraldisplayt t0dsˆ/Delta1B(t,s)|ρ(s)/angbracketright/angbracketright + |ξB(t)/angbracketright/angbracketright, where ˆ/Delta1B(t,s)≡−γ/summationdisplay m,n;α,βνmνnˆXm α(t)ˆXn β(s)Jmn αβ(t−s). (A25)By applying the definitions of the quantities ˆXm α(t),νm, and Jmn αβ(t), one can verify that the first term on the right-hand side of Eq. ( A25) is identical to the master equation for |ρ(t)/angbracketright/angbracketright in the Born approximation [ 9] [see text above Eq. ( 9)i n the main text for the single-channel case]. Hence, the Bornapproximation is equivalent to neglecting the term |ξ B(t)/angbracketright/angbracketrightin the above, and we identify |ξB(t)/angbracketright/angbracketrightas the error induced by the Born approximation. Note that the Born-approximated masterequation for |ρ(t)/angbracketright/angbracketright[Eq. ( A25) with the correction |ξ B(t)/angbracketright/angbracketright neglected] can also be obtained through other approachesthan the one we use here. For example, this result may alsobe obtained using the Nakajima-Zwanzig equation (see, e.g.,Refs. [ 9,44]). We now seek a bound for the norm of |ξ B(t)/angbracketright/angbracketright, i.e., the norm of the error in ∂t|ρ(t)/angbracketright/angbracketrightinduced by the Born approximation. 115109-16UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) Matching Eqs. ( A24) and ( A25), we see that |ξB(t)/angbracketright/angbracketright = iγ/summationdisplay m,n;α,βνmνn/integraldisplayt t0dsJmn αβ(t−s)ˆXm α(t) ×/integraldisplayt sdt/prime/angbracketleft/angbracketleftIB|ˆH(t/prime)ˆU(t/prime,s)ˆXn β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright. (A26) To obtain a bound for /bardbl|ξB(t)/angbracketright/angbracketright/bardbl, we take the norm on both sides of Eq. ( A26) above. Using the triangle inequality and submultiplicativity of the superoperator norm, along with/bardblˆX m α(t)/bardbl=1, we find /bardbl|ξB(t)/angbracketright/angbracketright/bardbl/lessorequalslantγ/summationdisplay m,n;α,β/integraldisplayt t0ds|Jmn αβ(t−s)|/integraldisplayt sdt/primeqn β(t,t/prime,s), (A27) where qn β(t,t/prime,s)≡/bardbl /angbracketleft /angbracketleft IB|ˆH(t/prime)ˆU(t/prime,s)ˆXn β(s)ˆU(s,t0)|ρB/angbracketright/angbracketright|ρ0/angbracketright/angbracketright/bardbl. Following the same line of arguments that showed that the number kn β(t,s) in Appendix A3was bounded by 1, one can verify that 0/lessorequalslantqn β(t,t/prime,s)/lessorequalslant/Gamma10, (A28) where /Gamma10was defined in Eq. ( A21). Substituting this result into Eq. ( A27) and evaluating the integral over t/prime, we find /bardbl|ξB(t)/angbracketright/angbracketright/bardbl/lessorequalslant4γ/Gamma10/integraldisplayt t0ds/bardblJ(t−s)/bardbl1·|t−s|, (A29) where the matrix norm /bardbl·/bardbl 1was defined in Appendix A3. Extending the lower limit of integration to −∞ and using the definition of /Gamma10in Eq. ( A21), we obtain /bardbl|ξB(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma12 0τ0,τ 0≡/integraltext∞ 0dt t/bardblJ(t)/bardbl1/integraltext∞ 0dt/bardblJ(t)/bardbl1. (A30) The timescale τ0, which was also identified in Ref. [ 17], can be seen as a measure for the characteristic decay timescaleof correlations in the bath, and we expect it to typically becomparable to the timescale τfrom the main text (see Ap- pendix Band Sec. II Bin the main text for further discussion). Importantly, in Appendix Bwe show that /Gamma1 0τ0/lessorequalslant/Gamma1τ. Using the above results, along with /Gamma10/lessorequalslant/Gamma1/2 (see Ap- pendix A3), we conclude that /bardbl|ξB(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma12τ/2. (A31) Recalling that |ξB(t)/angbracketright/angbracketrightgives the correction to the Born- approximated master equation for the system [Eq. ( A25)], we conclude that the Born approximation induces an error in theexpression for ∂ t˜ρ(t) whose spectral norm is no greater than /Gamma12τ/2. 5. Error induced by the Markov approximation We now implement the Markov approximation, which is the second approximation necessary to derive the Bloch-Redfield equation. Below, we show that Markov approxi-mation induces an error in the expression for |∂ tρ(t)/angbracketright/angbracketright[i.e., ∂t˜ρ(t) in the main text] whose spectral norm is bounded by /Gamma12τ/2. This bound is identical to the error bound we obtainedfor the Born approximation in Appendix A4. In this sense, the Markov approximation is valid on an equivalent level ofapproximation as the Born approximation: the validity of oneapproximation by our arguments implies the validity of theother. To implement the Markov approximation, we insert |ρ(s)/angbracketright/angbracketright = |ρ(t)/angbracketright/angbracketright +[|ρ(s)/angbracketright/angbracketright − |ρ(t)/angbracketright/angbracketright] into Eq. ( A25), thereby obtaining ∂ t|ρ(t)/angbracketright/angbracketright =/integraldisplayt t0dsˆ/Delta1B(t,s)|ρ(t)/angbracketright/angbracketright + |ξM(t)/angbracketright/angbracketright + |ξB(t)/angbracketright/angbracketright, (A32) where |ξM(t)/angbracketright/angbracketright =/integraldisplayt t0dsˆ/Delta1B(t,s)[|ρ(s)/angbracketright/angbracketright − |ρ(t)/angbracketright/angbracketright]. (A33) We note that neglecting the terms |ξB(t)/angbracketright/angbracketrightand|ξM(t)/angbracketright/angbracketrightin Eq. ( A32) results in a Markovian master equation for the system. Recalling that |ξB(t)/angbracketright/angbracketrightarises from the Born approx- imation, we hence identify |ξM(t)/angbracketright/angbracketrightas the error induced by the Markov approximation. The Bloch-Redfield equation [ 9] [see Eq. ( 10) in the main text for the single-channel case] is obtained by neglecting these two terms, and subsequentlytaking the limit t 0→− ∞ , i.e., using our assumption that t0was in the remote past. In Appendix A6we discuss the physical justification for this assumption, and provide a boundfor the correction that arises when this limit is not taken. To obtain an upper bound for the error induced by the Markov approximation |ξ M(t)/angbracketright/angbracketright, we take the norm on both sides in Eq. ( A33) and use the triangle inequality. Recalling from Appendix A3that/bardbl∂t|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma10,w eh a v e /bardbl|ρ(s)/angbracketright/angbracketright − |ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma10|t−s|, where /Gamma10was defined in Eq. ( A21). Moreover, we note /bardblˆ/Delta1B(t,s)/bardbl/lessorequalslant4γ/bardblJ(t−s)/bardbl1; this can be shown using the triangle inequality in Eq. ( A25). Combining these inequalities, we find /bardbl|ξM(t)/angbracketright/angbracketright/bardbl/lessorequalslant4/Gamma10γ/integraldisplayt t0ds/bardblJ(t−s)/bardbl1|t−s|. (A34) Extending the lower limit of integration to −∞, and using the definitions of /Gamma10andτ0in Eqs. ( A21) and ( A30), we conclude that/bardbl|ξM(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma12 0τ0.Recalling that /Gamma12 0τ0/lessorequalslant/Gamma12τ/2 (see Appendixes A4andB), we thus find /bardbl|ξM(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma12τ/2. (A35) The result in Eq. ( A35) shows that the error in the expression for∂t|ρ(t)/angbracketright/angbracketright[corresponding to ∂t˜ρ(t) in the main text] induced by the Markov approximation |ξM(t)/angbracketright/angbracketright, has spectral norm no greater than /Gamma12τ/2, as we claimed. Based on the derivation above, we conclude that the density matrix of the system evolves according to the Markovianmaster equation ∂ t|ρ(t)/angbracketright/angbracketright =/integraldisplayt t0dsˆ/Delta1B(t,s)|ρ(t)/angbracketright/angbracketright + |ξ(t)/angbracketright/angbracketright, (A36) where ˆ/Delta1B(t,s)i sg i v e ni nE q .( A25), and |ξ(t)/angbracketright/angbracketright ≡ |ξB(t)/angbracketright/angbracketright + |ξM(t)/angbracketright/angbracketrightdenotes the error induced by the Born-Markov ap- proximation. From our results above that |ξB(t)/angbracketright/angbracketright,|ξM(t)/angbracketright/angbracketright/lessorequalslant /Gamma12τ/2, we hence conclude that the total error induced by the Markov and Born approximations is bounded by /Gamma12τ,a sw e claimed in the main text. 115109-17FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) 6. Transient correction from initialization at t0 As a final step in our derivation, here we show that when t0 is in the remote past, the error induced by extending t0to−∞ in Eq. ( A36) is negligible compared to the error induced by the Born-Markov approximation, |ξ(t)/angbracketright/angbracketright ∼O(/Gamma12τ). Specifically, we show that the spectral norm of this error is bounded by/Gamma1τ/(t−t 0), and hence is negligible when t−t0/greatermuch/Gamma1−1.B y setting t0→− ∞ in Eq. ( A36), we obtain ∂t|ρ(t)/angbracketright/angbracketright = ˆDR(t)|ρ(t)/angbracketright/angbracketright + |ξ(t)/angbracketright/angbracketright, (A37) where ˆDR(t)≡/integraltextt −∞dsˆ/Delta1B(t,s). This is the Bloch-Redfield equation [ 9] [including the error induced by the Born-Markov approximation, see Eq. ( 10) in the main text for the single- channel case]. To establish a bound for the error induced by setting t0→ −∞ in Eq. ( A36), we rewrite Eq. ( A36)a sf o l l o w s : ∂t|ρ(t)/angbracketright/angbracketright =[ˆDR(t)+ˆDT(t)]|ρ(t)/angbracketright/angbracketright + |ξ(t)/angbracketright/angbracketright, (A38) where ˆDT(t)≡−/integraltextt0 −∞dsˆ/Delta1B(t,s).This term can be seen as the transient correction to the BR equation induced by theabsence of system-bath correlations in our assumed initialstate at time t 0,|ρSB(t0)/angbracketright/angbracketright = |ρ0/angbracketright/angbracketright|ρB/angbracketright/angbracketright. This “correction” is thus an artifact of our choice of initial state (see Sec. II A). Below, we show that /bardblˆDT(t)|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma1τ/(t−t0). Thus, when t−t0/greatermuch/Gamma1−1, i.e., after a time long enough for weak correlations to be established between the system and the bath,the transient correction ˆD T(t) is negligible compared to the bound we obtained for the error induced by the Born-Markovapproximation /Gamma1 2τ. As a result, the BR equation [Eq. ( A37)] accurately describes the system’s evolution in this limit. To show that /bardblˆDT(t)|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma1τ/(t−t0), we consider the superoperator norm of ˆDT(t)[ s e eE q .( A3)]. Noting that /bardblˆ/Delta1B(t,s)/bardbl/lessorequalslant4γ/bardblJ(t−s)/bardbl1[this can be shown using the triangle inequality in Eq. ( A25)], we find /bardblˆDT(t)/bardbl/lessorequalslant4γ/integraldisplayt0 −∞ds/bardblJ(t−s)/bardbl1. (A39) Using the fact that t>t0, we have that |t−s|/greaterorequalslant|t−t0|for alls/lessorequalslantt0. Thus, /bardblˆDT(t)/bardbl/lessorequalslant4γ/integraldisplayt −∞ds/bardblJ(t−s)/bardbl1|t−s| |t−t0|. (A40) Changing variables of integration and using the definitions of /Gamma10andτ0in Eqs. ( A21) and ( A30), we conclude /bardblˆDT(t)/bardbl/lessorequalslantτ0/Gamma10 t−t0. (A41) Using the fact that that /Gamma10τ0/lessorequalslant/Gamma1τ(see Appendix B) along with the definition of the superoperator norm, we concludethat/bardblˆD T(t)|ρ(t)/angbracketright/angbracketright/bardbl/lessorequalslant/Gamma1τ t−t0,as we claimed. APPENDIX B: RELATIONSHIP BETWEEN BATH TIMESCALES In this Appendix we discuss the relationship between the bath timescales /Gamma1andτintroduced in Eq. ( 26)o ft h em a i n text, and the timescales /Gamma1−1 0,τ0identified in Eqs. ( A21) and(A30) of Appendix A: /Gamma10=4γ/integraldisplay∞ 0dt/bardblJ(t)/bardbl1,τ 0=/integraltext∞ 0dt t/bardblJ(t)/bardbl1/integraltext∞ 0dt/bardblJ(t)/bardbl1,(B1) where J(t) denotes the matrix-valued bath correlation func- tion (see Sec. III C), and /bardblM/bardbl1≡/summationtext αβ|Mαβ|refers to the entrywise matrix 1-norm of a matrix Mwith elements {Mαβ} (see Appendix A). The above timescales /Gamma1−1 0andτ0were also identified in Ref. [ 17]. Like the timescales /Gamma1−1andτ,/Gamma1−1 0andτ0serve as measures for the characteristic timescales for bath-inducedevolution, and the decay bath correlations, respectively. Incontrast to /Gamma1 −1andτ, which are defined in terms of the “jump correlator” g(t) [see Eqs. ( 3) and ( 25)o ft h em a i nt e x t ] , the timescales /Gamma1−1 0andτ−1 0above are defined directly from the bath correlation function J(t). However, as discussed in Sec. II B and demonstrated in Fig. 1, we expect these two distinct ways of characterizing the timescales of the bath togive comparable results in most cases. Further supporting thispoint, in this Appendix, we rigorously prove the followinginequalities between the timescales {/Gamma1 −1 0,τ0}and{/Gamma1,τ}: /Gamma10/lessorequalslant/Gamma1/2 and /Gamma10τ0/lessorequalslant/Gamma1τ. (B2) These inequalities were used in Appendix A. We first show that /Gamma10/lessorequalslant/Gamma1/2. We note from the defini- tion of J(t)i nE q .( 24) that J(t)=J†(−t). Using /bardblM/bardbl1= /bardblM†/bardbl1, we thus have /bardblJ(t)/bardbl1=/bardbl J(−t)/bardbl1. Using this result in Eq. ( B1), we find /Gamma10=2γ/integraldisplay∞ −∞dt/bardblJ(t)/bardbl1. (B3) To obtain a bound for /bardblJ(t)/bardbl1we note that J(t) is related to the jump correlator g(t) through the convolution J(t)=/integraltext∞ −∞dsg(t−s)g(s).This result follows from the definition of g(t) in Sec. III C, and is the multichannel generalization of the result quoted above Eq. ( 15) in the main text. Using the triangle inequality, we obtain /bardblJ(t)/bardbl1/lessorequalslant/integraldisplay∞ −∞ds/bardblg(t−s)g(s)/bardbl1. (B4) To rewrite the integrand above, we now prove that, for any two matrices Aand B, /bardblAB/bardbl1/lessorequalslant/bardblA†/bardbl2,1/bardblB/bardbl2,1, (B5) where the matrix norm /bardbl·/bardbl 2,1was defined below Eq. ( 26) in the main text: /bardblM/bardbl2,1≡/summationtext β(/summationtext α|Mαβ|2)1/2for a matrix Mwith elements {Mαβ}. To prove Eq. ( B5), we recall that /bardblAB/bardbl1=/summationtext αβγ|AαβBβγ|.We consider the sum over the in- dexβfirst. Using the Cauchy-Schwartz inequality, we find /summationdisplay β|AαβBβγ|/lessorequalslant⎛ ⎝/summationdisplay β|Aαβ|2⎞ ⎠1/2⎛ ⎝/summationdisplay β/prime|Bβ/primeγ|2⎞ ⎠1/2 .(B6) Using this inequality in the expression for /bardblAB/bardbl1in Eq. ( B5), we conclude /bardblAB/bardbl1/lessorequalslantcAcB,where cA=/summationtext α(/summationtext β|Aαβ|2)1/2 andcB=/summationtext γ(/summationtext β/prime|Bβ/primeγ|2)1/2. Recalling the definition of the norm/bardbl·/bardbl 2,1, we identify cA=/bardbl A†/bardbl2,1andcB=/bardbl B/bardbl2,1. Thus Eq. ( B5) holds. 115109-18UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) Combining Eqs. ( B3)–(B5) we obtain /Gamma10/lessorequalslant2γ/integraldisplay∞ −∞ds/integraldisplay∞ −∞dt/bardblg†(t−s)/bardbl2,1/bardblg(s)/bardbl2,1. (B7) The Hermiticity of g(ω)i m p l i e st h a t g(t)=g†(−t) (see Sec. III C). Using this result in the above and shifting the variables of integration, we obtain /Gamma10/lessorequalslant2γ/bracketleftbigg/integraldisplay∞ −∞dt/bardblg(t)/bardbl2,1/bracketrightbigg2 . (B8) Comparing this result with the definition of /Gamma1in Eq. ( 26)i n the main text, we conclude that /Gamma10/lessorequalslant/Gamma1/2. We now prove the second inequality in Eq. ( B2),/Gamma10τ0/lessorequalslant /Gamma1τ. Using the fact that /bardblJ(t)/bardbl1=/bardbl J(−t)/bardbl1[see text above Eq. ( B3)], along with the definitions of /Gamma10andτ0in Eq. ( B1), we have /Gamma10τ0=2γ/integraldisplay∞ −∞dt|t|·/bardblJ(t)/bardbl1. (B9) Using Eqs. ( B4) and ( B5) along with g(t)=g†(−t) [see text above Eq. ( B8)], we obtain /Gamma10τ0/lessorequalslant2γ/integraldisplay∞ −∞dt/integraldisplay∞ −∞ds|t|k(s−t)k(s), (B10) where we introduced the shorthand k(t)≡/bardbl g(t)/bardbl2,1.U s i n g that|t|/lessorequalslant|s−t|+|s|and shifting variables of integration, one can then verify that /Gamma10τ0/lessorequalslant2γ/integraltext∞ −∞dt/prime/integraltext∞ −∞ds(|t/prime|+ |s|)k(t/prime)k(s).Exploiting the symmetry of this expression un- der exchange of t/primeands, we find /Gamma10τ0/lessorequalslant4γ/integraldisplay∞ −∞dt/prime/integraldisplay∞ −∞ds|t/prime|k(t/prime)k(s). (B11) Recalling that k(t)≡/bardbl g(t)/bardbl2,1, and comparing with the defi- nitions of /Gamma1andτin Eq. ( 26), we identify the right-hand side above as /Gamma1τ. Thus, /Gamma10τ0/lessorequalslant/Gamma1τ, and Eq. ( B2) holds. This was what we wanted to prove and concludes this Appendix. APPENDIX C: DERIV ATION OF THE ULE In this Appendix we rigorously derive the universal Lind- blad equation (ULE) in the interaction picture [Eq. ( 27)o ft h e main text]. As in Appendixes AandB, we consider here the case of arbitrary system-bath coupling Hint, such that the system and bath are connected through multiple noise channels (seeSec. IIin the main text). In the main text we heuristically de- rived the ULE for the case of a single noise channel [Eq. ( 21)]. This result is a special case of the more general result that werigorously prove here, and hence this Appendix also serves asa proof of Eq. ( 21). As discussed in the main text, the ULE [Eq. ( 27)] holds for a modified density matrix ρ /prime(t) whose spectral norm distance to the exact density matrix ˜ ρ(t) (in the interaction picture) remains bounded by /Gamma1τat all times. Here the bath timescales /Gamma1−1andτwere defined in Eq. ( 26) in the main text. In the Markovian limit, /Gamma1τ/lessmuch1, which is required for the ULE to be valid (see Secs. II A 2 andIII A in the main text), the modified density matrix ρ/primeis thus nearly identical to the true density matrix ˜ ρ, and accurately describes the state of the system.Our derivation below proceeds in three steps. In Appendix C1we define the modified density matrix ρ/prime(t)[ s e eE q .( C8)] and prove that its spectral norm distance to ˜ ρ(t) remains bounded by /Gamma1τat all times. Subsequently, in Appendix C2 we show that ρ/prime(t) evolves according to the master equation ∂tρ/prime(t)=L(t)[ρ/prime(t)]+ξ/prime(t), L(t)≡/integraldisplay∞ −∞ds/integraldisplay∞ −∞ds/primeF(s,t,s/prime), (C1) where the spectral norm of ξ/prime(t) is bounded by 2 /Gamma12τ, and, for any operator A, we have defined F(s,t,s/prime)[A]=γ/summationdisplay α,β,λθ(s−s/prime)(gαλ(s−t)gλβ(t−s/prime) ×[˜Xα(s),A˜Xβ(s/prime)]+g∗ αλ(s−t)g∗ λβ(t−s/prime) ×[˜Xβ(s/prime)A,˜Xα(s)]). (C2) Note that the definitions above generalize the superoperators L(t) andF(s,t,s/prime) in Sec. III A to cases with multiple noise channels [ 45]. As the third and final step of our derivation, in Appendix C3we show that the superoperator L(t) takes the Lindblad form in Eq. ( 27). Thereby we reach the goal of this Appendix, proving that ρ/prime(t) evolves according to the Lindblad-form master equation in Eq. ( 27)o ft h em a i nt e x t . 1. Modified density matrix Here we define the modified density matrix ρ/prime(t), and prove that /bardblρ/prime(t)−˜ρ(t)/bardbl/lessorequalslant/Gamma1τat all times. Our approach is to identify a transformation ρ/prime(t)≡[1+M(t)]˜ρ(t) such that, if ˜ ρ(t) satisfies the Bloch-Redfield equation [Eq. ( A36)], then, up to an error of order /Gamma12τ,ρ/prime(t) evolves according to Eq. ( C1) (which can then be expressed in Lindblad form). We will bound the norm distance between ρ/prime(t) and ˜ρ(t)u s i n gt h e explicit form of this transformation. To motivate our definition of ρ/prime(t), we note that the multichannel Bloch-Redfield (BR) equation [Eq. ( A36)i n Appendix A] can be written as ∂t˜ρ(t)=/integraldisplay∞ −∞ds/prime/integraldisplay∞ −∞dsF(t,s,s/prime)[˜ρ(t)]+ξ(t),(C3) where ξ(t) denotes the error induced by the Born-Markov approximation, with norm bounded by /Gamma12τ. The expression above generalizes the single-channel result in Eq. ( 15)i n the main text to the case of multiple noise channels. It isconvenient to rewrite the right-hand side above in terms ofthe superoperator G(t,s)≡/integraldisplay ∞ −∞ds/primeF(t,s,s/prime). (C4) Specifically, we express the BR equation [Eq. ( C3)] as ∂t˜ρ(t)=/integraldisplay∞ −∞dsG(t,s)[˜ρ(t)]+ξ(t). (C5) Similarly, we may rewrite Eq. ( C1) (our target equation of motion for the modified density matrix ρ/prime)a s ∂tρ/prime(t)=/integraldisplay∞ −∞dsG(s,t)[ρ/prime(t)]+ξ/prime(t). (C6) 115109-19FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) We will identify the precise form of the correction ξ/prime(t)i nt h e derivation below. Note that, when neglecting the corrections ξ(t) andξ/prime(t), the only difference between Eqs. ( C5) and ( C6) is the order of the arguments in the superoperator G. The modified density matrix ρ/prime(t) is obtained from a (time-local) linear operation on ˜ρ(t) that transforms Eq. ( C5) into Eq. ( C6). As we show in Appendix C2below, such a linear transformation is generated by the superoperator [1 +M(t)], where M(t)≡/integraldisplay∞ tds/integraldisplayt −∞ds/prime[G(s,s/prime)−G(s/prime,s)]. (C7) Specifically, we define ρ/prime(t) as follows: ρ/prime(t)=[1+M(t)][˜ρ(t)]. (C8) In Appendix C2we show that ρ/prime(t), as defined above, evolves according to Eq. ( C6). Before proving this, we show here that ρ/primedeviates from ˜ ρby a correction whose spectral norm is bounded by /Gamma1τat all times: /bardblρ/prime(t)−˜ρ(t)/bardbl/lessorequalslant/Gamma1τ. To show that /bardblρ/prime(t)−˜ρ(t)/bardbl/lessorequalslant/Gamma1τ, we prove below that, for any operator A, /bardblM(t)[A]/bardbl/lessorequalslant/Gamma1τ/bardblA/bardbl. (C9) By the definition of ρ/prime(t)i nE q .( C8), this result in particular implies that /bardblρ/prime(t)−˜ρ(t)/bardbl/lessorequalslant/Gamma1τ, since /bardbl˜ρ(t)/bardbl/lessorequalslant1. We will also use Eq. ( C9) for other purposes in Appendix C2. To prove Eq. ( C9), we use the triangle inequality in Eq. ( C7) to obtain /bardblM(t)[A]/bardbl/lessorequalslant/integraldisplay∞ tds/integraldisplayt −∞ds/prime{/bardblG(s,s/prime)[A]/bardbl+ /bardblG(s/prime,s)[A]/bardbl}. (C10) Using the triangle inequality in Eq. ( C4), we have /bardblG(t,s)[A]/bardbl/lessorequalslant/integraldisplay∞ −∞ds/prime/bardblF(t,s,s/prime)[A]/bardbl. (C11) From the definition of Fin Eq. ( C2), using the triangle inequality and the submultiplicativity of the spectral norm,one can verify that /bardblF(t,s,s /prime)[A]/bardbl/lessorequalslant4γ/bardblg(t−s)g(s− s/prime)/bardbl1/bardblA/bardblθ(t−s),where the 1-matrix norm /bardbl·/bardbl 1is defined in Appendix A3, and we used that /bardbl˜Xα(t)/bardbl/lessorequalslant1. In Appendix B we established that /bardblg(t)g(s)/bardbl1/lessorequalslant/bardblg(−t)/bardbl2,1/bardblg(s)/bardbl2,1, where the matrix norm /bardbl·/bardbl 2,1is defined in Sec. III C. Thus, /bardblF(t,s,s/prime)[A]/bardbl/lessorequalslant4γ/bardblg(s−t)/bardbl2,1/bardblg(s−s/prime)/bardbl2,1θ(t−s/prime)/bardblA/bardbl. Using this result in Eq. ( C11), we find /bardblG(t,s)[A]/bardbl/lessorequalslantG(t−s)/bardblA/bardbl, (C12) where G(t)≡4γ/bardblg(−t)/bardbl2,1/integraldisplayt −∞ds/bardblg(−s)/bardbl2,1. (C13) Using Eq. ( C12)i nE q .( C10) we obtain /bardblM(t)[A]/bardbl/lessorequalslant/integraldisplay∞ tds/integraldisplayt −∞ds/prime[G(s−s/prime)+G(s/prime−s)]/bardblA/bardbl. (C14) To rewrite Eq. ( C14), we note that, for any function f(s),/integraltext∞ tds/integraltextt −∞ds/primef(s−s/prime)=/integraltext∞ 0dssf (s) (this can be verifiedby change of integration variables). Using this result in Eq. ( C14), we obtain /bardblM(t)[A]/bardbl/lessorequalslant/integraldisplay∞ −∞dt|t|G(t)/bardblA/bardbl. (C15) We now seek a convenient bound for G(t). Extending the upper limit of integration in Eq. ( C13)t o∞, and using the definition of /Gamma1in Eq. ( 26), we obtain G(t)/lessorequalslant/radicalbig 4γ/Gamma1/bardblg(−t)/bardbl2,1. (C16) Using this result in Eq. ( C15) along with the definitions of /Gamma1andτin Eq. ( 26), we conclude that the right-hand side of Eq. ( C15) is bounded by /Gamma1τ/bardblA/bardbl. Thus, Eq. ( C9) holds. By the arguments below Eq. ( C9), we hence conclude /bardblρ/prime(t)− ˜ρ(t)/bardbl/lessorequalslant/Gamma1τ. This was what we wanted to show. 2. Master equation for modified density matrix We now show that ρ/prime(t), as defined in Eq. ( C8), evolves according to the master equation in Eq. ( C1). To establish this result, we explicitly take the time derivative of ρ/prime(t)i n Eq. ( C8), obtaining ∂tρ/prime(t)=∂t˜ρ(t)+∂tM(t)[˜ρ(t)]+M(t)[∂t˜ρ(t)],(C17) where we exploited the linear dependence of M(t)[˜ρ]o n˜ρ. We consider the second term first in the above. Using thedefinition of M(t)i nE q .( C7), one can verify by explicit computation that ∂ tM(t)=/integraldisplay∞ −∞dsG(s,t)−/integraldisplay∞ −∞dsG(t,s), (C18) Inserting this result into Eq. ( C17), and using Eq. ( C5) along withL(t)=/integraltext∞ −∞dsG(s,t) [see Eqs. ( C1) and ( C4)], we obtain ∂tρ/prime(t)=L(t)[˜ρ(t)]+ξ/prime 1(t)+ξ(t), (C19) where ξ/prime 1(t)≡M(t)[∂t˜ρ(t)]. Noting that /bardbl∂t˜ρ(t)/bardbl/lessorequalslant/Gamma1/2 (see Sec. II B), and that /bardblM(t)[A]/bardbl/lessorequalslant/Gamma1τ/bardblA/bardbl[Eq. ( C9)], we conclude that /bardblξ/prime 1(t)/bardbl/lessorequalslant/Gamma12τ/2. As the final step in our derivation, we show that we may replace the argument ˜ ρ(t)o fL(t)i nE q .( C19)b yρ/prime(t), at the cost of a correction ξ/prime 2(t) whose spectral norm is bounded by /Gamma12τ/2. To show this, we exploit the linearity of L(t) to write L(t)[˜ρ(t)]=L(t)[ρ/prime(t)]+L(t)[/Delta1ρ(t)], (C20) where /Delta1ρ(t)≡˜ρ(t)−ρ/prime(t). We now show that /bardblL(t)[A]/bardbl/lessorequalslant /Gamma1/bardblA/bardbl/2,such that the second term in Eq. ( C20) is bounded by/Gamma12τ/2 [recall that /bardbl/Delta1ρ(t)/bardbl/lessorequalslant/Gamma1τ, see Appendix C1]. To prove this result, we use L(t)=/integraltext∞ −∞dsG(s,t) along with Eq. ( C12) to obtain /bardblL(t)[A]/bardbl/lessorequalslant/integraldisplay∞ −∞dt G(t)/bardblA/bardbl, (C21) where G(t) was defined in Eq. ( C13). By explicit computa- tion, using the definition of /Gamma1in Eq. ( 26), one can verify that/integraltext∞ −∞dt G(t)/lessorequalslant/Gamma1/2. Thus /bardblL(t)[A]/bardbl/lessorequalslant/Gamma1/bardblA/bardbl/2. We con- clude that L(t)[˜ρ(t)]=L(t)[ρ/prime(t)]+ξ/prime 2(t), (C22) where /bardblξ/prime 2(t)/bardbl/lessorequalslant/Gamma12τ/2. 115109-20UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) Using the relation in Eq. ( C22)i nE q .( C19), we con- clude that ρ/prime(t), as defined in Eq. ( C8), evolves according to Eq. ( C1), with ξ/prime(t)=ξ/prime 1(t)+ξ/prime 2(t)+ξ(t). Since the spectral norm of ξ(t) is bounded by /Gamma12τ, while the spectral norms ofξ/prime 1(t) and ξ/prime 2(t) are both bounded by /Gamma12τ/2, we con- clude that /bardblξ/prime(t)/bardbl/lessorequalslant2/Gamma12τ. Note that the bound for the error ξ/prime 1(t)+ξ/prime 2(t) induced by the modified Markov approximation described above is identical to the bound for the error inducedby the Born-Markov approximation ξ(t).3. Lindblad form of master equation As the final step in our derivation, we now show that the right-hand side of the master equation for ρ/primein Eq. ( C1) is identical to the right-hand side of the universal Lindbladequation [Eq. ( 27) in the main text]. To prove this result, we first modify the expression for the superoperator L(t) that was defined in Eqs. ( C1) and ( C2). By decomposing the step function in Eq. ( C2) into its symmetric and antisymmetric components: θ(s−s /prime)=1 2[1+sgn(s−s/prime)], we find L(t)=LS(t)+LA(t), Li(t)≡/integraldisplay∞ −∞ds/prime/integraldisplay∞ −∞dsFi(s,t,s/prime),i={S,A}. (C23) For any density matrix ρ, we have defined FS(s,t,s/prime)[ρ]=−γ 2/summationdisplay α,β,λ[gαλ(s−t)˜Xα(s),gλβ(t−s/prime)˜Xβ(s/prime)ρ]+H.c., (C24) FA(s,t,s/prime)[ρ]=−γ 2/summationdisplay α,βφαβ(s−t,s/prime−t)[˜Xα(s),˜Xβ(s/prime)ρ]+H.c., (C25) where {φαβ(s,t)}denote the matrix elements of the N×N matrix φ(t,s)≡g(t)g(−s)sgn( t−s) that was defined below Eq. ( 29). Below, we show that the superoperator LSin Eq. ( C23) generates the dissipative component of the ULE, while LAgenerates the Lamb shift. We consider the term LSfirst. By direct computation, one can verify that LS(t)[ρ]=−1 2/summationdisplay λ[˜L† λ(t),˜Lλ(t)ρ]+H.c., (C26) where ˜Lλ(t) denotes the interaction picture jump operator defined in Eq. ( 28) in the main text. Here we used that ˜L† λ(t)=√γ/integraltext∞ −∞ds/primegαλ(s−t)˜Xα(s),which follows from the relation g(t)=g†(−t), along with the definition of ˜Lλ(t). Writing out all terms in Eq. ( C26), we obtain LS(t)[ρ]=/summationdisplay λ/bracketleftbigg ˜Lλ(t)ρ˜L† λ(t)−1 2{˜L† λ(t)˜Lλ(t),ρ}/bracketrightbigg .(C27) Hence LS(t) is in the Lindblad form and generates the dissi- pative part of the ULE. Next, we consider the term LAin Eq. ( C23). By expanding the commutator in Eq. ( C25), we obtain FA(s,t,s/prime)[ρ]= T1(s,t,s/prime)−T2(s,t,s/prime)+H.c., where T1(s,t,s/prime)≡γ 2/summationdisplay α,βφαβ(s−t,s/prime−t)˜Xβ(s/prime)ρ˜Xα(s), T2(s,t,s/prime)≡γ 2/summationdisplay α,βφαβ(s−t,s/prime−t)˜Xα(s)˜Xβ(s/prime)ρ. We now show that T1(s,t,s/prime)=−T† 1(s/prime,t,s). This im- plies that the net contribution to LA(t)f r o m T1and its Hermitian conjugate vanishes:/integraltext∞ −∞ds/prime/integraltext∞ −∞ds[T1(s,t,s/prime)+ T† 1(s,t,s/prime)]=0, and hence [see Eq. ( C23)] LA(t)[ρ]=−/integraldisplay∞ −∞ds/primeds[T2(s,t,s/prime)+T† 2(s,t,s/prime)].(C28)To prove that T1(s,t,s/prime)=−T† 1(s/prime,t,s), we note that φ(t,s)=−φ†(s,t) [this follows from the definition of φbe- low Eq. ( C25) along with g(t)=g†(−t)]. Using this identity in the definition of T1above, we find, after a relabeling of summation variables, T1(s,t,s/prime)=−γ 2/summationdisplay α,βφ∗ αβ(s/prime−t,s−t)˜Xα(s/prime)ρ˜Xβ(s). We identify the right-hand side as −T† 1(s/prime,t,s) (see defini- tion of T1above). Thus, T1(s,t,s/prime)=−T† 1(s/prime,t,s), and hence Eq. ( C28) holds. We finally note that/integraltext∞ −∞ds/prime/integraltext∞ −∞dsT 2(s,t,s/prime)=i˜/Lambda1(t)ρ, where ˜/Lambda1(t)=γ 2i/integraldisplay∞ −∞ds/integraldisplay∞ −∞ds/prime/summationdisplay αβ˜Xα(s)˜Xβ(s/prime)φαβ(s−t,s/prime−t) (C29) denotes the Lamb shift from Eq. ( 23) in the main text. Hence the antisymmetric component LAgenerates the Lamb shift in the ULE, as we claimed: LA(t)[ρ]=−i[˜/Lambda1(t),ρ]. (C30) Combining Eqs. ( C23), (C27), and ( C30), we obtain L(t)[ρ]=− i[˜/Lambda1(t),˜ρ(t)] +/summationdisplay λ/bracketleftbigg ˜Lλ(t)˜ρ(t)˜L† λ(t)−1 2{˜L† λ(t)˜Lλ(t),˜ρ(t)}/bracketrightbigg . (C31) Thus, the superoperator L(t) is in the Lindblad form. Using this result in Eq. ( C1), we conclude that the modified density matrix ρ/prime(t), as defined in Eq. ( C8), evolves according to the ULE in Eq. ( 27), with the correction term ξ/prime(t) being bounded by 2/Gamma12τ. Proving this was the goal of this Appendix. 115109-21FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) APPENDIX D: LAMB SHIFT FOR STATIC HAMILTONIANS In this Appendix we derive the expression for the Lamb shift in Eq. ( 34) of the main text, which holds for cases where the system Hamiltonian HSis time independent. Equation ( 34) is most conveniently derived in the interac- tion picture. We recall from Eq. ( 23) that, in the interaction picture, the Lamb shift is given by ˜/Lambda1(t)=γ 2i/integraldisplay∞ −∞ds/prime/integraldisplay∞ −∞ds/summationdisplay αβ˜Xα(s)˜Xβ(s/prime)φαβ(s−t,s/prime−t), (D1) where {φαβ(s,s/prime)}denote the elements of the matrix φ(t,s)≡ g(t)g(−s)sgn( t−s),and g(t) denotes the matrix-valued jump correlator defined in Eq. ( 25) in the main text. As a first step in our derivation, we decompose the time- evolved system operator ˜Xα(t) in terms of the eigenstates {|n/angbracketright} and energies {En}of the system Hamiltonian HS: ˜Xα(t)=/summationdisplay m,nX(α) mne−iEnmt|m/angbracketright/angbracketleftn|, (D2) where, as in the main text, X(α) mn≡/angbracketleftm|Xα|n/angbracketright, while Enm≡ En−Em. Inserting Eq. ( D2) into Eq. ( D1), shifting variables of integration, and using Elm+Enl=Enmalong with the definition of φαβ(t,s), we obtain ˜/Lambda1(t)=/summationdisplay mnl;αβX(α) mlX(β) lnfαβ(Elm,Enl)e−iEnmt|m/angbracketright/angbracketleftn|, (D3) where {fαβ(p,q)}denote the elements of the matrix f(p,q)=γ 2i/integraldisplay∞ −∞dt/integraldisplay∞ −∞dssgn(s−t)e−i(pt+qs)g(t)g(−s). (D4) Note that f(p,q) is the Fourier transform of φ(t,s), up to a constant prefactor. We now express the jump correlator in terms of its Fourier transform: g(t)=/integraltext∞ −∞dωe−iωtg(ω). After factoring out the integrals over tands, we obtain f(p,q)=γ 2i/integraldisplay∞ −∞dω/integraldisplay∞ −∞dω/primeg(ω)g(ω/prime)k(p+ω,q−ω/prime), (D5) where k(p,q)≡/integraltext∞ −∞ds/prime/integraltext∞ −∞dssgn(s−s/prime)e−i(ps+qs/prime).By ex- plicit computation, one can verify that k(p,q)=−4πiδ(p+q)Re/parenleftbigg1 p−i0+/parenrightbigg , (D6) where δ(x) denotes the Dirac delta function. Using this result in Eq. ( D5), integrating out ω/prime, and subsequently shifting variables of integration, we find f(p,q)=−2πγ/integraldisplay∞ −∞dωg(ω−p)g(ω+q)Re/parenleftbigg1 ω−i0+/parenrightbigg . We can rewrite this to the following: f(p,q)=−2πγP/integraldisplay∞ −∞dωg(ω−p)g(ω+q) ω, (D7) where Pdenotes the Cauchy principal value. As a final step in our derivation, we use the expression for ˜/Lambda1in Eqs. ( D7) and ( D3) to compute the Lamb shift in the Schrödinger picture /Lambda1. We recall that /Lambda1=U(t)˜/Lambda1(t)U†(t)[see below Eq. ( 31) in the main text], where U(t)=e−iHSt denotes the unitary evolution operator generated by the system Hamiltonian HS. Noting that for time-independent system Hamiltonians, U(t)|m/angbracketright/angbracketleftn|U†(t)=eiEnmt|m/angbracketright/angbracketleftn|, this implies that /Lambda1=/summationdisplay m,n,l/summationdisplay αβX(α) mnX(β) nlfαβ(Emn,Enl)|m/angbracketright/angbracketleftl|, (D8) where the matrix f(p,q) is defined in Eq. ( D7). This was the result quoted in the main text. We note that the above line of arguments can be general- ized to periodically driven systems with a few modifications.However, for the sake of brevity, we do not provide such aderivation here. APPENDIX E: CONDITIONS FOR SLOW TIME DEPENDENCE In this Appendix we identify the conditions on the time dependence of the system Hamiltonian HS(t), under which the Schrödinger picture jump operators {Lλ(t)}and Lamb shift /Lambda1(t) can be computed from the eigenstates and energies of the instantaneous Hamiltonian HS(t), using Eq. ( 33)o ft h em a i n text. To show this explicitly for the jump operator Lλ(t), we note that U(t,s)=e−i(t−s)HS(t)+O[v(t−s)2], where v= sups/lessorequalslantt/prime/lessorequalslantt/bardbl∂tHS(t/prime)/bardbldenotes the maximal rate of change of HS[46]. Using this form of U(t,s)i nE q .( 32)o ft h em a i n text, along with the results from Sec. IV A , we conclude that Lλ(t) can be computed from the spectrum and eigen- states of HS(t) through Eq. ( 34), up to a correction of order√ /Gamma1v(τ2)2[note from Eq. ( 1) that the jump operators have units of (Energy)1/2]. Here ( τ2)2≡/integraltext∞ −∞dt/bardblg(t)t2/bardbl2,1/N, where N≡√/Gamma1/4γ(see Sec. III C in the main text for the definition of the matrix norm /bardbl·/bardbl 2,1). The timescale τ2gives the square root of the second moment of the normalizeddistribution /bardblg(t)/bardbl 2,1/N[see definition of /Gamma1in Eq. ( 26)], and we expect it to typically be comparable to the first moment τ. Thus, when HS(t) changes slowly on the correlation timescale of the bath τ, i.e.,∂tHS(t)(τ2)2/lessmuch1, the jump operators of the system {Lλ(t)}can be computed from the instantaneous Hamiltonian HS(t)u s i n gE q .( 33). A similar result holds for t h eL a m bs h i f t /Lambda1(t). APPENDIX F: CALCULATION OF TRANSPORT PROPERTIES Here we define the heat and magnetization currents com- puted for the nonequilibrium spin chain in Sec. V. The average heat current ¯IEcan be identified from the equation of motion for the energy in the spin chain: ∂t/angbracketleftE(t)/angbracketright=Tr[H∂tρ]. Using the universal Lindblad equation [Eqs. ( 30) and ( 31)], along with [ H,H]=0, we find ∂t/angbracketleftE(t)/angbracketright=/summationtext λ/angbracketleftI(λ) E/angbracketright, where I(λ) E= L† λHLλ−1 2{L† λLλ,H}.F o rλ=1,2, we identify I(λ) Eas the heat current flowing into the system from bath λ. Since the energy of the chain is bounded, the time-averaged heat currentfrom bath 1 must exactly compensate the average heat currentfrom bath 2. Hence, we identify ¯I Eas the time-averaged expectation value of −I(1) E. The magnetization current ¯IMcan be obtained similarly from the equation of motion for themagnetization M,u s i n g[ H,M]=0. 115109-22UNIVERSAL LINDBLAD EQUATION FOR OPEN QUANTUM … PHYSICAL REVIEW B 102, 115109 (2020) [1] M. Scully and M. S. Zubairy, Quantum Optics (Akademie, Berlin, 1996). [2] N. G. Van Kampen, Stochastic Processes in Physics and Chem- istry (North Holland, Amsterdam, 2007). [3] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2010). [4] H. Feshbach, Ann. Phys. 5, 357 (1958) . [5] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997) . [6] M. Bourennane, M. Eibl, S. Gaertner, C. Kurtsiefer, A. Cabello, and H. Weinfurter, P h y s .R e v .L e t t . 92, 107901 (2004) . [7] F. Verstraete, M. M. Wolf, and J. Ignacio Cirac, Nat. Phys. 5, 633 (2009) . [8] S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nat. Phys. 7, 971 (2011) . [9] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002). [10] C. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2004). [11] S. Nakajima, Prog. Theor. Phys. 20, 948 (1958) . [12] R. Zwanzig, J. Chem. Phys. 33, 1338 (1960) . [13] R. K. Wangsness and F. Bloch, Phys. Rev. 89, 728 (1953) . [14] A. G. Redfield, Adv. Magn. Opt. Res. 1, 1 (1965) . [15] E. B. Davies, Commun. Math. Phys. 39, 91 (1974) . [16] C. Majenz, T. Albash, H.-P. Breuer, and D. A. Lidar, Phys. Rev. A88, 012103 (2013) . [17] E. Mozgunov and D. Lidar, Quantum 4, 227 (2020) . [18] G. Lindblad, Commun. Math. Phys. 48, 119 (1976) . [19] V . Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976) . [20] J. Dalibard, Y . Castin, and K. Mølmer, P h y s .R e v .L e t t . 68, 580 (1992) . [21] R. Dum, P. Zoller, and H. Ritsch, P h y s .R e v .A 45, 4879 (1992) . [22] H. Carmichael, An Open Systems Approach to Quantum Optics (Springer, Berlin, 1993). [23] G. Kiršanskas, M. Franckié, and A. Wacker, Phys. Rev. B 97, 035432 (2018) . [24] E. Kleinherbers, N. Szpak, J. König, and R. Schützhold, Phys. Rev. B 101, 125131 (2020) . [25] F. Nathan, Ph.D. thesis, University of Copenhagen, 2018.[26] Here the spectral norm is defined as the maximum singular value norm: /bardblX/bardbl=sup ψ,φ|/angbracketleftψ|X|φ/angbracketright|, where the supremum is taken over all normalized states. We note that, to apply ourframework to a system where Xmay be unbounded, some additional physically justified truncation of the Hilbert space isneeded. [27] The square root is introduced for convenience, since the “bare” system-bath coupling√ γonly appears in even powers in the master equations we obtain. As a result of this parametrization,Bhas dimensions of [Energy] 1/2. These units of Bare a natural choice when the bath has a continuous energy spectrum [ 9], such as is the case for the Ohmic bath in Sec. II B. [28] To be precise, a general Lindblad form allows the time deriva- tive of ρto be given by a sum of multiple terms on the form in Eq. ( 1), where the Lamb shift and each jump operator may be time dependent.[29] In Appendix Awe obtain a stricter bound, namely /bardblξ(t)/bardbl/lessorequalslant 2/Gamma10τ0,w h e r e /Gamma1−1 0andτ0are distinct, but typically compara- ble, measures for the characteristic timescales of bath-inducedevolution and bath correlations. These quantities are definedin Appendix B, where we also show that 2 /Gamma1 2 0τ0/lessorequalslant/Gamma12τ.T h e timescales /Gamma1−1 0andτ0were also identified in Ref. [ 17], where analogous bounds for the trace norm of the correction ξ(t) were derived in terms of these timescales. While we could have usedthe timescales /Gamma1 −1 0andτ0to express the bound for /bardblξ(t)/bardblin the BR equation, the steps leading to the ULE induce errors whosebounds we can only express in terms of /Gamma1andτ.T os i m p l i f yt h e discussion, in the main text we therefore use the (looser) bound/Gamma1 2τin Eq. ( 12). [30] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Phys. Rev. Lett. 110, 050403 (2013) . [31] S. Deffner and S. Campbell, J. Phys. A: Math. Theor. 50, 453001 (2017) . [32] Specifically, the relative weight of the jump correlator |g(t)| beyond a particular time t,/integraltext∞ tdt/prime|g(t/prime)|/C, is bounded by τ/t [this is straightforward to verify from Eq. ( 4)]. Note also that for many physically relevant cases, such as for the Ohmic bathdiscussed below, the jump correlator decays much faster thanby this power law (often exponentially). In particular, if thebath spectral function is smooth in a way such that, for somen,I n≡1√ 2π/integraltext∞ −∞dω|(∂ω)n√J(ω)|is a finite number, one can show that |g(t)|is always bounded by In/|t|n. This can be straightforwardly shown by using the definition of g(t) from Eq. (3), along with the triangle inequality. [33] In principle, the error induced by neglecting ξ/prime(t)i nt h eU L E may accumulate over time and result in inaccurate values ofρ /prime(t)f o r t/greaterorsimilar(/Gamma12τ)−1. While the bound we obtained for /bardblξ/prime(t)/bardbl can be used to infer rigorous results for the error to ρ/prime(t) (using, e.g., the spectral gap of the Liouvillian), such a discussion isbeyond the scope of this paper. We expect it is often a goodstrategy to simply compare the correction bound 2 /Gamma1 2τto the other relevant energy scales of the physical model and from thiscomparison determine whether the correction ξ /prime(t) can safely be neglected using physical arguments. We expect this approachwill include a much wider range of models than those allowedby rigorous mathematical results. [34] C. Ding, D. Zhou, X. He, and H. Zha, in Proceedings of the 23rd International Conference on Machine Learning(Association for Computing Machinery, New York, 2006),pp. 281–288. [35] Here the Floquet states and quasienergies define the unique set of stationary solutions to the Schrödinger equation of the form|ψ(t)/angbracketright=e −iεnt|φn(t)/angbracketright,s e eR e f .[ 42] for more details. [36] F. Nathan, I. Martin, and G. Refael, Phys. Rev. B 99, 094311 (2019) . [37] R. Blümel, A. Buchleitner, R. Graham, L. Sirko, U. Smilansky, and H. Walther, P h y s .R e v .A 44, 4521 (1991) . [38] S. Kohler, J. Lehmann, and P. Hänggi, Phys. Rep. 406, 379 (2005) . [39] D. W. Hone, R. Ketzmerick, and W. Kohn, P h y s .R e v .E 79, 051129 (2009) . [40] Here we assume that /Gamma1is finite; this condition is already required for the universal Lindblad equation to be valid. [41] M. I. K. Munk, J. Schulenborg, R. Egger, and K. Flensberg, arXiv:2004.02123 . 115109-23FREDERIK NATHAN AND MARK S. RUDNER PHYSICAL REVIEW B 102, 115109 (2020) [42] J. H. Shirley, Phys. Rev. 138, B979 (1965) . [43] V . V . Albert, B. Bradlyn, M. Fraas, and L. Jiang, P h y s .R e v .X 6, 041031 (2016) . [44] M. W. Evans, P. P. Grigolini, and G. P. Parravicini, Memory Function Approaches to Stochastic Problems in CondensedMatter , V ol. 62 of Advances in Chemical Physics (John Wiley & Sons, New York, 1985). [45] See Appendix A1for a general discussion of superoperators. [46] This can be proven by going to the interaction picture with respect to HS(t), and using /bardblHS(s)−HS(t)/bardbl/lessorequalslant|s−t|v. 115109-24

PhysRevB.102.041120.pdf

PHYSICAL REVIEW B 102, 041120(R) (2020) Rapid Communications Disorder recovers the Wiedemann-Franz law in the metallic phase of VO 2 Lei Jin,1,2Steven E. Zeltmann,1,3Hwan Sung Choe ,1,2Huili Liu,1,2Frances I. Allen ,1,3 Andrew M. Minor,1,3and Junqiao Wu1,2,* 1Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3National Center for Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 28 May 2020; revised 3 July 2020; accepted 6 July 2020; published 22 July 2020) At temperatures higher than 341 K, vanadium dioxide (VO 2) is a strongly correlated metal with resistivity exceeding the Mott-Ioffe-Regel limit. Its electronic thermal conductivity is lower than that predicted by theWiedemann-Franz (WF) law, and can be explained by nonquasiparticle transport where heat and chargecurrents follow separate diffusive modes. In contradiction, the Wiedemann-Franz law is a direct consequenceof quasiparticle transport where charge carriers are elastically scattered. In this work, we enhance elasticelectron scattering in VO 2by introducing atomic disorder with ion irradiation. A gradual and eventually full recovery of the WF law is observed at high defect densities. This observation provides an example that connectshydrodynamic quasiparticle transport to nonquasiparticle transport in metallic systems. DOI: 10.1103/PhysRevB.102.041120 The Wiedemann-Franz (WF) law predicts that the elec- tronic thermal conductivity of a metal ( κe) is proportional to the product of its electrical conductivity ( σ) and absolute temperature ( T). The proportionality constant in this relation is the Lorenz number ( L), and is equal to the Sommerfeld value, L0=π2 3(kB e)2, when the conducting charge carriers can be described as long-lived quasiparticles experiencingpredominantly elastic scattering. These conditions are wellsatisfied in normal metals above their Debye temperatures,such as in Cu and Al, where quasielastic electron-phonon orelastic electron-impurity scattering dominates [ 1]. A deviation ofLaway from L 0usually implies the existence of inelastic electron scattering processes, such as small-angle electron-phonon scattering at low temperatures, electron-electron scat-tering in the hydrodynamic regime, or a complete failure ofthe quasiparticle model where non-Fermi-liquid physics mustbe considered [ 2–8]. An example of the latter is the metallic (M) phase of vanadium dioxide (VO 2)[3], a strongly cor- related electron material with a well-known metal-insulatortransition (MIT) at T=341 K. The Mphase of VO 2is a “bad” metal [ 9,10], where the quasiparticle Drude model fails and conduction electrons are scattered by inelastic yetmomentum-conserving electron-electron interactions. It hasbeen experimentally observed that charge and heat currentsin the material are transported according to separate diffusivemodes and that the Lorenz number is suppressed by nearlyone order of magnitude below the Sommerfeld value [ 3]. On the other hand, the WF law has been shown to be restored in Fermi liquids by adding elastic, electron-impurityscattering to suppress the originally dominating inelastic,electron-phonon scattering [ 11]. This is consistent with the fact that to experimentally observe the breakdown of the WF *wuj@berkeley.edulaw in the hydrodynamic regime, samples must be ultrapurewith a negligible density of impurities [ 5,12]. It is intriguing to extend this defect effect of quasiparticle transport to “bad”metals with nonquasiparticle transport. In this work, we showthat even in a system without long-lived quasiparticles, the ar-tificially introduced point defects scatter electrons elasticallyto restore the Lorenz number toward the Sommerfeld value,the hallmark of quasiparticle transport. Here, VO 2nanowires were grown using the vapor transport method published previously [ 13]. The nanowires are single crystalline and have a rectangular cross section of thicknessranging from tens of nanometers to a few micrometers. Pointdefects were introduced into the VO 2nanowires by He ion irradiation using a Zeiss Orion NanoFab He ion microscopeoperated at 30 kV . The point defects (vacancies and intersti-tials of V and O) were efficiently created in the VO 2through the nuclear stopping process. The VO 2nanowires chosen for this study have thicknesses less than the projected range ofthe He ions of 160 nm, as predicted by Monte Carlo simula-tions, using the Stopping and Range of Ions in Matter ( SRIM ) program [ 14]. The SRIM simulations show that most ( >80%) of the He ions transmit the VO 2nanowires leaving behind a uniform distribution of point defects. The ratio of the numberof trapped He ions to the number of irradiation-inducedvacancies in the nanowires is less than 1%. During the irra-diation runs, care was taken to use low ion currents ( <1p A ) in order to minimize any thermal effects and hydrocarbondeposition. The thermal and electrical conductivity of the VO 2 nanowires were measured using suspended micropad devicesas shown in Fig. 1(a). The device consists of two SiN xpads that are suspended by long and flexible thin SiN xarms, with Pt coils patterned onto both micropads to functionas a microscale resistive heater and thermometer. A VO 2 nanowire was dry transferred using a micromanipulator ontothe two pads to bridge them. Focused ion beam (FIB) induced 2469-9950/2020/102(4)/041120(6) 041120-1 ©2020 American Physical SocietyLEI JIN et al. PHYSICAL REVIEW B 102, 041120(R) (2020) FIG. 1. Electrical and thermal conductivity of VO 2nanowires across the metal-insulator transition. (a) Optical image of two sus-pended micropads bridged by a VO 2nanowire. Scale bar is 10 μm. Inset: SEM image of the nanowire bonded to the electrode by FIB- induced Pt deposition. Scale bar is 200 nm. (b) Electrical and (c) ther-mal conductivity of a 140-nm-thick VO 2nanowire irradiated with He ions at accumulated doses of 0, 5, 9, and 15 ×1015ions/cm2. The data collected during both heating (filled symbols) and cooling(open symbols) are shown for each dose. deposition was then used to deposit a small amount of Pt onto the nanowire at four locations to bond it to the underlying fourelectrodes. A scanning electron microscope (SEM) image ofone of these Pt deposits is shown in the inset of Fig. 1(a).B o t h electrical and thermal contact resistance are found to be negli-gible after the Pt deposition [ 3]. Heat currents flow through the VO 2nanowire from the heating pad to the sensing pad, raising the temperature of the latter, which was used to determine thethermal conductivity of the nanowire after careful calibration.The measurements were taken in high vacuum with a pressureless than 10 −6Torr. Thermal conductance due to radiative and convective heat transfer is negligible compared to that of heatconduction through the nanowire. Electrical conductivity was measured simultaneously with the four-probe configurationusing the electrodes prepatterned onto the suspended pads,which also allow measurement of the Seebeck coefficient ( S). Further details on the fabrication process and measurementsusing the suspended micropad devices can be found else-where [ 3,15]. The experimental setup described maximally eliminates external strain, which strongly affects the MIT ofVO 2[16,17], and more importantly, both the heat and charge currents are measured along the same direction of the sample(the [100] direction in monoclinic, namely, the [001] directionin the rutile coordinates [ 13]), which is critically important for the evaluation of the Lorenz number. The electrical and thermal conductivities of the irradiated VO 2nanowires are shown in Figs. 1(b) and 1(c), respec- tively. Although many more nanowires were measured (shown later), the data in Figs. 1(b) and 1(c) were collected from a single nanowire undergoing multiple steps of irradiation.The measurements were taken immediately after the He ionirradiation to minimize sample degradation such as oxidationof the defected VO 2. The MIT persists in VO 2even after 1.5×1016ions/cm2dose of irradiation, although the sharp- ness of the MIT and conductivity ratio between the metallicand insulating ( I) phases are reduced. The temperature of the MIT ( T MIT) is reduced from 341 K for the pristine state to near 300 K for the heavily irradiated state. In addition, the TMIT values obtained from the electrical and thermal measurements are in very good agreement. It can be seen that the electricalconductivity ( σ)o ft h e Iphase increases while that of the Mphase decreases as a result of the irradiation. This trend is consistent with many other works investigating effects of oxygen vacancies in VO 2[18–23]. As O vacancies are created by irradiation and are expected to be electron donors in VO 2, σincreases in the Iphase. In the Mphase, such electron doping has negligible effect; instead, scattering of conductiveelectrons by the newly created O vacancies and other point de-fects reduces σ(discussed in further detail below). In contrast toσ, the thermal conductivity ( κ) is reduced by irradiation in both the IandMphases. After irradiation, the slope of κ in the Iphase over temperature becomes less negative. This implies stronger phonon-impurity scattering arising with moreirradiation-induced defects, as the impurity scattering rate isalmost Tindependent compared to the umklapp scattering. The surface scattering in transport measurements is negligibleas the mean free path (MFP) of both electron and phonon (discussed later) in VO 2is much less than the thicknesses of our nanowires. To determine the irradiation damage in the VO 2nanowires, selected area electron diffraction (SAED) patterns were ac-quired using an FEI Themis transmission electron microscope(TEM) operated at 300 kV . For this, single nanowires wereplaced onto lacey carbon TEM supports and targeted areaswere irradiated to the desired doses. Figure 2(a) shows a low-resolution TEM image of a nanowire with two regionsmarked top and bottom that were irradiated using doses of5×10 16and 1×1015ions/cm2, respectively (the higher dose corresponds to the maximum irradiation dose investigated inthis study). The central portion of the nanowire was not irradi-ated. The SAED results corresponding to all three regions arepresented in Figures 2(b)–2(d) and show constantly sharp and 041120-2DISORDER RECOVERS THE WIEDEMANN-FRANZ LAW IN … PHYSICAL REVIEW B 102, 041120(R) (2020) FIG. 2. (a) Low-resolution TEM image showing a VO 2nanowire irradiated with He ions to doses of 1 ×1015and 5×1016ions/cm2 in the regions marked at the bottom and top. respectively. The central region remains pristine. SAED patterns are shown for the (b) pristine, (c) 1 ×1015ions/cm2irradiated, and (d) 5 ×1016ions/cm2 irradiated regions of the VO 2nanowire. The survival of the sharp SAED pattern to the highest irradiation dose indicates that the He ion irradiation of VO 2damages the lattice by generating point defects, without degrading the crystallinity. clean patterns in all cases, indicating that the VO 2nanowires remain fully single crystalline for all the irradiation dosesused in this work. This confirms that the irradiation createspoint defects without generating extended defects or causingamorphization. Moreover, SRIM simulations show that the defect density is very small (around 0.45 at.%) in irradiatedVO 2nanowires at the highest dose (5 ×1016ions/cm2). In Ref. [ 3], using inelastic x-ray scattering combined with first-principles calculation, Lee et al. showed that the lattice component of thermal conductivity ( κl)f o rV O 2remains almost unchanged when the material transitions between theIand the Mphases at the MIT. That is, κ I l≈κM lnear TMIT. This is because, although the Mphase has a more anharmonic lattice [ 24] and hence shorter phonon mean free path for phonon modes <∼10 meV , the Iphase has many more optical phonon modes with very low group velocities due to itsdoubled unit cell [ 25], lowering the MFPs. As a result, the average MFPs for the MandIphases are 5.18 and 5.86 nm, respectively, very close to each other. The approximation κ I l≈ κM limproves further when boundary and impurity scattering of phonons are included via Matthiessen’s rule [ 3], because such scattering of thermal phonons is not drastically differentin the Mand Iphases. This effect provides a convenient method to evaluate the Lorenz number for the Mphase of VO 2, because according to κ=κl+κeand the fact that κe is nearly zero in the Iphase, κM eatTMITcan be estimated by κM e=κM−κM l≈κM−κI. That is, the jump in the measured FIG. 3. Normalized Lorenz number L/L0[as defined in Eq. ( 1)] for three VO 2nanowires of different thicknesses (labeled) for dif- ferent irradiation doses (in units of ions /cm2) as a function of σ for the Mphase just above the MIT, showing the recovery of L upon introduction of defects toward the Sommerfeld value ( L0= 2.44×10−8W/Omega1K−2) of a normal metal. The error range of the L/L0data points is about ±0.1. The MIR limit of σis indicated with an arrow, proving that all data exceed the σMIR. total thermal conductivity at TMITis primarily attributed to the electronic thermal conductivity in the Mphase. We will adopt this method to estimate κM ein all the VO 2samples, and use it to evaluate the Lorenz number near TMITaccording to L L0=κM e L0σMT≈κM−κI L0σMT. (1) Figure 3shows the measured L/L0values at the respec- tive TMIT’s for three different VO 2nanowires of different thicknesses, each of which was irradiated with doses rangingfrom zero to 1 .5×10 16ions/cm2. Two data points are shown for each measurement, calculated from data collected in theheating and cooling steps, respectively. L/L 0all increase as a function of irradiation dose for all the nanowires. Moreover,plotting L/L 0as a function of σMatTMITreveals a gener- ally linear trend for all the data, regardless of the differentsamples and different doses. L/L 0approaches 1, namely, the WF law is fully recovered with L=L0, at the highest irradiation dose (1 .5×1016ions/cm2). Interestingly, this dose is also the maximum dose under which the MIT can survive:Irradiation at higher doses completely destroys the MIT (i.e.,no longer showing an abrupt change in electrical conductivity,not shown here), even though according to the TEM results thesample is still crystalline. The coincidence of L=L 0with the disappearance of the MIT points to a fundamental connectionbetween the violation of the WF law and electron correlationin VO 2. TheMphase of VO 2hosts electrical conduction that is be- yond the Mott-Ioffe-Regel (MIR) resistivity limit [ 26] derived from the Heisenberg uncertainty principle for momentum andposition [ 27]. In electronic transport carried by quasiparticles, the MIR limit requires that the MFP of quasiparticles is 041120-3LEI JIN et al. PHYSICAL REVIEW B 102, 041120(R) (2020) FIG. 4. Effects of impurity scattering in irradiated VO 2. (a) The electronic figure of merit ( S2/L)a t TMITdecreases with L/L0.T h e nonquasiparticle transport in the Mphase of VO 2manifests in both the unusually high S2/Lvalue as well as the lower-than-unity L/L0 value, in contrast to normal metals such as Cu. (b) M-phase Seebeck coefficient of a 140-nm-thick VO 2nanowire after irradiation to different doses, compared with that of W-doped VO 2nanowires adopted from Lee et al. [3]. The upward shift in the Seebeck coefficient upon irradiation reflects increasingly stronger scattering of itinerant electrons by ionized impurities. longer than their de Broglie wavelength. Long-lived quasipar- ticles cannot exist in systems with resistivity higher than theMIR limit, and non-Fermi-liquid physics must be considered.Resistivity linear in temperature beyond the MIR limit hasbeen reported in VO 2[9,10]. Given the electron concentration ofn∼1×1023cm−3obtained by Hall-effect measurements from bulk single-crystalline VO 2[28], we estimate the MIR limit of M-phase VO 2to beσMIR∼7×105S/mu s i n gt h e formula given by Hartnoll [ 27].σMof all VO 2nanowires are beyond (i.e., lower than) the MIR limit, as shown in Fig. 3, clear evidence of nonquasiparticle electronic transport. For nonquasiparticle transport, it has been theoretically shown that Lcan be much smaller than L0, because in this case the charge and heat currents are carried with separate diffusivemodes, and the ratio of their conductivities has no reason tobe equal to L 0[4,27]. Indeed, as shown in Fig. 3,L/L0is measured to be much lower than 1 in pristine VO 2, consistent with the previous report by Lee et al. [3]. Another parameter that indicates the lack of quasiparticle transport is that thedimensionless electronic figure of merit, S 2/L=S2σMT/κM e, is unusually large. The parameter S2/Lis theorized [ 1]t ob e very low for quasiparticle transport in a Fermi liquid, as Sis suppressed by the factor kBT/EF. Indeed, it is on the order of 10−4for conventional metals such as Cu [ 29]. In contrast, in the metallic phase of VO 2, values of S2/Lare all between 10−2and 10−1, as shown in Fig. 4(a). In the framework of quasiparticle transport, charge currents can only be degraded in momentum-nonconserving scatteringprocesses of charge carriers, whereas electronic heat currentscan also be degraded by energy-nonconserving (i.e., inelastic)scattering of charge carriers [ 6,30]. Therefore, the WF law can be violated with L/L 0<1 at low temperatures where inelastic electron-phonon scattering becomes significant [ 30]. The breakdown of the WF law has also been reported inquasiparticle transport in hydrodynamic Fermi liquids [ 6,12]. In the hydrodynamic regime, electron-electron interactionsdominate, and the Lorenz number is lowered. This is becausethe charge current is not relaxed by the electron-electroninteractions (as they are momentum conserving) whereas the heat current is (because this can be inelastic). For these twoscenarios of quasiparticle transport where L<L 0, it has been shown that disorder raises Ltoward L0and restores the WF law [ 11,12,30], because the MFP of both charge and heat transport are simultaneously and proportionally degraded bythe additional, momentum-nonconserving, impurity scatteringof electrons. However, such theory of disorder effects in nonquasipar- ticle transport has not been established [ 12]. Despite many fundamental differences, strong electron-electron interactionsexist in both quasiparticle transport of hydrodynamic Fermiliquids and nonquasiparticle transport of non-Fermi liquids.We thus “borrow” the results obtained when treating disorderin hydrodynamic Fermi liquids [ 12] to discuss defect scat- tering in nonquasiparticle transport. In this case, it has beenshown that the Lorenz ratio is approximated by L L0≈/Gamma1 /Gamma1+γ, (2) where γis the electron-electron scattering rate, and /Gamma1is the electron-defect scattering rate. We also adopt the relaxationtime approximation and relate the scattering rate to electricalconductivity as σ M∝τ, where τis the relaxation time and is the reciprocal of the scattering rate. It is straightforward toshow that L L0≈1−σM σMpure, (3) where σM pureis the ideal electrical conductivity in defect-free samples. As shown in Fig. 3,L/L0has a nearly linear depen- dence on σMconsistent with Eq. ( 3). Therefore, the disorder indeed restores the WF law, and L/L0can be viewed as a measure of the fraction of disorder scattering in this “bad”metal. It is noted that the average separation between neighboring point defects from the He ion irradiation is about a fewnanometers according to simulations using the SRIM program. 041120-4DISORDER RECOVERS THE WIEDEMANN-FRANZ LAW IN … PHYSICAL REVIEW B 102, 041120(R) (2020) This is much longer than the MFP of charge carriers in the Mphase of pristine VO 2, which is about 1 ∼2 Å estimated assuming the Drude model. It is surprising that such sparselydistributed point defects can significantly reduce σofM- phase VO 2, by more than 50% (Fig. 3). The MFP resulting from defect scattering is inversely proportional to not onlythe density of defects, but also to the scattering cross section.It has been reported that atomic impurities show a largerscattering cross section area in “bad” metals than in normalmetals [ 31]. The unusually large electron-defect scattering cross section in a “bad” metal may only be explained bythe correlated nature of the interacting electrons [ 32]. Con- sequently, the significant reduction in σ Mby irradiation im- plies anomalously strong electron-defect interactions in the M phase of VO 2. Further evidence of significant change in the electron scat- tering mechanism by defects is the behavior of the Seebeckcoefficient. Derived from the linearized Boltzmann transportequation for parabolic bands, the Seebeck coefficient of ametallic system can be written as |S|=k B eπ2 3kBT EF/parenleftbigg3 2+r/parenrightbigg , (4) where a power law energy dependence for the relaxation time is assumed: τ(E)∝Er, and ris the power index [ 33–35]. For elastic scattering, rvaries from −1/2 for acoustic phonon scattering to 3 /2 for ionized impurity scattering. For inelastic scattering [ 34], the power law energy dependence of τfails and ris not a constant. As Fig. 4(b) shows, the values ofSagree very well with Lee et al. [3] for pristine VO 2, but increase for irradiated samples while retaining the lineardependence on T. This is in stark contrast to the behavior ofSfor W-doped VO 2measured by Lee et al. In the lattercase, Scollapses onto the same Tdependence as that of the pristine VO 2, suggesting that W dopants do not change the electron scattering mechanism in VO 2. In the absence of a defect theory for nonquasiparticle transport, we discuss thebehavior of Sin the framework of Eq. ( 4) in the relaxation time approximation. In Fig. 4(b), the overall upward shifting in S(T) after irradiation implies higher rvalues, which is consis- tent with the increased importance of defects to provide moreionized impurity scattering in the electron scattering process. In conclusion, we show empirical validity of a quasiparticle transport model in a system where long-lived quasiparti-cles are clearly absent. The evidence is restoration of theelectronic thermal conductivity to the quasiparticle value ina metallic system where it is originally suppressed by theabsence of quasiparticles. We introduce atomic-level pointdefects as elastic scattering centers for conduction electronsusing energetic particle irradiation. The increased weight ofelastic scattering fully recovers the Wiedemann-Franz lawat high defect densities. This finding bridges the decoupledelectrothermal transport in the nonquasiparticle regime to acoupled one in the more conventional, quasiparticle regime.In addition, ion irradiation is shown to be an effective tool fornot only modulating the conductivities, but also controllingthe conduction mechanism of electronic and thermal currents. This work was supported by U.S. NSF Grant No. DMR- 1608899. The device fabrication part was supported by theCenter for Energy Efficient Electronics Science (NSF AwardNo. 0939514). Work at the Molecular Foundry was supportedby the Office of Science, Office of Basic Energy Sciences,of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. 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PhysRevB.79.014510.pdf

Bistability in voltage-biased normal-metal/insulator/superconductor/insulator/normal-metal structures I. Snyman1,2and Yu. V . Nazarov3 1National Institute for Theoretical Physics, Private Bag X1, 7602 Matieland, South Africa 2Instituut-Lorentz, Universiteit Leiden, P .O. Box 9506, 2300 RA Leiden, The Netherlands 3Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands /H20849Received 27 August 2008; revised manuscript received 29 October 2008; published 15 January 2009 /H20850 As a generic example of a voltage-driven superconducting structure, we study a short superconductor connected to normal leads by means of low transparency tunnel junctions with a voltage bias Vbetween the leads. The superconducting order parameter /H9004is to be determined self-consistently. We study the stationary states as well as the dynamics after a perturbation. The system is an example of a dissipative driven nonlinearsystem. Such systems generically have stationary solutions that are multivalued functions of the system pa-rameters. It was discovered several decades ago that superconductors outside equilibrium conform to thisgeneral rule in that the order parameter as a function of driving may be multivalued. The main differencebetween these previous studies and the present work is the different relaxation mechanisms involved. This doesnot change the fact that there can be several stationary states at a given voltage. It can however affect theirstability as well as the dynamics after a perturbation. We find a region in parameter space where there are twostable stationary states at a given voltage. These bistable states are distinguished by distinct values of thesuperconducting order parameter and of the current between the leads. We have evaluated /H208491/H20850the multivalued superconducting order parameter /H9004at given V,/H208492/H20850the current between the leads at a given V, and /H208493/H20850the critical voltage at which superconductivity in the island ceases. With regards to dynamics, we find numericalevidence that only the stationary states are stable and that no complicated nonstationary regime can be inducedby changing the voltage. This result is somewhat unexpected and by no means trivial, given the fact that thesystem is driven out of equilibrium. The response to a change in the voltage is always gradual even in theregime where changing the interaction strength induces rapid anharmonic oscillations of the order parameter. DOI: 10.1103/PhysRevB.79.014510 PACS number /H20849s/H20850: 74.40. /H11001k, 74.78.Fk, 74.25.Fy, 74.78.Na I. INTRODUCTION Electron transport devices combining superconducting /H20849S/H20850, insulating /H20849I/H20850, and normal-metal /H20849N/H20850elements are known as superconducting heterostructures. Often such heterostruc-tures are more than the sum of their parts. 1,2Phenomena that are not present in bulk S,I,o r Nsystems appear when a device contains junction between these components. The fol-lowing examples are well known: /H208491/H20850the conductance of a high transparency NSjunction does not equal the conduc- tance of the normal metal on its own, as one might naivelyexpect. If the normal metal is free of impurities, the conduc-tance is higher than that of the normal metal. 3This surprising effect is due to a process known as Andreev reflection.4Dur- ing Andreev reflection at an NSinterface, an electron im- pinging on the interface from the Nside is reflected back as a hole while a Cooper pair propagates away from the inter-face on the Sside. /H208492/H20850In Josephson junctions, the simplest of which is perhaps the SISheterostructure, 5a dc current can flow at zero-bias voltage. This happens when the supercon-ducting phase difference across the junction is nonzero. 6 The above examples can be understood in terms of equi- librium properties of the heterostructure. When a supercon-ducting device is perturbed outside equilibrium, yet moreinteresting effects can occur, 7for instance, oscillations under stationary nonequilibrium conditions. An elementary ex-ample: if a Josephson junction is biased with a dc /H20849i.e., fixed /H20850 voltage, an ac /H20849i.e., oscillating /H20850current flows through the junction. 6Another example of the kind has been investigatedin the context of cold Fermi gases in optical traps. In these systems, the interaction between atoms can be tuned andchanged by means of a so-called Feshbach resonance. If theinteraction is attractive, the gas forms a BCS condensate.Recent studies 8,9have considered what happens if the value of the attractive pairing interaction is changed abruptly. Itwas discovered that, depending on the ratio between the ini-tial and final values of the interaction strength, the conden-sate order parameter can perform anharmonic oscillationsthat do not decay in time. The initial motivation for the research presented in this paper came from the study of Keizer et al. , 10where the au- thors investigated the suppression of the superconducting or-der parameter by a voltage applied to a superconductingwire. It was assumed that /H9004remains stationary. However, this assumption does not seem well justified: the stationaryvoltage could induce periodic oscillations of /H20841/H9004/H20841or even richer chaotic dynamics. Thus prompted, we wanted to ad-dress the validity of this assumption for a decidedly simplerNISIN structure, namely, a short superconductor connected to normal leads by means of tunnel junctions. The structureis biased with a voltage V. We require that /H208491/H20850the dominant energy relaxation mechanism in the superconductor is the tunneling of elec-trons to the leads, and /H208492/H20850spatial variations in the supercon- ducting order parameter inside the superconductor are negli-gible. To meet the first requirement, the superconductor musthave dimensions smaller than the inelastic-scattering lengthof quasiparticles. This is not an unrealistic requirement givenPHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 1098-0121/2009/79 /H208491/H20850/014510 /H2084913/H20850 ©2009 The American Physical Society 014510-1current experimental techniques. To meet the second require- ment, the superconductor should contain impurities or havean irregular shape so that the electron wave functions of theisolated island are isotropic on the scale of the superconduct-ing coherence length. 11Furthermore, the tunnel junctions connecting it to the leads should have a bigger normal-stateresistance than that of the superconductor proper. In thiscase, opening up the system by connecting leads does notreintroduce spatial anisotropy of wave functions inside theisland. The study of NISIN structures has a long history. 12,13Our study complements several previous studies.10,14,15These dealt with quasi-one-dimensional superconducting wires be-tween normal leads. Setups where either the superconductorwas impurity free or the transparency of the NSinterfaces was high were considered. For these setups, spatial varia-tions in the order parameter, specifically the spatial gradientof the superconducting phase, can be large. Including these spatial variations in the description of the superconductorsignificantly complicates matters. Hence these studies fo-cused on numerical calculations and assumed that the super-conducting order parameter and all other quantities of inter-est were stationary. It should also be mentioned thatasymmetric couplings, where the superconductor is coupledmore strongly to one lead than the other, did not receivedetailed analysis. The only asymmetric setup considered con-sisted of one interface with tunable transparency and theother perfectly transparent. 15One of the main conclusions of these studies is that, if the bias voltage is large enough, thesystem switches to the normal state. Some evidence for abistable region, where, depending on the history of the sys-tem, either the superconducting or the normal state can occurat a given voltage, was reported. 10 The absence of spatial variations in the system we study allows us to perform analytical calculations, provided weassume stationarity. Results are obtained for an arbitrary ra-tio of the coupling strengths to the leads. We derive transcen-dental equations relating the superconducting order param-eter to the bias voltage and derive an explicit formula for thecurrent between the leads. As mentioned, the assumption ofstationarity is however not a priori justified. As was seen in the examples mentioned at the beginning of this introduction,nonequilibrium conditions in superconductors often go handin hand with nonstationary behavior of observable quantities.Indeed, the NISIN junction that we study is a nonlinear sys- tem subjected to a driving force /H20849and to damping /H20850. Nonlin- earity here means that the dynamical equations for one-particle Green’s functions are not linear in the Green’sfunctions. This is due to the existence of a nonzero super-conducting order parameter. The driving force is provided bythe voltage /H20849and the damping by tunneling of electrons from the island into the leads /H20850. Nonlinear driven systems /H20849think of the nonlinear pendulum /H20850often have chaotic dynamics. The assumption of stationarity would miss this. We thereforesupplement our analytical calculation with numerical calcu-lations that study the dynamics in real time. Our main results are the following: the stationary states that we found analytically are stable. Furthermore, there is aparameter region where two different stationary states arestable at the same voltage. /H20849This multivaluedness of the orderparameter is by no means a new phenomenon. It is a com- mon feature of superconductors outside equilibrium. 16,17/H20850For a symmetric coupling to the source and drain leads, one ofthe two states we find is superconducting /H20849characterized by a nonzero order parameter /H20850and the other is normal. Since we are in the regime of high tunnel barriers, at a given voltage,the superconducting island allows less current to flow be-tween the leads than the island in the normal state. 3This current is a directly measurable quantity and allows one todistinguish between superconducting and normal states. Forsome asymmetric couplings however, both the stable statesare superconducting. We have calculated the current thatflows between the leads at a given voltage and at arbitraryasymmetry of the coupling to the two leads. We find that thevalue of the current also allows one to distinguish betweendifferent stable superconducting states at a given voltage. The time-dependent calculations revealed that, once the bias voltage becomes constant in time, the system alwaysrelaxes into one of the stationary states. Nonstationary be-havior of physical quantities always decays in time, unlike inthe case of a dc-biased Josephson junction. /H20849Despite it being a nonlinear system, a superconductor driven by a voltage istherefore fundamentally different from a nonlinear pendulumdriven by an external force. /H20850If the bias voltage is changed slowly, an initial stationary state evolves adiabatically. Bychanging the voltage slowly we have observed the expectedhysteresis associated with the existence of two stable statesat some voltages. The study presented here is complementary to earlier studies of bulk nonequilibrium superconductors. There, thesuperconducting order parameter is very sensitive to the de-tails of the quasiparticle distribution function. 18Owing to long inelastic relaxation times in the bulk superconductors,driving by either microwave radiation or tunnel quasiparticleinjection may result in sufficient modification of the distri-bution function. General reasoning predicts multiple super-conducting states in this situation, and indeed they have beenfound in Refs. 16and 17. The stability of these states strongly depend on the nature of the quasiparticle distribu-tion function and the relaxation mechanism. 16,17,19Proper ac- count of superconducting fluctuations may be required to understand the transitions between the stable states and even-tually to recover adiabaticity of the superconductor dynamicsin the low-frequency limit. 20In distinction from previous work, we assume that the relaxation is provided by tunnelingto/from the leads rather than by inelastic processes in thebulk. The relaxation time is therefore /H11229/H6036/E Thand may be- come comparable with inverse of the energy scales /H9004and eV. The latter forbids the use of the Boltzmann equation for the distribution function implemented in earlier studies.16,17 Therefore our work is based on a Green’s function technique. The rest of the paper is structured as follows. In Sec. IIwe specify the model to be studied, and present the equationsthat determine its state. In Sec. IIIwe solve these equations analytically, assuming that the system is in a stationary state.We analyze the stationary states we find and calculate the I-V characteristic of the system. In Sec. IVwe establish that the stationary states are the only stable states of the dc-biasedsystem. We do so by studying the dynamics of the systemafter a perturbation. In Sec. Vwe summarize our main re- sults.I. SNYMAN AND YU. V. NAZAROV PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-2II. MODEL As stated in Sec. I, we consider a superconducting island connected to two normal leads by means of low transparencytunnel barriers. The superconducting order parameter istaken to be spatially isotropic inside the island. The physicalrequirements for this condition to hold have already beendiscussed in Sec. I. We assume that the dominant energy relaxation mechanism for the superconductor is tunneling ofelectrons to the leads. For given barrier transparencies, thisrestricts the size of the superconductor to less than theinelastic-scattering length of quasiparticles inside the super-conductor. Our analysis of the system is based on the Keldysh Green’s function technique. 21–23We start our discussion of the equations governing the system by defining the necessaryGreen’s functions. A. Definition of Green’s functions The Green’s functions are expectation values of products of the Heisenberg operators am/H11006†/H20849t/H20850andam/H11006/H20849t/H20850that create and annihilate electrons in levels of the isolated island. Heremlabels single-particle levels. The /H11006index accounts for Kramer’s degeneracy. As we are dealing with a problem in-volving superconductivity, all Green’s functions are 2 /H110032 matrices in Nambu space. It is useful to define Nambu spacematrices /H9257j,j=0, ... ,3 such that /H92570is the identity matrix, and/H92571,/H92572, and/H92573are the standard Pauli matrices. We also define matrices /H9257/H11006=/H20849/H92571/H11006i/H92572/H20850/2. The retarded /H20849R/H20850, Keldysh /H20849K/H20850, and advanced /H20849A/H20850Green’s functions of each level are defined as24 Rm/H20849t,t/H11032/H20850=−i/H92573/H20883/H20873/H20853am+/H20849t/H20850,am+†/H20849t/H11032/H20850/H20854 /H20853am+/H20849t/H20850,am−/H20849t/H11032/H20850/H20854 /H20853am−†/H20849t/H20850,am+†/H20849t/H11032/H20850/H20854 /H20853am−†/H20849t/H20850,am−/H20849t/H11032/H20850/H20854/H20874/H20884 /H11003/H9258/H20849t−t/H11032/H20850, /H208492.1a /H20850 Km/H20849t,t/H11032/H20850=−i/H92573/H20883/H20873/H20851am+/H20849t/H20850,am+†/H20849t/H11032/H20850/H20852 /H20851am+/H20849t/H20850,am−/H20849t/H11032/H20850/H20852 /H20851am−†/H20849t/H20850,am+†/H20849t/H11032/H20850/H20852 /H20851am−†/H20849t/H20850,am−/H20849t/H11032/H20850/H20852/H20874/H20884, /H208492.1b /H20850 Am/H20849t,t/H11032/H20850=/H92573Rm/H20849t/H11032,t/H20850†/H92573. /H208492.1c /H20850 The Green’s functions are grouped into a matrix Gm/H20849t,t/H11032/H20850=/H20873Rm/H20849t,t/H11032/H20850Km/H20849t,t/H11032/H20850 0 Am/H20849t,t/H11032/H20850/H20874. /H208492.2 /H20850 This further 2 /H110032 matrix structure is referred to as Keldysh space. As with Nambu space, it is useful to define matrices /H9270j,j=0, ... ,3. The matrix /H9270jis the same as the matrix /H9257jbut now operating in Keldysh space. We also carry over the defi-nition of /H9270/H11006from Nambu space. A basis for the 4 /H110034 matri- ces that results from combining Keldysh and Nambu indicesis constructed by means of a tensor product /H9270j/H20002/H9257k, with the /H9270’s always acting in Keldysh space and the /H9257’s in Nambu space. The quantities that we calculate, namely, the order param- eter/H9004/H20849t/H20850and the current I/H20849t/H20850, are collective in the sense thatthey result from the sum of the contributions of all the indi- vidual levels. Accordingly a formalism exists that does notrequire knowledge of the Green’s functions of individual lev-els but only the sums 22,25–28 G/H20849t,t/H11032/H20850=i/H9254s /H9266/H20858 mGm/H20849t,t/H11032/H20850,G=G,R,K,A, /H208492.3 /H20850 which are known as quasiclassical Green’s functions. Here /H9254s is the mean level spacing of the island. We will work with the quasiclassical Green’s functions throughout the present section. The advantage of doing so isthat the theory can be formulated with the least amount ofclutter. When doing time-dependent numerics in Sec. IV however, we find it more convenient to work with theGreen’s functions of the individual levels. B. Equations of motion The equations that determine the Green’s functions can be derived from the circuit theory of nonequilibriumsuperconductivity. 26–28Viewed as a matrix in time, Nambu, and Keldysh indices, the Green’s function Gsatisfies the commutation relation29 /H20851H−/H9018,G/H20852=0 . /H208492.4 /H20850 Here Hdescribes the dynamics of the isolated supercon- ductor: H/H20849t,t/H11032/H20850=/H92700/H20002/H92573/H9254/H20849t−t/H11032/H20850/H20851i/H11509t−h/H20849t/H20850/H20852, /H208492.5a /H20850 h/H20849t/H20850=/H20873−/H9262s/H20849t/H20850/H9004/H20849t/H20850 /H9004/H20849t/H20850/H11569/H9262s/H20849t/H20850/H20874. /H208492.5b /H20850 The matrix h/H20849t/H20850is a remnant of the Bogoliubov–de Gennes Hamiltonian.11Bearing in mind that we consider a nonequi- librium setup, we must allow the order parameter /H9004/H20849t/H20850and the chemical potential /H9262s/H20849t/H20850of the superconductor to be time dependent. Their values at each instant in time are deter-mined by imposing self-consistency. The time derivative standing to the right of Gin the term GHof Eq. /H208492.4 /H20850can be shifted to act on the second time argument of Gat the cost of a minus sign, i.e., /H20885dt˜G/H20849t,t˜/H20850/H11509t˜/H9254/H20849t˜−t/H11032/H20850=−/H20885dt˜/H11509t˜G/H20849t,t˜/H20850/H9254/H20849t˜−t/H11032/H20850=−/H11509t/H11032G/H20849t,t/H11032/H20850. /H208492.6 /H20850 The self-energy contains a term corresponding to each lead, i.e., /H9018=/H9018/H20849l/H20850+/H9018/H20849r/H20850, /H208492.7 /H20850 landrreferring to the left and right leads, respectively. The leads act as reservoirs, broadening the island levels to a finitelifetime and determining their filling. The self-energy of lead jis/H9018 /H20849j/H20850=−i/H9003jG/H20849j/H20850, where Green’s function G/H20849j/H20850of lead jis defined similarly to the Green’s function of the supercon-ductor /H20851Eq. /H208492.3 /H20850/H20852with the sum now running over states in the lead. Here /H9003 jis the tunneling rate from any island level to lead j./H20849For simplicity, we take the rates associated withBISTABILITY IN VOLTAGE-BIASED NORMAL- … PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-3different levels to be the same. /H20850The leads are large compared to the superconductor, and therefore Gjdoes not depend on the state of the superconductor. Furthermore, since the leadsare normal, the off-diagonal Nambu space matrix elementsof the lead Green’s functions are zero. Explicitly then, theGreen’s function for lead j=l,rhas the form G /H20849j/H20850/H20849t,t/H11032/H20850=/H20873R/H20849j/H20850/H20849t,t/H11032/H20850K/H20849j/H20850/H20849t,t/H11032/H20850 0 A/H20849j/H20850/H20849t,t/H11032/H20850/H20874, /H208492.8 /H20850 with R/H20849j/H20850/H20849t,t/H11032/H20850=/H9254/H20849t−t/H11032/H20850/H92573=−A/H20849j/H20850/H20849t,t/H11032/H20850, /H208492.9a /H20850 K/H20849j/H20850/H20849t,t/H11032/H20850=2/H20873/H9268j/H20849t,t/H11032/H20850 0 0/H9268j/H20849t,t/H11032/H20850/H11569/H20874. /H208492.9b /H20850 The function /H9268jdescribes the distribution of particles in lead j. In general it is given by /H9268j/H20849t,t/H11032/H20850=/H20885dE 2/H9266e−iE/H20849t−t/H11032/H20850/H208511−2 fj/H20849E/H20850/H20852e−i/H20851/H9278j/H20849t/H20850−/H9278j/H20849t/H11032/H20850/H20852, /H208492.10 /H20850 where fj/H20849E/H20850is the filling factor of states at energy Ein lead j. The phase /H9278jsets the time-dependent chemical potential /H9262j/H20849t/H20850=/H11509t/H9278j/H20849t/H20850in lead j. The time-dependent bias voltage be- tween the leads is V/H20849t/H20850=/H20851/H9262l/H20849t/H20850−/H9262r/H20849t/H20850/H20852/e, /H208492.11 /H20850 where eis the electron charge. It is convenient to define the total inverse lifetime or Thouless energy ETh=/H9003l+/H9003rand a dimensionless symmetry parameter /H9253=/H20849/H9003l−/H9003r/H20850/ETh. For a perfectly symmetric coupling to the leads, /H9253=0 while /H9253 =/H110061 corresponds to the island being coupled to only one of the two leads. The commutator Eq. /H208492.4 /H20850on its own is not enough to specify Guniquely. Indeed what Eq. /H208492.4 /H20850says is that Ghas the same eigenstates as H−/H9018but it does not say anything about the eigenvalues of G. Additional to Eq. /H208492.4 /H20850there is also a relation between the eigenvalues of Gand those of H−/H9018.30Let /H20841/H9261/H20856be a simultaneous eigenstate of H−/H9018andG, such that its eigenvalue with respect to H−/H9018is/H9261. Then its eigenvalue with respect to Gis sgn /H20851Im/H20849/H9261/H20850/H20852./H20849One can show that the eigenvalues of H−/H9018come in complex-conjugate pairs and that there are no purely real eigenvalues. /H20850Hence G squares to unity, i.e., G2=I. /H208492.12 /H20850 C. Gauge invariance At this point we have defined three different Fermi ener- gies, namely, that of the superconductor /H9262s/H20849t/H20850and those of the leads /H9262j/H20849t/H20850with j=l,r. Since the reference point from which energy is measured is arbitrary, there is some redun-dancy. This redundancy is encoded in a symmetry of theequations for the Green’s function and boils down to gaugeinvariance. Consider a transformation on the Green’s func-tionG→G ˜=UGU†, /H208492.13a /H20850 U/H20849t,t/H11032/H20850=/H9254/H20849t−t/H11032/H20850/H92700/H20002exp /H20851i/H92573/H9011/H20849t/H20850/H20852. /H208492.13b /H20850 As is easily verified, G˜obeys equations of the same form as G, with chemical potentials and the order parameter trans- formed according to /H9262j/H20849t/H20850→/H9262˜j/H20849t/H20850=/H9262j/H20849t/H20850+/H11509t/H9011/H20849t/H20850,j=s,l,r, /H208492.14a /H20850 /H9004/H20849t/H20850→/H9004˜/H20849t/H20850=/H9004/H20849t/H20850exp /H208512i/H9011/H20849t/H20850/H20852. /H208492.14b /H20850 When considering stationary solutions we will fix the gauge by demanding that /H9004is time independent. When considering nonstationary solutions we will fix the gauge such that thereference point from which chemical potentials are measuredis halfway between the chemical potentials of the reservoirs,i.e., /H9262r/H20849l/H20850/H20849t/H20850=+ /H20849−/H20850eV/H20849t/H20850/2. D. Self-consistency of /H9004 The value of the order parameter is set by the self- consistency condition /H9004/H20849t/H20850=g/H9254s/H20858 m/H20855am−/H20849t/H20850am+/H20849t/H20850/H20856=−/H9266g 2Tr/H20851/H9257−K/H20849t,t/H20850/H20852, /H208492.15 /H20850 where g/H110220 is the dimensionless pairing interaction strength. This self-consistency equation suffers from the usual loga-rithmic divergence which requires regularization by intro-ducing a large energy cutoff E co. We define /H90040as the order parameter of an isolated superconductor at zero temperaturefor given gandE co, /H90040=Eco sinh1 g⇒1 g=/H20885 0Eco dE /H20881E2+/H900402. /H208492.16 /H20850 This definition then allows us to express /H9004in Eq. /H208492.15 /H20850in terms of /H90040rather than in terms of Ecoandg. E. Current and chemical potential The current from the superconductor into reservoir jis31 Ij/H20849t/H20850=/H9266 2eGj/H20885dt/H11032Tr/H20853/H9270−/H20002/H92573/H20851G/H20849t,t/H11032/H20850G/H20849j/H20850/H20849t/H11032,t/H20850 −G/H20849j/H20850/H20849t,t/H11032/H20850G/H20849t/H11032,t/H20850/H20852/H20854, Gl/H20849r/H20850=/H208511+ /H20849−/H20850/H9253/H20852ETh /H9254se2 /H20851/H6036/H20852. /H208492.17 /H20850 Here Gjis the tunneling conductance of the tunnel barrier between lead jand the superconductor, and we have indi- cated in square brackets a factor of /H6036which equals unity in the units we use throughout the paper. The total rate ofchange in the charge in the superconductor is equal to thenegative of the sum of the currents to the leads, i.e.,I. SNYMAN AND YU. V. NAZAROV PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-4−d dtQ/H20849t/H20850=Il/H20849t/H20850+Ir/H20849t/H20850. /H208492.18 /H20850 The charge in the superconductor is related to the chemical potential /H9262sby means of the capacitance Cof the supercon- ductor so that /H9262shas to obey 1 ed dt/H9262s/H20849t/H20850=Cd dtQ/H20849t/H20850. /H208492.19 /H20850 When the system is not stationary, this equation sets the value of /H9262s/H20849t/H20850at each instant in time since dQ /H20849t/H20850/dtcan be calculated directly from G/H20849t,t/H11032/H20850. F. Summary In summary then, our task is to find the Green’s function Gas defined in Eq. /H208492.3 /H20850of the superconductor. In general, the procedure for doing this is as follows: we make an ansatzfor the order parameter /H9004/H20849t/H20850and the chemical potential /H9262s/H20849t/H20850. We then diagonalize the operator H−/H9018/H20849that depends on /H9004 and/H9262/H20850. The Green’s function Gis constructed in the eigen- basis of H−/H9018, according to the prescription of Sec. II B. Subsequently we judge the correctness of the ansatz for /H9004/H20849t/H20850 and/H9262/H20849t/H20850by inquiring whether Eqs. /H208492.15 /H20850and /H208492.19 /H20850are satisfied. III. STATIONARY SOLUTIONS We consider a time-independent bias voltage between the left and right reservoirs. In this case the chemical potentials /H9262land/H9262rof the reservoirs are time independent. We make the ansatz that the chemical potential /H9262sand the order pa- rameter /H9004of the superconductor are also time independent. The Green’s function G/H20849t,t/H11032/H20850only depends on the time dif- ference t−t/H11032. It is convenient to work with the Fourier- transformed Green’s function G/H20849E/H20850which is related to G/H20849t,t/H11032/H20850by G/H20849t,t/H11032/H20850=/H20885dE 2/H9266e−iE/H20849t−t/H11032/H20850G/H20849E/H20850. /H208493.1 /H20850 It is also convenient to construct a traceless operator M=H −/H9018−/H92700/H20002/H92570/H9262swith Keldysh structure M=/H20873MRMK 0MA/H20874. /H208493.2 /H20850 In the energy representation, the components of Mhave the explicit form MR/H20849E/H20850=/H20873E+iETh −/H9004 /H9004/H11569−E−iETh/H20874, /H208493.3a /H20850 MA/H20849E/H20850=/H20873E−iETh −/H9004 /H9004/H11569−E+iETh/H20874, /H208493.3b /H20850 MK/H20849E/H20850=2iETh/H20873/H9268/H20849E/H20850 0 0/H9268/H20849−E/H20850/H20874, /H208493.3c /H20850/H9268/H20849E/H20850=1−/H9253 2/H9268l/H20849E/H20850+1+/H9253 2/H9268r/H20849E/H20850. /H208493.3d /H20850 We take the left and right leads to be in local zero- temperature equilibrium at Fermi energies /H9262l=/H9262+eV/2 and /H9262r=/H9262−eV/2 so that the filling factors in both reservoirs is a step function fj/H20849E/H20850=/H9258/H20849−E/H20850and from Eq. /H208492.10 /H20850follows /H9268l/H20849E/H20850= sgn /H20849E−/H9262−eV/2/H20850, /H208493.4a /H20850 /H9268r/H20849E/H20850= sgn /H20849E−/H9262+eV/2/H20850, /H208493.4b /H20850 where /H9262is the average chemical potential /H20849/H9262r+/H9262l/H20850/2 in the leads, in the gauge where the phase of the order parameter istime independent. The value of /H9262will later be determined by requiring self-consistency of the order parameter /H9004. The Green’s function G/H20849E/H20850obeys /H20851M/H20849E/H20850,G/H20849E/H20850/H20852=0. The retarded, advanced, and Keldysh components of this equation are /H20851MR/H20849E/H20850,R/H20849E/H20850/H20852=/H20851MA/H20849E/H20850,A/H20849E/H20850/H20852=0 , /H208493.5a /H20850 MR/H20849E/H20850K/H20849E/H20850+MK/H20849E/H20850A/H20849E/H20850−R/H20849E/H20850MK/H20849E/H20850−K/H20849E/H20850MA/H20849E/H20850=0 . /H208493.5b /H20850 With the aide of the prescription below Eq. /H208492.11 /H20850for choosing the eigenvalues of G, one then readily finds for the retarded and advanced Green’s functions R/H20849E/H20850=1 c/H20849E/H20850/H20873E+iETh −/H9004 /H9004/H11569−E−iETh/H20874, /H208493.6a /H20850 A/H20849E/H20850=1 c/H20849−E/H20850/H20873E−iETh −/H9004 /H9004/H11569−E+iETh/H20874, /H208493.6b /H20850 c/H20849E/H20850=/H20881/H20849E+iETh/H208502−/H20841/H9004/H208412. /H208493.6c /H20850 The function c/H20849E/H20850, which we will frequently encounter, is defined with branch cuts along the lines E/H11006=/H11006/H20841/H9004/H20841/H11006x −iEThwhere xis real and positive. The branch with limE/H33528R→/H11006/H11009c/H20849E/H20850/E=1 is taken. Considered as a function of realE, the real part of c/H20849E/H20850is odd, and the imaginary part is even and positive so that c/H20849E/H20850/H11569=−c/H20849−E/H20850. /H208493.7 /H20850 The real and imaginary parts of c/H20849E/H20850are plotted for real Ein Fig.1. Note that R/H20849E/H208502=A/H20849E/H208502=/H92570as required by Eq. /H208492.12 /H20850.I n general the superconducting density of states is /rho1/H20849E/H20850 =Tr/H92573/H20851R/H20849E/H20850−A/H20849E/H20850/H20852/2/H9254sso that we find from the solutions forRandA/H20851Eq. /H208493.6 /H20850/H20852 /rho1/H20849E/H20850=2 /H9254sRe/H20875E+iETh c/H20849E/H20850/H20876. /H208493.8 /H20850 The density of states for an isolated superconductor has sin- gularities at energies E=/H11006/H20841/H9004/H20841of the form 1 //H20881E2−/H20841/H9004/H208412. The coupling to the leads regularizes the singularities at an en-ergy scale of E Th. Furthermore, whereas the density of states of the isolated superconductor vanishes for energies /H20841E/H20841 /H11021/H20841/H9004/H20841, the coupling to the leads softens the gap so that there are some states for energies /H20841E/H20841/H11021/H20841/H9004/H20841as shown in Fig. 2.BISTABILITY IN VOLTAGE-BIASED NORMAL- … PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-5The next step is to solve Eq. /H208493.5b /H20850forK/H20849E/H20850. Here we use the fact that R/H20849E/H20850K/H20849E/H20850A/H20849E/H20850=−K/H20849E/H20850, which follows from the requirement that G2=I/H20851Eq. /H208492.12 /H20850/H20852. Note also that R/H20849E/H20850 =MR/H20849E/H20850/c/H20849E/H20850andA/H20849E/H20850=MA/H20849E/H20850/c/H20849−E/H20850. Using these identi- ties and multiplying Eq. /H208493.5b /H20850from the left by R/H20849E/H20850,w efi n d K/H20849E/H20850=1 c/H20849E/H20850+c/H20849−E/H20850/H20851MK/H20849E/H20850−R/H20849E/H20850MK/H20849E/H20850A/H20849E/H20850/H20852. /H208493.9 /H20850 After some algebra we obtain K/H20849E/H20850=/H20873K/H208491/H20850/H20849E/H20850 K/H208492/H20850/H20849E/H20850 −K/H208492/H20850/H20849E/H20850/H11569K/H208491/H20850/H20849−E/H20850/H20874, /H208493.10 /H20850 where K/H208491/H20850/H20849E/H20850=/H9254s/rho1/H20849E/H20850/H9268/H20849E/H20850−/H20841/H9004/H208412 ERe/H208751 c/H20849E/H20850/H20876/H20851/H9268/H20849E/H20850+/H9268/H20849−E/H20850/H20852, /H208493.11a /H20850 K/H208492/H20850/H20849E/H20850=−/H9004Re/H208751 c/H20849E/H20850/H20876/H20877/H20851/H9268/H20849E/H20850−/H9268/H20849−E/H20850/H20852 −iETh E/H20851/H9268/H20849E/H20850+/H9268/H20849−E/H20850/H20852/H20878. /H208493.11b /H20850 Having obtained K/H20849E/H20850we can find /H20841/H9004/H20841and/H9262from the self- consistency condition /H20851Eq. /H208492.15 /H20850/H20852. Below we write the real and imaginary parts of the self-consistency equation sepa-rately. The real part reads 0=/H20885dE/H20877Re/H208751 c/H20849E/H20850/H20876/H9268/H20849E/H20850−/H9268/H20849−E/H20850 2−1 /H20881E2+/H900402/H20878, /H208493.12 /H20850 while the imaginary part reads 0=/H20885dE1 ERe/H208751 c/H20849E/H20850/H20876/H20851/H9268/H20849E/H20850+/H9268/H20849−E/H20850/H20852. /H208493.13 /H20850 These integrals can be done explicitly. We use the identities /H20885 0E dE/H11032Re1 c/H20849E/H11032/H20850=FR/H20849E/H20850−FR/H208490/H20850, /H208493.14a /H20850/H20885 0E dE/H110321 E/H11032Re1 c/H20849E/H11032/H20850=1 /H20881ETh2+/H20841/H9004/H208412FI/H20849E/H20850, /H208493.14b /H20850 where FR/H20849E/H20850=l n/H20879E+iETh+c/H20849E/H20850 /H90040/H20879, /H208493.15a /H20850 FI/H20849E/H20850= arctan/H20875Re/H20851c/H20849E/H20850/H20852 /H20881ETh2+/H20841/H9004/H208412/H20876. /H208493.15b /H20850 Here the branch for which − /H9266/2/H11021arctan /H20849x/H20850/H11021/H9266/2 is im- plied. Thus we obtain the transcendental equations 0= /H208491−/H9253/H20850FR/H20873eV 2+/H9262/H20874+/H208491+/H9253/H20850FR/H20873/H9262−eV 2/H20874, /H208493.16a /H20850 0= /H208491−/H9253/H20850FI/H20873eV 2+/H9262/H20874+/H208491+/H9253/H20850FI/H20873/H9262−eV 2/H20874, /H208493.16b /H20850 that determine /H20841/H9004/H20841and/H9262for given V,ETh, and/H9253. Below we solve these equations analytically in certain limiting casesand numerically for more general cases. Only the amplitudeof/H9004is fixed by these equations. By choosing the appropriate gauge /H20851cf. Eq. /H208492.14b /H20850/H20852, we can set the phase of /H9004to any value. In the rest of this section we therefore drop the abso-lute value notation, and take /H9004as real and positive. Before explicitly finding /H9004and /H9262, we calculate the cur- rent from Eq. /H208492.17 /H20850and the solution for K/H20849E/H20850. Assuming that the self-consistency equation /H20851Eq. /H208493.13 /H20850/H20852is fulfilled, we find that the current Irfrom the superconductor to the right lead is equal to the negative of the current Ilfrom the supercon- ductor to the left lead, as it should. For the current I=Ir =−Ilfrom the left lead to the right lead, we find/Minus3 /Minus2 /Minus1 0 1 2 3012345 E/ |∆|δs/rho1 FIG. 2. The density of states of the superconducting island /H20851Eq. /H208493.8 /H20850/H20852for finite Thouless energy /H20849solid line /H20850. The dashed line shows the density of states of the isolated superconducting island with thesame /H20841/H9004/H20841while the horizontal dot-dashed line shows the density of states of the normal island. A value of E Th=0.1 /H20841/H9004/H20841was used./Minus2 /Minus1 0 1 2/Minus101 E/ |∆|c(E)/|∆| FIG. 1. The function c/H20849E/H20850as defined in Eq. /H208493.6c /H20850frequently appears in expressions associated with stationary solutions. Thesolid line represents the real part and the dashed line the imaginarypart. The Thouless energy was taken as E Th=0.1 /H20841/H9004/H20841.I. SNYMAN AND YU. V. NAZAROV PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-6I=/H208491−/H92532/H20850eETh 2/H20885 /H9262−eV/2/H9262+eV/2 dE/rho1/H20849E/H20850 =GN eRe/H20875c/H20873/H9262+eV 2/H20874−c/H20873/H9262−eV 2/H20874/H20876. /H208493.17 /H20850 Here GNis the series conductance of the tunneling barriers to the leads GN=/H20851Gl−1+Gr−1/H20852−1, /H208493.18 /H20850 andGlandGrare the junction conductances given in Eq. /H208492.17 /H20850. Now we investigate the transcendental equations /H20851Eqs. /H208493.16 /H20850/H20852for/H9262and/H9004. There are three parameters, namely, ETh, /H9253, and V, which determine the solution. Two of these, ETh and/H9253, are fixed for a given device while the voltage Vcan be varied for a given device. /H20849Recall that EThmeasures the over- all coupling to the leads while /H9253measures the degree of asymmetry between the two lead couplings. /H20850Hence it is natural to specify values for EThand/H9253, and then consider /H9004, /H9262, and Ias functions of V. In Fig. 3we show four curves of /H9004versus V, each corresponding to a different choice of pa- rameters EThand/H9253. In Fig. 4we show the corresponding curves of /H9262versus V. Let us first note the general trend that increasing EThleads to a smaller order parameter. The reason for this is that ETh−1is the typical time an electron remains in the superconductor.The shorter this time /H20849the larger ETh/H20850the harder it is for electrons to form Cooper pairs, and superconductivity is in-hibited. Second, note that at large enough E Ththe order pa- rameter is a decreasing function of V. We can therefore ob- tain the critical Thouless energy ETh/H20849c/H20850beyond which superconductivity vanishes by setting Vto zero and asking how large can we make EThbefore /H9004becomes zero. In the case of V=0, the self-consistency equations are solved by /H9262=0 and /H9004=/H20902/H90040/H208811−2ETh /H90040ETh/H11021/H9004 0/2 0 ETh/H11022/H9004 0/2/H20903. /H208493.19 /H20850 From this we conclude that the critical Thouless energy is ETh/H20849c/H20850=/H90040/2. Having established the range of EThin which supercon- ductivity persists, we now take a closer look at /H9004as a func- tion of V. We have chosen the parameters of the four solu- tions in Fig. 3to show all the different possible shapes that curve of /H9004versus Vcan take. We see that at a given voltage Vthere can be either zero, one, two, or three nonzero solu- tions/H9004. We note that this behavior is qualitatively similar /H20849as is to be expected /H20850to that of a driven bulk superconductor in which the electron-phonon interaction is responsible forrelaxation. 16,17As explained in Sec. I, quantitative differ- ences are accounted for by the different relaxation mecha-nism /H20849tunneling /H20850that operates in the present system. To characterize the different types of curve, we consider V as a function of /H9004on the interval /H9004/H33528/H208510,/H9004 0/H208811−2ETh//H90040/H20852.I n curves of the type Ain Fig. 3,Vis a monotonically decreas- ing function of /H9004. In contrast, curves of type B,C, and D have local extrema. A curve of type Bhas a local minimum at the left boundary /H9004=0 of the /H9004interval on which the function V/H20849/H9004/H20850is defined. Then the curve reaches a maximum at some intermediate value /H90041before dropping to zero at the right boundary /H9004=/H90040/H208811−2ETh//H90040. Curves CandDare dis- tinguished from curve Bby the fact that Vreaches a local maximum instead of a minimum at the left boundary /H9004=0 of the/H9004interval. There is another local maximum at interme- diate/H90041before Vdrops to zero at /H9004=/H90040/H208811−2ETh//H90040.I n curves of type C, the absolute maximum of Vas a function of/H9004is at the intermediate value /H90041while for curves of type Dthe absolute maximum of Vis at/H9004=0. Next we ask how the ETh-/H9253parameter space is divided into regions A,B,C, and Dcorresponding to the respective types of solution of the self-consistency equations. Specifi-cally, which regions share a mutual border? Assuming thatthe function V/H20849/H9004/H20850changes smoothly as E Thand/H9253are varied, the transitions A↔B,B↔C,C↔D, and D↔Aare possible. The transition A↔Cis not possible. Whenever one tries to smoothly deform a curve of type Ain Fig. 3to a curve of type C, one invariably reaches a curve of type BorDduring an intermediate stage of the deformation. Similarly the tran-sition B↔Dis impossible. A smooth deformation of a curve of type Binto one of type Dpasses through an intermediate stage where the curve is of types AorC. To illustrate these ideas we consider a polynomial equation of the form0.0 0.5 1.0 1.5 2.00.00.51.0∆/∆0 eV/∆0AA BB CCDD FIG. 3. The order parameter /H9004versus voltage V, for given ETh and/H9253. Curves A,B,C, and D, respectively, correspond to ETh =0.35/H90040and/H9253=0.2, ETh=0.2/H90040and/H9253=0.075, ETh=0.1/H90040and/H9253 =0.1, and ETh=0.01/H90040and/H9253=0.3. 0.0 0.5 1.0 1.5 2.00.00.20.40.6µ/∆0 eV/∆0AA BBCCDD FIG. 4. The chemical potential /H9262versus voltage V, for given EThand/H9253. Curves A,B,C, and Dcorrespond to the same parameter values as in Fig. 3.BISTABILITY IN VOLTAGE-BIASED NORMAL- … PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-7V˜ /H90040=V0 /H90040−1 6/H20873/H9004 /H90040/H208746 −a 4/H20873/H9004 /H90040/H208744 −b 2/H20873/H9004 /H90040/H208742 . /H208493.20 /H20850 We ask: what are the respective regions of the a-bplane in which V˜/H20849/H9004/H20850is a curve of type A,B,C, and D? Region A, where V˜/H20849/H9004/H20850is of type A, is given by a/H110220,b/H110220o r a /H110210,b/H11022a2/4. Region Bwhere V˜/H20849/H9004/H20850is of type Bconsists of all points /H20849a,b/H20850such that b/H110210. Region Cconsists of all points /H20849a,b/H20850such that a/H110210 and 0 /H11021b/H110213a2/16. Region D consists of all points /H20849a,b/H20850such that a/H110210 and 3 a2/16/H11021b /H11021a2/4. The regions and their borders are shown in the inset of Fig. 5. The most pertinent feature of the figure is that the four distinct regions meet in the single point a=b=0. Based on a combination of numerical and analytical re- sults, we have concluded that the ETh-/H9253parameter space has a very similar topology to this polynomial example. /H20849In prin- ciple it could have differed from the polynomial example byhaving disconnected regions of the same type, for instancetwo islands of region Dwith one embedded in a sea of re- gion Aand the other in a sea of region C./H20850Figure 5is a schematic diagram of how the E Th-/H9253parameter space is par- titioned into regions A,B,C, and D. The following features of the diagram are conjectures based on numerical evidence:/H208491/H20850the regions of types A,B,C, and Dare simply connected. /H208492/H20850The border between regions AandDstarts at the corner /H9253=1,ETh=0. Other features are deduced from analytical re- sults: /H208491/H20850the line /H9253=0,ETh/H11021/H9004 0/2/H208812 belongs to region B. /H208492/H20850The line /H9253=0,/H90040/2/H208812/H11021ETh/H11021/H9004 0/2 belongs to region A. /H208493/H20850ForETh/H11022/H9004 0/2 the system is in the normal state while it is superconducting for ETh/H11021/H9004 0/2./H208494/H20850The border of regions DandCmeets the border of region BandCatETh=0,/H9253 =0.In region D, superconductivity can persist up to voltages that are large compared to /H90040. For given EThand/H9253there is however always a critical voltage Vcbeyond which super- conductivity ceases. /H20849This is a second-order phase transi- tion. /H20850For the voltage Vcwe have obtained the following analytical result from Eq. /H208493.16 /H20850. At finite /H9253and for EThthat is sufficiently small, Vcobeys the power law V/H20849c/H20850=/H90040 2e/H208752ETh /H90040sec/H9266/H9261 2/H20876−1//H9261 ,/H9261=1− /H20841/H9253/H20841 1+ /H20841/H9253/H20841. /H208493.21 /H20850 This power law is valid as long as V/H20849c/H20850/H11271/H9004 0/e. It is from this result that we are able to conclude that the region of finite /H9253 and infinitesimal EThbelongs to region D. Another analytical result can be obtained for the case of perfectly symmetric coupling to the leads, i.e., /H9253=0. In this case, Eq. /H208493.16b /H20850is solved by /H9262=0 and the relation between /H9004andVcan be stated as eV=/H90040/H208731+/H90042 /H900402/H20874/H208811−4ETh2//H900402 /H208731−/H90042 /H900402/H208742. /H208493.22 /H20850 This result is plotted for several values of EThin Fig. 6.I ti s from this result that we are able to conclude that the line segment /H9253=0, 0/H11021ETh/H11021/H9004 0/2/H208812 belongs to region Bwhile the line segment /H9253=0,/H90040/2/H208812/H11021ETh/H11021/H9004 0/2 belongs to re- gion A. The ETh→0 limit of Eq. /H208493.22 /H20850can be obtained by considering a bulk superconductor and assuming a quasipar-ticle distribution function n/H20849E/H20850=/H20851 /H9258/H20849−eV/2−E/H20850+/H9258/H20849eV/2 −E/H20850/H20852/2. It is also worth noting that the same result is ob- tained for a Tjunction where the stem of the Tis a super- conductor and the bar a voltage-biased dirty normal-metalwire. 10 Finally, we consider the I-Vcurves associated with the solutions /H9004and/H9262of Figs. 3and4. The results are shown in Fig.7. From these curves we can infer the results that will be obtained in an experiment in which the voltage Vis swept adiabatically from zero to several /H90040/eand back to zero. In region Aof parameter space, there is a single current associ-/SolidCircle /SolidCircle/SolidCircle/SolidCircle 0.0 0.50.00.51.0 00γ ETh/∆0AA AABB BBCC CCDD DD NN 1/2√ 21/2√ 2QQab FIG. 5. Schematic diagram of the partitioning of the ETh-/H9253pa- rameter space into regions where the curve of /H9004versus Vis of the types A,B,C, and D/H20849Fig.3/H20850. The regions A,B,C, and Dmeet at point Q. The line ETh=/H90040/2 separates the normal and supercon- ducting regions of parameter space. The dots in the figure indicatethe parameter values that correspond to the curves in Fig. 3. The inset shows the regions A,B,C, and Din the parameter space a-b of the polynomial V˜/H20849/H9004/H20850of Eq. /H208493.20 /H20850. The topology of the ETh-/H9253 parameter space of the superconductor in the region of point Qcan be understood by considering the topology of the parameter spaceof the polynomial.0.0 0.5 1.0 1.5 2.00.00.51.0∆/∆0 eV/∆0 FIG. 6. The order parameter /H9004versus voltage Vfor symmetric coupling to the leads, i.e., /H9253=0, according to Eq. /H208493.22 /H20850. Different curves correspond to different ETh. From the top to bottom curve we took ETh//H90040=.01, 0.14, 0.26, 1 /2/H208812/H20849/H112290.35 /H20850, 0.42, and 0.47. The curve corresponding to ETh=1/2/H208812/H90040is plotted thicker than the others. For smaller ETh, curves are of type Bwith two nonzero values for /H9004at some voltages. For larger ETh, curves are of type A, with at most one nonzero /H9004at every voltage.I. SNYMAN AND YU. V. NAZAROV PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-8ated with each voltage. At some voltage V+of order /H90040/e, the system makes a phase transition to the normal state butthis does not lead to a discontinuity in the current versusvoltage curve. In contrast, in regions B,C, and D, the current will make discontinuous jumps as the voltage is swept. Hys-teresis will also be observed. Suppose the device is in regionBof parameter space, as Vis swept from zero upward, a voltage V +is crossed where the current makes a finite jump. After the jump, the system is in the normal state and thecurrent is G NV./H20851GNis the normal-state conductance of the setup, cf. Eq. /H208493.18 /H20850./H20852On the backward sweep from V/H11022V+to zero, the system remains normal when V+is reached. At some voltage V−/H11021V+the current jumps from its value GNV− in the normal state to a smaller value, signaling the onset of superconductivity. The behavior of the system in region Cof parameter space is similar. The upward sweep of the voltageproduces a jump in the current at a voltage V +. After the jump the system is normal and the current is given by I =GNV. The difference from region Bappears when the volt- age is swept back from V+to zero. At some voltage smaller than V+the current starts deviating from its value in the normal state but there is no discontinuous jump yet. Even so,the system has turned superconducting. When the jump incurrent now occurs at V −/H11021V+, the system switches between two different superconducting states. Finally, for parametersin region D, the voltage sweep produces results similar to that in region C. The difference between regions CandDis that in Dthe system also jumps between two superconduct- ing states at V +during the forward sweep. IV . DYNAMICS We concluded the previous section with a discussion of hysteresis in the current-voltage characteristic of the super-conducting island. The conclusions we drew rely on the as-sumption that after the system is perturbed by a change in thebias voltage, it relaxes into a stationary state. The validity ofthis assumption is by no means obvious since the system isdriven /H20849by the bias voltage /H20850and the stationary state is not an equilibrium state. Frankly, our own initial expectation wasthat the presence of a bias voltage would cause the dynamicsof /H20841/H9004/H20849t/H20850/H20841to be quasiperiodic or chaotic. We therefore did numerical simulations in order to investigate the dynamics of/H20841/H9004/H20849t/H20850/H20841in the presence of a bias voltage. Our main result is this: suppose the bias voltage assumes the constant value V f for times t/H11022tf, then /H20849contrary to our original expectations /H20850at t/H11271tfthe superconductor will always be found in one of the stationary states associated with Vf. This is true regardless of the history of the system prior to t/H11021tf. In particular, the time dependence of the bias voltage prior to tfdoes not matter. Nor does the state of the superconductor prior to tfmatter. Only when there is more than one nonzero stationary solu-tion associated with V fdoes the history of the system have anybearing on its final state. In this case, the history of the system determines which of the possible stationary stateseventually becomes the final state of the superconductor. Forslowly varying voltages, the predictions of Sec. IIIregarding hysteresis are confirmed. In this section we discuss the nu-merics that yielded the above results. For the purpose of numerics we find it advantageous not to take the sum over levels of the Green’s function as we didin the previous sections. Instead we work with the Green’sfunctions of each individual level. The advantage of thisscheme is that it allows us to work with ordinary differentialequations. From these differential equations it is straightfor-ward to construct a time series in which the next element canbe calculated if the present elements are known. As far as wecan see, no such “local in time” update equations exist forthe Green’s functions summed over levels. Naturally thereare disadvantages to working with the individual levelGreen’s functions as well; the number of equations to besolved numerically is increased significantly. As a result thecalculation is computationally expensive and therefore time-consuming. The Green’s functions of the individual levels obey the equations /H20849H m−/H9018/H20850Gm=Gm/H20849Hm−/H9018/H20850=I. /H208494.1 /H20850 Here Hmdiffers from the operator Hthat appeared in Eq. /H208492.5 /H20850in that it contains the energy /H9255mof level m. It is explic- itly given by Hm/H20849t,t/H11032/H20850=/H92700/H20002/H92573/H9254/H20849t−t/H11032/H20850/H20851i/H11509t−hm/H20849t/H20850/H20852, /H208494.2a /H20850 hm/H20849t/H20850=/H20873/H9255m−/H9262s/H20849t/H20850/H9004/H20849t/H20850 /H9004/H20849t/H20850/H11569/H9262s/H20849t/H20850−/H9255m/H20874. /H208494.2b /H20850 The operator hm/H20849t/H20850is the time-dependent Bogoliubov–de Gennes Hamiltonian.11The self-energy /H9018is the same as in Sec. II B. We measure energies from a point halfway between the chemical potentials of the leads. As a result the phases /H9278j/H20849t/H20850 that appear in the reservoir self-energies are /H9278r/H20849l/H20850/H20849t/H20850 =+ /H20849−/H20850/H9278/H20849t/H20850/2 where /H9278is related to the voltage VbyV/H20849t/H20850 =/H11509t/H9278/H20849t/H20850/e. We parametrize the Green’s functions in terms of a set of auxiliary functions. This eliminates some redundancies thatare present due to symmetries of the equations of motion. Westart by noting that, since the retarded and advanced Green’sfunctions are related by Eq. /H208492.1c /H20850, we do not need to con-0.0 0.5 1.0 1.5 2.00.00.51.01.52.0eI/GN∆0 eV/∆0AABBCCDD FIG. 7. The current Ithrough the superconductor versus the voltage Vacross it. Curves A,B,C, and Dcorrespond to the re- spective parameter values quoted in Figs. 3and4. The dashed line shows the current through the system in the absence ofsuperconductivity.BISTABILITY IN VOLTAGE-BIASED NORMAL- … PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-9sider both. We work with the retarded Green’s function. We define a matrix rm/H20849t,/H9270/H20850that is related to Rm/H20849t,t−/H9270/H20850by the equation Rm/H20849t,t−/H9270/H20850=i/H92573rm/H20849t,/H9270/H20850, /H208494.3a /H20850 rm/H20849t,/H9270/H20850=rm/H208490/H20850/H20849t,/H9270/H20850/H92570+irm/H20849t,/H9270/H20850·/H9257. /H208494.3b /H20850 Here the component rm/H208490/H20850/H20849t,/H9270/H20850ofrm/H20849t,/H9270/H20850is a scalar function whereas the other three components are grouped into a vec-torr m/H20849t,/H9270/H20850such that rm/H20849t,/H9270/H20850=/H20851rm/H208491/H20850/H20849t,/H9270/H20850,rm/H208492/H20850/H20849t,/H9270/H20850,rm/H208493/H20850/H20849t,/H9270/H20850/H20852. /H208494.4 /H20850 The vector /H9257=/H20849/H92571,/H92572,/H92573/H20850contains the Pauli matrices in Nambu space. Before the voltage bias between the leads is established /H20849i.e., for t/H113490 and all /H9270/H20850, the functions rm/H208490/H20850/H20849t,/H9270/H20850 and rm/H20849t,/H9270/H20850are real. When the equations of motion /H20851Eq. /H208494.1 /H20850/H20852for the retarded Green’s function are rewritten in terms r/H208490/H20850and r, we find that their reality is preserved at all times. Next we consider the Keldysh Green’s function. In order to calculate the time evolution of the order parameter, weonly need to know the Keldysh Green’s function at coincid-ing times. Here the parametrization K m/H20849t,t/H20850=i/H92573km/H20849t/H20850·/H9257, /H208494.5 /H20850 in terms of a real vector, km/H20849t/H20850=/H20851km/H208491/H20850/H20849t/H20850,km/H208492/H20850/H20849t/H20850,km/H208493/H20850/H20849t/H20850/H20852, /H208494.6 /H20850 is respected by the initial condition and preserved by the equations of motion. From the equations of motion /H20851Eq. /H208494.1 /H20850/H20852, we derive dif- ferential equations d dtrm/H20849t,/H9270/H20850=bm/H20849t/H20850rm/H20849t,/H9270/H20850−rm/H20849t,/H9270/H20850bm/H20849t−/H9270/H20850, /H208494.7 /H20850 d dtkm/H20849t/H20850+2bm/H20849t/H20850/H11003km/H20849t/H20850+2EThkm/H20849t/H20850=4EThfm/H20849t/H20850, /H208494.8 /H20850 for the matrix rm/H20849t,/H9270/H20850and the vector km/H20849t/H20850. Equation /H208494.8 /H20850 with ETh=0 was studied in Refs. 8and9. In these references the dynamics of the order parameter of an isolated supercon-ductor was calculated. We see that coupling the system toleads introduces two terms proportional to E Th. One /H20851on the left-hand side of Eq. /H208494.8 /H20850/H20852can be considered a damping term and is proportional to km/H20849t/H20850. The other /H20851on the right- hand side of Eq. /H208494.8 /H20850/H20852can be considered a driving or source term. In Eq. /H208494.8 /H20850,bm/H20849t/H20850is a matrix and bm/H20849t/H20850a vector such that bm/H20849t/H20850=ibm/H20849t/H20850·/H9257, /H208494.9a /H20850 bm/H20849t/H20850=/H20851Re/H9004/H20849t/H20850,− Im/H9004/H20849t/H20850,/H9262s/H20849t/H20850−/H9255m/H20852. /H208494.9b /H20850 The equation for km/H20849t/H20850contains a source term 4 EThfm/H20849t/H20850. The vector fm/H20849t/H20850is given byfm/H20849t/H20850=/H20885 0/H11009 d/H9270/H20851rm/H208490/H20850/H20849t,/H9270/H20850s/H20849t,/H9270/H20850−rm/H20849t,/H9270/H20850s/H208490/H20850/H20849t,/H9270/H20850−rm/H20849t,/H9270/H20850 /H11003s/H20849t,/H9270/H20850/H20852. /H208494.10 /H20850 In this equation the scalar function s/H208490/H20850/H20849t,/H9270/H20850and the vector s/H20849t,/H9270/H20850parametrize the Keldysh component of the self-energy as follows: /H9018K/H20849t,t/H11032/H20850=2ETh/H92573s/H20849t,/H9270/H20850, /H208494.11a /H20850 s/H20849t,/H9270/H20850=s/H208490/H20850/H20849t,/H9270/H20850/H92570+is/H20849t,/H9270/H20850·/H9257. /H208494.11b /H20850 Referring back to Sec. II, where /H9018Kis expressed in terms of the Fourier transform of the reservoir filling factors, we findexplicitly s /H208490/H20850/H20849t,/H9270/H20850=1 /H9266P/H208731 /H9270/H20874cos/H9278/H20849t/H20850−/H9278/H20849t−/H9270/H20850 2, /H208494.12a /H20850 s/H20849t,/H9270/H20850=−/H9253 /H9266/H9270/H208730,0,sin/H9278/H20849t/H20850−/H9278/H20849t−/H9270/H20850 2 /H20874. /H208494.12b /H20850 By imposing self-consistency, the order parameter /H9004/H20849t/H20850is expressed in terms of the components of km/H20849t/H20850as /H9004/H20849t/H20850=g/H9254s 2/H20858 m=−/H9024/H9024 km/H208491/H20850/H20849t/H20850−ikm/H208492/H20850/H20849t/H20850, /H208494.13 /H20850 where the number of levels on the island is 2 /H9024+1. This makes the differential equations nonlinear since they containterms in which /H9004/H20849t/H20850multiplies k mandr. We eliminate the dimensionless pairing strength gand the mean level spacing /H9254sin favor of /H9004eq, the order parameter of the island in equi- librium, by means of the equilibrium self-consistency rela-tion 2 g/H9254s=/H20858 m=−/H9024/H90241 /H9264m2 /H9266arctan/H9264m ETh, /H208494.14 /H20850 where /H9264m=/H20881/H9255m2+/H9004eq2. /H208494.15 /H20850 As with /H9004/H20849t/H20850, the chemical potential /H9262s/H20849t/H20850is determined by a self-consistency equation. The chemical potential takesinto account the work that must be performed against theelectric field of the excess charge on the superconductor inorder to add more charge. Thus /H9262s/H20849t/H20850is related to the charge of the island by /H9262s/H20849t/H20850=e/H20851Q/H20849t/H20850−Q0/H20852/C, where Cis the capaci- tance of the island. In this equation Q0represents the fixed positive background charge and Q/H20849t/H20850is the combined charge of all the electrons on the island. Since the differential Eqs./H208494.7 /H20850and /H208494.8 /H20850only depend on the difference /H9262s/H20849t/H20850−/H9262s/H20849t −/H9270/H20850, the positive background charge need not be specified. The charge Q/H20849t/H20850is related to the Keldysh Green’s function. Indeed, the average number nm/H20849t/H20850of electrons /H20849with spin degeneracy included /H20850in level mat time tis given by nm/H20849t/H20850 =/H208531−iTr/H20851Km/H20849t,t/H20850/H20852/H20854/2. Hence /H9262s/H20849t/H20850is related to kmby the equationI. SNYMAN AND YU. V. NAZAROV PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-10/H9262s/H20849t/H20850−/H9262s/H20849t−/H9270/H20850=e2 2C/H20858 m=−/H9024/H9024 km/H208493/H20850/H20849t/H20850−km/H208493/H20850/H20849t−/H9270/H20850./H208494.16 /H20850 Finally, we have to specify the initial conditions for rm/H20849t,/H9270/H20850and km/H20849t/H20850. We will assume for our simulations that the voltage between the reservoirs is zero and the system isin zero-temperature equilibrium for times t/H110210. The corre- sponding initial condition at t=0 is r m/H208490/H20850/H208490,/H9270/H20850=−/H9258/H20849/H9270/H20850e−ETh/H9270cos /H20849/H9264m/H9270/H20850, /H208494.17a /H20850 rm/H208490,/H9270/H20850=/H9258/H20849/H9270/H20850e−ETh/H9270sin/H20849/H9264m/H9270/H20850 /H9264m/H20849−/H9004eq,0,/H9255m/H20850,/H208494.17b /H20850 km/H208490/H20850=1 /H9264m2 /H9266arctan/H9264m ETh/H20849/H9004eq,0,−/H9255m/H20850. /H208494.17c /H20850 We are now ready to study the time evolution of /H9004/H20849t/H20850 when a nonzero bias voltage V/H20849t/H20850between the leads is present for times t/H110220. In the calculations we report on here, we worked with ETh=0.069 /H90040and three different /H9253, namely, /H9253=0.05, /H9253=0.1, and /H9253=0.2. These all correspond to points from regions CandDin the ETh-/H9253parameter space of Fig. 5. Hence, for each of the parameter choices, there is a biasvoltage interval /H20851V −,V+/H20852where there are more than one non- zero stationary solutions for /H20841/H9004/H20841. The three curves of station- ary /H20841/H9004/H20841versus V, corresponding to the different parameter choices, are plotted in the top panel of Fig. 8. For given EThand/H9253we did two numerical runs with different time-dependent voltages V/H20849t/H20850. The two voltages are plotted as functions of time in the bottom panel of Fig. 8.I n the first run we start by rapidly establishing a bias voltageV 1/H11021V−. Rapid here means dV/dt/H11271/H9004 0ETh/e. In this case V changes by an amount of order /H90040/e—the scale at which the stationary solution for /H20841/H9004/H20841depends on V—in a time that isshort compared to the relaxation time ETh−1./H20849Slow refers to the opposite limit. /H20850We then keep the voltage constant at V1for a length of time of several ETh. This time interval is long enough for any transient behavior induced by the rapidchange in V/H20849t/H20850to disappear. We then slowly increase the bias voltage until we reach a bias voltage V f/H33528/H20851V−,V+/H20852for which more than one nonzero stationary solutions exist. In the sec- ond run we start by rapidly establishing a bias voltage V2 /H11022V+. We keep the voltage fixed at V2for a time of several ETh−1. We then slowly decrease the voltage to Vf. The values of V1,V2, and Vfwere chosen V1=0.83/H90040,V2=1.76/H90040, and Vf=1.34/H90040. The calculations were performed with 501 equally spaced levels with level spacing /H9254s=0.018 /H90040and the capacitance was chosen C=0.1 e2//H90040. The resulting /H20841/H9004/H20841are plotted as functions of time in Fig. 9. They first show that after the initial rapid change in the biasvoltage the system always relaxes into a stationary state con-sistent with the new voltage. The relaxation takes a time of the order E Th−1. Second, if the system is in a stationary state and the bias voltage is changed slowly, then /H20841/H9004/H20849t/H20850/H20841adiabati- cally tracks the stationary solution corresponding to the in-stantaneous value of the voltage. This is seen most clearly inFig.10where we plot /H20841/H9004/H20849t/H20850/H20841as a function of V/H20849t/H20850and com- pare this to the stationary /H20841/H9004/H20841vs constant Vcurves. Our pre- diction about hysteresis is confirmed. Systems with differenthistories end up in different stationary states at the samevoltage bias. If the voltage is slowly swept from a smallinitial voltage to V f/H33528/H20851V−,V+/H20852, a stationary state with a large value for /H20841/H9004/H20841is reached. If the voltage is swept from a large initial voltage to Vf/H33528/H20851V−,V+/H20852, a stationary state is reached that corresponds to a small value of /H20841/H9004/H20841. We must mention here that we observe some slow drift /H20849too slow to be visible in Fig. 9/H20850in/H20841/H9004/H20841after the voltage has reached Vf. The value of /H20841/H9004/H20841seems to increase linearly at a rate d/H20841/H9004/H20841/dt/H1101110−4/H900402. Within the numerical accuracy of the calculation, this is neg-FIG. 8. Top panel: The stationary solutions for /H9004vs the bias voltage V, corresponding to the parameters used in generating Fig. 9. The outer curve /H20849dashed /H20850corresponds to ETh=0.069 /H90040and/H9253 =0.2. The middle curve /H20849solid /H20850corresponds to ETh=0.069 /H90040and /H9253=0.1. The inner curve /H20849dot dashed /H20850corresponds to ETh=0.069 /H90040 and/H9253=0.05. The vertical lines indicate VfandV2./H20849V1is beyond the left edge of the figure. /H20850Bottom panel: the time dependence of the voltage. The upper /H20849solid /H20850curve corresponds to the solid curves of /H9004vstin Fig. 9. The lower /H20849dashed /H20850curve corresponds to the dashed curves of /H9004vstin Fig. 9.FIG. 9. The amplitude of the order parameter as a function of time. All curves are for ETh=0.069 /H90040. The top, middle, and bottom panels correspond to /H9253=0.05, /H9253=0.1, and /H9253=0.2, respectively. The dashed curves correspond to a voltage that is increased from V1 =0.83/H90040toVf=1.34/H90040. The solid curves correspond to the voltage being decreased from V2=1.76/H90040toVf=1.34/H90040. The vertical lines indicate the time interval in which the voltage changes from eitherV 1orV2toVf. The thin horizontal lines correspond to the stationary values of /H20841/H9004/H20841for a bias voltage V=Vfas calculated from Eq. /H208493.16 /H20850.BISTABILITY IN VOLTAGE-BIASED NORMAL- … PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-11ligible and we believe the drift is simply an artifact of the numerics. In our data there is one exception to the rule of adiabatic evolution. In the middle panel of Fig. 9,/H20841/H9004/H20849t/H20850/H20841takes much longer that ETh−1to respond when the voltage is changed from V2toVf. Hence, /H20841/H9004/H20849t/H20850/H20841as a function of V/H20849t/H20850does not track the stationary solution in this instance. The reason is the follow-ing: for a voltage V=V 2, the only stationary solution has /H9004 =0. When the voltage is decreased to Vf, a nonzero station- ary solution for /H9004exists. However /H9004=0 is still a valid state, albeit unstable. The time it takes the system to diverge fromthe unstable state is not determined by E Thbut rather by small numerical errors that perturb the unstable state. One possible explanation for the observed stability of the stationary states is overdamping. According to this hypoth-esis, if we decrease the Thouless energy further, thereby de-creasing the damping, the stationary solutions will becomeunstable. Some evidence for the hypothesis might be visiblein Fig. 9. After the voltage is changed rapidly, we might expect /H20841/H9004/H20849t/H20850/H20841to perform damped oscillations while relaxing to the new stationary state. However in Fig. 9no such oscil- lations are visible, apparently implying that the relaxationrate is larger than the oscillation frequency. There is howeveranother possible explanation for the lack of oscillatory be-havior after an abrupt change in V. The argument is that an abrupt change in Vcannot be communicated to the system abruptly but only at a rate comparable to the damping rateE Th. This is because the superconductor learns of the change in voltage by the same mechanism as by which dampingoccurs, that is, by tunneling of particles between the leadsand the island. Hence the response of the order parameter isalways gradual.How do we test whether overdamping hypothesis is true or false? Ideally we would have liked to repeat the abovenumerical calculation with a smaller value of E Thand see if the stationary states are still stable. However, the value ETh =0.069 /H90040that we used above is close to the smallest value for which we can do reliable numerics in reasonable time.Since we cannot make E Thsmaller, we resolve the issue of overdamping as follows. We compare the dynamics of /H9004 after an abrupt change in the pairing interaction strength gat ETh=0.069 /H90040to the dynamics after a change in gatETh=0.32 We know that in the isolated system, /H20849ETh=0/H20850/H20841/H9004/H20841will per- form persistent oscillations.8,9The period of oscillation gives a typical time scale for the internal dynamics of /H9004. If, in the open system /H20849i.e.,ETh/HS110050/H20850, we observe a few damped oscil- lations /H20849the more the better /H20850in/H20841/H9004/H20841before the system relaxes to equilibrium, it means that damping occurs at a timescalelarger than that of the internal dynamics of the supercon-ductor. In this case the hypothesis of overdamping is discred-ited. In our numerical implementation of the above, we work with the following parameters: the initial pairing interactionis such that for t/H110210,/H9004=/H9004 i. The increased pairing interaction strength corresponds to an equilibrium value of the orderparameter /H9004 f=20/H9004i. The persistent oscillations of /H20841/H9004/H20849t/H20850/H20841in the isolated system are shown in the blue curve in Fig. 11. We repeat the calculation, which is now for a superconductorconnected to leads. We use a Thouless energy E Th=0.075 /H9004f. The result for /H20841/H9004/H20849t/H20850/H20841in the presence of leads is the solid curve in Fig. 11. We see that /H20841/H9004/H20849t/H20850/H20841eventually decays to a constant, as expected. The extent of the damping is such that severaloscillations are completed within the decay time. Hence weconclude that the numerical results that we obtained previ-ously are outside the regime of overdamping. It follows thatthe lack of oscillatory behavior in Fig. 9is due to the fact that the superconductor only gradually becomes aware of achange in the voltage. V . CONCLUSION We have studied a voltage-biased NISIN junction, i.e., a superconducting island connected to normal leads by meansFIG. 10. The amplitude of the order parameter /H20841/H9004/H20849t/H20850/H20841as a func- tion of voltage V/H20849t/H20850. The parameter values of the three panels are the same as those in Fig. 9, i.e., all curves are for ETh=0.069 /H90040. The top, middle, and bottom panels correspond to /H9253=0.05, /H9253=0.1, and /H9253=0.2, respectively. The thick dashed curves correspond to a volt- age that is increased from V1=0.83/H90040toVf=1.34/H90040. The thick solid curves correspond to the voltage being decreased from V2 =1.76/H90040toVf=1.34/H90040. The thin dashed lines represents the station- ary value of /H20841/H9004/H20841vsV, as calculated in Sec. IIIand plotted in Fig. 8.FIG. 11. The order parameter versus time after the pairing strength was increased from /H9004i=0.05/H9004fto/H9004fabruptly at t=0. The dashed curve is for an isolated superconductor while the solid curveis for a superconductor connected to leads. For this case a Thoulessenergy E Th=0.075 /H9004fwas used. The data was obtained using 501 equally spaced levels with level spacing /H9254s=0.02/H9004f. The capaci- tance was chosen C=0.1 e2//H9004f.I. SNYMAN AND YU. V. NAZAROV PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-12of tunnel junctions. We restricted ourselves to the regime where the dominant energy relaxation mechanism in the su-perconductor is the tunneling of electrons from the supercon-ductor to the leads. We also restricted ourselves to the regimeof low transparency junctions where the position dependenceof the order parameter inside the superconductor can be ne-glected. In Sec. IIIwe found the stationary states of the system. For these, the order parameter /H9004and the chemical potential are implicitly determined by Eq. /H208493.16 /H20850. We also found the current between the leads /H20851cf. Eq. /H208493.17 /H20850/H20852. The most striking feature of the stationary states is that, as is commonly the case in nonequilibrium superconductors and indeed moregenerally in many dissipative driven nonlinear systems, therecan be more than one stationary state at a given voltage.These are characterized by different values of /H20841/H9004/H20841and of the current as can be seen in the I-Vcurves of Fig. 7. Depending on system parameters, superconductivity can survive up tovoltages that are large compared to /H9004 0, which is the order parameter of the isolated superconductor. In this case, in-creasing the voltage eventually leads to a second-order phase transition to the normal state. We have found that the criticalvoltage at which the transition occurs obeys a power law /H20851cf. Eq. /H208493.21 /H20850/H20852. In Sec. IVwe studied time-dependent states of the system. In this way we were able to demonstrate the stability of thestationary states we have found in the previous section. Ourresults also indicate that a dc-biased system always relaxesinto a stationary state. In the parameter region of multiplestationary states we demonstrated bistability. Associated withthis are first-order phase transitions: there are critical volt-ages where /H9004/H20849and the current /H20850make finite jumps. Further- more, there is hysteresis of /H20841/H9004/H20841and the current associated with the bistability. ACKNOWLEDGMENT This research was supported by the Dutch Science Foun- dation NWO/FOM. 1C. J. Lambert and R. Riamondi, J. Phys.: Condens. Matter 10, 901 /H208491998 /H20850. 2C. W. J. Beenakker, in Mesoscopic Quantum Physics , edited by E. Akkermans, G. Montambaux, J.-L. Pichard, and J. Zinn-Justin/H20849North-Holland, Amsterdam, 1995 /H20850. 3G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25, 4515 /H208491982 /H20850. 4A. F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 /H208491964 /H20850/H20851Sov. Phys. JETP 19, 1228 /H208491964 /H20850/H20852. 5B. D. Josephson, Phys. Lett. 1, 251 /H208491962 /H20850. 6M. Tinkham, Introduction to Superconductivity /H20849McGraw-Hill, New York, 1996 /H20850. 7N. B. Kopnin, Theory of Nonequilibrium Superconductivity /H20849Clarendon, Oxford, 2001 /H20850. 8R. A. Barankov and L. S. Levitov, Phys. Rev. Lett. 96, 230403 /H208492006 /H20850; Phys. Rev. A 73, 033614 /H208492006 /H20850. 9E. A. Yuzbashyan, B. L. Altshuler, V. B. Kuznetsov, and V. Z. Enolskii, J. Phys. A 38, 7831 /H208492005 /H20850; Phys. Rev. B 72, 220503 /H20849R/H20850/H208492005 /H20850. 10R. S. Keizer, M. G. Flokstra, J. Aarts, and T. M. Klapwijk, Phys. Rev. Lett. 96, 147002 /H208492006 /H20850. 11P. G. de Gennes, Superconductivity of Metals and Alloys /H20849Ben- jamin, New York, 1966 /H20850. 12I. U. Giaver, U.S. patent 116427 /H208491963 /H20850. 13M. Hidaka, S. Ishizaka, and J. Sone, J. Appl. Phys. 74, 7402 /H208491993 /H20850. 14J. Sánchez-Cañizares and F. Sols, J. Low Temp. Phys. 122,1 1 /H208492001 /H20850; J. Phys.: Condens. Matter 7, L317 /H208491995 /H20850. 15A. Martin and C. J. Lambert, Phys. Rev. B 51, 17999 /H208491995 /H20850. 16V. M. Galitskii, V. F. Elesin, and Yu.V. Kopaev, in Nonequilib- rium Superconductivity , edited by D. N. Langenberg and A. I.Larkin /H20849Elsevier, Amsterdam, 1984 /H20850, p. 377. 17V. F. Elesin, Sov. Phys. JETP 46, 185 /H208491977 /H20850. 18G. M. Eliashberg, JETP Lett. 11,1 1 4 /H208491970 /H20850. 19Yu. M. Gal’perin, V. I. Kozub, and B. Z. Spivak, Sov. Phys. JETP 54, 1126 /H208491981 /H20850. 20Yu. V. Nazarov and G. M. Eliashberg, Fiz. Nizk. Temp. 9, 247 /H208491983 /H20850. 21L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 /H208491964 /H20850/H20851Sov. Phys. JETP 20, 1018 /H208491965 /H20850./H20852 22J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 /H208491986 /H20850. 23A. Brinkman, A. A. Golubov, H. Rogalla, F. K. Wilhelm, and M. Yu. Kupriyanov, Phys. Rev. B 68, 224513 /H208492003 /H20850. 24Several different conventions exist for the definition of the Keldysh structure of the Green’s functions. Here we use theconvention of Eqs. /H208492.26 /H20850and /H208492.27 /H20850in Ref. 22. 25W. Belzig, G. Schön, C. Bruder, and A. D. Zaikin, Superlattices Microstruct. 25, 1251 /H208491999 /H20850. 26Yu. V. Nazarov, in Handbook of Theoretical and Computational Nanotechnology , edited by M. Rieth and W. Schommers /H20849American Scientific Publishers, Stevenson Ranch, CA, 2006 /H20850. 27Yu. V. Nazarov, Superlattices Microstruct. 25, 1221 /H208491999 /H20850. 28Yu. V. Nazarov, Phys. Rev. Lett. 73, 1420 /H208491994 /H20850. 29To derive this equation from the full theory as presented in Ref. 27, the tunneling limit must be taken. This boils down to using Eq. /H2084914/H20850of Ref. 27instead of the more general Eq. /H2084936/H20850. 30A. L. Shelankov, J. Low Temp. Phys. 60,2 9 /H208491985 /H20850. 31cf. Eqs. /H2084936/H20850and /H2084937/H20850of Ref. 27. 32While our numerical scheme breaks down when 0 /H11021ETh/H11270/H9004 0,i t is again possible to do numerics when the system is perfectlyisolated, i.e., when E This strictly zero.BISTABILITY IN VOLTAGE-BIASED NORMAL- … PHYSICAL REVIEW B 79, 014510 /H208492009 /H20850 014510-13

PhysRevB.102.174423.pdf

PHYSICAL REVIEW B 102, 174423 (2020) Mn 1/4NbS 2: Magnetic and magnetotransport properties at ambient pressure and ferro- to antiferromagnetic transition under pressure S. Polesya,1S. Mankovsky ,1H. Ebert ,1P. G. Naumov,2,3M. A. ElGhazali,2W. Schnelle,2S. Medvedev ,2 S. Mangelsen ,4and W. Bensch4 1Department Chemie /Physikalische Chemie, Ludwig-Maximilians-Universität München, Butenandstr. 5-13, 81377 München, Germany 2Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany 3Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow 119333, Russia 4Inst. für Anorgan. Chemie, Christian-Albrechts-Universität Kiel, Olshausenstr. 40, 24098 Kiel, Germany (Received 27 March 2020; revised 9 September 2020; accepted 27 October 2020; published 12 November 2020) Transition-metal dichalcogenides (TMDCs) stand out with their high chemical stability and the possibility to incorporate a wide range of atoms and molecules between the layers. The behavior of conduction electronsin such 3 d-metal-inserted materials is closely related to their magnetic properties and can be sensitively controlled by external magnetic fields. Here, we study the magnetotransport properties of Mn-inserted NbS 2, Mn 1/4NbS 2, demonstrating a complex behavior of the magnetoresistance and of the ordinary and anomalous Hall resistivity. Application of high pressure as tuning parameter leads to the drastic changes of the magnetotransportproperties of Mn 1/4NbS 2exhibiting large negative magnetoresistance up to −65% at 7.1 GPa. First-principles electronic structure calculations indicate a pressure-induced transition from a ferromagnetic to antiferromagneticstate. Theoretical calculations accounting for the finite temperature magnetic properties suggest a field-inducedmetamagnetic ferromagnetic-antiferromagnetic transition as an origin of the large negative magnetoresistance.These results inspire the development of materials for spintronic applications based on 3 d-element-inserted TMDCs with a well controllable metamagnetic transition. DOI: 10.1103/PhysRevB.102.174423 I. INTRODUCTION The transition-metal dichalcogenides (TMDCs) have been the focus of various investigations for many years becausethey exhibit a broad spectrum of structure- and composition- dependent physical properties. Being nonmagnetic itself, some of the known TMDC materials allow insertion [ 1]o f magnetic 3 delements [ 2], having a tendency to the formation of ordered compounds when the concentration of insertedelements is close to 25% or 33%. This creates a family ofTMDC-based magnetic compounds exhibiting rather pecu-liar magnetic and transport properties, which, however, have not been well investigated so far. Among the properties of the 3 d-element-inserted TMDCs one can mention the high magnetic anisotropy [ 3,4] and anomalous Hall effect (AHE) [3,5]i nF e 1/4TaS 2observed experimentally and discussed re- cently on the basis of the first-principles calculations [ 6,7]. A large magnetoresistance (MR) has been found in disor-dered Fe xTaS 2single crystals, up to 60% at x=0.28 [8] and 140% at x=0.297 [ 9], that was attributed to spin disorder and strong spin-orbit coupling in the system. A long-periodhelimagnetic structure along the hexagonal caxis has been observed experimentally in Cr 1/3NbS 2, with the Cr magnetic moments aligned within the basal plane perpendicular to thecaxis [ 10,11]. Stabilization of the helimagnetic structure in Cr 1/3NbS 2has been confirmed theoretically by means of first principles calculations [ 12]. Similar to Cr 1/3NbS 2heli- magnetic properties have also been predicted for Mn 1/3NbS 2[12], while for Fe 1/3NbS 2stability of the magnetic structure referred to as an ordering of the third kind was demon-strated [ 12], in agreement with experimental results [ 13–15]. The calculations demonstrate a transition to the AFM statealso for Co 1/3NbS 2and Ni 1/3NbS 2[16], again in line with experiment [ 17]. Application of high pressure provides an appealing degree of freedom to manipulate the physical properties of solids,albeit only a few studies on the 3 d-element-inserted TMDC materials have been done so far. The pressure-induced elec-tronic structure modification can have either a direct impacton the physical properties or can lead first of all to a crys-tal structure transformation (see, e.g., Refs. [ 18,19]). As a direct impact one can mention the pressure-induced changesof the magnetic and transport properties of Co 1/3NbS 2[20] as well as an enhancement of the superconducting proper-t i e so fF e xNbS 2[21] reported recently. On the other hand, a suppression of the helimagnetic structure has been observedfor Cr 1/3NbS 2at comparatively low pressure ( ≈3–4 GPa) as a result of the structural transformations in the system [ 19]. In the present work we focus on the magnetic properties of Mn-inserted 2H-NbS 2, with 25% manganese concentra- tion, at zero and high pressures up to ≈10 GPa. First, we discuss the experimental results of magnetotransport mea-surements (magnetoresistance and Hall effect), because theyare crucial for inferring information about the interactionsbetween itinerant charge carriers and the magnetic degreesof freedom. To interpret the details of the experimental 2469-9950/2020/102(17)/174423(13) 174423-1 ©2020 American Physical SocietyS. POLESYA et al. PHYSICAL REVIEW B 102, 174423 (2020) observations, we discuss our theoretical results based on den- sity functional theory (DFT) calculations and Monte Carlosimulations. II. TECHNICAL DETAILS A. Experimental details 1. Sample preparation A mixture of 1 g in total of the elements (Mn, 99.95%, Alfa Aesar; Nb, 99.9% ChemPUR; S, 99.9995%, Alfa Aesar) withnominal stoichiometry Mn 0.265NbS 2were ground and loaded into an argon flushed silica ampoule. CBr 4(≈25 mg, 98%, Fluka) was added as transport agent. Due to the volatility ofCBr 4the lower end of the filled ampoule was cooled with liquid nitrogen prior to evacuation ( p/lessorequalslant1×10−4mbar), after which the ampoule was sealed. The ampoule was placed in thenatural gradient of a single tube furnace in inverse position forthe prereaction and cleaning transport. After heating to 450 ◦C during 6 h, the temperature was maintained there for 10 hbefore it was raised to 900 ◦C. After two days the ampoule was placed in a gradient 900◦C→≈ 800◦C and, within 14 d, crystals of several mm diameter in the basal plane grew. Theampoule was postannealed by slow cooling from 800 ◦Ct o room temperature within 24 h to allow for a good order of theMn ions which become disordered above ≈400 ◦C[22–24]. After opening the ampoules the crystals were washed withwater, a dilute solution of Na 2S2O3(97%, Grüssing), water, and acetone. The crystals have dimensions up to several mmin diameter and 1 mm in height and show a metallic silverluster. They were stored in an evacuated desiccator until use. 2. Characterization Composition analysis by energy-dispersive x-ray spec- troscopy (EDXS) was performed using a Philips XL30 ESEMequipped with an EDAX detector operated at 20 kV accel-eration voltage. At least two crystals from each batch wereanalyzed, on each crystal the composition was measured onthree spots with an integration time of 120 s each. T h es e r i e so f0 0 l-reflections were measured on a Pana- lytical X’Pert Pro-MPD (Cu-K αradiation, 1 /16◦divergence mask, Göbel mirror, primary Soller slit (0.04 rad) on theincident beam path, parallel plate collimator, and PIXcel 1Ddetector on the diffracted beam path) in Bragg-Brentano ge-ometry. For sample alignment and rocking curves a receivingslit was used. For collecting diffraction patterns in [001] orientation a thin slice ( /lessorequalslant100μm) was cut from a crystal with a thin razor blade. The crystal piece was mounted on adhesivetape and measured in transmission geometry on a PanalyticalEmpyrean diffractometer [Cu- K αradiation, 1 /2◦divergence mask, focussing mirror, primary Soller slit (0.04 rad) onthe incident beam path, secondary Soller slit (0.04 rad) andPIXcel one-dimensional (1D) detector on the diffracted beampath]. The instrumental broadening was derived from a mea-surement of LaB 6(NIST SRM 660c) via fitting the profile to a Thompson-Cox-Hastings Pseudo-V oigt profile with a Pawley-fit, and determination of the lattice parameters was done usingTopas-Academic Version 6.0 [ 25] via Pawley-fits.3. Ambient-pressure experiments The magnetization was measured with a MPMS3 (SQUID- VSM, Quantum Design) on a single crystal ( m=1.859 mg; flat platelet showing hexagon faces) with the magnetic fieldapplied along the [1000] (in-plane; IP) or along the [0001](out-of-plane; OP) direction, respectively. M(μ 0H) isotherms at selected temperatures were recorded after cooling in zerofield from T=200 K, temperature sweeps were taken during cooling in low fields. The heat capacity was determined onthe same crystal with the HC option of a Physical PropertyMeasurement System (PPMS-9, Quantum Design) at zerofield and with μ 0H=9 T applied along [0001] (OP). Transverse magnetoresistance ρ( T M R )a sw e l la sH a l l resistivity ρHdata as function of the magnetic field along [0001] (OP) were measured in a conventional four-wire and afive-wire (with external potentiometer) configuration, respec-tively. The electrical transport option of a PPMS-9 was usedto take data during magnetic-field sweeps at selected con-stant temperatures. For ambient pressure also the longitudinalmagnetoresistance (LMR) with the magnetic field appliedin-plane (IP) parallel to the current direction was measured. Symmetrization of the TMR and LMR and antisymmetriza- tion of the Hall data with respect to the applied field wasperformed. 4. High-pressure experiments For high-pressure experiments, a diamond-anvil cell (DAC) manufactured from the nonmagnetic alloy MP35N andequipped with Boehler-Almax design diamond anvils with500-μm culets was used. The tungsten gasket was insulated with a cubic BN-epoxy mixture. A single crystal sampleof suitable size (120 μm×120μm×10μm) was cut and placed into the central hole of the gasket filled with NaClas a pressure-transmitting medium along with a ruby chipfor pressure calibration. The electrical leads were fabricatedfrom 5- μm-thick Pt foil and attached to the sample in a van der Pauw configuration. Electrical resistivity was measured atdifferent pressures in the temperature range 1.8–300 K in amagnetic field up to 9 T. Magnetoresistance and Hall resis-tivity measurements at different pressures were performed byisothermal field sweeps, using the electronics and evaluationmethods as described above. High-pressure Raman spectra were recorded at room temperature using a customary confocal micro-Raman spec-trometer with a HeNe laser as excitation source and asingle-grating spectrograph with 1 cm −1resolution. Pressure was measured with an accuracy of ≈0.1 GPa using the ruby luminescence method. B. Computational details The first-principles electronic structure calculations have been performed using the spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) Green’s function method [ 26,27]. For the angular-momentum expansion of the Green’s functiona cutoff of l max=3 was applied. All calculations have been performed within the framework of the local spin-densityapproximation (LSDA) to density-functional theory (DFT)as well as beyond the level of LSDA by accounting for 174423-2Mn 1/4NbS 2: MAGNETIC AND MAGNETOTRANSPORT … PHYSICAL REVIEW B 102, 174423 (2020) correlation effects by means of the LDA +Uscheme [ 28,29]. The LSDA calculations used a parametrization for the ex-change and correlation potential as given by V osko et al. [30]. In the LDA +Ucalculations the so-called atomic limit expression was used for the double-counting correction inthe LDA +Ufunctional. The Hubbard parameters UandJ for Mn have been taken from the literature [ 31,32]. As these parameters depend to some extent on the specific materials,we used the appropriately “averaged” values U=3 eV and J=0.7e V . To investigate the equilibrium magnetic structure as well as finite-temperature magnetic properties, Monte Carlo simula-tions have been performed, which are based on the Heisenbergmodel using the exchange coupling parameters J ijcalculated here from first principles [ 33]. Within these simulations, en- ergy contributions due to the interaction with all neighborswithin a sphere with radius R max=2a(ais the lattice param- eter) have been taken into account. The temperature-dependent behavior of electronic resistiv- ity of the systems under consideration was investigated on thebasis of the Kubo-St ˇreda formalism in combination with the alloy analogy model. It allows us to account for thermal latticevibrations [ 34] as well as spin fluctuations [ 35], treating them within the adiabatic approximation [ 36]. First of all, the properties corresponding to ambient pres- sure have been calculated using the structure parametersobtained in experiment. Mn 1/4NbS 2exhibits a 2 ×2 super- structure in the Mn layers arranged within the Waals gap,leading to a structure with the space group P6 3/mmc (SG194), with a=6.67 and c=12.49 Å. This implies an occupation of the 2 aWyckoff positions by Mn atoms, and occupation of the 2 band 6 h(with x=0.5) positions by Nb atoms, and 4f(z=0.121) and 12 kpositions ( x=5/6,z=0.37) by S atoms. In the investigations of the pressure-dependent properties, the pressure-dependent structure parameters of Mn 1/4NbS 2 have not been measured. Therefore, auxiliary calculations have been performed using the V ASP package [ 37,38]i n order to determine the “relaxed” lattice parameters at am-bient pressure as well as for high pressures, keeping theexperimental c/aratio. In these calculations using the GGA density functional for the exchange and correlation potential,the PBE-parametrization scheme has been used as given byPerdew et al. [39]. As the van der Waals interactions may be important to describe correctly the pressure-dependentbehavior of TMDC-based system [ 40], these interactions have been taken into account using the DFT-D3 method forthe dispersion corrections as given by Grimme et al. [41]. The Monkhorst-Pack (8 ×8×8)k-point grid was used for the integration over the Brillouin zone. A plane-wave basis setup to a cutoff energy of 440 eV was used for the wave-functionrepresentation. As we know from our previous calculationson HfTe 2[40], such an approach gives reliable results for thep-Vrelation, which was found in good agreement with experiment. It is worth noting that, in contrast with HfTe 2,t h e Mn-inserted NbS 2system does not exhibit a strong change of the van der Waals gap when pressure is applied. The calcu-lated lattice parameters at ambient pressure is a=6.67 Å. The other parameters (i.e., the atomic positions within the cell)show only a minor deviation from the experimental values. Intensity1/2 (a.u.) ω (deg)6.8 6.9 7 7.1 7.2 7.3 7.4Intensity1/2 (a.u.) 2Θ (deg)10 20 30 40 50 60 70 80 90 100 110 120 130Yobs Ycalc Difference002 004006008 0010 0012 0014(b)Intensity1/2 (a.u.) 2Θ (deg)20 30 40 50 60 70 80 90YObs Ycalc Difference 100 110200 120 300220 130400 240(a) FIG. 1. Observed and calculated diffraction patterns for a sample of Mn 0.25NbS 2f o rt h es e to f( a ) hk0 reflections and (b) 00 lreflec- tions. In the inset of panel (b) a representative rocking curve of the 002 reflection is presented. III. RESULTS A. Experiment: Ambient pressure 1. Sample characterization The details of sample preparation are described in Sec. II A. The elemental composition was determined by means of EDXS to be Mn 0.25NbS 2within the limits of ex- perimental accuracy. The Mn ions are known to form asuperstructure of 2 a×2awith respect to the 2H-NbS 2host lattice. In fact, the average composition determined is veryclose to the ideal value for this type of superstructure. Tocheck this, x-ray diffraction (XRD) was carried out on a thinpiece cut from a crystal, which was oriented with reflectionsof the [001] zone axis being under diffraction condition. In thecorresponding diffraction pattern [Fig. 1(a)] only reflections of type h00 and hk0 are visible, which have a very narrow full width at half maximum (FWHM), indicating large coherentlydiffracting domains. This holds true both for the host structure(110 reflection) as well as for the domains of the superstruc-ture (100 reflection). To further ensure the crystal quality theseries of 00 lreflections was measured [see Fig. 1(b)]. Also here a very low FWHM can be observed, indicating a largedomain size along the caxis. The rocking curves measured on the 00 lreflections are of a very low average FWHM [0 .06 ◦, cf. Fig. 1(b)], underscoring the high quality of the crystals. 174423-3S. POLESYA et al. PHYSICAL REVIEW B 102, 174423 (2020) -1.0-0.8-0.6-0.4-0.20.2 00.40.60.81.0-1.0-0.8-0.6-0.4-0.20.2 00.40.60.81.0 MB PI) (MB PO) (T= 2 K 10 K 25 K 50 K 75 K 100 K 150 K 200 K -0.16-0.3 -0.2 -0.1 0.1 0.2 0.3 0-0.080.080.16 0 M) ((a) (b) 00.20.40.60.81.0 0 40 80 120(e)IP OP50 mT100 mT M) ( T(K) -0.9-0.6-0.30.30.60.9 0 -30 -20 -10 10 02 0 3 0(d) H(mT)M) ((c) -8 -6 -4 -2 0 2 4 68 0H(T) FIG. 2. Magnetic moment Mper formula unit of Mn 1/4NbS 2.( a ) MOPfor magnetic field along [0001]. The inset (b) shows the central part of minor loops up to ±0.4T .( c ) MIPfor field along [1000]. The inset (d) shows the strongly magnified central part of minor loops up to±0.2 T. The inset (e) displays magnetic moment obtained during cooling in low fields of 50 and 100 mT for both the IP and OP direction. The data sets are of high density and are thus displayed as lines. Only in the two lower insets is the individual data shown assymbols. From the XRD data presented here we can conclude that only the 2 a×2atype superstructure is formed by the Mn- ions with long-range order. This is evinced by the presenceof only one set of 00 lreflections and, in particular, by the diffraction pattern of the [001] zone axis, where only reflec-tions belonging to this type of superstructure are present.The lattice parameters are a=b=6.6715(4) Å and c= 12.4932(1) Å, in very good agreement with reports from lit- erature [ 13,42,43]. These measurements showed reproducible results on several crystals from the same batch. In summary,the samples can be described as nearly perfect Mn 1/4NbS 2. 2. Magnetization The magnetization of Mn 1/4NbS 2as a function of applied magnetic field (represented in μBper formula unit) is shown in Fig. 2. All curves in Figs. 2(a)–2(d) are isothermal five- segment loops of which none shows a visible field hysteresiseffect. Magnetization data obtained while cooling [Fig. 2(e)] or warming in low magnetic fields indicate ferromagnetic(FM) ordering at T C=104(2) K. From these temperature-dependent data as well as from the M(μ0H) isothermal loops it is evident that Mn 1/4NbS 2is a very soft easy-plane ferromagnet. For field IP the M(μ0H) curves for all temper- atures well below TCincrease within less than 10 mT to a temperature-dependent saturation value. Between μ0H=1T and 7 T (our maximum field) M(μ0H) increases by only 0.5%. In contrast, for the OP field direction, the magnetizationincreases distinctly slower with applied field and at a fielddetermined by the temperature finally also reaches a saturationvalue M sat. For the same temperature, the Msatvalues obtained atμ0H=7 T are quite similar for the OP and IP field direc- tions. At T=2.0 K for the IP as well as the OP orientations the saturation magnetization of 1 .05μBis attained, i.e., per Mn atom one has Msat=4.2μB. The observed behavior and the derived parameters are in fair agreement with a previousreport by ¯Onuki et al. [42] For OP fields up to ≈0.1 T [see Fig. 2(b)] a more pro- nounced increase of M(μ 0H) is observed than at higher fields. We will discuss the origin of this behavior in connection withrelated features in the magnetotransport in Sec. III A 5 . Only for temperatures well above T Cdoes the magne- tization become proportional to the applied field. Fits ofa Curie-Weiss law to the IP and the OP susceptibilitiesχ=M/Hbetween 200 and 300 K lead both to values of the effective magnetic moment per Mn atom μ eff≈5.5μB and a paramagnetic (PM) Curie-Weiss temperatures θCW≈ +128 K. The value for μeffis close to the value expected for af r e eM n2+ion with 3 d5configuration. Similar values for μeffwere reported previously [ 42,43]; however, θCWgiven in Ref. [ 42] differs due to the inclusion of a large nega- tive temperature-independent susceptibility χ0and a different range of the fit. For our data, we observe an isotropic magnetic behavior of Mn 1/4NbS 2at high T, which is due to the absence of orbital contributions for the Mn 3 d5electronic configuration. However, for T<200 K the magnetic moments are easily aligned by the application of a magnetic field. The polar-ization is still visible in the isothermal magnetization dataat 150 K [Figs. 2(a) and2(c)] which are slightly sublinear. A polarization by such relatively low magnetic fields is alsoobserved in the specific-heat and magnetoresistance data. 3. Specific heat The temperature dependence of the specific heat is pre- sented as cp(T)/Tin Fig. 3.cp(T) still increases at room temperature [ cp(298 K) =71.8Jm o l−1K−1] and the Dulong- Petit value 3 nR(Ris the ideal gas constant, n=3.25 the number of atoms) is not yet reached. For zero field, a smallstep-like phase transition is observed with its midpoint at104(2) K. No indications for latent heat could be detected inthe raw data, suggesting a magnetic long-range ordering andthe absence of a sizable lattice distortion at this temperature. Amagnetic field of μ 0H=9 T flattens out the peak and shifts its magnetic entropy towards higher T(up to ≈160 K), indicating a strong polarization of the magnetic moments of Mn in thismoderate applied field. At low temperatures (inset of Fig. 3)c p(T) in zero as well as in a 9 T magnetic field is described by the sum ofa linear and a cubic contribution, i.e., c p(T)=γT+βT3 174423-4Mn 1/4NbS 2: MAGNETIC AND MAGNETOTRANSPORT … PHYSICAL REVIEW B 102, 174423 (2020) 0 40 80 120 160 200 240 280 T(K)00.050.100.150.200.250.300.35 cTp/lom J(K) -1 -2Mn NbS1/4 2 cTp/lom Jm(K) -1-2 0 50 100 150 200010203040 T22(K )H=9 TH=0 T FIG. 3. Specific heat capacity cp/Tof Mn 1/4NbS 2for zero and 9 T magnetic field. The inset shows the data in a cp/TvsT2repre- sentation at low temperatures. in the temperature range 5–14 K. The cubic contribution is changed insignificantly by the applied field. It can be assignedto lattice phonons and a fit yields the Debye temperatureθ D≈330 K. The obtained values for the electronic contribu- tionγare 12.1(1) and 9.8(2) mJ mol−1K−2for zero and 9 T, respectively. The significant decrease of γwith field indicates a specific-heat contribution from magnetic fluctuations whichis (partially) quenched in a field of 9 T. Since the field ishigh enough to reach full magnetic saturation (see Fig. 2) the quenching is expected to be complete. Thus, the intrinsicSommerfeld parameter γfor the conduction electrons may be estimated to be ≈10 mJ mol −1K−2. 4. Electrical resistivity: Temperature dependence The electrical resistivity ρ(T) of a single-crystal (with measurement current in the abplane) is shown in Fig. 4.I t s temperature dependence measured at ambient pressure in zero 0.0 0.1 0.2 0.3 0.4 0 m( ytivitsiseR)mc temperature (K)Ambient pressure 5.5 GPa 7.1 GPa 10 GPa 13.5 GPa 16.5 GPa 19.1 GPa 22.2 GPa 50 100 150 200 250 300 FIG. 4. Electrical resistivity obtained experimentally for Mn 1/4NbS 2at zero magnetic field, plotted as function of temperature for different pressure values.field indicates that Mn 1/4NbS 2is a poor metallic conductor [ρ(300 K) ≈380μ/Omega1cm at 300 K]. The residual resistance ratio (RRR) of 30 demonstrates that the investigated crystalis of good quality (previously reported RRRs are <8[42] and≈15 [43]). The strongly curved increase of ρ(T)f o r T<T Cis the hallmark of a strong scattering of the charge carriers due to the spin disorder of the magnetic moments.An estimate [a linear fit to ρ(T) above 200 K] yields a rather small contribution from phonon and lattice defect scatteringof<100μ/Omega1cm at 300 K, i.e., the contribution from magnetic disorder scattering is dominant. The transition temperature T C is seen as a sharp change of slope. Some short-range order of the magnetic moments (at μ0H=0) is visible as the gradually decreasing negative curvature of ρ(T)u pt o T≈200 K. 5. Magnetoresistance The transverse magnetoresistance [ ρ(μ0H)−ρ(0)]/ ρ(0)×100% (TMR) and the Hall resistivity ρHas function of field applied out-of-plane (OP) are shown in Figs. 5(b) and 5(d), respectively. To rationalize the dependence of these properties on the corresponding OP magnetization, thederivative dM OP/dHis plotted in Fig. 5(a). The longitudinal magnetoresistance (LMR), with the field applied in-plane(IP) parallel to the current, needs to be compared with the IPmagnetization [Fig. 2(c)]. The field dependence of the TMR appears more complex than that of the LMR and the Hallresistivity. Three different regimes, dependent on temperature,applied field and field direction, may be distinguished. Asdetailed below, different scattering mechanisms govern thecarrier transport in these regimes. First we discuss the paramagnetic (PM) regime well above T C. TMR is small, negative, and almost quadratic with respect to the field, e.g., at T=200 K it is −1.9% at μ0H=9 T and a power-law fit results in an exponent of 1.915. At 300 K theTMR becomes slightly positive ( +0.3% at 9 T). The LMR is negative and has almost the same size as the TMR, e.g., at200 K it is −1.7% at 9 T and a power-law fit yields an ex- ponent 1.905. The Hall resistivity ρ His linear with respect to field (ordinary Hall effect, OHE) and results in Hall constants,R H=+2.3×10−9(at 200 K) and +1.9×10−9/Omega1mT−1 (at 300 K), respectively. Similar values have been reported previously ( +0.92×10−9[42] and +0.75×10−9/Omega1mT−1 [43]). Within a one-band model our values for T/greaterorequalslant200 K correspond to a hole concentration of 2 .7–3.5×1021cm−3. In the FM state, three regimes of magnetotransport behav- ior can be distinguished. To start with, the regime of saturatedmagnetization (FM-sat) is discussed. For IP fields, saturationis already achieved at fields below 10 mT, depending on thetemperature. Correspondingly, for all higher fields, the LMRshows only one type of field characteristics, which, however,depends on the temperature. LMR is negative and varies al-most linearly (slightly sublinearly) with field for T/greaterorequalslant10 K, e.g., it is −19% for T=50 K. For T=2 K the LMR in FM-sat regime shows clearly positive values. In contrast, forOP fields, saturation is achieved at high field values, whichcan be easily read from the derivative dM OP/dH[Fig. 5(a)]. In the FM-sat regime, also the TMR varies almost linearlywith field. It is negative for T=50 K but turns positive at some temperature above 10 K. The negative sign of both LMR 174423-5S. POLESYA et al. PHYSICAL REVIEW B 102, 174423 (2020) -8 -6 -4 -2 2 04 6 8 0H(T)(d)T=2 K 10 K 50 K 100 K 150 K 200 K 250 K2 K 10 K 50 K 100 K 150 K 200 K 250 K 300 KT=2 K 10 K 50 K 100 K 150 K 200 K 250 K 300 KT=2 K 10 K 50 K 100 K 150 K 200 K 250 K 300 K75 K (c)(b)(a) -3-2-10123 H)m 01(-8-20-15-10-50510 R ML %-20-15-10-505 R MT %00.10.20.30.40.5 Hd MdPO)eO lom/U ME( / FIG. 5. Comparison of magnetization and magnetotransport be- havior of Mn 1/4NbS 2: (a) Derivative dM OP/dHof the magnetization (field out-of-plane). (b), (c) Transverse (TMR) and longitudinal mag- netoresistance (LMR) in percent. (d) Hall resistivity ρH(μ0H). The dashed vertical lines mark characteristic changes with field for T= 2.0, 50, and 100 K. and TMR is connected to the slow increase of the Mn moment (polarization) enforced by an increasing field, which is evidentforT/greaterorequalslant25 K for both field directions (see Fig. 2). In theFM-sat regime, the slopes of all Hall resistivity curves are similar. This suggests a rather temperature-independent holecarrier concentration in Mn 1/4NbS 2. In the low-field FM regime (FM-low), MOP(μ0H) varies linearly with H[cf. Fig. 2(a)] due to continuous turning of the Mn moments out of plane. Thus, dM OP/dHtakes a con- stant high level and the regime borders can be determinedeasily (see vertical dashed lines in Fig. 5). Remarkably, in this regime, the Hall resistivity curves [Fig. 5(d)] are linear in field and the slopes are all the same for T=2 to 100 K. R HforT=2 K is calculated as 0 .97×10−9/Omega1mT−1, about half the value of the PM regime. In the FM-low regime theTMR is positive and linear for T=2 K, but the amplitude diminishes rapidly and goes to zero for T/greaterorequalslant50 K. For IP fields, the FM-low regime is limited to |μ 0H|<0.01 T and is only visible in the LMR curve for 2 K as a very sharp negativeMR cusp ( ≈− 3.5%) around zero field. Further interesting phenomena are observed at fields below 0.5 T. The much steeper initial increase of M OP[apparently forμ0H/lessorequalslant0.1 T, see Figs. 2(a) and2(b)]i sm i r r o r e db ya peak around zero field in the derivative [Fig. 2(a)]. From the derivative it becomes clear that MOPis anomalous up to fields of≈0.5 T in all curves from 2 to 75 K. This peak matches with a similarly wide peak in the TMR [Fig. 5(b)]f o r T=2K (and 10 K), which results in a relative resistance decrease of≈− 2.5%. Remarkably, there is no such feature in the Hall resistance [Fig. 5(d)]. For IP magnetic fields, the magnetization is saturated within ≈0.01 T [easy plane, see Fig. 2(d)]. A corresponding LMR peak is visible for T=2K [ F i g . 5(c)], where an initial decrease of resistivity by −3.4% within the same field range clearly stands out on the curve of overall positive LMR. Thisfeature is also visible at 10 K but not for the higher tempera-ture curves. We assign both low-field features to the existence of magnetic domains with different IP magnetization directions.When cooling the hexagonal crystal platelet in zero field toT<T C, domains with magnetization in different directions in-plane will form spontaneously. An IP field of ≈0.01 T is sufficient to form a single domain. It might be assumedthat the charge-carrier scattering on magnetic domain wallsis not strongly temperature dependent. Then the decrease inresistivity due to the removal of the domain walls will showup as a negative LMR [i.e., relative decrease /Delta1ρ/ρ (0)] only when the zero-field resistance is low, i.e., at T/lessorequalslant10 K. For magnetic fields applied OP, any small misalignment of the crystal platelet with respect to the applied field willalso lead to the selection of a particular domain which thengrows at the expense of the other domains. From a comparisonof IP and OP low-field M(T) measurements in field-cooling mode (see Fig. S1, Ref. [ 44]), a misalignment of 0 .8 ◦can be estimated, which is a realistic inaccuracy for the crystalmounted in our magnetometer for field OP. Due to such asmall misalignment, the IP component of the applied fieldis small and the corresponding OP magnetization and TMRfeatures will span a much larger range of field around zero(here, up to 0.5 T). Since the low-field features are observed only at low Tin LMR and TMR but not in the Hall effect, they neither origi-nate from thermal excitation nor from a finite Berry curvature. 174423-6Mn 1/4NbS 2: MAGNETIC AND MAGNETOTRANSPORT … PHYSICAL REVIEW B 102, 174423 (2020) Moreover, the aforementioned low-field MOPdata (Fig. S1 in the Supplemental Material [ 44]) are strictly proportional to the applied field, excluding a sizable out-of-plane canting ofthe ordered moment in Mn 1/4NbS 2. Magnetic moment canting (weak ferromagnetism) in the basically antiferromagnetic or-dered structure of Co 1/3NbS 2has been proposed recently to be the origin of a large anomalous Hall effect [ 45]. An alternative origin for the phenomena might be a small misaligned crystal-lite in our sample. However, in a microscopic inspection of thespecimen we could not see any indications for such a crystaldomain. In the field regions between the FM-low and FM-sat regimes, where M OPvaries sublinear with H, the slope of the Hall resistivity changes between the above-mentioned typicalvalues. The TMR, however, is negative due to the ongoingturning of Mn moments out-of-plane. For T=2 K this effect leads to the drastically negative TMR behavior for 3 .5T< |μ 0H|<5.7T .F o r T=100 K (just below TC) the increasing forced polarization of the Mn moments in both OP and IPmagnetic fields leads to large negative TMR and LMR. Bothvary sublinearly with Hand both reach around −24% at 9 T, i.e., a few K below T Cand for fields μ0H>3.5T ( F M - s a t regime) the magnetotransport and also the magnetization isalmost isotropic. At T=150 K both TMR and LMR show a behavior between normal paramagnetic behavior and theinfluence of increasing magnetic polarization. The size of theTMR and LMR at 9 T is almost the same ( ≈− 8.5%), i.e., the quenching of the spin fluctuations is similarly effective forfields applied OP or IP. This suggests that the spin fluctuationsare isotropic, as expected for the 3 d 5state of Mn2+. The magnetization as well as the TMR and Hall resistiv- ity curves do not show any hysteresis. For low temperatures(2–50 K), the linear extrapolation of the ρ Hcurves from the high-field FM-sat regime to zero field results essentially inzero intercepts (maximum −1.1×10 −9/Omega1ma t T=10 K). As this extrapolation allows us to identify the intrinsic contri-bution to the anomalous Hall resistivity in ordered materials,one can conclude that it is quite small for Mn 1/4NbS 2.H o w - ever, for the curve at T=100 K an intercept (anomalous Hall effect) of +15.9×10−9/Omega1m is obtained. For this temperature right below TCthe ordered in-plane moment is very small. A perpendicular field of ≈1.0 T readily turns the moment OP [cf. vertical dashed line in Figs. 5(a) and5(d)], leading to a huge Hall resistance at this field. The slope of ρHabove this field is between those in the PM and FM-low regimes.Interestingly, for T=150 K and μ 0H=9 T. the quite small OP polarization leads to an even slightly larger anomalous ρH than that at 100 K. This observation demonstrates that the AHE in Mn 1/4NbS 2is entirely associated with the extrinsic contribution due to spin fluctuations. B. Experiment: High pressure As one can see in Fig. 4, the resistivity in zero magnetic field for all pressures decreases upon cooling the sampledown from room temperature, changing the slope at a criticaltemperature T c, indicating a transition to some magnetically ordered state. At ambient pressure, this corresponds to a Curietemperature T C≈104 K as Mn 1/4NbS 2exhibits FM order at lower temperatures. A pressure increase up to 10 GPa results FIG. 6. (a) Pressure evolution of Raman spectra and (b) pressure dependence of the frequencies of the observed Raman peaks. in a continuous decrease of the critical temperature down to ≈75 K at 10 GPa. However, it increases abruptly to ≈135 K at 13.5 GPa, while a further pressure increase results in a de-crease of the critical temperature, reaching 40 K at a pressureof 22.2 GPa; the highest pressure attained in this study. The sudden increase of the critical temperature at 13.5 GPa might indicate a change of the crystal structure of Mn 1/4NbS 2 above 10 GPa. To monitor the possible structural phase transi- tion, Raman spectra were recorded to monitor changes underpressure, as shown in Fig. 6(a). The Raman spectra at am- bient pressure are qualitatively similar to that of the relatedisostructural compound Fe 0.239NbS 2[46]. All Raman reso- nances observed at ambient pressure are persistent in theRaman spectra up to the pressures beyond 20 GPa [Fig. 6(a)] while the frequency of these peaks shown in Fig. 6(b) shows normal increase with pressure without any discontinuities, in-dicating no structural phase transition in this pressure regime.These observations allow us to conclude that the pressure-induced modification of the transport properties observed inexperiment has to be attributed to the changes either of theelectronic or magnetic structure of Mn 1/4NbS 2. This applies in particular to the changes of the electrical resistivity observedabove 10 GP mentioned above. However, we will focus belowon the pressure effect in the region up to ≈10 GPa, while the results obtained at higher pressure need more detailedinvestigations. The transverse magnetoresistance (TMR) and Hall resis- tivity measured in the DAC at the lowest applied pressure of0.25 GPa (see Fig. S2(b) in the Supplemental Material [ 44]) demonstrate qualitatively a similar behavior compared withambient pressure. However, application of a higher pressurehas a significant impact on the shape of the Hall resistivityρ H(μ0H) and TMR( μ0H). The low-temperature magneto- transport data [Fig. 7, as well as Figs. S2(b), S2(d), S2(f), and S2(h), Ref. [ 44]] indicate that the pressure-induced change of the TMR can be associated with the modification of magneticproperties of Mn 1/4NbS 2. Discussing first the data at T=2K in Fig. 7(a), the positive TMR observed for T/lessorequalslant10 K in the FM-low regime at ambient pressure is continuously sup-pressed while the absolute value of the negative TMR at the 174423-7S. POLESYA et al. PHYSICAL REVIEW B 102, 174423 (2020) FIG. 7. Pressure- and temperature-dependent (a) transverse mag- netoresistance and (b) Hall resistivity at T=2K f o r M n 1/4NbS 2 plotted as function of external magnetic field. highest fields continuously increases. For p=5.5G P a ,T M R is negative for all applied magnetic fields, for all temperatureregimes (see Fig. S2(d), Ref. [ 44]). Approaching the pressure 7.1 GPa, negative TMR in the high-field regime increases upto 65% at μ 0H=9T[ F i g . 7(a)]. At p=10 GPa, TMR dras- tically drops to 3.2% at 9 T although it remains negative withnearly parabolic dependence. Above 10 GPa, TMR suddenlydecreases to very small positive values (only about 0.1% at9 T at 13.5 GPa). Hall resistivity curves ρ H(μ0H) at pressures up to 5.5 GPa at a temperature of 2 K [Fig. 7(b); Figs. S2(a), S2(c), S2(e), and S2(g), Ref. [ 44]] are qualitatively similar to that for ambient pressure demonstrating two regimes. These are stillpersistent at further pressure increase as can be seen from theρ H(μ0H) curve at 7.1 GPa [Fig. 7(b)]. For the magnetically ordered FM state ( T<TC)o fM n 1/4NbS 2, one can clearly see a nonlinear behavior of the Hall resistivity as function of themagnetic field. It changes with temperature and is associatedwith the extraordinary contribution vanishing together withthe mean magnetization in the magnetically disordered state[Figs. S2(a), S2(c), S2(e), and S2(g), Ref. [ 44]], leading to a linear dependence of the Hall resistivity on the magneticfield,ρ H(μ0H). At p=10 GPa, the Hall resistivity ρH(μ0H)is sublinear, while above 10 GPa ρH(μ0H) is linear at all tem- peratures (above and below the critical temperature derivedfrom the temperature dependence of resistivity), indicating thesuppression of the AHE, which can be attributed to a transitionto the AFM state characterized by a zero net magnetization.Note also that in this pressure regime, ρ H(μ0H) has a neg- ative slope revealing a dominating electron conductivity inMn 1/4NbS 2. In addition, one can point out that ρAHE mea- sured below TCchanges its sign twice upon pressure increase when the temperature approaches the Curie temperature. Thisobservation will be discussed below. To get more insight into the observed transport proper- ties, detailed information about the magnetic structure andits pressure and temperature dependence is required. At highpressure, however, we could not apply the same spectrumof experimental methods as at ambient pressure. A tiny flat-prismatic Mn 1/4NbS 2crystal was selected and positioned on the diamond culet of the DAC. Presently, with this DAC andcryostat, the magnetic field can be applied only parallel to thecrystallographic caxis. We performed TMR and Hall-effect measurements on the sample. However, quantitative magne-tization measurements are extremely difficult to realize ina DAC. Therefore the DFT-based theoretical investigationsare performed to interpret experimental data, suggesting apossible scenario for the pressure-dependent modifications ofelectronic structure and corresponding magnetic and transportproperties. This, however, should be confirmed by furtherexperimental investigations. C. Theory To allow for a detailed interpretation of the experimen- tal results, theoretical investigations based on first-principleselectronic structure calculations have been performed by us-ing the SPR-KKR band structure method [ 26,27]. The DFT calculations based on the local spin-density approximation(LSDA) for FM Mn 1/4NbS 2at ambient pressure lead to an underestimation of the Mn spin magnetic moment (3 .3μBvs experimental 4 .2μBper Mn atom), and to a suppression of the Mn spin magnetic moment at high pressure. Therefore,the calculations have been extended treating correlation ef-fects beyond LSDA, using the LSDA +Uapproach with the Hubbard Uparameter for Mn of U=3 eV. This leads to an increase of the spin magnetic moment of Mn at ambientpressure up to 3 .79μ Bas well as to a stabilization of the finite Mn spin magnetic moment on the Mn atom when the pressureincreases up to 11 GPa. The pressure-volume relation ( p-V) determined for ordered Mn 1/4NbS 2by using the V ASP package [ 37,38] (Fig. S3, Ref. [ 44]), shows a linear p-Vdependence up to a pressure of≈9 GPa. It should be noted that the lattice parameters of the system corresponding to p=0 GPa, that resulted from these calculations, slightly differ from experiment, as waspointed out above. However, this change in volume leadsto only minor changes of electronic structure and magneticproperties. In particular, the Mn spin moment is 3 .72μ B, while the orbital moment of Mn is 0 .021μBvs 0.019μBobtained for the experimental lattice parameters. Therefore, to be coherent,all results presented for ambient pressure have been obtainedfor the system with calculated lattice parameters. 174423-8Mn 1/4NbS 2: MAGNETIC AND MAGNETOTRANSPORT … PHYSICAL REVIEW B 102, 174423 (2020) At higher pressures one can expect an instability of the original crystalline or magnetic phase. This is in line withthe experimental results, which show significant changes ofthe TMR when the pressure increases above 10 GPa. On theother hand, the experimental Raman spectra do not exhibitany evidence for a structural phase transition in this pressureregime. As it follows from the electronic structure calcula-tions within the LSDA +Uapproach, a pressure increase up top=8 GPa leads to a decrease of the Mn spin magnetic moment from m=3.8μ Bat ambient pressure down to m= 3.2μBatp=8G P a , m=3.0μBatp=9 GPa and to 2 .21μB atp=10.5 G P a .T h u s ,u pt o p=9 GPa, one can see a slow decrease of the spin magnetic moment caused by a continuousbroadening of the Mn dbands accompanied by a decrease of the exchange splitting of the majority- and minority-spinstates and by an occupation of the top of the majority-spinband (see Fig. S4, Ref. [ 44]). In the pressure interval between 9 and 10.5 GPa, one can see a rather quick decrease of the Mnspin magnetic moment. Therefore, one may speculate about apossible pressure-induced transition to the low-spin magneticstate. As it follows from the Mn DOS for four pressure values,shown in Figs. S4 and S5, Ref. [ 44], a broadening of Mn d states at p=10.5 GPa leads in addition to a partial occupation of the minority-spin states of Mn, resulting in a quick decreaseof the Mn spin magnetic moment. However, more detailed in-vestigations on the properties of Mn 1/4NbS 2, both theoretical and experimental, are required for this pressure regime. Inconnection with our experimental data we focus here on thelow-pressure phase exhibiting a linear p-Vdependence (see p-Vdependence in Fig. S3, Ref. [ 44]). The properties of the Hall resistivity are determined by the features of the electronic structure. To demonstrate this, Fig. 8shows the calculated Bloch spectral functions (BSFs) A(/vectork,E) [Figs. 8(a) and8(b)], and A(/vectork ||,EF) [Figs. 8(c),8(d) and8(e),8(f)] representing the electronic band structure and Fermi surface cut with the (100) and (001) planes for thetwo pressures p=0 and 8 GPa, respectively. Because the resistivity measurements have been performed with current in the crystal abplane, we focus on the details of the electronic structure within the ( k x,ky) plane. A positive slope of the Hall resistivity ρH(μ0H) observed experimentally at ambient and small pressures implies a dom-inating hole-type character of the electric carriers in line withthe hole-like pockets around the Kpoint of the Brillouin zone [BZ; see Fig. 8(a)] created by the unoccupied top of the minority-spin bands [Fig. S6(a), Ref. [ 44]). This is also seen in the Fermi surface cut with the (001) plane, shown in Fig. 8 representing the Fermi surface for the minority-spin states[Fig. 8(d)], as well as by the hole-like pockets for the majority- spin states [Fig. 8(c); see also Fig. S6(a), Ref. [ 44]] along the /Gamma1-MandK-Mhigh-symmetry directions. When the pressure increases, the minority-spin pockets at the Kpoint disappear as the corresponding energy bands move down in energy, ascan be seen in Fig. 8(b), as well as in Fig. S6(b), Ref. [ 44], for p=8.0 GPa. At this pressure, the shape of the Fermi surface has essentially electron-like features [see Figs. 8(e) and8(f)], leading to an electron-like character reflected by the ordinaryHall effect (OHE), although the hole-like pockets created bythe majority-spin states around the /Gamma1point still survive at this pressure. FIG. 8. Total Bloch spectral function for Mn 1/4NbS 2under pres- sure for (a) p=0 and (b) 8.0 GPa. (c), (d) The Bloch spectral functions A(ky,kz,EF)( l e f t )a n d A(kx,ky,EF) (right) at the Fermi level for p=0, for the majority-spin and minority-spin states, re- spectively; the BSF’s A(ky,kz,EF)( l e f t )a n d A(kx,ky,EF) (right) in panels (e) and (f) correspond to the majority- and minority-spin states, respectively, for 8.0 GPa. The TMR is a characteristic of the change of the elec- trical resistivity ρ(T,μ0H) in the presence of a transversal magnetic field. The corresponding resistivities ρ(T,μ0H)a r e calculated for the single domain system, as a function of tem-perature. Therefore, in the case of FM-ordered Mn 1/4NbS 2, these results should be compared with the experimentaldata for an applied magnetic field strong enough to ensure 174423-9S. POLESYA et al. PHYSICAL REVIEW B 102, 174423 (2020) magnetization saturation in the system, i.e., in the regime we call FM-sat in Sec. III A 5 . It is worth noting that the magnetotransport properties of Mn 1/4NbS 2are associated with the dominating contribution of the electric carriers originating from the sandpstates of S and Nb. Due to a strong spin-dependent hybridization ofthese states with the dstates of Mn (Fig. S7, Ref. [ 44]), the resistivity of this material is very sensitive to the magneticconfiguration in the Mn sublattice and in turn strongly de-pends on the external magnetic field. The temperature dependent electrical resistivity of magnet- ically ordered metallic systems is determined by thermallyinduced lattice vibrations and spin fluctuations. Because ofa weak dependence of the lattice vibrations on the magneticfield, their contribution to the TMR can be neglected and wewill focus on the transverse spin fluctuations that are stronglyaffected by the temperature and dependent on the magneticstructure in the system. Monte Carlo (MC) simulations basedon the Heisenberg model give access to the dependence ofthe magnetization M(T,μ 0H) on temperature and external field providing in particular information on the amplitudeof the transverse spin fluctuations and as a result on theelectron scattering due to thermal spin fluctuations. As thesystem considered here is metallic, difficulties associated inparticular with the long-distance exchange coupling parame-ters and their dependence on the magnetic configuration mayoccur [ 47,48]. Despite that, the approach can be successfully applied to metallic materials, as was demonstrated by manygroups (see, e.g., Refs. [ 48–52]). The simulations use the exchange coupling parameters obtained within first-principlescalculations performed for the FM reference state. Positivefirst- and second-neighbor exchange coupling parameters J ij at low pressure [see Fig. 9(a)] guarantee the FM ground state of the system. The M(T,μ0H) dependence obtained within MC simulations at ambient pressure is shown in Fig. 9(b) for μ0H=0 and 4 T, respectively. The resistivities as a function of temperature calculated on the basis of M(T,0) and M(T,μ0H=4 T) lead to a negative magnetoresistance TMR( T) plotted in Fig. 10(b) . In this case the TMR is governed by the mechanism rather common forFM metals [ 53]: alignment of the spin magnetic moments along the magnetic field reduces the electron scattering anddecreases the resistivity ρ(T,μ 0H=4 T) with respect to ρ(T,0), leading to a negative TMR in the FM ordered system [54,55]. The impact of the field on the magnetization has a maximum around the Curie temperature, leading to a maxi-mum of negative TMR in this temperature region [minimumof corresponding curve in Fig. 10(b) ]. This is in good agree- ment with the experimental findings [Fig. 10(a) ]. When the pressure increases, the calculated exchange cou- pling parameters J ijexhibit significant changes, as shown in Fig. 9(a). The first-neighbor interaction parameters corre- sponding to an interaction between the Mn ions located inneighboring Mn layers in Mn 1/4NbS 2become negative when the pressure is approaching p=8 GPa. This should lead to an antiferromagnetic alignment of the magnetic moments ofthese layers if no other interactions are taken into account. Thesecond-neighbor parameters characterizing the Mn-Mn inter-actions within the layers are positive, stabilizing the FM orderwithin the layers, although they decrease with increasing pres-11 . 52 Rij (Units of lattice parameter)-8-6-4-2024Jij (meV)P = 0.0 GPa (FM) P = 3.5 GPa (FM) P = 5.1 GPa (FM) P = 8.0 GPa (FM) P = 9.0 GPa (FM) P = 8.0 GPa (AFM)(a) 05 0 1 0 0 1 5 0 Temperature (K)00.20.40.60.81M(T)/M(0) μ0H = 0.0 T μ0H = 4.0 Tp = 0 GPa(b) 05 0 1 0 0 1 5 0 Temperature (K)00.20.40.60.81M(T)/M(0) μ0H = 0 T (Ref. AFM) μ0H = 2 T (Ref. AFM) μ0H = 0 T (Ref. FM)p = 8 GPa(c) 0 50 100 150 Temperature (K)00.020.040.060.080.10.12M(T)/M(0) μ0H = 0. T μ0H = 4.0 Tp = 9 GPa(d)cenergy pressure 1/λ(spin modulation) pFM AFM 0 000 0(e) FIG. 9. (a) Pressure-dependent exchange interactions for Mn 1/4NbS 2. Open symbols represent the results calculated for the FM reference states, closed symbols the AFM reference state. The MC results for the temperature-dependent magnetization M(T)o f Mn 1/4NbS 2for different pressure: M(T) calculated for μ0H=0 (closed circles) and 4.0 T (open circles) at (b) p=0.0, (c) p=8.0 and (d) p=9.0 GPa. (e) Schematic representation of the total energy modification under pressure, plotted as a function of spin modulation characterized by period λ. 174423-10Mn 1/4NbS 2: MAGNETIC AND MAGNETOTRANSPORT … PHYSICAL REVIEW B 102, 174423 (2020) 00 . 5 1 1.5 2 T/TC-70-60-50-40-30-20-100% TMR p = 0 GPa p = 7.1 GPaExperiment, μ0H = 9 T(a) 00 . 5 11 . 5 2 T/TC-50-40-30-20-100% TMR p = 0 GPa p = 8 GPa(b) Theory FIG. 10. (a) The temperature-dependent magnetoresistance of Mn 1/4NbS 2: (a) Experimental results for the magnetic field of 9 T and the two pressures 0 and 7.1 GPa; (b) theoretical results for the magnetic field of 4 T used in the case of p=0a n d5G P a .T h er e s u l t s forp=8 GPa are obtained using the resistivities for the FM and AFM states, TMR =[ρ(T,FM)−ρ(T,AFM)] /ρ(T,AFM). sure. Despite negative first-neighbor interactions, the FM state turns out to be stable within the MC simulations [see Fig. 9(c), open diamonds] due to the contributions of all neighboringsites within a sphere with radius R=2a(with athe lattice parameter). However, the DFT-based total-energy calculationsforp=8 GPa give the difference E FM−EAFM=36 meV per formula unit, indicating stability of the AFM state. Giventhat the exchange coupling parameters can depend on thereference state, as was pointed out previously [ 47,48,56], the calculations of J ijfor Mn 1/4NbS 2atp=8 GPa have been performed also for the AFM reference state with thelayer-by-layer antiferromagnetic alignment of the magneticmoments. The resulting Mn-Mn exchange coupling parame-ters [Fig. 9(a), filled triangles], stabilize indeed the AFM state in line with the ground state deduced from total-energy calcu-lations. This is shown by MC simulations giving the M(T) dependence [Fig. 9(c), filled circles]. This implies that, at p≈8 GPa, the system has two energy minima corresponding to FM and AFM ordering (see as an example the discussionsof the two energy minima occurring for Fe xMn 1−xalloys, within spin-spiral calculations [ 57]). From the DFT total-energy calculations one can see that the AFM minimum is the global one, while the FM minimum is local. This is illustratedin Fig. 9(e), representing schematically the pressure-induced modification of the total energy as a function of the inversespin modulation period, demonstrating that way a continuoustransition from the FM state at ambient pressure to the AFMstate at high pressure. The formation of two minima in the vicinity of the critical pressure p cimplies the exchange parameters calculated for the FM and AFM reference states which allow for a localstability of these FM and AFM states, respectively. If theenergy difference between the FM and AFM states is rathersmall (keeping in mind a continuous transition under pressurefrom the FM to the AFM state), we expect that the appliedmagnetic field pushes the magnetic moments towards the FMalignment [open circles in Fig. 9(c)], stabilizing that way the FM state due to the modification of the Mn-Mn exchangeinteraction parameters mentioned above. When the stabilizingmagnetic field is switched off, any fluctuations or defects canbring the system back to the low-energy AFM state. How-ever, the details of the dynamics of these transitions needsfurther investigations beyond the present study. This concernsin particular the pressure-dependent energy barrier betweentwo energy minima. It should be noted that, for p=0G P a , the exchange parameters calculated both for the FM as wellas for the AFM reference state indicate a stability for theFM state, while for p=9 GPa—the stability of the AFM state is ensured again by both sets of exchange parameterscalculated for the FM and the AFM reference states. Thus, onecan expect a field-induced AFM-FM transition at p=8G P a . MC simulations for the pressure p=9 GPa demonstrate the stability of the layer-by-layer AFM state [see Fig. 9(d)]f o r which the field-induced metamagnetic transition is not possi-ble anymore [see Fig. 9(d), open circles]. To discuss the behavior of the magnetoresistance TMR( T) of Mn 1/4NbS 2atp=8 GPa, the change of the resistivity during the AFM-to-FM transition was calculated as a differ-ence of the electrical resistivities for the AFM (i.e., withoutmagnetic field) and for the FM (with magnetic field) statesof the system. The AFM state was approximated by thelayer-by-layer AFM structure, i.e., with two sublattices hav-ing antiparallel alignment of the magnetic moments. For thesake of simplicity, the temperature-dependent magnetizationfor each sublattice, M(T), was taken to be the same as for the FM state calculated for p=8 GPa [open diamonds in Fig. 9(c)]. The resulting TMR is shown in Fig. 10(b) by circles as a function of reduced temperature T/T C. A crucial result following from these calculations is the maximum of theTMR at low temperature due to weak thermal disorder in thesystem. With increasing temperature the TMR decreases andvanishes at the critical temperature due to a transition to theparamagnetic state. A similar behavior of the TMR has beenobtained experimentally, as shown in Fig. 10(a) by circles. Finally, it is worth to discuss briefly the pressure-dependent behavior of the nonconventional contribution to the Hall resis-tivity. As shown above, the high-field ρ H(μ0H) extrapolated to the H=0 T limit gives the anomalous Hall resistivity assuming FM order in the system, which exhibits a nonmono-tonic behavior changing sign twice upon the pressure increaseup to 10 GPa [see Figs. S2(a), S2(c), and S2(e), Ref. [ 44]]. 174423-11S. POLESYA et al. PHYSICAL REVIEW B 102, 174423 (2020) However, no indication for a sign change of the anomalous Hall resistivity has been found in calculations for the FM stateof the system. In contrast, in the intermediate pressure regime, before the layer-by-layer AFM state is stabilized, a noncollinear AFMstructure is expected as a result of competition between theFM and AFM interatomic exchange interactions. Moreover,in this case the Dzyaloshinskii-Moriya interactions (DMI)should have a crucial role for the formation of a chiral mag-netic texture, despite its magnitude ( ≈0.2 meV) being smaller by an order of magnitude when compared with the compet-ing isotropic exchange interactions. As a result, an additionaltopological contribution to the Hall resistivity [ 58,59] occurs in the presence of an external magnetic field, ρ THE[topologi- cal Hall effect (THE)] [ 59–61] ρxy=ρOHE+ρAHE+ρTHE, (1) withρAHE∼MzandρTHE∼Heff, where Mzis the magnetiza- tion component along the zdirection and Heffis the emergent magnetic field having topological origin and being nonzero inthe magnetic textures characterized by finite scalar chirality.Note also that a competition of the DMI with the isotropicexchange interactions and the applied magnetic field can alsolead to the formation of more complicated topologically non-trivial magnetic textures (e.g., skyrmions), as was predictedrecently for Fe 1/4TaS 2[62]. IV . SUMMARY To summarize, we studied the magnetic and magne- totransport properties of Mn 1/4NbS 2at ambient and high pressure both experimentally and by first-principles calcu-lations. The compound exhibits soft ferromagnetism withan easy plane of magnetization at ambient pressure. Theresults from magnetic-field-dependent measurements of theLMR and TMR below the Curie temperature show a complexdependency on magnetic field and temperature. Comparingthese with the M(H,T) data we identified multiple regimes. Spin reorientation and suppression of spin fluctuations play amajor role in the emergence of a strong negative LMR andTMR effects in this compound. The AHE seems to be drivenalmost completely by spin fluctuations as well. Upon application of pressure up to 18.8 GPa, no hints for a structural phase transitions are evident from the Ramanspectra. In contrast, the magnetotransport properties show remarkable changes pointing towards a change of either theelectronic or magnetic structure. The sign of the ordinaryHall effect changes from positive to negative upon pressureincrease from 5.5 to 7.1 GPa, which indicates a crossover fromholes to electrons as majority charge carriers. This findingwas confirmed by calculations of the BSFs at 0 and 8 GPa,where coexisting holes and electrons at the Fermi surfacewith accordingly altered balance was found. The maximumnegative TMR shows an impressive increase up to ≈65% at 7.1 GPa with an almost step-like increase beyond ≈4T o f magnetic field. Upon further pressure increase up to 16.5,the TMR is rapidly diminishing. The AHE can be tracedfor pressures up to 10 GPa, above which it vanishes. Thisindicates an overall crossover from ferro- to antiferromagneticorder. The underlying mechanism of the peculiar behavior at a pressure of 7.1 GPa was further examined by means of first-principle electronic structure calculations and Monte Carlosimulations. From the results we suggest a field-induced meta-magnetic transition, giving rise to the exceptionally largenegative MR effect. The results motivate further research, e.g.,on anion- or cation-substituted Mn 1/4NbS 2towards control- lable AFM to FM switching at ambient pressure. This mightbe feasible by substituting Mn for Fe, as Fe 1/4NbS 2shows AFM order [ 14] and one can expect a continuous change in dominant magnetic exchange. Possibly this competitionof exchange energies can also be exploited to induce chiralmagnetic textures by application of a magnetic field. ACKNOWLEDGMENTS The work in Dresden was supported by the German Re- search Foundation (DFG) under Projects No. ME 3652 /3-1 and No. GRK 1621 and by the ERC Advanced Grant No.742068 “TOPMAT.” P.G.N. acknowledges the support by theRussian Science Foundation (Project No. 17-72-20200). Fi-nancial support by the German Research Foundation (DFG)under the Project No. BE 1653 /35-1 and by the state of Schleswig-Holstein is gratefully acknowledged for the workcarried out in Kiel. S.P. and S.M. (München) acknowledgefinancial support from the DFG via SFB 1277 and via the DFGpriority program EB154 /36-1. 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PhysRevB.82.184412.pdf

Spin polarization and exchange coupling of Cu and Mn atoms in paramagnetic CuMn diluted alloys induced by a Co layer M. Abes, *D. Atkinson, and B. K. Tanner Department of Physics, University of Durham, South Road, Durham DH1 3LE, United Kingdom T. R. Charlton and Sean Langridge† ISIS, Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Oxon OX11 0QX, United Kingdom T. P. A. Hase Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom M. Ali, C. H. Marrows, and B. J. Hickey School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom A. Neudert‡and R. J. Hicken School of Physics, University of Exeter, Exeter EX4 4QL, United Kingdom D. Arena National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973-5000, USA S. B. Wilkins Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA A. Mirone European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France S. Lebègue Laboratoire de Cristallographie, Résonance Magnétique, et Modélisations, CRM2, UMR CNRS 7036, Institut Jean Barriol, Nancy Université, BP 239, Boulevard des Aiguillettes, 54506 Vandoeuvre-lès-Nancy, France /H20849Received 12 May 2010; revised manuscript received 4 October 2010; published 9 November 2010 /H20850 Using the surface, interface, and element specificity of x-ray resonant magnetic scattering in combination with x-ray magnetic circular dichroism, we have spatially resolved the magnetic spin polarization, and theassociated interface proximity effect, in a Mn-based high-susceptibility material close to a ferromagnetic Colayer. We have measured the magnetic polarization of Mn and Cu 3 delectrons in paramagnetic CuMn alloy layers in /H20851Co /Cu/H20849x/H20850/CuMn /Cu/H20849x/H20850/H20852 20multilayer samples with varying copper layer thicknesses from x=0 to 25 Å. The size of the Mn and Cu L2,3edge dichroism shows a decrease in the Mn-induced polarization for increasing copper thickness indicating the dominant interfacial nature of the Cu and Mn spin polarization. TheMn polarization is much higher than that of Cu. Evidently, the Mn moment is a useful probe of the local spindensity. Mn atoms appear to be coupled antiferromagnetically with the Co layer below x=10 Å and ferro- magnetically coupled above. In contrast, the interfacial Cu atoms remain ferromagnetically aligned to the Colayer for all thicknesses studied. DOI: 10.1103/PhysRevB.82.184412 PACS number /H20849s/H20850: 72.25.Mk, 75.25. /H11002j, 85.75. /H11002d I. INTRODUCTION Spin-dependent electron-transport phenomena such as gi- ant magnetoresistance, domain-wall magnetoresistance, andmagnetization-reversal processes induced by transverse spininjection, 1,2in ferromagnetic/nonmagnetic structures, show intriguing characteristics associated with interfacial phenom-ena. Devices based on spin transport effects have great ad-vantages over conventional electronic devices because of theadditional spin functionality. 3This functionality is often in- terface dominated. Of utmost relevance for spin transport isthe length scale over which the electron spin retains its initialpolarization direction. This is known as the spin diffusionlength /H20849/H9261 sf/H20850and its magnitude has been the subject of somedebate.4,5Clearly to understand such systems requires a de- tailed quantitative knowledge of the spatial distribution ofthe magnetization. This is experimentally challenging giventhat the polarization may be small, located at a buried inter-face and in proximity to material with a much larger magne- tization. Nonetheless, experimentally determining the spatialpolarization has widespread significance in areas such asRuderman-Kittel-Kasuya-Yoshida /H20849RKKY /H20850 interlayer coupling, 6–8induced proximity magnetism9and in the case of spin injection, the driven accumulation10,11of spin at an interface. Understanding the mechanisms behind the equilib-rium and out-of-equilibrium magnetization becomes impor-tant to the future realization and technological exploitation ofPHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 1098-0121/2010/82 /H2084918/H20850/184412 /H2084911/H20850 ©2010 The American Physical Society 184412-1spin-electronic devices, in particular, the lengthscale over which the spin functionality can be transported. In several cases /H20849RKKY , proximity magnetization, and spin accumulation /H20850the magnitude of the induced magnetiza- tion is small, typically of order 0.01 /H9262Bper atom with a spatial extent localized close to the interfacial region. Byestablishing a methodology to observe these small polariza-tions in an equilibrium /H20849proximity /H20850system we expect that such ideas can be extended to the, out-of-equilibrium, drivencase of spin accumulation. We have undertaken x-ray resonant magnetic scattering /H20849XRMS /H20850measurements combined with x-ray magnetic circu- lar dichroism /H20849XMCD /H20850to study the proximity effect by de- termining the moment of delta-doped 12high-susceptibility impurities in close proximity to a ferromagnetic layer. Theinterface sensitivity of the XRMS technique is ideal for char-acterizing the spatial extent of the spin polarization. 13–17As part of this work we have used hard x-ray reflectivity toprovide a structural characterization of the sample. To enhance the interfacial sensitivity we chose to study the polarization of Mn atoms in a paramagnetic CuMn alloywith a 50% Mn concentration. Mn is an ideal choice as elec-tronic band-structure calculations have shown that the Mnatoms possess almost a 100% spin polarization at the Fermilevel. 18,19 In this paper, we focus on the interface effect which, through the contact between Co, Cu, and Mn atoms, favors adirect hybridization and therefore induces ferromagnetism inMn and Cu. We investigate the Cu and Mn 3 dmagnetic pro- files in a /H20851Co /CuMn /H2084920 Å /H20850/H20852 20multilayer /H20849ML/H20850where there is direct contact between the ferromagnet and the paramag-netic layers. 19To spatially resolve the magnetic profile we have also studied the Cu and Mn magnetic profiles in/H20851Co /Cu/H20849x/H20850/CuMn /Cu/H20849x/H20850/H20852MLs. Here a delta layer of Cu /H20849xÅ thick /H20850has been inserted at the interfaces to separate the Mn and Co atoms. The magnetic profile is then modeled byfitting the x-ray data using a calculation based on the Fresnelformalism. 20 This study is organized as follows: the outline theory of XRMS is presented in Sec. II. The experimental details and growth of the multilayers are summarized in Sec. III. The experimental and theoretical results are compared with thecalculated reflectivity spectra and presented in Sec. IVand discussed in Sec. V. II. X-RAY RESONANT MAGNETIC REFLECTIVITY During the last three decades, magnetic soft x-ray spec- troscopy techniques have become an increasingly valuabletool in the characterization of the magnetic properties in ma-terials. These activities were stimulated by the observation ofstrong dichroic effects in absorption, transmission, or scatter-ing experiments in the vicinity of the 2 pedges of the 3 d transition metals. 21,22Among these techniques, x-ray reso- nant magnetic scattering has received considerable attentionand offers several specific advantages compared to absorp-tion or transmission measurements. It combines the benefitsof magnetic dichroism with a scattering technique and en-ables the determination of the spatial profile of the magneti-zation through the layers. The dichroic effects can exceed the magnitude of corresponding effects in photoabsorption andbenefit from a much larger probe depth. 21However, since the reflectivity depends on the experimental geometry and themorphology of the sample /H20849e.g., layer thicknesses and rough- nesses /H20850, a detailed analysis of spectroscopic reflectivity data requires a more sophisticated numerical treatment than thatassociated with the analysis of photoabsorption data. Never-theless, experiment and simulation demonstrate the promis-ing potential of reflectivity analysis. 23,24 At the Cu and Mn L2,3edges, the dichroic signal results from electric dipole transitions from the 2 patomic core level to the unoccupied 3 dstates which support the magnetic mo- ment. As with XMCD, the magnetic sensitivity arises fromthe exchange splitting of the unoccupied 3 dstates induced by their magnetic polarization and from the spin polarizationof the photoelectron which is related to the spin-orbit cou-pling in the 2 pcore level. Here we only cite the pertinent expressions relevant to the interpretation of our experimentalresults. The resonant x-ray atomic scattering factor of a magnetic atom can be written in the dipole approximation as 25 f/H20849E/H20850=/H20849/H9255ˆf/H11569·/H9255ˆi/H20850F0/H20849E/H20850−i/H20849/H9255ˆf/H11569/H11003/H9255ˆi/H20850·yˆF1/H20849E/H20850/H20849 1/H20850 with the charge- and magnetization-dependent scattering am- plitudes F0/H20849E/H20850=f0+f/H11032/H20849E/H20850+if/H11033/H20849E/H20850, /H208492/H20850 F1/H20849E/H20850=m/H11032/H20849E/H20850+im/H11033/H20849E/H20850, /H208493/H20850 respectively. Here Eis the energy of the incident x rays, /H9255iˆ and/H9255fˆare the unit polarization vectors of the incident and scattered x rays, respectively, and yˆis the magnetization unit vector. f0is the tabulated atomic scattering factor26andf/H11032/H20849E/H20850 and f/H11033/H20849E/H20850are the real and imaginary parts of the complex resonant anomalous scattering factor, respectively. In Eq. /H208493/H20850, m/H11032/H20849E/H20850and m/H11033/H20849E/H20850are the real and imaginary parts of the resonant magnetic scattering factor. Beyond the region fortotal external reflection we can define complex charge F/H20849q /H6023,E/H20850and magnetic structure Mˆ/H20849q/H6023,E/H20850factors23 F/H20849q/H6023,E/H20850=/H20858 jF0/H20849E/H20850eiq/H6023.r/H6023j, /H208494/H20850 Mˆ/H20849q/H6023,E/H20850=/H20858 jyˆjF1/H20849E/H20850eiq/H6023·r/H6023j. /H208495/H20850 In order to determine the profile of the magnetization of a layer, particularly one whose polarization is strongly thick-ness dependent, the number of atomic planes, their interpla-nar distances, and concentrations are obtained directly fromthe analysis of conventional, nonresonant hard x-ray diffrac-tion, and reflectivity data. Once the chemical/electronicstructure has been determined to sufficient precision, it ispossible to fit the element-specific XRMS in order to extractthe magnetic profile of the Mn and Cu layers. The intensitiesobserved in elastic scattering are then related to the square ofthe total structure factor to yield cross terms that representthe resonant magnetic-charge interference scattering. ThisABES et al. PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-2magnetic cross term can be accessed by taking the difference of the observed intensity, /H20849I/H11006/H20850, by either changing the helic- ity of the incident circularly polarized photons or reversingthe magnetization within the sample I +−I−=−2 /H20849kˆ+kˆ/H11032cos 2/H9258/H20850/H20849F/H11032Mˆ/H11032+F/H11033Mˆ/H11033/H20850, /H208496/H20850 where FandMˆare written as sums of the real and imaginary parts F/H20849q/H6023,E/H20850=F/H11032+iF/H11033, /H208497/H20850 Mˆ/H20849q/H6023,E/H20850=Mˆ/H11032+iMˆ/H11033, /H208498/H20850 where F/H11032,F/H11033,M/H11032ˆ, and M/H11033ˆare real quantities for centrosym- metric structures. In our measurements the magnetic fieldwas applied parallel and antiparallel to the direction definedby the sample plane and the scattering plane as shown in Fig.1. In Fig. 1, the y-zplane is parallel to the plane of x-ray scattering, defined by the incident and scattered wave vectors k /H6023andk/H11032/H6023, and the + zaxis is parallel to the outward normal of the multilayer surface. In specular reflection, k/H6023andk/H11032/H6023make angles − /H9258and/H9258to the + yaxis, respectively, which is along k/H6023+k/H11032/H6023.The scattered intensity was modeled by adapting Parratt’s recursion formula27for nonmagnetic specular reflectivity from a multilayer. The refractive index for layer jisnj/H11006=1 −/H9254j/H11006+i/H9252j/H11006for each of the applied magnetic field directions /H20849/H11006/H20850. The refractive indices are related to fandmfor the forward scattering by /H9254/H11006=/H208732/H9266n0re k2/H20874/H20851f0+f/H11032/H20849E/H20850/H11007m/H11032/H20849E/H20850cos/H9258/H20852, /H208499/H20850 /H9252/H11006=/H208732/H9266n0re k2/H20874/H20851f/H11033/H20849E/H20850/H11007m/H11033/H20849E/H20850cos/H9258/H20852, /H2084910/H20850 where n0is the number of atoms per unit volume and reis the electron classical radius. In practice it is often convenientto decompose the measured quantities into the magnetic andnonmagnetic contributions /H9254/H11006=/H92540/H11007/H9004/H9254, /H2084911/H20850 /H9252/H11006=/H92520/H11007/H9004/H9252. /H2084912/H20850 The absorptive charge and magnetic components, /H92520and/H9004/H9252, are then proportional to the measured linear absorption coef-ficient /H9262/H11006/H20849E/H20850through the optical theorem. III. EXPERIMENTAL DETAILS Polycrystalline samples were prepared at the University of Leeds, by dc magnetron sputtering operating at a basepressure of 5 /H1100310 −8Torr at room temperature. In order to enhance the /H20849111/H20850texturing the multilayers were grown on Ta buffer layers deposited onto silicon substrates and weresputtered at growth rates of 2.3 Å /s, 4.2 Å /s, and 3.9 Å /s for Co, Cu 50Mn 50, and Cu, respectively, in an argon atmo- sphere of 2.5 mTorr. The samples were protected from oxi-dation by a 15 Å Al capping layer. Six samples were grown in the same vacuum cycle with nominal structures, Si/H20849100/H20850/Ta/H20849200 Å /H20850//H20851Co/H2084940 Å /H20850/Cu/H20849x/H20850/Cu50Mn 50/H2084920 Å /H20850/Cu/H20849x/H20850/H2085220/Al/H2084915 Å /H20850 with x=0, 5, 10, 15, 20, and 25 Å. We shall refer to these samples as Co /Cu/H20849x/H20850/CuMn. All samples had an in-plane easy magnetization axis. The Co layers saturate below 400Oe with a remanent in-plane magnetization at nearly 100%of the saturation magnetization. XRMS and XMCD experiments were performed on a two-circle diffractometer with a vertical scattering geometryin a high vacuum chamber on station 5U1 at the DaresburySRS. Energies in the range of 200–1400 eV with a resolutionof/H11015150 meV and flux of typically 10 10photons/sec/100mA were available. A magnetic field was applied in the scattering plane along the sample surface. This corresponds to the ge-ometry employed in the longitudinal magneto-optical Kerr effect. The maximum field of /H11006500 Oe was large enough to fully saturate the Co layer in all samples. By tuning the pho-ton energy to the absorption edge of interest, a scan of thereflected intensity is performed as a function of scatteringvector. For each scattering vector an asymmetry ratio can bedefined R=/H20849I +−I−/H20850//H20849I++I−/H20850. A measurement of the energy- dependent asymmetry ratio at a fixed scattering vector q/H6023is sufficient to separate the structural and magnetic contribu-tions to the scattered intensity. For the measurements of theabsorptive parts /H20849XMCD /H20850, the photon incident angle was set at 70°.ykq=k'-k xzk' m 2θ θ FIG. 1. /H20849Color online /H20850Schematic of the longitudinal geometry used in the XRMS measurements. k/H6023andk/H11032/H6023are the wave vectors of the incoming and outgoing circularly polarized x rays, respectively. m/H6023is the magnetization of the sample. The applied field H/H6023is parallel to the yaxis.SPIN POLARIZATION AND EXCHANGE COUPLING OF Cu … PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-3The analysis of the multilayer magnetization profile is performed within the pythonic program for multilayers /H20849PPM /H20850 software.28In order to determine the structure of the samples, hard x-ray reflectometry measurements were re-corded on X22B at the National Synchrotron Light Source,Brookhaven National Laboratory at an energy of 8.9 keV .Specular scans /H20849q zscan /H20850provide information on the near- surface electron density and total interface width. They donot distinguish between compositional grading and trueroughness as there is no component of scattering vector inthe surface. Transverse diffuse scans /H20849q xscan at fixed qz/H20850allow the total interface width /H20849/H9268rms/H20850, determined from the specular scans to be subdivided into topological roughness and com-positional grading components. 29The transverse diffuse scan is sensitive to the in-plane structure of the interfaces and bysimulating the diffuse data, the in-plane correlation length, /H9264, and the fractal parameter hcan be determined. The hparam- eter can take values between 0 and 1 and is related to thefractal dimension of the surface and describes the jaggednessof the interface. Off-specular /H20849longitudinal diffuse /H20850scans were performed in which the scattering geometry is the sameas that of the specular scan but with a small offset /H20849−0.1° /H20850in the sample angle. These longitudinal scans in reciprocalspace, which probe the diffuse scatter close to the specularridge, are necessary in order for the true specular scatter tobe obtained by subtraction of the diffuse scatter from themeasured specular scatter. They also permit us to determinethe degree of conformal roughness, Kiessig fringes, andBragg peaks in the diffuse scatter arising only from the pres-ence of roughness that replicates through the multilayerstructure. IV . RESULTS A. Hard x-ray reflectometry A series of grazing incidence true specular scans for the MLs are shown in Fig. 2/H20849a/H20850. The Bragg peaks due to the periodicity of the MLs are clearly visible in the specularreflectivity spectra at /H110150.1 Å −1, which agree with the nomi- nal sample periodicity. In addition to the Bragg peaks thepresence of well-defined finite-thickness oscillations /H20849Kiessig fringes /H20850clearly indicates a well-ordered layered structure. The off-specular Bragg peaks remain /H20849not shown /H20850, indicating that the out-of-plane correlation is retained within themultilayer. The specular scatter of the multilayers was mod-eled using the Bede REFS-MERCURY software which uses a genetic algorithm to refine a model structure to the experi-mental data by an iterative cominimization process. 29,30Val- ues of thickness and interface width were first found usingthis code for the specular data and these parameters thenused to fit the diffuse scatter manually /H20851Fig.2/H20849b/H20850/H20852. Best fits to the experimental data are included in the figures. In all caseswe were able to obtain good fits between the simulated andexperimental data, the results for the MLs being shown inTable I. Transverse diffuse data taken at the second Bragg peak for all samples /H20851as in example Fig. 2/H20849b/H20850on the Co/ CuMn multilayers /H20852and the associated simulations show that there is no or very small detectable interdiffusion contribu-tion to the interface width between the CuMn/Co, Co/CuMn, Cu/Co, and Co/Cu layers in any of the samples and all of theeffective roughness corresponds to rms topological rough-ness. The in-plane correlation length of the roughness mor-phology was determined to be in the region of a few hundredAngstroms and the fractal parameter is /H110150.5. The rms topo- logical roughness values are approximately 5 Å for allCuMn/Co, Co/CuMn, Cu/Co, Co/Cu, CuMn/Cu, and Cu/CuMn interfaces. The amplitude of the in-plane correlationlength and the topological roughness are in agreement withprevious observations in Co/Cu multilayers grown by mag-netron sputtering 31,32while the jaggedness of the interfaces shows no significant difference. B. Soft x-ray reflectometry The incident-photon-flux normalized transmission x-ray absorption spectroscopy /H20849XAS /H20850spectra of cobalt, manganese, and copper are taken from aCo/H2084950 Å /H20850/Cu/H208495Å /H20850/CuMn /H2084920 Å /H20850sample. The applied magnetic field direction was along the intersection of theincidence plane and the sample surface plane. The relativeabsorption cross sections from the transmission spectra havebeen deduced using the method of Chen et al. 33After taking into account the incident photon angle of 70° and the degreeof circular polarization of 100% /H20849i.e., multiplying /H9262+−/H9262−by 1 cos/H2084970/H20850while keeping /H9262++/H9262−the same /H20850, the resulting /H9262/H11006/H20849E/H20850, /H9262+/H20849E/H20850−/H9262−/H20849E/H20850/H20849the XMCD difference spectra /H20850, and /H9262+/H20849E/H20850 +/H9262−/H20849E/H20850/H20849the XAS sum spectra /H20850are deduced. The resulting /H20849electronic /H20850/H92520and/H9004/H9252/H20849magnetic /H20850contributions to the imagi-FIG. 2. /H20849Color online /H20850/H20849a/H20850Specular reflectivities of the Co /Cu/H20849x/H20850/CuMn multilayers. The data sets are offset on the ordi- nate axis for display purposes. /H20849b/H20850Representative transverse scan of the Co/CuMn /H20849x=0/H20850sample measured on the second Bragg peak. The solid line /H20849best fit /H20850is calculated using h=1 /2,/H9268=5.5 Å, and /H9264=140 Å. The experimental data are shown as symbols and the calculated reflectivity is represented by the solid lines. The mea-surements are realized with hard x rays /H20849E=8.9 keV /H20850.ABES et al. PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-4nary part of the refractive index are shown as solid lines in Fig. 3for Mn and Co. The real parts /H92540and/H9004/H9254/H20849shown as dashed lines in Fig. 3/H20850have been obtained using the Kramers-Kronig transformation /H20849KKT /H20850. Note that the /H9004/H9252 contribution of the Mn has been smoothed since the experi- mental data was noisy. According to the sum rules, the or-bital and spin magnetic moments can be determined from theXAS and XMCD spectra. The experimental values deducedfrom the sum rules are 1.50 /H110060.10 /H9262Band 0.21 /H110060.04/H9262B corresponding to the Co spin and orbital moments, respec- tively. A small dichroism signal is detected at the L2,3edges of the manganese while no dichroism is observed at the L2,3 edges of the copper. In order to simulate our data from the XRMS, the dichroic signals at the Cu L2,3edges are taken from Co /Cu/H2084910 Å /H20850multilayers.34From the XMCD and XAS spectra the value of the spin magnetic moment can, inprinciple, be obtained via the sum rules. The Cu moment isextracted from the data following Samant et al. 34This results in a derived Cu moment of 0.10 /H110060.02/H9262B. However, the sum-rule analysis becomes problematic for Mn because the2p-3delectrostatic interactions are relatively large compared to the 2 pspin-orbit interaction. This causes the manifolds of the 2 p3/2and 2 p1/2levels to overlap strongly and, conse- quently, there is a substantial amount of mixing betweenthese two jlevels. The correction factor needed to extract a meaningful magnetic moment from the sum rules is /H9273=1.5.35,36From the correction of these sum rules, the Mn moment extracted is 0.05 /H110060.01/H9262B. The dichroic signal is generally rather weak and noisy resulting in a rather largeambiguity on the final Mn moment. We can compare oursum-rules analysis with the XMCD amplitudes available inthe literature. The amplitude of the L 3XMCD signal ampli- tude in our case represents 0.03% of the absorption signal. Inthe case of fcc Mn/Co /H20849001/H20850system, a 36% amplitude of the XMCD signal was assigned to 4.5 /H9262B,/H20849or 8% XMCD //H9262B/H20850.37For the bcc Mn/Fe /H20849001/H20850systems, the follow- ing percentages of XMCD //H9262Bhave been reported: 8%,38 9.2%,39and 7.4%.40Assuming that an average value of 8/H110061%L 3XMCD signal corresponds to 1 /H9262B, we obtain from our spectra a magnetic moment for the Mn atoms of/H208490.04/H110060.02 /H20850 /H9262B. This value is in agreement with our more rigorous sum-rules analysis and given the uncertainties inapplying this analysis to Mn it provides confidence for theextracted value for Mn in a wide range of environments. Figure 4displays the specular x-ray resonant reflectivityTABLE I. Best-fit parameters for the specular reflectivity of the Co /Cu1 /H20849x/H20850/CuMn /Cu2 /H20849x/H20850 multilayers. xThickness Interface morphology Co,/H110061Å Cu1,/H110060.5 Å CuMn, /H110060.5 Å Cu2,/H110060.5 Å /H9268topological ,/H110060.5 Å /H9268grading ,/H110060.5 Å /H9264,/H1100620 Å h,/H110060.1 0 42 23.5 5.5 1.0 140 0.5 5 41 6.5 22.5 5.5 4.0 0.5 170 0.5 10 41 12.0 19.5 11.5 5.0 1.0 220 0.515 43 16.5 23.0 17.5 5.5 0.5 180 0.520 39 22.5 20 21.0 5.5 0.5 180 0.525 43 27.0 22.5 28.5 6.0 1.0 250 0.5 FIG. 3. /H20849Color online /H20850Optical constants of the Co and Mn lay- ers in the vicinity of the 2 pedges determined by means of XAS and XMCD /H20849black solid lines /H20850and subsequent KKT /H20849dashed lines /H20850as described in the text. The top left panel shows the Co nonmagnetic/H20849purely electronic /H20850imaginary, /H92520and real part /H92540of the index of refraction. The magnetic imaginary /H9004/H9252and real contribution /H9004/H9254are shown in the lower left panel. The right panel presents the equiva-lent data for Mn.FIG. 4. /H20849Color online /H20850Reflectivity of the Co /Cu/H20849x/H20850/CuMn mul- tilayers measured with circularly polarized soft x rays /H20849E =780 eV /H20850as a function of copper thickness /H20849see legend /H20850. The ex- perimental data are shown as the open black circles and the calcu-lated reflectivity is represented by the solid line.SPIN POLARIZATION AND EXCHANGE COUPLING OF Cu … PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-5measurements for Co /Cu/H20849x/H20850/CuMn /Cu/H20849x/H20850multilayers sit- ting just above the Co L3edge /H20849E=780 eV /H20850where the mag- netic contribution of the absorptive part is maximum. Thesolid lines passing through the reflectivity data are the best fitand although the fit of the individual curves is not perfect,the intensity changes from Bragg peak to Bragg peak havebeen satisfactorily reproduced. In contrast to the fitting strat-egy for the hard x-ray data where the layers were consideredas individual fitting elements, in these simulations, we havedivided the layer into slices along the out-of-plane zdirection approximately one atomic plane in thickness /H20849/H110152.5 Å /H20850. The interfacial structures were then modeled by varying the rela-tive densities of Co, Cu, and CuMn densities through thesample. Note that for the specular reflectivity, we cannot dis-tinguish between a topologically rough and a composition-ally graded interface. The relative densities of the Co, Cu,and CuMn within the bilayer or trilayer are shown in panels/H20849a/H20850,/H20849c/H20850,/H20849e/H20850,/H20849g/H20850,/H20849i/H20850, and /H20849k/H20850of Fig. 5for multilayers with a Cu spacer thickness of x= 0Å ,5Å ,1 0Å ,1 5Å ,2 0Å , and 25 Å, respectively. The density profiles show extendedregions on either side of the central CuMn layer, which con-sist of a mixture of Co and CuMn atoms at x=0 Å and Cu and Co atoms at x/H110220. Error bars in these figures are esti-mated at 0.1 along the ordinate axis. These alloy-type re- gions account for the effects of interfacial roughness andinterdiffusion that can alter the profile of the magnetization. We collected energy-dependent XRMS data at the Cu and MnL 2,3edges from Co /Cu/H20849x/H20850/CuMn multilayers for a fixed scattering vector. The energy dependence of the asymmetryratios recorded at the Cu and Mn L 2,3edges are shown in Figs. 6and7, respectively, for all samples, at a fixed scatter- ing vector corresponding to a position of a maximum in theasymmetry ratio at a fixed energy. The asymmetry ratios atthe Cu and Mn L 2,3edges exhibit a dichroism effect, unam- biguously demonstrating a magnetic polarization of the Cuand Mn 3 delectrons. The dichroism effect is observed for all samples but weakens with increasing Cu thickness. We notethat no dichroism is observed at the Cu and Mn L 2,3edges for a pure CuMn /H20849x=/H11009/H20850sample as shown in Figs. 6and7 indicating that the magnetic polarization observed at the Cuand Mn L 2,3edges for Co /Cu/H20849x/H20850/CuMn /Cu/H20849x/H20850multilayers samples are induced by the Co layer. The effective field fromthe Co induces a small Mn ferromagnetic componentthrough a partial alignment of the Mn paramagnetic arrange-ment. For Cu spacing between x=0 and 5 Å, this effective field is negative inducing an antiparallel alignment betweenFIG. 5. /H20849Color online /H20850Profiles of the relative Co, Cu, and CuMn densities as a function of CuMn layer depth are shown in panels /H20849a/H20850,/H20849c/H20850, /H20849e/H20850,/H20849g/H20850,/H20849i/H20850, and /H20849k/H20850. The errors in the relative density are estimated as /H110060.1. Panels /H20849b/H20850,/H20849d/H20850,/H20849f/H20850,/H20849h/H20850,/H20849j/H20850, and /H20849l/H20850present the profiles of the Cu and Mn polarizations, given in units of /H9262B/Cu and /H9262B/Mn.ABES et al. PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-6the Mn and Co atoms. For x/H1135010 Å, the effective field is positive inducing a parallel alignment between the Mn andCo atoms. As the XRMS technique is sensitive to both thestructural and magnetic contributions the observed asymme-try ratios can be different even though the magnetic structureis similar /H20849see, for example, Fig. 7/H20850. Direct evidence of the change in coupling with Cu spacer thickness is confirmed by the measurements of the XRMShysteresis loops at the Co and Mn L 2edges shown in Fig. 8. The Mn loops fit remarkably well to the Co loops clearlyindicating that the Co magnetization is responsible for thespin polarization of the Mn. The Co moment is alwaysaligned parallel to the applied field indicating that the mag-netization of the Co layers is greater than that induced in theMn. The hysteresis loops also show the reversal of the signof the coupling between the Co and Mn. The Mn atoms arecoupled antiferromagnetically with the Co layer below x =10 Å and ferromagnetically coupled above. In particular,we note the small size of the measured Cu dichroic effect,indicating the extreme sensitivity of the XRMS technique tosmall magnetic moments although it was not possible to ob-serve a hysteresis loop for Cu. In order to provide a quanti-tative explanation and extract the magnetic moment from thereflectivity, the asymmetry data have been analyzed using themodel explained in Sec. III. The spectra were calculated us- ing a magnetic multiplying factor for each slice of the inter-facial structures containing Mn and Cu given that the XMCDsignal is directly proportional to the magnetic moment. This magnetic factor is used to reduce or increase the magneticcontribution to the reflectivity in order to fit our data. Figures6and7show the simulations of the asymmetry ratio as a function of the energy at the Cu and Mn L 2,3edges for all multilayers. The model reproduces the main features of theexperimental data /H20849solid lines in Figs. 6and7/H20850. The resultant profile of the induced magnetization within the Mn and Cucomponent of all the multilayers are shown in panels /H20849b/H20850,/H20849d/H20850, /H20849f/H20850,/H20849h/H20850,/H20849j/H20850, and /H20849l/H20850of Fig. 5and are scaled to the relative densities in each slice. The profiles have then been normal-ized to the total magnetic moment to give the induced Mnand Cu magnetization in units of /H9262B/atom. We observe that the polarization occurs mainly when the Mn and Cu atomsare close to the cobalt layer and the induced magnetization ofthe Mn and Cu falls off rapidly away from the Co layer. Atthe interface, the magnetic moment of the Cu is found to be/H110150.05 /H9262B, this value is equivalent to the value found in the literature in Co/Cu multilayers.34The enhanced Cu dmo- ment near the interface is the result of a considerable hybrid-ization of the Cu and Co 3 dorbitals near the interface. 34For the Mn, the value of the magnetic moment is /H110150.3/H9262Band decreases with increasing copper thickness indicating thedominant interfacial nature of the Mn spin polarization /H20849Fig. 5/H20850. Furthermore, the Mn polarization switches from an anti- parallel alignment with the Co to parallel as the copper layerthickness exceeds x/H1135010 Å. This value is consistent with the RKKY coupling period in Co/Cu multilayers. 7 To better understand our observations we performed a theoretical study of the polarization of cobalt, copper, andmanganese in idealized Co/Cu/CuMn multilayers which con-sisted of a sharp interface region and a lattice-matched su-perstructure. To calculate the relevant quantities, we haveused the density-functional theory /H20849DFT /H20850/H20849Refs. 41and42/H20850 in the framework of the projector augmented wave /H20849PAW /H20850 method, 43as implemented in the Vienna ab initio simulation package /H20849V ASP /H20850.44The PAW method is a very powerful tool for performing electronic-structure calculations within theframework of the DFT. It takes advantage of the simplicityFIG. 6. /H20849Color online /H20850Comparison of the experimental asym- metry ratios /H20849R/H20850measured at the Cu L2,3edges for the multilayers /H20849open circles /H20850and simulation /H20849solid lines /H20850. FIG. 7. /H20849Color online /H20850Comparison of the experimental asym- metry ratios /H20849R/H20850measured at the Mn L2,3edges for the multilayers /H20849open circles /H20850and simulation /H20849solid lines /H20850.FIG. 8. /H20849Color online /H20850Hysteresis loops at the Co and Mn L3 edges /H20849778 eV and 639 eV , respectively /H20850as a function of Cu spacer thickness. Solid lines and symbols are the loops of Co and Mn,respectively.SPIN POLARIZATION AND EXCHANGE COUPLING OF Cu … PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-7of pseudopotential methods but describes correctly the wave function in the augmentation regions. Since this computa-tional method has been described at length before, 43it is not presented here, but it is worth mentioning that it isa full potential and all-electron method, used widely to in-vestigate the magnetic properties of materials. ThePerdew-Burke-Ernzerhof 45variant of the generalized gradi- ent approximation /H20849GGA /H20850was used for the exchange- correlation potential. Strong correlation effects were takeninto account by adding a Hubbard term to the GGA potential,known as the DFT+ Umethod. 46Values of U=4 eV and J =0.9 eV were used, which are standard values in the frame-work of this method. 47Convergence of the total energy was ensured by using a cutoff of 500 eV for the plane-wave ex-pansion of the wave function. To simulate the experimentalsystem as closely as possible within our computational re-sources, we have built a supercell containing four hexagonallayers of cobalt, stacked in an AB-ABfashion, and 11 hex- agonal layers of copper, stacked in an ABC -ABC fashion /H20849like fcc copper /H20850. Then, copper atoms were substituted by manganese at various positions /H20849see below /H20850. For each system studied, the geometry of the cell was completely relaxed/H20849volume, cell shape, and positions of the atoms within the cell/H20850so that interface reconstruction, lattice mismatch, and substitutional effects on the electronic structure are fullytaken into account. The Brillouin zone was meshed by an8/H110038/H110031 mesh during the relaxation procedure, then a final run using the relaxed geometry and a 16 /H1100316/H110031 mesh was performed in order to obtain precise total energies and den-sity of states. Before studying the Co-CuMn system, we made prelimi- nary calculations on the CuMn system. We setup a supercellwith four atoms of Cu and four atoms of Mn in a fcc geom-etry, and relaxed completely the system. All the parametersused are the same as described above. We found that theground-state magnetic configuration is antiferromagnetic.The Mn dorbitals are populated with slightly more than five electrons so that each atom has a spin moment of /H110153.8 /H9262B. As a result of the polarization of the Mn atoms, a magneticmoment is induced on the Cu atoms, of /H110150.05 /H9262Bon the d orbitals. Then, we studied the Co/Cu system without any Mn.We found that the ferromagnetically ordered cobalt layersinduce a polarization of the copper atoms near the interface.The magnetic moment on the cobalt atoms have an averagedvalue of 2.0 /H9262Band the induced magnetic moment on the first layer of copper /H20849next to the cobalt /H20850is/H110150.02/H9262Bon the dorbitals and is parallel with the cobalt moment /H20849ferromag- netic ordering /H20850. The magnetic moment of the second layer is decreased and reaches almost zero for the third layer. Theextracted moment and spatial decay are consistent with ourexperimental observations. Note that the absolute values ofthe induced magnetic moments on the copper atoms are quitedifficult to state since they depend on the fine details of thecalculations, in particular, the radius of the sphere aroundeach atom used to integrate the density is arbitrary /H20849in our case we have used the default values provided by the code /H20850. Finally, we have introduced two Mn atoms in our system, by replacing two Cu atoms, and we have varied the distancebetween them by intercalating from zero to four layers ofcopper. For each configuration, three possible magnetic or-ders were checked: either the two manganese atoms are coupled to each other antiferromagnetically or they arecoupled ferromagnetically, with the magnetic moments in thesame direction as those of the Co atoms, or in the oppositedirection. Over these three possibilities, we found that themagnetic moments carried by the Mn atoms are ordered pref-erably in an antiferromagnetic way. In Fig. 9, we show our computed total density of states of the system /H20849top plot /H20850,a s well as the partial density of states /H20849PDOS /H20850for the cobalt atom at the interface /H20849second top plot /H20850, the copper atom at the interface /H20849second picture from bottom /H20850, and a manganese atom /H20849bottom picture /H20850. The data presented in Fig. 9are for an antiferromagnetic coupling of the two Mn atoms and repre-sent the lowest total energy for the system. We show the MnPDOS for the atom aligned parallel to the Co atom. ThePDOS for the Mn aligned antiparallel to the Co is very simi-lar, i.e., the spin-up PDOS of one of the Mn atoms being thespin-down PDOS of the second Mn. The Fermi level is set at0 eV . In this case, the two Mn atoms are in neighboringpositions. As expected, the total density of states shows alarge magnetic moment for the cell because of the ferromag-netically ordered cobalt layers. More interesting is the mag-netic moment carried by the dorbitals of copper atom near the interface, which is /H110150.02 /H9262B, and parallel to the Co magnetic moment. Therefore, the presence of Mn atoms doesnot modify significantly the interaction between the cobaltatoms and the layer of copper at the interface. As for Mn inthe CuMn system /H20849see above /H20850the majority spin direction of Mn is completely filled by electrons while the minority spindirection is partially filled /H20849see Fig. 9/H20850. From these calcula- tions, the Mn atoms are predicted to be antiferromagneticFIG. 9. /H20849Color online /H20850Computed total density of states of the Co-CuMn system /H20849top plot /H20850for an antiferromagnetic alignment be- tween the Mn atoms. The remaining panels display the PDOS forthe cobalt atom at the interface /H20849second top plot /H20850, the copper atom at the interface /H20849second picture from bottom /H20850, and a manganese atom /H20849bottom picture /H20850.ABES et al. PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-8with a magnetic moment of /H110153.9/H9262B, a value close for a single atom /H208495/H9262B/H20850, but were found to be considerably larger than those determined experimentally using superconductingquantum interference device /H20849SQUID /H20850magnetometry /H208490.8 /H9262B/H20850with a maximum magnetic field of 5 T and T =10 K. This zero-temperature calculation is consistent with both the absence of the magnetization along the external appliedfield direction during the XRMS measurements for a pureCuMn sample and the induced magnetization by the cobaltlayer. The latter seems to be antiferromagnetically coupledfrom the interface to within four atomic planes and ferro-magnetically in the central CuMn layer. The antiferromag-netic coupling between Co and Mn is experimentally sup- ported by 55Mn NMR from Co/Mn-sputtered multilayers.48 The spatial dependence of the Cu-induced magnetic moment is predicted to fall away very quickly from the maximumvalue so that within two atomic planes /H20849/H110155Å /H20850, it is almost zero. In this respect, the calculations and experimental dataare in good agreement for the copper and manganese. V . DISCUSSION First, the structural profiles shown in Fig. 5using soft x rays are consistent with the fits using the hard x-ray tech-nique indicating the robustness of the analysis procedure inextracting the interfacial profiles of the Co/CuMn and Co/Cuinterfaces. The Gaussian distribution of the roughness usedin the model from the hard x rays show that the resultantprofile is found to be identical to that from the fitting strategyof the soft x rays. Significantly, the introduction of the CuMnlayers does not alter the physical interfacial structure. It is informative to look more closely at the spatial and spectroscopic structure which is evident in Figs. 5–7. The Cu polarization is seen to decay rapidly away from the interface/H20849in agreement with our DFT calculations /H20850and to change sign. While the magnitude of the Cu polarization is small awayfrom the interface the change in sign is consistent with theshort-period oscillation expected from the Cu Fermi-surfaceneck spanning vector. 49Unfortunately, the present experi- mental resolution prevents us from observing further oscilla-tions in the extracted Cu profile. Closer inspection of the CuXMCD /H20849Fig.6/H20850reveals structure in the experimental data not replicated by the simulation. Given the quality of Cu opticalconstants used in the simulation this is not surprising. Inprinciple, there is also the possibility of separating out the CuXMCD contribution from pure Cu and CuMn. Furthermore,a higher quality determination of the Cu optical constants, asobtainable at a third generation x-ray source, would presum-ably allow a refinement of the simulation to capture anymore subtle physical effects. The Mn moment decays rapidly away from the interface with increasing Cu thickness. Also, the Mn polarization isapproximately symmetric about the middle of the spacerlayer as would be expected for a ferromagnetic alignment ofthe Co layers. The value of the magnetic moment of themanganese atoms experimentally extracted from the XRMSmeasurements is small compared to the maximum momentpossible for the pure 3 d 5high-spin ground state.50–53The lowmagnetic moment could suggest that the ground state is not a high but low-spin state.54This is supported by the shape of Mn XAS spectrum /H20849Fig. 3/H20850which does not show the multi- plet structure associated with a pure high-spin d5state. The DFT calculations also predict a large Mn moment /H20849antiferro- magnetically coupled /H20850. This high-spin d5state gives a sig- nificantly larger moment compared to 0.8 /H9262B/Mn that we determined experimentally using SQUID magnetometry witha maximum magnetic field of 5 T and T=10 K. Kouvel’s 55 model and Smit et al.56from an experimental analysis inter- preted their results in terms of a compensation of certain Mnmoments which are antiferromagnetically coupled withnearest-neighbor Mn atoms. Smit et al. have shown clear evidence for magnetic clusters and short-range antiferromag-netic interactions where the fraction of the Mn atoms partici-pating in the strongest antiferromagnetic interaction actingfrom 25 T is estimated to be /H1101560%. According to this study less than 20% of the Mn atoms would be saturated when thesample was subject to a maximum external magnetic field of5 T. This would correspond to a magnetic value/H110153.9 /H9262B/5/H110150.8/H9262B, a value very close to our value found using bulk magnetometry. Also, Cable57found that for 10–20 %Mn in Co alloys, a Mn moment of /H110150.3/H9262B aligned antiparallel to the Co. This is consistent with our results for both the magnitude and orientation of the Mn inproximity to the Co. The weak observed value of the Mn magnetic moment is not surprising since it is averaging over still largely disor-dered moments that are only slightly preferentially alignedby the exchange field of the cobalt. We can estimate simplythe magnitude of this effective indirect exchange field fromthe bulk room-temperature susceptibility of CuMn. 19The calculated field is then a factor of /H110155 lower than that ob- served in Co/Cu multilayers7with a 9.3 Å Cu spacer. Given the idealized simplicity of the calculation and that our spacerthickness was /H1101511–12 Å /H20849see Table I/H20850, this is a reasonable agreement and would be consistent with the half-filled d states and high-spin configuration. The DFT calculations reproduce many of the features ob- served in our experiments: namely, the rapid decay in Cu andMn polarization with distance from the Co layer, the sign andmagnitude of the Cu polarization and the large high-spinmoment is consistent with SQUID magnetometry. The DFTcalculations predict antiferromagnetic coupling between theMn atoms which is consistent with our SQUID observationsat low temperature. The XRMS measurements were per-formed at room temperature where bulk CuMn is known tobe paramagnetic. 19This is also confirmed by SQUID mea- surements. In both cases, the effective field from the Co in-duces a small Mn ferromagnetic signal through a partialalignment of the Mn paramagnetic arrangement. It is worthnoting that in this experiment the Mn is intended simply as asensing layer but it is clear that there exists substantial scopefor further experimental and theoretical study of the interfa-cial ordering of such alloys. VI. CONCLUSIONS Magnetic-charge interference scattering has been used to determine the spatial profile of the magnetic polarization ofSPIN POLARIZATION AND EXCHANGE COUPLING OF Cu … PHYSICAL REVIEW B 82, 184412 /H208492010 /H20850 184412-9the Mn and Cu atoms induced by proximity to a Co layer. Agreement between the calculation and experiment could beachieved through the introduction of extended rough regionsat the interfacial boundaries. The Mn and Cu polarizationsare predominantly at the low Cu and Mn concentrations endof the interface, in close proximity to the Co and decaysrapidly as a function of depth toward the center of the CuMnlayer. The Mn polarization is much higher than that of Cu.The Mn atoms are coupled antiferromagnetically with the Colayer below x=10 Å and ferromagnetically coupled above. In contrast, the interfacial Cu atoms remain ferromagneti-cally coupled to the Co layer for all thicknesses studied assupported by DFT calculations but change sign for largerdistances from the polarizing Co layer. Evidently the Mnsensing scheme is a very useful probe for spatially observingsmall induced moments in layered systems. Having estab-lished the element-specific polarization profile in the proxim- ity case /H20849and the experimental sensitivity /H20850it is now possible to consider extending these measurements to the spin accu-mulation state for which the polarization and lengthscalemay be comparable. 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PhysRevB.83.104502.pdf

PHYSICAL REVIEW B 83, 104502 (2011) Three energy scales characterizing the competing pseudogap state, the incoherent, and the coherent superconducting state in high- Tccuprates Y . Okada,1T. Kawaguchi,1M. Ohkawa,2K. Ishizaka,2T. Takeuchi,3S. Shin,2and H. Ikuta1 1Department of Crystalline Materials Science, Nagoya University, Nagoya 464-8603, Japan 2Institute for Solid State Physics (ISSP), The University of Tokyo, Kashiwa 277-8581, Japan 3EcoTopia Science Institute, Nagoya University, Nagoya 464-8603, Japan (Received 9 December 2010; published 10 March 2011) We have studied the momentum dependence of the energy gap of Bi 2(Sr,R)2CuO yby angle-resolved photoemission spectroscopy (ARPES), particularly focusing on the difference between R=La and Eu. By comparing the gap function and characteristic temperatures between the two sets of samples, we show that thereexist three distinct energy scales, /Delta1 pg,/Delta1sc0,a n d/Delta1eff sc0, which correspond to T∗(pseudogap temperature), Tonset (onset temperature of fluctuating superconductivity), and Tc(critical temperature of coherent superconductivity). The results not only support the existence of a pseudogap state below T∗that competes with superconductivity, but also the duality of competition and superconducting fluctuation at momenta around the antinode below Tonset. DOI: 10.1103/PhysRevB.83.104502 PACS number(s): 74 .72.Kf, 74 .62.−c, 74.40.−n I. INTRODUCTION One of the significant differences between high- Tccuprates and conventional superconductors is the presence in the formerof a pseudogap state above T c. Whether the pseudogap state is a precursor to superconductivity or a state that competeswith it has been a matter of long-standing debate. 1–8To address this problem, angle-resolved photoemission spec-troscopy (ARPES) is one of the most powerful techniques,since the momentum dependence of the energy gap is directlylinked to the pseudogap issue. Many ARPES experimentshave investigated this problem, however, the data and theirinterpretations are still controversial. If the momentum dependence of the gap function is constituted by only one component, the pseudogap state can be regarded as a precursor to the superconducting state.This picture has been supported by some of the ARPESexperiments, which concluded that the energy gap has pre-dominantly a d-wave symmetry. 9–14An intimate relation between pseudogap and superconductivity has been suggestedalso by high-frequency conductivity measurements, 15the enhanced Nernst signal,16enhanced diamagnetism,17and the observation of the quasiparticle interference pattern suggestinga phase incoherent pairing gap above T c.18On the other hand, other ARPES experiments suggested the existence oftwo gap components that depend differently on momentum,temperature, and carrier doping. 19–28If the gap function consists of two components, the presence of an additional order other than the d-wave superconductivity must be assumed. This picture was supported further by a recent experimentproviding evidence for the existence of a density wave state inhigh-T ccuprates.29,30Here, if the co-existing state competes with superconductivity and suppresses Tcby reducing the number of paired electrons, the superconducting order wouldsignificantly fluctuate as has been suggested. 4In this case, both competition andsuperconducting fluctuation should be consistently accounted for above Tc. It is known that Tccan be controlled both by the element R andxin Bi 2Sr2−xRxCuO y(R=rare-earth elements).27,31–33 Using R=La and Eu single crystals of this system, it wasdemonstrated in our earlier works that three characteristic temperatures, T∗(pseudogap temperature), Tonset (onset tem- perature of fluctuating superconductivity), and Tc, can be defined, which behave differently on the phase diagram withchange of both Randx. 34,35To approach the pseudogap issue further in the present study, we probed the momentumdependence of the energy gap and compared the characteristic energy scales to the above three temperatures, focusing on the same system as in the previous study. All the experimentalresults shown in this paper consistently point to the existenceof three distinct energy and temperature scales arising fromthe competition between the two states in high- T ccuprates. Moreover, the duality of competition and superconductingfluctuation around the antinodal region is suggested to beimportant. II. EXPERIMENT Single crystals of Bi 2Sr2−xRxCuO y(R=La and Eu) were grown by the floating zone method.34,36The bulk sensitive ARPES spectra with an ultraviolet laser (6.994-eV photons)were taken by a Scienta R4000 hemispherical analyzer atthe Institute of Solid State Physics (ISSP). 37In this study, the total-energy resolution of a photoemission spectrometer(/Delta1E) is defined by fitting the Au spectrum with the Fermi Dirac function; its intensity at each energy is broadenedby the Gaussian (full width at half maximum /Delta1E). The energy resolutions of all the ARPES experiments with the6.994-eV photons shown in this paper were better than2.2 meV , and all the measurements were performed at pres-sures below 5 ×10 −11Torr. Prior to the ARPES measurements, we carefully evaluated the doping levels of the crystals withc-axis lattice constant, thermopower, and/or inductive coupling plasma (ICP) spectroscopy. 34 III. RESULTS AND DISCUSSIONS A./Delta1sc0: Energy scale of pairing at the antinode Figure 1shows the ARPES results obtained at 5 K with 6.994-eV photons on the optimally doped Bi 2Sr2−xRxCuO y 104502-1 1098-0121/2011/83(10)/104502(6) ©2011 American Physical SocietyY . OKADA et al. PHYSICAL REVIEW B 83, 104502 (2011) -0.03-0.02-0.010.00EEF(eV)M (a. u.) 40 30 20 100T (K) La-OP (Tc=33 K) Eu-OP (Tc=18 K) (d1) -0.03-0.02-0.010.00E-EF(eV) 1.0 0.8 0.6high low 30 25 20 15 10 5 0Gap size, EEF (meV) 0.6 0.4 0.2 0.0 |cos2( kxa)-cos2( kya) |/20.00 -0.021 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.02 -0.021 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.00 -0.021 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.02 -0.021 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 30 25 20 15 10 5 0Gap size, EEF (meV) 0.6 0.4 0.2 0.0 |cos2( kxa)-cos2( kya) |/2(g2) (g1) La-OP ( Tc=33 K) at 5 KEu-OP ( Tc=18 K) at 5 Kcut1 cut3 EEF(eV) Intensity (arb. units) EEF(eV)Intensity (arb. units)(e2) (f2) Eu-OPEu-OP (e1) (f1) La-OP La-OPmomentum -0.4-0.3-0.2-0.10.00.1EEF(eV) high low -0.4-0.3-0.2-0.10.00.1EEF(eV) high lowEu-OP(c2)La-OP(c1) momentum momentumcut1cut1 (d2)momentum cut1 cut2cut2 cut3 7 (,0)(,) 81 1011 1432 45 9 12 (0,0)6 1315 1617cut1 cut2 cut3antinode node 6 (,0)(,) 791 11 12 1432 45 10 (0,0)1315 16cut1 cut3cut2 8antinode node(a) (b1) (b2) La-OP Eu-OP FIG. 1. (Color online) ARPES data obtained with 6.994-eV photons at 5 K for optimally doped Bi 2Sr2−xRxCuO ywithR= La (La-OP, Tc=33 K) and R=Eu (Eu-OP, Tc=18 K) are shown. (a) Temperature dependence of magnetization of the crystals. (b1) and (b2) Mapping of the Fermi momentum kFfor La-OP and Eu-OP, respectively. Here, the Fermi surface that was determinedwith 21.214-eV photons in our previous study (Ref. 38) is shown with dotted lines. (c1) and (d1) [(c2) and (d2)] show the dispersion images along the momentum shown in (b1) [(b2)] for La-OP (Eu-OP). Weshow also momentum distribution curves at E Fin the upper part of (c1) and (c2). (e1) and (f1) [(e2) and (f2)] show the energy distribution curves and their symmetrized spectra at kFfor La-OP (Eu-OP), respectively. (g1) [(g2)] shows the intensity map of (f1) [(f2)] together with the gap size as a function of |cos(kxa)−cos(kya)|/2 for La-OP (Eu-OP). The intensity is normalized to unity at the gap energy, andthe color scales are the same for both figures. withR=La (La-OP) and R=Eu (Eu-OP). As shown in Fig. 1(a),t h eTcof the La-OP and Eu-OP samples was 33and 18 K, respectively. Figures 1(b1) and 1(b2) show the Fermi momenta kFwhere the ARPES spectra were taken. The Fermi surfaces determined with 21.214-eV photons inour previous work using samples with similar doping 38are also shown in Figs. 1(b1) and 1(b2). Figures 1(c1) and 1(d1) [Figs. 1(c2) and 1(d2)] show the momentum dependence of the spectral intensity of La-OP (Eu-OP) along the cuts shownin Fig. 1(b1) [Fig. 1(b2)]. The energy distribution curves at k F and the spectra that were symmetrized about EFare shown in Figs. 1(e1)and 1(e2), and 1(f1) and 1(f2), respectively. We determined the energy gap by fitting the symmetrized spectrawith the phenomenological spectral function, 39,40which has been used in many other reports.9,11–13,22 The gap size with La-OP and Eu-OP is plotted as a function of |cos(kxa)−cos(kya)|/2i nF i g s . 1(g1) and 1(g2), respectively. Since a d-wave gap is expressed as /Delta1= /Delta1sc0|cos(kxa)−cos(kya)|/2, Figs. 1(g1) and 1(g2) show that the gap has a pure d-wave form around the node for both La-OP and Eu-OP. On the other hand, the data points deviatedfrom the d-wave form near the antinode. This deviation is accompanied by a huge broadening of the spectral linewidthas is evident from the image plot of the ARPES spectrashown in the same figure. We determined /Delta1 sc0by fitting the linear part of the data, which gave 14.1 and 12.0 meVfor La-OP ( T c=33 K) and Eu-OP ( Tc=18 K), respectively. Hence the value of /Delta1sc0changed together with Tc. However, the difference in /Delta1sc0is not as large as the change of Tc since the ratio of /Delta1sc0(14.1/12.0≈1.23) is much smaller than the Tcratio 33 /18≈1.8. This is in strong contrast to conventional superconductors, for which Tcscales with the binding energy of the paired electrons /Delta1sc0, and suggests the possible existence of an energy scale other than /Delta1sc0 corresponding to Tc. B./Delta1ef f sc0: Energy scale related to Tc The temperature evolution of the symmetrized spectrum of La-OP and Eu-OP across Tcis shown in Figs. 2(a1)– 2(a4) and2(b1)– 2(b4) for various momenta. Figures 2(c) and2(d) show the temperature dependence of the gap size (left axis)together with the gap depth (right axis; see the caption for thedefinition) for the momentum that is closest to the antinodeamong the data shown in Fig. 2. From this figure, a sudden change in the gap depth was observed across T c, although the gap size for this momentum did not show obvious change.On the contrary, a more dramatic change happened across T c at momenta around the node: The energy gaps all collapsed simultaneously at Tc. Based on this experimental observation, we can define a characteristic energy scale /Delta1eff sc0, which is indicated by the arrows in Figs. 2(c) and 2(d) on the left axes. When the energy gap at T=0 was smaller than this characteristic energy /Delta1eff sc0, it decreased abruptly to zero at Tc, while the energy gap remained finite above Tcif it was larger than/Delta1eff sc0atT=0. The existence of such a characteristic energy is consistent with other recent reports.13,22,41–43Here, /Delta1eff sc0of La-OP ( Tc=33 K) and Eu-OP ( Tc=18 K) are 5.6±1.1 meV and 3.5 ±1.1 meV , respectively. The ratio of/Delta1eff sc0between the two samples is about 1.6 ±0.5, which is close to the ratio of Tc(≈1.8) within experimental error. Moreover, the values of 2 /Delta1eff sc0/kBTcfor La-OP (3.9 ±0.8) 104502-2THREE ENERGY SCALES CHARACTERIZING THE ... PHYSICAL REVIEW B 83, 104502 (2011) -0.02 0.00 0.02kF 11 (11=50º) -0.02 0.00 0.02kF 13 (13=55º) -0.02 0.00 0.02kF 15 (15=61º) -0.02 0.00 0.02kF 17 (17=66º) -0.02 0.00 0.02kF 11 (11=51º) -0.02 0.00 0.02kF 14 (14=61º) -0.02 0.00 0.02kF 12 (12=54º) -0.02 0.00 0.02kF 13 (13=57º)Intensity(arb. units) Intensity(arb. units)(a1) (a2) (a3) (a4) (b1) (b2) (b3) (b4)EEF(eV)35 25 20 10 T=5 K50 25 2010 T=5 K35 25 20 10 T=5 K35 25 20 10 T=5 K35 25 19 14 T=5 K925 19 14 T=5 K925 19 14 T=5 K919 14 T=5 K9 EEF(eV) 12 10 8 6 4 2 0Gap size (meV) 50 40 30 20 10 0 Temperature (K)0.7 0.6 0.5 0.4 0.3 0.2 0.1 1-I(0)/I(Δ)Tc=33 K 11=50º12=52º13=55º14=58º 15=61º16=64ºkF17, 17=66ºkF17, 17=66º12 10 8 6 4 2 0Gap size (meV) 30 25 20 15 105 0 Temperature (K)0.6 0.5 0.4 0.3 0.2 0.1 0.0 1-I(0)/I(Δ)Tc=18 K 11=51º12=54º13=57ºkF14, 14=61ºkF14, 14=61º(d) (c) FIG. 2. (Color online) Evolution of the energy gap across Tc of La-OP ( Tc=33 K) and Eu-OP ( Tc=18 K) measured with laser ARPES (6.994-eV photons). (a1)–(a4) and (b1)–(b4) show the temperature dependence of the symmetrized ARPES spectra forLa-OP and Eu-OP, respectively. Here, the index of k Fcorresponds to the numbers in Figs. 1(b1) and 1(b2), where θis also defined. (c) and (d) show the temperature evolution of the gap size (left axis) at variousmomenta across T cfor La-OP and Eu-OP, respectively. The hatched area shows roughly the range where the energy gap was strongly temperature dependent and collapsed at Tc. The characteristic energy /Delta1eff sc0is indicated by the arrows on the left axes of (c) and (d). The temperature dependence of 1 −I(0)/I(/Delta1) (right axis) calculated from the spectra measured at point 17 (14) for the R=La (Eu) sample is also plotted, where I(0) and I(/Delta1) are the intensity at EF and at the gap edge, respectively. and Eu-OP (4.5 ±1.4) were close to the value observed by Andreev reflection experiments on a wide range of cuprateswith various T c’s.45These quantitative comparisons indicate that/Delta1eff sc0can be attributed to the energy scale corresponding toTc.C./Delta1pg: Energy scale related to a competing pseudogap state The question to be addressed next is why /Delta1eff sc0is much lower than /Delta1sc0. Figure 3shows the momentum dependence of the energy gap of the La-OP and Eu-OP samples atT=5 K and T/greaterorsimilarT c. The energy gap in the antinodal region obtained with 21.214-eV photons (at 5 K with lessthan 20 meV resolution) in our previous studies 38,45are also included. In contrast to the gap around the node,the antinodal gap /Delta1 pgis clearly larger for Eu-OP ( Tc= 18 K) than La-OP ( Tc=33 K) showing that the nodal and antinodal gaps depend differently on Tc, which is qualitatively consistent with scanning tunnel microscopy (STM) /scanning tunnel spectroscopy (STS) results.46Previously, we observed that the coherent part of the remnant Fermi surface, whereclear peaks were observed in the ARPES spectra at thesuperconducting state, narrowed with increasing /Delta1 pg.38This observation suggested that /Delta1pgshrinks the coherent part of the remnant Fermi surface, which naturally decreasesthe superfluid density. All our experimental observationssuggest that the antinodal pseudogap state characterizedby/Delta1 pgcompetes with superconductivity and supresses Tc, resulting in the deviation of /Delta1eff sc0from/Delta1sc0.T h e T=5K data of Fig. 3is the indication of the co-existence of the competing state with superconductivity. This supportsthe existence of two different momentum dependent gapcomponents 19–24,26–28; that is, the antinodal gap /Delta1pghas its origin in a competing state with no direct relation to the d-wave superconductivity.47 60 50 4021.214 (eV) photon at 8K15 106.994 (eV) photon La-OP(5 K) Eu-OP(5 K) La-OP(35 K) Eu-OP(19 K)pg(Eu-OP)Δ 20 10 0Gap size (meV) 1.0 0.8 0.6 0.4 0.2 0.0 2|cos( kxa)-cos( kya)|/20 0.6 0.4 0.2 0.0 sc0(Eu-OP)Δsc0(La-OP)Δpg(La-OP)Δ eff sc0(Eu-OP)Δeff sc0(La-OP)Δ FIG. 3. (Color online) Comparison of the momentum depen- dence of the energy gaps of La-OP ( Tc=33 K) and Eu-OP ( Tc= 18 K) at T=5 K, which is well below Tc,a n da t T/greaterorsimilarTc.T h et h r e e data points closest to the antinode (indicated by arrows) are obtained using 21.214-eV photons both for La-OP and Eu-OP (at 5 K with lessthan 20 meV resolution) (Refs. 38and45). The three characteristic energy scales for both La-OP and Eu-OP are shown on the right axis. The inset is an enlarged plot around the node to show more clearlythe neighborhood of the node and /Delta1 eff sc0. 104502-3Y . OKADA et al. PHYSICAL REVIEW B 83, 104502 (2011) 7 6 5 4 3 2 1 0Gap size (meV) 0.4 0.3 0.2 0.1 0.0|cos( kxa)-cos( kya)|/2La-LUD (Tc=30 K, x=0.42)7 6 5 4 3 2 1 0Gap size (meV) 0.4 0.3 0.2 0.1 0.0 |cos( kxa)-cos( kya)|/2La-UD (Tc=23 K, x=0.54)7 6 5 4 3 2 1 0Gap size (meV) 0.4 0.3 0.2 0.1 0.0 |cos( kxa)-cos( kya)|/2La-LOD (Tc=31 K, x=0.22) 7 6 5 4(meV)Eu-LUD (Tc=13 K, x=0.40)7 6 5 4(meV)Eu-LUD (Tc=16 K, x=0.35)7 6 5 4(meV)Eu-LOD (Tc=17 K, x=0.25)(a1) (a2) (a3) (b1) (b2) (b3)400 300 200 100 T*(K)80 60 40 20T* R=La R=Eu R=La ( cal.) R=Eu ( cal.) 80 60,TcMF(K)15 10TTcMF R=La R=Eu Tonset R=La R=Eu(c)Energy scale Δ(meV) La-OP ,Eu-OP3 2 1 0Gap size ( 0.4 0.3 0.2 0.1 0.0 |cos( kxa)-cos( kya)|/23 2 1 0Gap size ( 0.4 0.3 0.2 0.1 0.0 |cos( kxa)-cos( kya)|/23 2 1 0Gap size ( 0.4 0.3 0.2 0.1 0.0 |cos( kxa)-cos( kya)|/240 20 0Tc,Tonset, 1.0 0.8 0.6 0.4 0.2 La, Eu content x5 0Tc R=La R=Eu R=La ( cal.) R=Eu ( cal.) FIG. 4. (Color online) Energy gaps around the node of Bi 2Sr2−xRxCuO ywithR=La and Eu samples determined by the laser-ARPES measurements (6.994-eV photons) are shown in (a1)–(a3) and (b1)–(b3), respectively. All data shown here were measured at T/lessorequalslant5K ,w h i c h is well below Tc. (c) Phase diagram of the three characteristic temperatures. The data of T∗,TMF c,a n dTcplotted with solid symbols were calculated from /Delta1pg,/Delta1sc0,a n d/Delta1eff sc0assuming 2 /Delta1/k BT=4.3. The right axis of (c) gives the energy scale /Delta1that is connected to the temperature scale (left axis) by the above relation. The Tc,Tonset,a n dT∗data shown with empty symbols are from our previous studies (Refs. 32–34). D. Intimate relation between /Delta1sc0and Tonset Figures 4(a1)– 4(a3) and 4(b1)– 4(b3) show the momentum dependence of the energy gap around the node of variousBi 2Sr2−xRxCuO ycrystals with different xforR=La and Eu, respectively. The results indicate that the slope of theenergy gap as a function of |cos(k xa)−cos(kya)|/2 did not change much with Rorxdespite the large variation of Tc. This relatively insensitive behavior of /Delta1sc0mimics that of Tonset, the temperature below which the Nernst signal starts to be enhanced with decreasing temperature.34To address this similarity more quantitatively, we calculated the mean-fieldtransition temperature ( T MF c) based on the weak-coupling theory of d-wave superconductivity (2 /Delta1sc0/kBTMF c=4.3). Figure 4(c) shows TMF ccalculated from the data shown in Figs. 4(a1)– 4(a3) and 4(b1)– 4(b3) together with Tc,Tonset, andT∗reported in our previous studies.34–36Interestingly, we found that TMF cagrees quite well with Tonset. We think that this agreement implies that the energy scale /Delta1sc0is related to the onset pairing temperature Tonset.T h el a r g e difference between Tonset andTc(/Delta1sc0and/Delta1eff sc0) indicates that there exists a large superconducting fluctuation. Thephenomenological explanation of the existence of a largesuperconducting fluctuation is due to weak perturbation ofthe pairing energy /Delta1 sc0by stabilization of the competing state (increasing /Delta1pg). We think that this is consistent with the existence of a relatively homogeneous gap despite the largevariation of the pseudogap in real space as was revealed byrecent STM experiments. 47,48 Note that while /Delta1sc0is the pairing energy scale at the antinode, the energy gap observed at this momentum is not/Delta1sc0but/Delta1pg. As shown in Fig. 4(c), the characteristic temper- ature scale /Delta1sc0is related to Tonset. Therefore the observed relation between Tonset and/Delta1sc0suggests the existence of both competition and superconducting fluctuation at momentaaround the antinode below T onset. E. Three energy and temperature scales in high- Tccuprates In Fig. 4(c), we plot all the experimentally obtained energy and temperature scales changing both xandRin Bi2Sr2−xRxCuO y. This phase diagram clearly shows the existence of three energy ( /Delta1eff sc0,/Delta1sc0, and /Delta1pg) and tem- perature ( Tc,Tonset, andT∗) scales connected by the relation 2/Delta1/k BT=4.3. The natural consistent picture led by the phase diagram of Fig. 4(c) is that the pseudogap state (characterized by/Delta1pgandT∗) suppresses coherent superconductivity ( /Delta1eff sc0 andTc) while keeping the pairing strength ( /Delta1sc0andTonset) similar.49Therefore the competing state kills superconduc- tivity mainly by enhancing fluctuation of superconductingorder through reducing superfluid density (phase stiffness).In other words, one may say that the competition enhances thesuperconducting fluctuation . We think the conclusion in this paper can be extended more or less to all the high- T c cuprates, including systems that have a comparable /Delta1sc0and /Delta1pg, such as Bi 2Sr2CaCu 2Oy. IV . SUMMARY In summary, we compared the momentum dependence of the gap function and the characteristic temperature scalesof Bi 2Sr2−xRxCuO y(R=La and Eu). All the experimental 104502-4THREE ENERGY SCALES CHARACTERIZING THE ... PHYSICAL REVIEW B 83, 104502 (2011) results point toward the existence of three distinct energy and temperature scales corresponding to the competing pseudogapstate and the incoherent and coherent superconducting states.Accounting for all these three phenomena consistently wouldbe crucial for understanding the pseudogap issue in high- T c cuprates.ACKNOWLEDGMENTS We thank E. Hudson and V . Madhavan for useful discus- sions. We also thank Y . Hamaya and S. Arita for experimental assistance. Y .O. thanks the Japan Society for the Promotion of Science for financial support. 1T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999). 2M. R. Norman, D. Pines, and C. Kallin, Adv. Phys. 54, 715 (2005). 3S. Huffner, M. A. Hossain, A. Damascelli, and G. A. Sawatzky, Rep. Prog. Phys. 71, 062501 (2008). 4V . J. Emery and S. A. Kivelson, Nature (London) 374, 434 (1995). 5C. M. Varma, P h y s .R e v .B 55, 14554 (1997). 6S. Chakravarty, R. B. Laughlin, D. K. 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Oda, R. M. Dipasupil, N. Momono, and M. Ido, J. Phys. Soc. Jpn.69, 983 (2000). 104502-5Y . OKADA et al. PHYSICAL REVIEW B 83, 104502 (2011) 42T. Kurosawa, T. Yoneyama, Y . Takano, M. Hagiwara, R. Inoue, N. Hagiwara, K. Kurusu, K. Takeyama, N. Momono, M. Oda, andM. Ido, P h y s .R e v .B 81, 094519 (2010). 43A. Kanigel, U. Chatterjee, M. Randeria, M. R. Norman, S. Souma, M. Shi, Z. Z. Li, H. Raffy, and J. C. Campuzano, Phys. Rev. Lett. 99, 157001 (2007). 44G. Deutscher, Nature (London) 397, 410 (1998), and references therein. 45Probing the electronic structure around the antinode with 6.994-eVphotons was difficult due to the matrix element effect. However,there was a smooth connection between the gap sizes obtained with6.994- and 21.214-eV photons. Also, the deviation from the d-wave gap for Eu-OP was stronger than for La-OP in both measurements.Hence the data with different photon energies are consistent witheach other. 46A. Sugimoto, S. Kashiwaya, H. Eisaki, H. Kashiwaya, H. Tsuchiura,Y . Tanaka, K. Fujita, and S. Uchida, P h y s .R e v .B 74, 094503 (2006).47M. C. Boyer, W. D. Wise, K. Chatterjee, M. Yi, T. Kondo, T. Takeuchi, H. Ikuta, and E. W. Hudson, Nat. Phys. 3, 802 (2007). 48A. Pushp, C. V . Parker, A. N. Pasupathy, K. K. Gomes, S. Ono,J. Wen, Z. Xu, G. Gu, and A. Yazdani, Science 324, 1689 (2009). 49As shown in this paper, ARPES directly gives us the importantthree energy scales characterized around the node (probably ratherhomogeneous) and antinode (probably inhomogeneous) leading toa deeper understanding of the pseudogap issue in high- T ccuprates. Most of our discussions and conclusions in this paper are drawnin the momentum space, however, we think that our conclusioncan be reconciled with the real-space picture of an inhomogeneouselectronic structure. Especially the origin of the duality betweencompetition and superconducting fluctuation around the antinodeis possibly related to the inhomogeneous quasiparticle excitationsin real space. 104502-6

PhysRevB.77.104108.pdf

Nitrogen-induced interface defects in Si oxynitride Eun-Cheol Lee Division of Bio-Nano Technology, Kyungwon University, Gyeonggi 461-701, Korea and Gachon Bio-Nano Research Institute, Gyeonggi 461-701, Korea /H20849Received 25 August 2007; revised manuscript received 12 February 2008; published 10 March 2008 /H20850 Based on first-principles density-functional calculations, we propose N-associated defects at Si /SiO 2inter- face, which behave as dominant interface states in Si oxynitrides instead of conventional Pbcenters in pure SiO 2. These defects involve an N interstitial, and their energy levels lie in the broad range, from the midgap to the conduction-band edge of Si, depending on local oxidation status near N, while electrical states from Pb-type centers are removed by substitutional N atoms. Our results provide a mechanism for degradations related to the interface defects in Si oxynitrides which are used in state-of-the-art metal-oxide-semiconductordevices. DOI: 10.1103/PhysRevB.77.104108 PACS number /H20849s/H20850: 61.72. /H11002y, 71.55. /H11002i, 73.20. /H11002r, 85.30. /H11002z Defects at Si /SiO 2interface have been intensively studied for several decades because they act as an interface chargetrap which degrades performances and reliabilities of metal-oxide-semiconductor /H20849MOS /H20850transistors. It is well identified by experiments and theories 1–5that dominant interface de- fects, the so-called Pbcenters, are isolated dangling bonds from Si substrate near the interface. Unfortunately, a further study on the interface traps be- comes required, as Si oxynitride /H20849or lightly nitrided oxide /H20850 has replaced pure SiO 2in state-of-the-art MOS transistors that are used in commercial semiconductor devices. The ni-tridation of SiO 2is essential for suppressing boron penetra- tion from p+-polysilicon electrodes.6In addition, it also im- proves electrical properties such as charge trapping,7and theoretical models for this effect have already been given.8,9 On the other hand, the N incorporation at the interface is known to increase densities of interface traps,10causing per- formance degradations and reliability problems.11–14 Recent experiments clearly show that new interface de- fects are created and act as dominant species in Si oxyni-trides instead of P bcenter. Stathis et al. showed that the interface states are located near the conduction-band edge ofSi with a broad energy distribution, through low-voltage stress-induced leakage current /H20849LV-SILC /H20850measurements, while pure oxide shows negligible interface state generationat the conduction-band edge. 15,16On the other hand, the de- fect density at midgap is much larger in SiO 2, as compared to that in oxynitride.15,16Campbell et al. also observed a new defect center in oxynitride which is fundamentally differentfrom P b0/Pb1defects,17through electron spin resonance measurements. Understanding the N-related interface defectson an atomistic level is very important to modern MOS tech-nology, but these defects are not clearly identified yet. In this paper, we present an atomic model for N-induced defects at the interface between Si and Si oxynitride basedon first-principles density-functional calculations. We findthat N interstitials at the interface create defect levels in theSi band gap, which move upward from the midgap to theconduction-band edge, as the interactions with oxygen atomsin SiO 2matrix are enhanced. On the other hand, the deep defect level caused by a Pb-like center, i.e., a threefold- coordinated Si, is removed by substitution of an N atom. Ourmodel for N-associated interface defects explains the en-hanced interface state generation near the conduction-band edge and the reduction of midgap defect density in Si oxyni-trides. First-principles pseudopotential calculations are per- formed using density-functional theory /H20849DFT /H20850within the generalized gradient approximation. 18We use ultrasoft pseudopotentials for O, N, and H /H20849Ref. 19/H20850and a norm- conserving pseudopotential for Si.20A converged plane wave basis set is generated with energy cutoffs of 30 Ry for wavefunctions and 180 Ry for charge densities, respectively. Todescribe Si /SiO 2interface, we construct the interface with Si crystal and tridymite SiO 2in a supercell containing 128 at- oms, similar to the previous theoretical studies.21,22This su- percell geometry is a periodic slab structure with seven Silayers for bulk Si and two Si and four O layers for SiO 2, and the Si dangling bonds at both surfaces are passivated by Hatoms. The Si band gap of this interface is found to be1.05 eV, similar to previous calculations. 21This value is larger than for usual DFT band gap of bulk Si because of thequantum confinement effect in the slab geometry. 21 First, we study electronic and atomic structures of an N interstitial /H20849Ni/H20850and its complexes with oxygen ions at the interface. When an N interstitial is incorporated deep into the Si layers in our interface model, it is found that a /H20849100/H20850-split- interstitial geometry is the most stable, similar to previouscalculations on N in Si. 23,24A symmetric split-interstitial structure at the interface has difficulty maintaining its stabil-ity because of lattice strain near the interface. The optimizedstructure of the N iat the interface is illustrated in Fig. 1/H20849a/H20850. This defect also includes a trivalent N and Si atom, similar tothe /H20849100/H20850split interstitial, and the threefold Si atom /H20849Si I/H20850 protrudes into the SiO 2side. Si Iis undercoordinated and, thus, a chemically active atom in the defect, while the va-lence shell of N is completely filled. Due to the asymmetricstructure, the Si I-SiIIIdistance /H208493.15 Å /H20850is longer by 0.52 Å than the Si I-SiIIdistance /H208492.63 Å /H20850. A charge density analysis indicates that no bonding occurs between Si Iand Si II, al- though the distance between them is only longer by 0.3 Åthan the Si-Si bond length in crystalline Si. The Si-N bondlengths are found to be 1.73–1.85 Å, and nearest-neighborSi-Si bonds from N are elongated by 0.08 Å to compensatefor lattice strain caused by these short Si-N bonds. If posi-PHYSICAL REVIEW B 77, 104108 /H208492008 /H20850 1098-0121/2008/77 /H2084910/H20850/104108 /H208494/H20850 ©2008 The American Physical Society 104108-1tions of the Si Iand N atom are changed in Fig. 1/H20849a/H20850, that geometry would be unrealistic due to the unstable N-O bond,which are not supported by theoretical studies. 25,26In fact, we investigated a number of geometries of N iat other Si sites at the interface for various bonding directions and didnot find any significant difference from the structure in Fig.1/H20849a/H20850. For partially oxidized interface, it is energetically favor- able that N iforms a complex with O ions. We model partial oxidation of the interface by using O interstitials /H20849Oi/H20850incor- porated at the Si-Si bonds because our supercell does not include O atoms at the interface, i.e., the top layer of Si. ForN i-Oicomplex, the O interstitial prefers the nearest Si-Si bond sites from the N i, as shown in Fig. 1/H20849b/H20850; the total energy is lowered by 0.62 eV than for an isolated O iat the same Si layer and an isolated N ibecause of a coordinate covalent bonding of Si Iwith the O i. Thus, nearest-neighbor Si-Si bonds from N are expected to behave as attracting centers forO ions in partially oxidized interfaces. The Si Iatom is shifted away by 0.37 Å from its initial position to reduce repulsiveCoulomb interaction with the lone-pair orbital of O I. When the second O ion /H20849OII/H20850is incorporated at another nearest- neighbor bond from N, Si Iand O Iatoms become fourfold and threefold coordinated, respectively, by forming theSi I-OIbond /H20851see Fig. 1/H20849d/H20850/H20852. We refer to this structure as Ni-Oi-Orcomplex, where O rmeans a threefold-coordinated O atom. The complex is more stable by 1.04 eV than anisolated N iand two isolated O I. This structure is similar to that of the O i-O2rcomplex, which is shallow thermal donor in Si,27,28if N is replaced by an O atom. The N i-O2istruc- ture, in which Si I-OIis broken, is found to be unstable in the same O ipositions because of lattice strain near the interface, although the similar defect, N i-O2iwith C2vsymmetry, is suggested to be stable in bulk Si.23,24On the other hand, the Ni-O2icomplex becomes stable, when O IImoves to the second-nearest-neighbor Si-Si bond, as shown in Fig. 1/H20849c/H20850. This defect is less stable by 0.16 eV than for N i-Oi-Or.W e find that an isolated O IIat the first Si layer is more stable by 0.18 eV than an isolated O Iat the second Si layer. Comparedto N i-Oiplus an isolated O II, the binding energy of N i-O2iis found to be 0.08 eV. It increases to 0.26 eV, as referenced toan isolated O Iand N i-Oi. The Si I-OIIdistance, 3.47 Å, is not much different from other Si-O distances lying in the rangeof 3.32–3.36 Å except for the Si I-OIdistance of N i-Oi-Or /H20849see Table I/H20850. The interaction between Si Iand O IIis also found in the defect level, shown in Fig. 2/H20849a/H20850, where the lone- pair orbital of O IIis hybridized. Because of exothermic bind- ing energies, N interstitials might form complexes with oxy-gen ions at the interface. The main features found here areexpected to be unchanged even for amorphous SiO 2because the geometries of N i-related defects are mainly determined at the silicon side of the interface, as exhibited in Figs.1/H20849a/H20850–1/H20849d/H20850. Here, we find that energy levels of N i-associated interface defects are shifted upward in Si band gap, as interactionswith O ions are enhanced. The shifts of defect levels do notresult from direct interactions between N iand O i, but from the interactions of threefold Si atoms created by N iwith O i. To deal with charge capture process via nonequilibrium tun-neling process, we analyze the single-particle defect levels ofneutrally charged defects. The calculated defect levels for theN i-related defects are listed in Table I, as referenced to the bottom of the conduction band /H20849Ec/H20850. For N i, a deep defect level is found at Ec−0.4 eV. The level does not arise from the N atom itself but from the dangling bond of the threefoldSi atom, similar to the N i-O2icharacterized in Fig. 2/H20849a/H20850. For Ni-Oi, the defect level is elevated by 0.11 eV because of the Coulombic repulsion between the lone-pair orbital of O Iand the dangling bond of Si I. Contrary to our results, it is re-TABLE I. Single-particle defect levels /H20849EKS/H20850, defect transition levels /H20849E−/0/H20850between neutral and negatively charged states, and Si-O distances for N i-related defects. Ecindicates the conduction- band minimum of Si. DefectEKS /H20849eV/H20850E−/0 /H20849eV/H20850Distance /H20849Å/H20850 SiI−OI SiI−OII Ni Ec−0.4 Ec−0.31 Ni-Oi Ec−0.29 Ec−0.20 3.36 Ni-Oi−O r Ec Ec 1.85 3.33 Ni-O2i Ec−0.19 Ec−0.12 3.32 3.47 (a) (b) (c) (d)SiI SiIOIO SiISiIOIOII OIOII N NN NNiNi-Oi Ni-O2iNi-Oi-OrSiIISiIII FIG. 1. Atomic structures of the /H20849a/H20850Ni,/H20849b/H20850Ni-Oi,/H20849c/H20850Ni-O2i, and /H20849d/H20850Ni-Oi-Ordefects at the interface. (a) (b) NSiNOII OISiI SiIOI OII FIG. 2. Charge densities are plotted for the defect levels of the /H20849a/H20850Ni-O2iand /H20849b/H20850Ni-Oi-Ordefects for an isosurface of 0.002 a.u.EUN-CHEOL LEE PHYSICAL REVIEW B 77, 104108 /H208492008 /H20850 104108-2ported that the defect state becomes deeper for bulk Si after the formation of N i-Oicomplex;24this is attributed to the fact that the dangling bond orbital of N i-Oibecomes more localized after the symmetry of the N iis broken. For the Ni-Oi-Ordefect, it is found that a shallow-level interface state near Ecis created. The physical origin of the shallow interface state can be explained as follows; as shown in Fig.1/H20849d/H20850, this defect contains no undercoordinated Si atom and thereby does not have the defect level associated with the Sidangling bond. Instead, one excess electron is left after fill-ing the valence shell of overcoordinated O I, and this electron may be transferred to an antibonding orbital of Si Iand O I. However, since this antibonding orbital /H20851Fig.2/H20849b/H20850/H20852lies above the conduction band, the electron may occupy the conductionband and thus the creation of the shallow level is expected bythe effective mass theory. The key component of theshallow-level defect is the threefold-coordinated oxygenatom. If O Iin N i-Oi-Oris replaced b y a N atom, the defect has no undercoordinated or overcoordinated atoms; the Si,N, and O atoms are fourfold, threefold, and twofold coordi- nated, respectively. Thus, this defect might be an electricallyinactive complex with no defect levels in the Si band gap.Although the chemical structure of N i-Oi-Oris almost iden- tical to that of this inactive defect, the threefold O atom /H20849Or/H20850 has one more electron than the N atom. Since N i-Oi-Orhas no defect levels in the band gap, it is expected that the shal-low level just below the conduction band is created, similarto the phosphorus defect in Si. Other interface defects in-volving a stable threefold O atom might also have shallowdefect levels near the conduction band. For the N i-O2icom- plex, the defect level moves down to 0.19 eV below theconduction-band minimum, as the bond of the Si Iand the O II atoms is broken. However, it is still affected by O II, as dis- cussed earlier, and thereby higher by 0.1 eV than for N i-Oi. We also analyze thermodynamic defect transition levels,which are obtained from the energy difference of two relaxedgeometries with different charges. The transition levels be-tween neutral and negative charge states are calculated be-cause these levels describe the electron tunneling from inter-face traps below the n-well Fermi level into the p +-polysilicon electrode, which leads to the LV-SILC in p-type metal-oxide-semiconductor field-effect transistor.15 As listed in Table I, the calculated transition levels are pushed upward from Ec−0.31 eV to Ec, as the interaction with O ion is enhanced, similar to the single-particle levels.Thus, our results show that energy levels of N i-related de- fects possibly lie in the broad energy range from the midgapto conduction-band edge, depending on the degree of inter-actions with O ions. For amorphous SiO 2, more broad level distributions are expected because of more various local con-figurations of oxygen atoms. On the other hand, we find that a substitutional N atom possibly removes electrical activity of an interfacial Si dan-gling bond, the so-called P bcenter, as reported in previous theoretical calculations,29although the N i-related species act as an interface trap. In our previous calculations, we suggestthat most of the substitutional N atoms /H20849N Si/H20850exist as an N pair at interface, while single N is unstable due to the cre- ation of a dangling bond.8However, single substitutional N is stabilized, when it replaces a threefold-coordinated Si withan unpaired dangling bond. To validate this argument, we first construct a model geometry for Pbcenter as in Fig. 3/H20849a/H20850. This model is similar to the dimer model for the Pb1defect in the previous calculations,5but our conclusions are not af- fected even if other interface models are used. We find thatundercoordinated Si is possibly replaced by a threefold Natom without any remaining dangling bond orbital, as shownin the N Sidefect /H20851see Fig. 3/H20849b/H20850/H20852. This defect is an electrically inactive center with no defect level in the band gap, while adeep level is found at E c−0.4 eV for our Pbcenter model. We investigate the stability of N Sidefect via the reaction, Pb center + N i/H20849at the interface /H20850→NSi/H20849at the interface /H20850+Si /H20849of perfect crystal /H20850, and find that this reaction is exothermic by 1.93 eV. Thus, the annihilation of the Pbcenter by substitu- tion of N is energetically favorable in Si oxynitride. The experimentally reported interface states and sup- pressed Pbcenters in Si oxynitrides are explained by our proposed models for N-interstitial-related and substitutionalN defects, respectively. In LV-SILC and gated diode mea-surements, the interface states in oxynitrides are found to bedistributed over a broad energy range in the upper portion ofthe Si band gap. 15We showed that the defect levels for the Ni-related species are possibly located at various positions between the midgap and the conduction-band edge, depend-ing on interactions with neighboring O atoms, in good agree-ment with the experiments. Among the defects distributedover the broad range, it is expected that the concentrations ofconduction-band edge defects such as N i-Oi-Orare higher than for midgap defects due to the following reasons. Theisolated N interstitial produces a deep interface state, but it isonly stabilized at the oxygen-free interface like Fig. 1/H20849a/H20850, where the top layer of Si bears no oxygen atom. For a par-tially oxidized interface, which is usually expected in normaloxidation process, N imight attract O atoms because of the exothermic binding energies, and thus the defect levels arepushed upward by the interactions with O atoms. Especially,defects involving trivalent O atoms might greatly enhanceLV-SILC, since they possess defect levels close to the con-duction band. The suppression of midgap states 16is ex- plained by the removal of Pb-center-like defect by substitu- tional N atoms. However, electrical improvement of theinterface is not guaranteed by the removal of P bcenters, since the N i-associated species possibly degrade interface properties. Our N-defect models only involve N /H11013Si3spe- cies, i.e., one N bonded to three Si atoms, in good agreement (a) (b) SiO SiO NH H FIG. 3. Atomic models of the /H20849a/H20850Pbcenter and /H20849b/H20850NSiat the interface. The arrow in the figure indicates a threefold-coordinatedSi.NITROGEN-INDUCED INTERFACE DEFECTS IN Si … PHYSICAL REVIEW B 77, 104108 /H208492008 /H20850 104108-3with x-ray photoelectron spectroscopy measurements for in- terfacial N atoms, while the chemical bond of N away from the interface is possibly to be either N bonded to two Si andone O atom /H20849O-NvSi 2/H20850or N bonded to two Si atoms /H20849NvSi2/H20850.25,26,30–33Thus, our calculations successfully ex- plain the recent experiments for the N-related species in ul- trathin Si oxynitrides which have not been fully understood. In conclusion, we investigate the N-induced defects at the Si/SiO 2interface through ab initio pseudopotential calcula- tions. We find that the N interstitials at the interface createvarious defect levels in the Si band gap, which range fromthe midgap to the conduction band of Si. The level positionsare dependent on the configuration of oxygen atoms around the N interstitial. On the other hand, the midgap level causedbyP bcenter is possibly removed by substitution of an N atom for a threefold-coordinated Si atom in the defect. Ourcalculations explain why the interface state generation is en-hanced in Si oxynitride, especially near the conduction-bandedge of Si, although the density of P bcenter is reduced. This work was supported by the Kyungwon University Research Fund in 2007. Calculations in this work have been done using the QUANTUM-ESPRESSO package,34and all fig- ures were generated by XCRYSDEN program.35 1P. Caplan, E. Poindexter, B. Deal, and R. Razouk, J. Appl. Phys. 50, 5847 /H208491979 /H20850. 2F. C. Rong, J. F. Harvey, E. H. Poindexter, and G. J. Gerardi, Appl. Phys. Lett. 63, 920 /H208491993 /H20850. 3H. J. von Bardeleben, M. Schoisswohl, and J. L. Cantin, Colloids Surf., A 115, 277 /H208491996 /H20850. 4A. Stesmans and V. V. Afanas’ev, J. Appl. Phys. 83, 2449 /H208491998 /H20850. 5A. Stirling, A. Pasquarello, J.-C. Charlier, and R. Car, Phys. 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PhysRevB.77.125107.pdf

Surface dissipation in nanoelectromechanical systems: Unified description with the standard tunneling model and effects of metallic electrodes C. Seoánez and F. Guinea Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E28049 Madrid, Spain A. H. Castro Neto Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA /H20849Received 5 December 2007; revised manuscript received 22 January 2008; published 7 March 2008 /H20850 By modifying and extending recent ideas /H20851C. Seoánez et al. , Europhys. Lett. 78, 60002 /H208492007 /H20850/H20852, a theoretical framework to describe dissipation processes in the surfaces of vibrating micro- and nanoelectromechanicaldevices, thought to be the main source of friction at low temperatures, is presented. Quality factors as well asfrequency shifts of flexural and torsional modes in doubly clamped beams and cantilevers are given, showingthe scaling with dimensions, temperature, and other relevant parameters of these systems. Full agreement withexperimental observations is not obtained, leading to a discussion of limitations and possible modifications ofthe scheme to reach a quantitative fitting to experiments. For nanoelectromechanical systems covered withmetallic electrodes, the friction due to electrostatic interaction between the flowing electrons and static chargesin the device and substrate is also studied. DOI: 10.1103/PhysRevB.77.125107 PACS number /H20849s/H20850: 03.65.Yz, 62.40. /H11001i, 85.85./H11001j I. INTRODUCTION The successful race for miniaturization of semiconductor technologies2manifests itself spectacularly in the form of nanoelectromechanical systems /H20849NEMS /H20850,3–6which are ma- chines in the micron and submicron scales whose mechanicalmotion, integrated into electrical circuits, has a wealth oftechnological applications, including control of currents atthe single-electron level, 7single-spin detection,8subattonew- ton force detection,9mass sensing of individual molecules,10 high-precision thermometry,11orin vitro single-molecule biomolecular recognition.12 These mechanical elements /H20849such as cantilevers or beams, see Fig. 1/H20850are also the focus of attention and intensive re- search as experiments are approaching the quantumregime, 13–15where manifestations of quantized mechanical motion of a macroscopic degree of freedom such as theircenter of mass should become apparent. Several schemes toprepare the mechanical oscillator in a nonclassical state andobserve clear signatures of its quantum behavior have beenrecently suggested. 16–21 A key figure of merit of the mechanical oscillation is its quality factor Q=/H9275//H9004/H9275, where/H9004/H9275is the measured line- width of the corresponding vibrational eigenmode of fre-quency /H9275. To reach the quantum regime, as well as for most practical applications, where the measured shifts of the reso-nant frequency /H9275constitute the detectors principle, a very high Qis compulsory. Imperfections and the environment surrounding the oscillator result in both a finite linewidth /H9004/H9275 and a frequency shift /H9254/H9275with respect to the ideal case. Therefore, several works have been devoted to the analysisof the different sources of dissipation present in microelec-tromechanical systems and NEMS, 22–32trying to determine the dominant damping mechanisms and the ways to mini-mize them. Among the different mechanisms affectingsemiconductor-based NEMS, the most important and diffi-cult to avoid are /H20849i/H20850clamping losses 33,34through the transferof energy from the resonator mode to acoustic modes at the contacts and beyond, up to the substrate, /H20849ii/H20850thermoelastic damping,35–37and /H20849iii/H20850friction processes taking place at the surfaces.38–40At low temperatures and for decreasing sizes, the prevailing mechanism is the last one,6as indicated by the linear decrease of the quality factor of flexural modes withdecreasing size /H20849see Fig. 2/H20850or the sharp increase of Qwhen the resonator is annealed. 38,41Excitation of adsorbed mol- ecules, movement of lattice defects, or configurational rear-rangements irreversibly absorb energy from the excitedeigenmode and redistribute it among the rest of the degreesof freedom of the system. A theoretical quantitatively accurate description of surface dissipation proves therefore challenging, as many differentdynamical processes and actors come into play, some ofwhom are not yet well characterized, so simplifications needto be done to provide a unified framework for all of them. InRef. 1, such a scheme was given, based on the following considerations: /H20849i/H20850experimental observations indicate that surfaces of otherwise monocrystalline resonators acquire a hLw t FIG. 1. /H20849Color online /H20850Sketch of the systems considered in the text. The inset shows a doubly clamped beam, while the main figureshows a cantilever characterized by its dimensions, width /H20849w/H20850, thickness /H20849t/H20850, and length, /H20849L/H20850, where w/H11011t/H11270L. The height above the substrate is h. A schematic view of the surface is given, highlighting imperfections such as roughness and adsorbates, which dominatedissipation at low temperatures.PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 1098-0121/2008/77 /H2084912/H20850/125107 /H2084914/H20850 ©2008 The American Physical Society 125107-1certain degree of roughness, impurities, and disorder, resem- bling an amorphous structure.42/H20849ii/H20850In amorphous solids, the damping of acoustic waves at low temperatures is success-fully explained by the standard tunneling model, 43–46which couples the acoustic phonons to a set of two-level systems/H20849TLSs /H20850representing the low-energy spectrum of all the de- grees of freedom /H20849DOFs /H20850able to exchange energy with the strain field associated with the vibration. These DOFs corre-spond to impurities or clusters of atoms within the structurewhich have two energy minima separated by an energy bar-rier in their configurational space /H20849similar to the dextro and/or levo configurations of the ammonia molecule /H20850, which are modeled as a DOF tunneling between two potentialwells. At low temperatures, only the two lowest eigenstates have to be considered, characterized by the bias /H9004 0zbetween the wells and the tunneling rate /H90040xthrough the barrier. The standard tunneling model specifies the properties of the set of TLSs in terms of a probability distribution P/H20849/H90040x,/H90040z/H20850which can be inferred from general considerations43,44and is sup- ported by experiments.46Note, however, that below a certain temperature the model breaks down due to the increasingrole played by interactions among the TLSs /H20849not shown /H20850. 47,48 In Ref. 1, a description of the attenuation of vibrations in nanoresonators, due to their amorphouslike surfaces, in termsof an adequate adaptation of the standard tunneling modelwas intended to be given. An estimate for Q −1/H20849T/H20850was pro- vided, reproducing correctly the weak /H11011T1/3temperature de- pendence as well as the order of magnitude observed in re-cent experiments. 30,32However, the way the standard tunneling model was adapted was not fully correct, and arevision of the results for Q −1/H20849T/H20850obtained therein was man- datory. In this work, we will /H20849i/H20850discuss in detail some important issues hardly mentioned in, Ref. 1/H20849ii/H20850modify some points to obtain a theory fully consistent with the standard tunnelingmodel and which includes all possible dissipative processesdue to the presence of TLSs, /H20849iii/H20850extend the results and give expressions for the quality factor of cantilevers as well asdamping of torsional modes and frequency shifts associated,/H20849iv/H20850compare with available experimental data by discussing the validity of the model, range of applicability, and aspectsto be modified and/or included to reach an accurate quanti-tative fit to experiments, and /H20849v/H20850study the dissipative effects associated with the presence of metallic electrodes frequentlydeposited on top of the resonators. Section II starts with a brief summary of the model de- scribed in Ref. 1, discussing the approximations involved, while the different dissipative mechanisms due to the pres-ence of TLSs are presented in Sec. III. Section IV analyzesand compares the two main mechanisms, namely, relax-ational processes associated with asymmetric /H20849biased /H20850TLSs, and nonresonant damping of symmetric TLSs. The extensionof the results to cantilevers and torsional modes is given inSec. V . A brief account of the frequency shift expressions isfound in Sec. VI. In Sec. VII, a comparison with experimentsand a discussion of the applicability, validity, and furtherextensions of the model are made. Section VIII discussesdissipation effects due to the existence of metallic leadscoupled to the resonator, which can be coupled to the elec-trostatic potential induced by trapped charges in the device.Under some circumstances, these effects cannot be ne-glected. Finally, the main conclusions of our work are givenin Sec. IX. Details of some calculations are provided in theAppendixes. We will not analyze here the dissipation due to clamping and thermoelastic losses, which may dominate dissipation inthe case of very short beams and strong driving, respectively.We will also not consider direct momentum exchange pro-cesses between the carriers in the metallic circuit and thevibrating system. 49 II. SURFACE FRICTION MODELED BY TWO-LEVEL SYSTEMS A. Hamiltonian We will consider a rod of length L, width w, and thickness t/H20849see Fig. 1/H20850, and we use units such that /H6036=1= kB. As de- scribed in Ref. 1, the vibrating resonator with imperfect sur- faces is represented by its vibrational eigenmodes coupled toa collection of noninteracting TLSs, assuming that the maineffect of the strain caused by the phonons is to modify the bias/H9004 0zof the TLSs50as follows: H=/H20858 k,j/H9275k,jak,j†ak,j+/H20858 /H90040x,/H90040z/H20877/H90040x/H9268x +/H20875/H90040z+/H20858 k,j/H9261k,j/H20849bk,j†+bk,j/H20850/H20876/H9268z/H20878. /H208491/H20850 The index jrepresents the three kinds of modes present in a thin beam geometry:51flexural /H20849bending /H20850, torsional, and compression modes. These modes will be present for wave-lengths/H9261/H11022t, while for shorter ones, the system is effectively three-dimensional /H208493D/H20850, with the corresponding 3D modes. The main effect of these high-energy 3D modes is to renor-malize the tunneling amplitude, so we will take that renor- malized value as our starting point /H9004 0xand forget about the 3D modes in the following. The sum over TLSs is character- ized by the probability distribution P/H20849/H90040x,/H90040z/H20850=P0//H90040x.43,44FIG. 2. /H20849Color online /H20850Evolution of reported quality factors in monocrystalline mechanical resonators with size, showing a de-crease with linear dimension, i.e., with increasing surface-to-volume ratio, indicating a dominant role of surface-related losses/H20849Ref. 6/H20850.SEOÁNEZ, GUINEA, AND CASTRO NETO PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-2This result follows, as explained in Refs. 44and45, due to /H20849i/H20850the exponential sensitivity of /H90040xto the properties of the energy barrier of the two-well potential, giving rise to the TLS description of the system at low T/H20851resulting in a 1 //H90040x dependence of P/H20849/H90040x,/H90040z/H20850/H20852, and /H20849ii/H20850the characteristic energy scale of the distribution of asymmetries /H90040z, much bigger than 1 K, which is the temperature at which experiments are per-formed and which, thus, fixes the scale of the bias of a TLS if it is to contribute significantly to dissipation, /H9004 0z/H333551K . /H20851Therefore, the relevant TLSs have values of /H90040zlying in a very narrow energy range around /H90040z=0 as compared to the variance of their probability distribution, allowing us to consider that P/H20849/H90040x,/H90040z/H20850to a first approximation does not depend on/H90040z./H20852Unphysical divergencies do not appear, as /H90040x/H11022/H9004 min, with/H9004minfixed by the typical time scale of the experiment, which is given by the time needed to obtain aspectrum around the resonance frequency of the excited vi- brational eigenmode of the resonator, and /H9255=/H20881/H20849/H90040x/H208502+/H20849/H90040z/H208502 /H11021/H9255 max, which is estimated to be of the order of 5 K.46For typical amorphous insulators, P0/H110111044J−1m−3. To see to which low-energy modes the TLSs are more coupled, inducing a more effective dissipation at low tem-peratures, the spectral function J/H20849 /H9275,j/H20850/H11013/H20858 k/H20841/H9261k,j/H208412/H9254/H20849/H9275−/H9275k,j/H20850 characterizing the evolution of the strength of the coupling for each type of mode can be computed. Due to their non- linear dispersion relation /H9275=/H20881EI//H9267twk2, with I=t3w/12,E the Young modulus, and /H9267the mass density, flexural modes show a sub-Ohmic behavior Jflex/H20849/H9275/H20850=/H9251b/H20881/H9275co/H20881/H9275, with /H9251b/H20881/H9275co= 0.3/H92532 t3/2w/H208491+/H9263/H20850/H208491−2/H9263/H20850 E/H208493−5/H9263/H20850/H20873/H9267 E/H208741/4 , /H208492/H20850 where/H9253/H110115 eV is a coupling constant appearing in /H9261k,j,/H9263is Poisson’s ratio, and /H9275co/H11229/H20881EI//H9267tw/H208492/H9266/t/H208502is the high-energy cutoff of the bending modes. A detailed derivation of Jflex/H20849/H9275/H20850 is given in Appendix A. Even though the length Lof our system is finite, and thus the vibrational spectrum is discrete,a continuum approximation like this one will hold ifkT/H11271/H6036 /H9275fund, with/H9275fundbeing the frequency of the fundamen- tal mode. The bending modes prevail over the other, Ohmic-like, modes as a dissipative channel at low energies, thanks totheir weaker J flex/H20849/H9275/H20850/H11011/H92751/2dependence. One may ask at what frequency do the torsional and compression modes be- gin to play a significant role, and a rough way to estimate itis to see at what frequency do the corresponding spectralfunctions have the same value, J flex/H20849/H9275*/H20850=Jcomp,tors /H20849/H9275*/H20850. By using the expressions in Ref. 1, namely, Jcomp,tors /H20849/H9275/H20850 =/H9251c,t/H9275, with/H9251c=/H20849/H9253/H90040x//H90040/H208502/H208492/H92662/H9267tw/H20850−1/H20849E//H9267/H20850−3/2and/H9251t =C/H20849/H9253/H90040x//H90040/H208502/H208498/H92662/H9262tw/H9267I/H20850−1/H20849/H9267I/C/H208503/2, the results are /H9275* /H1101130/H208491+/H9263/H208502/H208491−2/H9263/H208502/H20849E//H9267/H208501/2//H20851t/H208493−5/H9263/H208502/H20852for the case of com- pression modes and /H9275*/H11011300 /H208491−2/H9263/H208502/H20849E//H9267/H208501/2//H20851t/H208493−5/H9263/H208502/H208491 +/H9263/H20850/H20852for the torsional modes. By comparing these frequen- cies to that at the onset of 3D behavior, /H9275co, the frequencies are found to be similar, justifying a simplified model whereonly flexural modes are considered.B. Two-level system dynamics The interaction between the bending modes and the TLSs affects both of them. When a single mode is externally ex-cited, as is done in experiments, the coupling to the TLSswill cause an irreversible energy flow from this mode to therest of the modes through the TLSs, as depicted in Fig. 3/H20849a/H20850. The dynamics of the TLSs in the presence of the vibrationalbath determines the efficiency of the energy flow and thusthe quality factor of the excited mode. By taking a given TLSplus the phonons, its dynamics is characterized by the Fou-rier transform of the correlator /H20855 /H9268z/H20849t/H20850/H9268z/H208490/H20850/H20856, the spectral function A/H20849/H9275/H20850, which at T=0 reads A/H20849/H9275/H20850/H11013/H20858 n/H20841/H208550/H20841/H9268z/H20841n/H20856/H208412/H9254/H20849/H9275−/H9275n+/H92750/H20850, /H208493/H20850 where /H20841n/H20856is an excited state of the total system TLS plus vibrations. In Ref. 1, an analysis of A/H20849/H9275/H20850was made for the case of a symmetric TLS /H20849/H90040z=0/H20850, concluding that /H20849i/H20850if/H90040x /H11270/H9251b2/H9275co, the tunneling amplitude is basically suppressed and the TLS does not participate in dissipative processes. Forreasonable system dimensions, the coupling constant is verysmall, /H9251b/H112701, so this effect can be ignored. /H20849ii/H20850Around the resonance at/H9275=/H90040x, a broadening appears, /H9003/H20849/H90040x/H20850, which, for /H9003/H20849/H90040x/H20850/H11270/H90040x, is given by the Fermi golden rule result /H9003/H20849/H90040x/H20850 =16/H9251b/H20881/H9275co/H20881/H90040x/H20851forT=0 K; at T/H110220, Eq. /H208497/H20850applies /H20852./H20849iii/H20850 The coupling to phonons of all energies provides the“dressed” TLS with tails far from resonance, A/H20849 /H9275/H20850 /H11008/H9251b/H20881/H9275co/H9275//H20849/H90040x/H208502for/H9275/H11270/H90040xand A/H20849/H9275/H20850/H11008/H9251b/H20881/H9275co/H20849/H90040x/H208502/H9275−7/2 for/H9275/H11271/H90040x/H20851see left side of Fig. 3/H20849b/H20850/H20852./H20849iv/H20850The main effect of the asymmetry /H90040zis to suppress the TLSs dynamics, so the TLSs playing an active role in dissipation satisfy /H90040x/H11022/H20841/H90040z/H20841. Excited modeEnsemble of TLS Rest of vibrational modesEE Excited modeEEnsemble of “dressed” TLSa) b) A( )/CID1 /CID1BiasSpectrum of 1 dressed TLS /CID1A( )tot/CID1Spectrum of ensemble /CID3 FIG. 3. /H20849Color online /H20850/H20849a/H20850Schematic representation of the irre- versible flow of energy from the externally excited mode to the TLSensemble, and from the ensemble to the rest of the vibrationalmodes. This process can be viewed as a flow of energy from theexcited mode to an ensemble of dressed TLSs, with their dynamicsmodified by the presence of the vibrational modes. /H20849b/H20850Left: Spec- tral function A/H20849 /H9275/H20850of a single dressed TLS that is weakly damped /H20849/H9003/H11021/H90040x/H20850. A peak around /H9275=0 arises if the system is biased, corre- sponding to the relaxational dissipation mechanism. Right: Totalspectral function A tot/H20849/H9275/H20850of the ensemble of dressed TLSs.SURFACE DISSIPATION IN NANOELECTROMECHANICAL … PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-3Finally, for those low-energy overdamped TLSs such that /H9003/H20849/H90040x/H20850/H11271/H90040x, an incoherent decay exp /H20849−t//H9270/H20850with time, with /H9270/H20849/H90040x/H20850=/H9003/H20849/H90040x/H20850−1, was assumed. Two points that Ref. 1missed, and now will be consid- ered, are the following: First, the effect of the asymmetry isnot simply the one stated. An underdamped biased TLS de- velops an additional peak around /H9275=0, as shown in Fig. 3/H20849b/H20850, which corresponds to the relaxation mechanism that dominates the dissipation of acoustic waves in amorphoussolids 46/H20851see Eq. /H208499/H20850/H20852. In Ref. 1, the relaxation mechanism associated with Eq. /H208499/H20850was misinterpreted to correspond to friction due to overdamped TLSs, while in this work it willbe linked to the presence of biased underdamped TLSs, con-sistent with the standard tunneling model. It will be de-scribed in more detail in the next section and taken intoaccount in the computation of the total dissipation. The second point missed by Ref. 1is the following. One can estimate, using the probability distribution P/H20849/H9004 0x,/H90040z/H20850, the total number of overdamped TLSs in the vol- ume fraction of the resonator having amorphous features, Vamorph . With Vamorph /H11011Vtot/10 and using /H9003/H20849E,T/H20850=/H9003/H20849E,T =0/H20850coth /H20851E/2T/H20852/H110152T/H9003/H20849E,T=0/H20850/E, the number of over- damped TLSs, /H90040x/H33355/H9003 /H20849/H90040x,T/H20850→/H90040x/H33355/H2085130/H9251b/H20881/H9275coT/H208522/3,i s N/H11015P0twL /H2085130/H9251b/H20881/H9275coT/H208522/3, which for typical resonator sizes L/H110111/H9262m and t,w/H110110.1/H9262m is less than 1 for T/H110211K . Therefore, unless the resonator is bigger, and hence, P0too, this contribution to dissipation can be safely neglected, aswill be done in the following. III. DISSIPATIVE MECHANISMS From the features of the spectral function Atot/H20849/H9275/H20850 =/H20858/H90040x,/H90040zA/H20849/H90040x,/H90040z,/H9275/H20850of the ensemble of dressed TLSs, one can classify into three kinds the dissipative mechanisms af- fecting an externally excited mode /H92750. A. Resonant dissipation Those TLSs with their unperturbed excitation energies close to/H92750will resonate with the mode, exchanging energy quanta, with a rate proportional to the mode’s phonon popu-lation n /H92750to first order. For usual excitation amplitudes /H110111 Å, the vibrational mode is so populated /H20849as compared with the thermal population /H20850that the resonant TLSs become saturated and their contribution to the transverse /H20849flexural /H20850 wave attenuation becomes negligible, proportional to n/H92750−1/2.52 B. Dissipation of symmetric nonresonant two-level systems We show first that a correct description of the dissipation due to weakly damped TLSs is given by the approximateHamiltonian used in Ref. 1, starting from Eq. /H208491/H20850,i fo n e considers in some way that the presence of the relaxational peak at /H9275=0 is due to a finite bias /H90040z/HS110050. We focus the attention on a given TLS plus the vibrations /H20849spin-boson model /H20850,H=/H90040x/H9268x+/H90040z/H9268z+Hint+Hvibr,Hint=/H9268z/H20858 k/H9261k2 /H20881/H9275k/H20849ak†+ak/H20850, /H208494/H20850 with/H9261defined in Eq. /H20849A4/H20850. In Ref. 1, a change of basis to the unperturbed eigenstates of the TLS was performed, obtaining H=/H9255/H9268z+/H20851/H20849/H90040x//H9255/H20850/H9268x+/H20849/H90040z//H9255/H20850/H9268z/H20852/H20858k/H9261k2 /H20881/H9275k/H20849ak†+ak/H20850+Hvibr. Then the term /H20849/H90040z//H9255/H20850/H9268zwas ignored /H20851remember/H9255 =/H20881/H20849/H90040x/H208502+/H20849/H90040z/H208502/H20852, arguing that the key role in dissipation is played by fairly symmetrical TLSs, /H20841/H90040z/H20841/H11270/H9255. This allowed us to simplify the spin-boson Hamiltonian to that of a symmet-ric TLS, which is much easier to analyze, with tunneling amplitude/H9255instead of/H9004 0xand coupling/H9261/H20849/H90040x//H9255/H20850instead of/H9261. The consistency was kept by restricting the sums over the TLS ensemble to those such that /H20841/H90040z/H20841/H33355/H90040x. One can check this consistency by going back to the origi- nal basis, where the approximation of the Hamiltonian reads H/H11015/H90040x/H9268x+/H90040z/H9268z+/H20851/H20849/H90040x//H9255/H208502/H9268z−/H20849/H90040z/H90040x//H92552/H20850/H9268x/H20852/H20858k/H9261k2 /H20881/H9275k/H20849ak†+ak/H20850 +Hvibr. Restricting the application of this Hamiltonian to those TLSs such that /H20841/H90040z/H20841/H33355/H90040xseems to be a fairly good approximation, but a price has been paid; namely, the spec-tral weight at /H9275=0 due to the bias has been lost in this effectively symmetric spin-boson approximation. Thisweight cannot be ignored, and it has to be added as a differ-ent mechanism /H20849the relaxational mechanism /H20850, which will be done in the next section. Once this issue has been taken careof, all the dissipative processes due to the presence of TLSsare correctly included in this framework, and we can proceeddescribing nonrelaxational friction due to nonresonant,weakly damped, weakly biased TLSs. As mentioned before, the coupled system TLSs /H11001vibra- tions can be viewed, taking the coupling as a perturbation,from the point of view of the excited mode /H92750as a set of TLSs with a modified absorption spectrum. The TLSs,dressed perturbatively by the modes, are entities capable ofabsorbing and emitting over a broad range of frequencies,transferring energy from the excited mode /H92750to other modes. The contribution to the value of the inverse ofthe quality factor, Q −1/H20849/H92750/H20850, of all these nonresonant TLSs will be proportional to Aoff-restot/H20849/H92750/H20850=/H20858/H90040x=0/H92750−/H9003/H20849/H92750/H20850A/H20849/H90040x,/H90040z,/H92750/H20850 +/H20858/H92750+/H9003/H20849/H92750/H20850/H9255max A/H20849/H90040x,/H90040z,/H92750/H20850/H110152P/H9251b/H20881/H9275co//H92750, a quantity measur- ing the density of states which can be excited through Hintat frequency/H92750/H20849see Appendix B for details /H20850. For an excited mode /H92750populated with n/H92750phonons, Q−1/H20849/H92750/H20850is given by Q−1/H20849/H92750/H20850=/H9004E/2/H9266E0, where E0is the energy stored in the mode per unit volume, E0/H11229n/H92750/H6036/H92750/twL, and/H9004Eis the energy fluctuations per cycle and unit volume. /H9004Ecan be obtained from the Fermi golden rule as follows: /H9004Eoff-restot/H112292/H9266 /H92750/H6036/H927502/H9266 /H6036n/H92750/H20873/H9261k02 /H20881/H92750/H208742 Aoff-restot/H20849/H92750/H20850, /H208495/H20850 and the inverse quality factor of the vibration follows. For finite temperatures, the calculation of Aoff-restot/H20849/H92750,T/H20850is done in Appendix B. The result, valid for temperatures below the breakdown of the TLS approximate description of the two-well potential /H20851T/H110115K /H20849Ref. 46/H20850/H20852, is different from the oneSEOÁNEZ, GUINEA, AND CASTRO NETO PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-4given in Ref. 1, because there the value of Q−1/H20849/H92750,T/H20850was interpreted as corresponding to the net energy loss of the studied mode /H20849subtracting emission processes from absorp- tion ones /H20850, while in experiments the observed linewidth is due to the total amount of fluctuations , the addition of emis- sion and absorption processes. So in this context, dissipationmeans fluctuations and not net loss of energy. The contribu-tion of these kind of processes is thus /H20849Q −1/H20850off-restot/H20849/H92750,T/H20850/H1122910P0t3/2w/H20873E /H9267/H208741/4/H9251b2/H9275co /H92750cotanh/H20875/H92750 T/H20876. /H208496/H20850 C. Contribution of biased two-level systems to the linewidth: Relaxation absorption This very general friction mechanism arises due to the phase delay between stress and imposed strain rate. In ourcontext, for a given TLS, the populations of its levels take afinite time to readjust when a perturbation changes the en-ergy difference between its eigenstates. 46,53This time/H9270cor- responds to the inverse linewidth and is given, for not toostrong perturbations, by the Fermi golden rule result /H9003/H20849/H9255,T/H20850=1 6 /H9251b/H20881/H9275co/H20881/H9255coth /H20851/H9255/2T/H20852, /H208497/H20850 where/H9255=/H20881/H20849/H90040x/H208502+/H20849/H9004z/H208502is the energy difference between the levels. Notice that /H9004zis not just the bare /H90040zappearing in the Hamiltonian of the system, but the net bias including themodification due to the coupling to the vibrational modes, /H9004 z=/H90040z+/H9264k/H11509uk/H20849/H9264kis the corresponding coupling constant with the proper dimensions and /H11509ukis a component kof the deformation gradient matrix, defined recalling Eq. /H208494/H20850as /H11509uk/H11011/H20855k,nk/H20841/H20849k//H20881/H9275k/H20850/H20849ak†+ak/H20850/H20841k,nk/H20856, associated with a vibra- tional mode /H20841k,nk/H20856/H20850. Equation /H208497/H20850is valid in the range of applicability of the TLS description of the two-well poten-tials /H20849T/H333555K /H20850and for values of /H9251bsuch that/H9003/H20849/H9255,T/H20850/H11021/H9255. Therefore, the energy levels /H92551,2=/H110071 2/H20881/H20849/H90040x/H208502+/H20849/H9004z/H208502 =/H110071 2/H20881/H20849/H90040x/H208502+/H20849/H90040z+/H9264k/H11509uk/H208502depend on /H11509uk, and to first order, the sensitivity of these energies to an applied strain is pro- portional to the bias /H11509/H9255i /H11509/H20849/H11509uk/H20850=/H20849/H90040z/H11007/H9264k/H11509uk/H20850 /H9255i/H9264k=/H9004ztot /H9255i/H9264k, /H208498/H20850 with a response of the TLS /H11008/H20849/H9004z//H9255/H208502. In Ref. 46, a detailed derivation is given, and the imaginary part of the response, corresponding to Q−1, is also/H11008/H9270//H208491+/H92752/H92702/H20850, which, in terms ofA/H20849/H9275/H20850, is the Lorentzian peak at /H9275=0 of Fig. 3/H20849b/H20850. The mechanism is most effective when /H9270/H110111//H9275, where/H9275 is the frequency of the vibrational mode; then, along a cycle of vibration, the following happens /H20849see Fig. 4/H20850: When the TLS is under no stress, the populations, due to the delay /H9270in their response, are being still adjusted as if the levels corre-sponded to a situation with maximum strain /H20851and therefore, of maximum energy separation between them; cf. /H9255 1,2/H20849uk/H20850/H20852, so that the lower level becomes overpopulated. As the strain is increased to its maximum value, the populations are stilladjusting as if the levels corresponded to a situation withminimum strain, thus overpopulating the upper energy level. Therefore, in each cycle, there is a net absorption of energyfrom the mechanical energy pumped into the vibrationalmode. If, on the other hand, /H9270/H112711//H9275, the TLS level populations are frozen with respect to that fast perturbation, while in theopposite limit, /H9270/H112701//H9275, the levels’ populations are always in instantaneous equilibrium with the variations of /H92551,2and there is no net absorption of energy. For an ensemble of TLSs, the contribution to Q−1is46 Qrel−1/H20849/H9275,T/H20850=P0/H92532 ET/H20885 0/H9255max d/H9255/H20885 umin1 du/H208811−u2 u1 cosh2/H20849/H9255/2T/H20850 /H11003/H9275/H9270 1+ /H20849/H9275/H9270/H208502. /H208499/H20850 The derivation of Eq. /H208499/H20850only relies on the assumptions of the existence of well defined levels that need a finite time /H9270 to reach thermal equilibrium when a perturbation is applied and the existence of bias /H20841/H90040z/H20841/H110220. This implies that such a scheme is applicable also to our one-dimensional /H208491D/H20850vibra- tions, but is valid only if the perturbation induced by the bathon the TLSs is weak, so that the energy levels are still welldefined. Therefore, we will limit the ensemble to under-damped TLSs where /H9003/H20849/H9255,T/H20850/H11021/H9255.I nE q . /H208499/H20850, the factor cosh −2/H20849/H9255/2T/H20850imposes an effective cutoff /H9255/H11021T, so that in Eq. /H208497/H20850, one can approximate coth /H20851/H9255/2T/H20852/H110112T//H9255, resulting in /H9003/H20849/H9255/H20850/H110111//H20881/H9255and thus the underdamped TLSs will satisfy /H9255 /H11350/H2085130/H9251b/H20881/H9275coT/H208522/3. For T/H11271/H2085132/H9251b/H20881/H9275co/H208522, which is fulfilled for typical sizes and temperatures /H20849see Appendix C for details /H20850, Qrel−1/H20849/H92750,T/H20850/H1101520P0/H92534 t3/2w/H208491+/H9263/H20850/H208491−2/H9263/H20850 E2/H208493−5/H9263/H20850/H20873/H9267 E/H208741/4/H20881T /H92750./H2084910/H20850 Here, Vamorph /H11011Vtot/10 was assumed. IV. COMPARISON BETWEEN CONTRIBUTIONS TO Q−1: RELAXATION PREVALENCE It is useful to estimate the relative importance of the con- tributions to Q−1coming from the last two mechanisms. For that sake, we particularize the comparison to the case of thefundamental flexural mode, which is the one usually excitedand studied, of a doubly clamped beam, with frequency /H92750 /H110156.5/H20849E//H9267/H208501/2t/L2/H20849for a cantilever, these considerationst FIG. 4. /H20849Color online /H20850Schematic representation of the levels’ population evolution of an ensemble of five identical TLSs with aresponse time /H9270/H110111//H9275, where/H9275is the frequency of the excited bending mode, whose evolution is also depicted in the lower part ofthe figure. The delay /H9270plus the bias/H90040zgive rise to the relaxational energy loss mechanism of the mode /H20849see text /H20850.SURFACE DISSIPATION IN NANOELECTROMECHANICAL … PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-5hold, with only a slight modification of the numerical pref- actors; the conclusions are the same /H20850. The result is /H20875Qrel−1/H20849T/H20850 Qoff-res−1/H20849T/H20850/H20876 fund/H11015300t1/2 L/H208493−5/H9263/H20850 /H208491+/H9263/H20850/H208491−2/H9263/H20850E /H92671 T1/2./H2084911/H20850 For a temperature T=1 K, the result is as big as 106even for a favorable case; t=1 nm, L=1/H9262m,E=50 GPa,/H9263=0.2, and /H9267=3 g /cm3. So for any reasonable temperature and dimen- sions, the dissipation is dominated by the relaxation mecha-nism, so that the prediction of the limit that surfaces set onthe quality factor of nanoresonators is, within this model forthe surface, Eq. /H2084910/H20850as follows: Q surface−1/H20849/H92750,T/H20850/H11015Qrel−1/H20849/H92750,T/H20850/H11011T1/2 /H92750. /H2084912/H20850 For typical values L/H110111/H9262m,t,w/H110110.1/H9262m,/H9253/H110115 eV, P0Vamorph /Vtot/H110111044J/m3, and Tin the range 1 mK–0.5 K, the estimate for Qsurface−1/H1101110−4gives the observed order of magnitude in experiments such as Ref. 30and also predicts correctly a sublinear dependence, but with a higher expo-nent, 1 /2 versus the experimental fit 0.36 in Ref. 30or 0.32 in Ref. 32/H20849see Fig. 5for an example /H20850. V. EXTENSIONS TO OTHER DEVICES A. Cantilevers, nanopillars, and torsional oscillators The extrapolation from doubly clamped beams to cantilevers54and nanopillars7is immediate, the only differ- ence between them being the allowed (k,/H9275/H20849k/H20850)values due to the different boundary conditions at the free end /H20851and even this difference disappears as one considers high frequencymodes, where in both cases one has k n/H11015/H208492n+1/H20850/H9266/2L/H20852. All previous results apply, and one has just to take care in the expressions corresponding to the Q−1of the fundamental mode, where there is more difference between the frequen- cies of both cases, the cantilever one being /H92750cant /H11015/H20849E//H9267/H208501/2t/L2as compared to the doubly clamped case, /H92750clamped/H110156.5/H20849E//H9267/H208501/2t/L2.B. Effect of the flexural modes on the dissipation of torsional modes The contribution from the TLSs /H11001sub-Ohmic bending mode environment to the dissipation of a torsional mode of agiven oscillator can be also estimated. We will study theeasiest /H20849and experimentally relevant 55/H20850case of a cantilever. For paddle and double paddle oscillators, the geometry ismore involved, modifying the moment of inertia and otherquantities. When these changes are included, the analysisfollows the same steps we will show. 1. Relaxation absorption We assume, based on the previous considerations on the predominant influence of the flexural modes on the TLSsdynamics, as compared with the influence of the othermodes, that the lifetime /H9270=/H9003−1of the TLSs is given by Eq. /H208497/H20850. The change in the derivation of the expression for Q−1is shown in Eq. /H208498/H20850, where the coupling constant /H9264ktorsis differ- ent, which translates simply, in Eq. /H208499/H20850, into substituting /H92532↔/H9253tors2, and the corresponding prediction for Qrel−1is as follows: Qrel−1/H20849/H92750,T/H20850/H1101520P0/H9253tors2/H92532 t3/2w/H208491+/H9263/H20850/H208491−2/H9263/H20850 E2/H208493−5/H9263/H20850/H20873/H9267 E/H208741/4/H20881T /H92750, /H2084913/H20850 where now/H92750is the frequency of the corresponding flexural mode. The range of temperatures and sizes for which thisresult applies is the same as in the case of an excited bendingmode. 2. Dissipation of symmetric nonresonant two-level systems The modified excitation spectrum of the TLS’s ensemble, Aoff-restot/H20849/H92750,T/H20850, remains the same, and the change happens in the matrix element of the transition probability of a mode /H20841k0,n0/H20856appearing in Eq. /H208495/H20850,/H20849/H9261k02//H20881/H92750/H208502. The operator yield- ing the coupling of the bath to the torsional mode which causes its attenuation is the interaction term of the Hamil-tonian, which for twisting modes is /H20851see Appendix D for the derivation of Eqs. /H2084914/H20850and /H2084915/H20850/H20852 H inttors=/H6036/H9268z/H20858 k/H9253/H20881C 8/H9262tw/H208811 2/H9267I/H6036Lk /H20881/H9275k/H20849ak†+ak/H20850,/H2084914/H20850 where/H9262=E//H208512/H208491+/H9263/H20850/H20852is a Lande coefficient and C =/H9262t3w/3 is the torsional rigidity. Again, Q−1/H20849/H9275j/H20850 =/H9004E/2/H9266E0, where the energy E0stored in a torsional mode /H9278j/H20849z,t/H20850=Asin/H20851/H208492j−1/H20850/H9266z/2L/H20852sin/H20849/H9275jt/H20850per unit volume is E0 =A2/H9275j2/H9267/H20849t2+w2/H20850/48/H20849zis the coordinate along the main axis of the rod /H20850. Expressing the amplitude Ain terms of phonon number nj, the energy stored in mode /H20841kj,nj/H20856is E0/H20849kj,n/H20850=1 2/H6036/H9275j /H20849t3w+w3t/H20850L/H20849t2+w2/H20850/H208492nj+1/H20850, /H2084915/H20850 the energy fluctuations in a cycle of such a mode isT (K)Q (T)-1 QT/CID1-1 rel1/2QT/CID1 exp T/CID1 FIG. 5. /H20849Color online /H20850Example of fit of Qrel−1/H20849/H92750,T/H20850/H20851Eq. /H2084910/H20850/H20852to experimental data. The data correspond to a Si resonator vibratingat /H92750=12.028 MHz /H20849see Ref. 30/H20850. Although the predicted order of magnitude for the dissipation and the sublinear temperature depen-dence is observed, the experimental trend follows a weaker tem-perature dependence than the prediction T 1/2. For comparison, a linear temperature dependence is shown /H20849dotted line /H20850. See the text for more details.SEOÁNEZ, GUINEA, AND CASTRO NETO PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-6/H9004E=2/H9266 /H9275j/H6036/H9275j2/H9266 /H6036/H92532/H6036/H9275j 16Ltw/H9262njAoff-restot/H20849/H9275j,T/H20850, /H2084916/H20850 and the inverse quality factor /H20849Q−1/H20850off-restot/H20849/H92750,T/H20850/H110150.04/H92534P t3/2w/H92671/4/H208491+/H9263/H208502/H208491−2/H9263/H20850 E9/4/H208493−5/H9263/H20850 /H110031 /H20881/H92750cotanh/H20875/H92750 T/H20876. /H2084917/H20850 For sizes and temperatures as the ones used for previous estimates, the relaxation contribution dominates dissipation. VI. FREQUENCY SHIFT Once the quality factor is known, the relative frequency shift can be obtained via a Kramers–Kronig relation /H20849valid in the linear regime /H20850, because both are related to the imaginary and real parts, respectively, of the acoustic susceptibility.First, we will demonstrate this, and afterwards, expressionsfor beam and cantilever will be derived and compared toexperiments. A. Relation to the acoustic susceptibility In the absence of sources of dissipation, the equation for the bending modes is given by − /H2084912/H9267/t2/H20850/H115092X//H11509t2=E/H115094X//H11509z4. The generalization in the presence of friction is −12/H9267 t2/H115092X /H11509t2=/H20849E+/H9273/H20850/H115094X /H11509z4, /H2084918/H20850 where/H9273is a complex-valued susceptibility. Inserting a solution of the form X/H20849z,t/H20850=Aei/H20849kx−/H9275t/H20850, where kis now a complex number, one gets the dispersion relation /H9275 =/H20881t2/H20849E+/H9273/H20850//H2084912/H9267/H20850k2. Now, assuming that the relative shift and dissipation are small, implying Re /H20849/H9273/H20850/H11270Eand Im /H20849k/H20850 /H11270Re/H20849k/H20850, the following expressions for the frequency shift and inverse quality factor are obtained in terms of /H9273: Q−1=/H9004/H9275//H9275=−I m /H20849/H9273/H20850/E, /H9254/H9275//H9275=R e /H20849/H9273/H20850/2E. /H2084919/H20850 Therefore, a Kramers–Kronig relation for the susceptibility can be used to obtain the relative frequency shift as follows: /H9254/H9275 /H9275/H20849/H9275,T/H20850=−1 2/H9266P/H20885 −/H11009/H11009 d/H9275/H11032Q−1/H20849/H9275/H11032,T/H20850 /H9275/H11032−/H9275, /H2084920/H20850 where Phere means the principal value of the integral. B. Expressions for the frequency shift Relaxation processes of biased, underdamped TLSs domi- nate the perturbations of the ideal response of the resonator,as we have already shown for the inverse quality factor. For most of the frequency range, /H9275/H33356/H2085130/H9251b/H20881/H9275coT/H208522/3,Q−1/H20849/H9275,T/H20850 /H11015A/H20881T//H9275, with Adefined by Eq. /H2084910/H20850. The associated pre- dicted contribution to the frequency shift, using Eq. /H2084920/H20850,i s/H9254/H9275 /H9275/H20849/H9275,T/H20850/H11015−A 2/H9266/H20881T /H9275log/H20875/H208791−/H9275 /H2085130/H9251b/H20881/H9275coT/H208522/3/H20879/H20876. /H2084921/H20850 For low temperatures, /H9275/H11271/H2085130/H9251b/H20881/H9275coT/H208522/3, the negative shift grows toward zero as /H9254/H9275//H9275/H20849/H9275,T/H20850/H11011/H20881Tlog/H20851T2/3//H9275/H20852, reaching at some point a maximum value. For high temperatures, /H9275/H11021/H2085130/H9251b/H20881/H9275coT/H208522/3, the negative shift decreases as /H9254/H9275//H9275/H20849/H9275,T/H20850/H110111/T1/6. Even though the prediction of a peak in/H9254/H9275//H9275/H20849T/H20850qualitatively matches the few experimental re- sults currently available,30,32it does not fit them quantita- tively. VII. APPLICABILITY AND FURTHER EXTENSIONS OF THE MODEL: DISCUSSION As mentioned, the predictions obtained within this theo- retical framework do qualitatively match the experimentalresults in terms of observed orders of magnitude for Q −1/H20849T/H20850, weak sublinear temperature dependence, and the presence of a peak in the frequency shift temperature dependence. How-ever, quantitative fitting is still to be reached. On the experi-mental side, more experiments need to be done at low tem-peratures to confirm the, until now, few results. 30 The several simplifications involved in the model put cer- tain constraints, some of which are susceptible to improve-ment. We enumerate them first and discuss some of them afterwards: /H20849i/H20850The probability distribution P/H20849/H9004 0x,/H90040z/H20850, bor- rowed from amorphous bulk systems, may be different for the case of the resonator’s surface. /H20849ii/H20850The assumption of noninteracting TLSs, only coupled among them in an indi-rect way through their coupling to the vibrations, breaksdown at low enough temperatures, where also the discrete-ness of the vibrational spectrum affects our predictions. /H20849iii/H20850 When temperatures rise above a certain value, high-energyphonons with 3D character dominate dissipation, the two-state description of the degrees of freedom coupled to thevibrations is not a good approximation, and thermoelasticlosses begin to play an important role. /H20849iv/H20850For strong driv- ing, anharmonic coupling among modes has to be consideredand some steps in the derivation of the different mechanisms,which assumed small perturbations, must be modified. Thiswill be the case of resonators driven to the nonlinear regime,where bistability and other phenomena take place. The solution to issue /H20849i/H20850is intimately related to a better knowledge of the surface and the different physical processestaking place there. Recent studies try to shed some light onthis question, 40and from their results, a more realistic P/H20849/H90040x,/H90040z/H20850could be derived, which is left for future work. Before that point, it is easier to wonder about the conse- quences of a dominant kind of dissipative process which corresponds to a set of TLSs with a well defined value of /H90040x and a narrow distribution of /H90040z’s of width/H90041, as was sug- gested for single-crystal silicon.56Following Ref. 56a Q−1/H20849T/H20850/H11011/H20881Tbehavior is obtained for low temperatures T/H11021/H9004 1if/H9003/H20849/H90040x,T/H20850/H11021/H9275, and Q−1/H20849T/H20850/H110111//H20881Tif/H9003/H20849/H90040x,T/H20850/H11022/H9275, while at high temperatures T/H11022/H9004 1, a constant Q−1/H20849T/H20850/H11011Q0is predicted for both cases. These predictions do not match bet-SURFACE DISSIPATION IN NANOELECTROMECHANICAL … PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-7ter with experiments than the results obtained with P/H20849/H90040x,/H90040z/H20850/H11011P0//H90040x, so issue /H20849i/H20850remains open. From the point of view of those who attribute the origin of the low-energy TLSs to the long range interaction between localized defects, a possible source of change for P/H20849/H90040x,/H90040z/H20850may be the de- crease in size of the resonator below the correlation length of this interaction. We try now to have a first estimate of the temperature for which interactions between TLSs cannot be ignored. Follow-ing the ideas presented in Ref. 46, we will estimate the tem- perature T *at which the dephasing time /H9270intdue to interac- tions is equal to the lifetime /H9270/H20849T/H20850=/H9003−1/H20849T/H20850defined in Eq. /H208497/H20850, for the TLSs that contribute most to dissipation, which are those with/H9255=/H20881/H20849/H90040x/H208502+/H20849/H90040z/H208502/H11011/H90040x/H11011T. For them,/H9270−1/H20849T/H20850 =/H9003/H20849T/H20850/H1101540/H9251b/H20881/H9275co/H20881T. The interactions between the TLSs are dipolar, described by Hint=/H20858i,jU1,2/H92681z/H92682z, with U1,2=b12/r123, b12verifying /H20855b12/H20856/H110150, and /H20855/H20841b12/H20841/H20856/H11013U0/H11015/H92532/E.46,47From the point of view of a given TLS, the interaction affects its bias, /H9004jz=/H20849/H90040z/H20850j+/H20858iUij/H9268jz, causing the fluctuations of its phase /H9254/H9255j/H20849t/H20850/H20849where there means time and not thickness /H20850, which have an associated /H9270intdefined by/H9254/H9255j/H20849/H9270int/H20850/H9270int/H110111. These fluc- tuations are caused by those TLSs which, within the time /H9270int, have undergone a transition between their two eigenstates,affecting the value of the bias of our TLS through the inter-action H int. At a temperature T, the most fluctuating TLSs are those such that /H9255=/H20881/H20849/H90040x/H208502+/H20849/H90040z/H208502/H11011/H90040x/H11011T, and their density can be estimated, using P/H20849/H90040x,/H90040z/H20850,a s nT/H11015P0kT. They will fluctuate with a characteristic time /H9270/H20849T/H20850/H11015/H2085140/H9251b/H20881/H9275co/H20881T/H20852−1, so for a time t/H11021/H9270/H20849T/H20850, the amount of these TLSs that have made a transition is roughly n/H20849t/H20850/H11015P0kTt//H9270/H20849T/H20850. For a dipolar interaction like the one described above, the average energy shift is related to n/H20849t/H20850by57/H9254/H9255/H20849t/H20850/H11015U0n/H20849t/H20850. Substituting it in the equation defining /H9270intand imposing/H9270int/H20849T*/H20850=/H9270/H20849T*/H20850give the transition temperature T*/H11015/H208516/H9251b/H20881/H9275co//H20849U0P0/H20850/H208522. For ex- ample, for a resonator such as the silicon ones studied in Ref. 30, with L=6/H9262m,t=0.2/H9262m, and w=0.3/H9262m, the estimated onset of interactions is at T*/H1101510 mK. An upper limit Thighof applicability of the model due to high-energy 3D vibrational modes playing a significant role can be easily derived by imposing Thigh=/H9275min3D. The frequency /H9275min3Dcorresponds to phonons with a wavelength comparable to the thickness tof the sample /H20849do not confuse with time /H20850, /H9275min3D=2/H9266/H20881E//H9267/t. The condition is very weak, as the value, for example, for silicon resonators reads Thigh/H11015400/t, with t given in nanometers and Thighin Kelvin. At much lower tem- peratures, the two-state description of the degrees of freedomcoupled to the vibrations ceases to be realistic, with a hightemperature cutoff in the case of the model applied to amor-phous bulk systems of T/H110115K . VIII. DISSIPATION IN A METALLIC CONDUCTOR Many of the current realizations of nanomechanical de- vices monitor the system by means of currents appliedthrough metallic conductors attached to the oscillators. Thevibrations of the device couple to the electrons in the metal-lic part. This coupling is useful in order to drive and measurethe oscillations, but it can also be a source of dissipation. We will apply here the techniques described in Refs. 58and59 /H20849see also Ref. 60/H20850in order to analyze the energy loss pro- cesses due to the excitations in the conductor. We assume that the leading perturbation acting on the electrons in the metal are offset charges randomly distributed throughout the device. A charge qat position R ˜interacts with an energy V/H20849r˜,t/H20850/H11013q2 /H92800/H20841R˜/H20849t/H20850−r˜/H20849t/H20850/H20841, /H2084922/H20850 with another charge qat a position r˜inside the metal /H20849see Fig. 6/H20850. As the bulk of the device is an insulator, this poten- tial is only screened by a finite dielectric constant, /H92800. The oscillations of the system at frequency /H92750modulate the rela- tive distance /H20841R˜/H20849t/H20850−r˜/H20849t/H20850/H20841, leading to a time-dependent poten- tial acting on the electrons of the metal. The probability per unit time of absorbing a quantum of energy/H92750can be written, using second order perturbation theory, as58,59 /H9003=/H20885dr˜dr˜/H11032dtdt /H11032V/H20849r˜,t/H20850V/H20849r˜,t/H11032/H20850Im/H9273/H20851r˜−r˜/H11032,t−t/H11032/H20852ei/H92750/H20849t−t/H11032/H20850, /H2084923/H20850 where Im/H9273/H20851r˜−r˜/H11032,t−t/H11032/H20852is the imaginary part of the response function of the metal. A. Charges in the oscillating part of the device We write the relative distance as R˜/H20849t/H20850−r˜/H20849t/H20850=R˜0−r˜0 +/H9254R˜/H20849t/H20850−/H9254r˜/H20849t/H20850and expand V/H20849r˜,t/H20850, whose time-dependent part is approximately V/H20849r˜,t/H20850/H11015q2/H20851R˜0−r˜0/H20852·/H20851/H9254R˜/H20849t/H20850−/H9254r˜/H20849t/H20850/H20852 /H92800/H20841R˜0−r˜0/H208413. /H2084924/H20850 For a flexural mode, /H20851R˜0−r˜0/H20852·/H20851/H9254R˜/H20849t/H20850−/H9254r˜/H20849t/H20850/H20852has turned into /H20851X0−x0/H20852/H20851/H9254X/H20849t/H20850−/H9254x/H20849t/H20850/H20852/H11011 tAsin/H20849/H9275t/H20850, where t=tins+tmetalis the thickness of the beam and Ais the amplitude of vibration of the mode. Thus, the average estimate for this case for thecorrection of V/H20849r ˜,t/H20850is =V intezx FIG. 6. /H20849Color online /H20850Sketch of the distribution of charges in the device. When the system oscillates, these charges induce time-dependent potentials which create electron-hole pairs in the metalliclayer deposited on top of the beam, absorbing part of the mechani-cal energy of the flexural mode. See text for details.SEOÁNEZ, GUINEA, AND CASTRO NETO PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-8/H9254V/H20849r˜,t/H20850/H11011q2 /H92800tAsin/H20849/H9275t/H20850 L3, /H2084925/H20850 Lbeing the resonator’s length. The integral over the region occupied by the metal in Eq. /H2084923/H20850can be written as an integral over r˜0andr˜0/H11032. The dielec- tric constant of a dirty metal in the random phase approxi-mation is 61 Im/H9273/H20849r˜−r˜/H11032,t−t/H11032/H20850=/H20885d/H9275dq˜ /H208492/H9266/H208504eiq˜/H20849r˜−r˜/H11032/H20850ei/H9275/H20849t−t/H11032/H20850/H20841/H9275/H20841 e4D/H9263/H20841q˜/H208412, /H2084926/H20850 where eis the electronic charge, D=/H6036vFlis the diffusion constant, vFis the Fermi velocity, lis the mean free path, and /H9263/H11015/H20849kFtmetal /H208502//H6036vFis the one-dimensional density of states. By combining Eqs. /H2084923/H20850and /H2084926/H20850and assuming that the position of the charge, R˜0, is in a generic point inside the beam and that the length scales are such that kF−1,tmetal,t /H11270L, we can obtain the leading dependence of /H9003in Eq. /H2084923/H20850 onLas follows: /H9003/H11015/H20841/H92750/H20841A2 D/H9263L/H20873t L/H208742 /H11015/H20841/H92750/H20841A2 lkF2L3/H20873t tmetal/H208742 , /H2084927/H20850 where we also assume that /H20841q/H20841=e. The energy absorbed per cycle of oscillation and unit volume will be /H9004E =/H208492/H9266//H92750/H20850/H6036/H92750/H9003ph/t2L=2/H9266/H6036/H9003ph/t2Land the inverse quality factor Qph−1/H20849/H92750/H20850will correspond to Qph−1/H20849/H92750/H20850=1 2/H9266/H9004E E0=/H6036/H9003ph twL1 1 2/H9267/H927502A2, /H2084928/H20850 where E0is the elastic energy stored in the vibration and Ais the amplitude of vibration. Substituting the result for /H9003, one obtains Q−1/H110152/H6036 lkF2L4t2/H9267/H92750/H20873t tmetal/H208742 . /H2084929/H20850 In a narrow metallic wire of width tmetal, we expect that l/H11011tmetal. Typical values for the parameters in Eq. /H2084929/H20850are kF−1/H110151Å , A/H110151Å , l/H11011tmetal /H1101510 nm /H11011102Å,t/H11015100 nm /H11015103Å, and L/H110151/H9262m/H11015104Å. Hence, each charge in the device gives a contribution to Q−1of order 10−20. The effect of all charges is obtained by summing over all charges in thebeam. If their density is n q, we obtain Q−1/H110152/H6036nq lkF2L3/H9267/H92750/H20873t tmetal/H208742 . /H2084930/H20850 For reasonable values of the density of charges, nq=lq−3,lq /H1140710 nm, this contribution is negligible, Q−1/H1135110−16. B. Charges in the substrate surrounding the device Many resonators, however, are suspended at distances much smaller than Lover an insulating substrate, which can also contain unscreened charges. As the Coulomb potentialinduced by these charges is long range, the analysis de-scribed above can be applied to all charges within a distance of order Lfrom the beam. Moreover, the motion of these charges is not correlated with the vibrations of the beam, so that now the value of /H20841 /H9254R˜/H20849t/H20850−/H9254r˜/H20849t/H20850/H20841has to be replaced by /H20841/H9254R˜/H20849t/H20850−/H9254r˜/H20849t/H20850/H20841 /H11015 Aei/H92750t, /H2084931/H20850 and the value of /H20841R˜0−r˜0/H20841/H11011L. Assuming, as before, a density of charges nq=lq3, the effect of all charges in the substrate leads to Q−1/H110152/H6036L l/H20849kFtmetal /H208502lq3t2/H9267/H92750/H110150.3/H6036L3 l/H20849kFtmetal /H208502lq3t3/H20881E/H9267, /H2084932/H20850 where the second result corresponds to the fundamental mode,/H92750/H110156.5/H20849t/L2/H20850/H20881E//H9267. For values L/H110151/H9262m,A/H110151Å , kF−1/H110151Å , l/H11011tmetal /H1101510 nm, and lq/H1101110 nm, we obtain Q−1 /H1101110−9. Thus, given the values of Q−1reached experimen- tally until now, this mechanism can be disregarded althoughit sets a limit to Q −1at the lowest temperatures. Note also that this estimate neglects cancellation effects betweencharges of opposite signs. IX. CONCLUSIONS Disorder and configurational rearrangements of atoms and adsorbed impurities at surfaces of nanoresonators dominatethe dissipation of their vibrational eigenmodes at low tem-peratures. We have given a theoretical framework to describein a unified way these processes, improving and extendingprevious ideas. 1Based on the good description of low tem- perature properties of disordered bulk insulators provided bythe standard tunneling model, 43,44,46and in particular, of acoustic phonon attenuation in such systems, we adapt it todescribe the damping of 1D flexural and torsional modes ofNEMS associated with the amorphouslike nature of their sur-faces. Correcting some aspects of Ref. 1, we have calculated the damping of the modes by the presence of an ensemble ofindependent TLSs coupled to the local deformation gradientfield /H11509iujcreated by vibrations. The different dissipation channels to which this ensemble gives rise to have been de-scribed, focusing the attention on the two most important:relaxation dynamics of biased TLSs and dissipation due tosymmetric nonresonant TLSs. The first one is caused by thefinite time it takes for the TLSs to readjust their equilibrium populations when their bias /H9004 0zis modified by local strains, with biased TLSs playing the main role, as this effect is /H11008/H20851/H90040z//H20881/H20849/H90040z/H208502+/H20849/H90040x/H208502/H208522. In terms of the excitation spectrum of the TLSs, it corresponds to a Lorentzian peak around /H9275=0. The second effect is due to the modified absorption spectrumof the TLSs caused by their coupling to all the vibrations,especially the flexural modes, whose high density of states atlow energies leads to sub-Ohmic damping. 62–64A broad in- coherent spectral strength is generated, enabling the“dressed-by-the-modes” TLSs to absorb the energy of an ex-cited mode and deliver it to the rest of the modes when theydecay.SURFACE DISSIPATION IN NANOELECTROMECHANICAL … PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-9We have given analytical expressions for the contributions of these mechanisms to the linewidth of flexural /H20851Eqs. /H208496/H20850 and /H2084910/H20850/H20852or torsional modes /H20851Eqs. /H2084913/H20850and /H2084917/H20850/H20852in terms of the inverse quality factor Q−1/H20849/H92750/H20850=/H9004/H92750//H92750, showing the de- pendencies on the dimensions, temperature, and other rel- evant parameters characterizing the device. We have com-pared the two mechanisms, concluding that relaxationdominates dissipation, with a predicted Q −1/H20849/H92750,T/H20850 /H11011T1/2//H92750. Expressions have been provided for damping of flexural modes in cantilevers and doubly clamped beams aswell as for damping of their torsional modes. Analytical predictions for associated frequency shifts have also been calculated /H20851Eq. /H2084921/H20850/H20852. Some important find- ings have been achieved, such as the qualitative agreementwith a sublinear temperature dependence of Q −1/H20849T/H20850, the pres- ence of a peak in the frequency shift temperature dependence /H9254/H9275//H9275/H20849T/H20850, or the observed order of magnitude of Q−1/H20849T/H20850in the existing experiments studying flexural phonon attenua- tion at low temperatures.30,32Nevertheless, the lack of full quantitative agreement has led to a discussion on the as-sumptions of the model, its links with the physical processesoccurring at the surfaces of NEMS, its range of applicability,and the improvements to reach the desired quantitative fit. Finally, we have also considered the contributions to the dissipation due to the presence of metallic electrodes depos-ited on top of the resonators, which can couple to the elec-trostatic potential induced by random charges. We haveshown that the coupling to charges within the vibrating partsdoes not contribute appreciably to the dissipation. Couplingto charges in the substrate, although more significant, stillleads to small dissipation effects /H20851Eq. /H2084932/H20850/H20852, imposing a limit at low temperatures of Q −1/H1101110−9, which is very small com- pared to the values reached in current experiments.30,32 ACKNOWLEDGMENTS C.S. and F.G. acknowledge funding from MEC /H20849Spain /H20850 through FPU grant /H20849Grant No. FIS2005-05478-C02-01 /H20850and the Comunidad de Madrid through the program CITEC-NOMIK /H20849No. CM2006-S-0505-ESP-0337 /H20850. A.H.C.N. is sup- ported through NSF Grant No. DMR-0343790. APPENDIX A: CALCULATION OF THE SPECTRAL FUNCTION FOR THE BENDING MODES The starting point is the Hamiltonian of Ref. 1,H=/H90040/H9268x +/H9253/H9268z/H11509iuj, where /H11509iujis a component of the deformation gra- dient matrix. In the case of the bending modes of a rod ofdimensions L,t, and w, and mass density /H9267, there are two variables X/H20849z/H20850,Y/H20849z/H20850/H20849transversal displacements of the rod as a function of the position along its length, z/H20850obeying51 EIy/H115094X /H11509z4=−/H9267tw/H115092X /H11509t2, EIx/H115094Y /H11509z4=−/H9267tw/H115092Y /H11509t2/H20849A1/H20850 /H20849where Iy=t3w/12 and Ix=w3t/12/H20850, so that there are plane waves X/H20849z,t/H20850,Y/H20849z,t/H20850/H11011ei/H20849kz−/H9275t/H20850, but with a quadratic disper-sion relation,/H9275j/H20849k/H20850=/H20881EIj /H9267abk2. One can thus express X/H20849z/H20850,Y/H20849z/H20850 in terms of bosonic operators, for example, X/H208490/H20850 =/H20858k/H20881/H6036 2/H9267twL/H9275k/H20849ak†+ak/H20850. We can relate this variables to the strain field /H11509iujthrough the free energy Fas follows: Frod=1 2/H20885dzEI y/H20873/H115092X /H11509z2/H208742 +EIx/H20873/H115092Y /H11509z2/H208742 =1 2/H20885dz/H20885dS1 2/H9261/H20858 iuii2+/H9262/H20858 i,kuik2 /H110151 2/H20885dz/H20885dS/H208733 2/H9261+9/H9262/H20874uij2, /H20849A2/H20850 extracting an average equivalence for one component uij, uij/H110152/H20881EIy//H208493/H9261+18/H9262/H20850tw/H115092X//H11509z2. The interaction term in the Hamiltonian is then Hint=/H6036/H9268z/H20858 k/H9261k2 /H20881/H9275k/H20849ak†+ak/H20850 =/H6036/H9268z/H20858 ij/H20858 k/H208752/H9253/H90040x /H90040/H20881EIy /H208493/H9261+1 8/H9262/H20850tw/H208811 2/H9267tw/H6036L/H20876 /H11003/H20849kij/H208502 /H20881/H9275kij/H20849akij†+akij/H20850. /H20849A3/H20850 So we have approximately nine times the same Hamiltonian, once for each uij, and the corresponding spectral function J/H20849/H9275/H20850will be nine times the one calculated for Hint=/H6036/H9268z/H20858 k/H9261k2 /H20881/H9275k/H20849ak†+ak/H20850 /H11229/H6036/H9268z/H20858 k/H208752/H9253/H90040x /H90040/H20881EIy /H208493/H9261+1 8/H9262/H20850tw/H208811 2/H9267tw/H6036L/H20876 /H11003k2 /H20881/H9275k/H20849ak†+ak/H20850. /H20849A4/H20850 For a Hamiltonian of the class H=/H90040/H9268x+/H6036/H9268z/H20858k/H9261k/H20849ak†+ak/H20850, the spectral function J/H20849/H9275/H20850is given by J/H20849/H9275/H20850=1 2/H9266/H20858k/H9261k2/H9254/H20849/H9275 −/H9275k/H20850, so that in our case the expression for it is J/H20849/H9275/H20850=1 2/H9266/H20858 k/H208752/H9253/H90040x /H90040/H20881EIy /H208493/H9261+1 8/H9262/H20850tw/H208811 2/H9267tw/H6036Lk2 /H20881/H9275k/H208762 /H11003/H9254/H20849/H9275−/H9275k/H20850. /H20849A5/H20850 Taking the continuum limit /H208491 L/H20858k→1 /H9266/H20848dk/H20850, J/H20849/H9275/H20850=2L /H208492/H9266/H208502/H20885 kminkmax dk/H208752/H9253/H90040x /H90040/H20881EIy /H208493/H9261+1 8/H9262/H20850tw/H208811 2/H9267tw/H6036L/H208762 /H11003k4 /H9275k/H9254/H20849/H9275−/H9275k/H20850. /H20849A6/H20850 Using the dispersion relation /H9275j/H20849k/H20850=/H20881EIj /H9267abk2=ck2, we express the integral in terms of the frequency as follows:SEOÁNEZ, GUINEA, AND CASTRO NETO PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-10J/H20849/H9275/H20850=L /H208492/H9266/H208502/H20885 /H9275min/H9275cod/H9275k /H20881c/H9275k/H208752/H9253/H90040x /H90040/H20881EIy /H208493/H9261+1 8/H9262/H20850tw /H11003/H208811 2/H9267tw/H6036L/H208762k4 /H9275k/H9254/H20849/H9275−/H9275k/H20850 =L /H208492/H9266/H208502/H208752/H9253/H90040x /H90040/H20881EIy /H208493/H9261+1 8/H9262/H20850tw/H208811 2/H9267tw/H6036L/H208762/H20881/H9275 c5/2. /H20849A7/H20850 Jflex/H20849/H9275/H20850is just nine times this /H20851Eq. /H208492/H20850/H20852. APPENDIX B: DISSIPATION FROM OFF-RESONANCE DRESSED TWO-LEVEL SYSTEMS We follow the method of Ref. 65. The form of A/H20849/H9275/H20850, the spectral function of a single TLS, for frequencies /H9275/H11270/H9004 and /H9275/H11271/H9004 can be estimated using perturbation theory. Without the interaction, the ground state /H20841s/H20856of the TLS is the sym- metric combination of the ground states of the two wells andthe excited state is the antisymmetric one, /H20841a/H20856. We will use Fermi’s golden rule applied to the sub-Ohmic spin-boson Hamiltonian, H=/H9004 /H9268x+/H6036/H9261/H9268z/H20858k/H20851k2 /H20881/H9275k/H20852/H20849ak†+ak/H20850+/H20858k/H6036/H9275/H20849k/H20850ak†ak, where akis the annihilation operator of a bending mode k. Considering only the low-energy modes /H9275/H20849k/H20850/H11270/H9004, to first order the ground state and a state with energy /H9275/H20849k/H20850are given by /H20841g/H20856/H11229/H20841 s/H20856−/H9261k2 /H20881/H9275k 2/H9004ak†/H20841a/H20856+¯, /H20841k/H20856/H11229ak†/H20841s/H20856−/H9261k2 /H20881/H9275k 2/H9004/H20841a/H20856+¯. /H20849B1/H20850 We estimate the behavior of A/H20849/H9275/H20850by taking the matrix ele- ment of/H9268zbetween these two states, obtaining /H20851remember that/H9275/H20849k/H20850/H11008k2/H20852 A/H20849/H9275/H20850/H11011/H6036/H9251b/H20881/H9275co/H20881/H9275 /H90042+¯,/H9275/H20849k/H20850/H11270/H9004. /H20849B2/H20850 The expression in the numerator is proportional to the spec- tral function of the coupling, J/H20849/H9275/H20850=/H9251b/H20881/H9275co/H20881/H9275. Now we turn our attention to the case /H9275/H20849k/H20850/H11271/H9004, where the ground state /H20841g/H20856 and an excited state /H20841k/H20856can be written as /H20841g/H20856/H11229/H20841 s/H20856−/H9261k2//H20881/H9275k /H6036/H9275k+2/H9004ak†/H20841a/H20856+¯, /H20841k/H20856/H11229ak†/H20841s/H20856+/H9261k2//H20881/H9275k /H6036/H9275k−2/H9004/H20841a/H20856+¯. /H20849B3/H20850 The matrix element /H208550/H20841/H9268z/H20841k/H20856is/H11011/H9261k2 /H20881/H9275k4/H9004 /H20849/H6036/H9275k/H208502, leading to A/H20849/H9275/H20850/H11011/H9251b/H20881/H9275co/H90042 /H60363/H92757/2+¯,/H9275/H20849k/H20850/H11271/H9004. /H20849B4/H208501. Value of Aoff-restot„/H92750… Now we will add the contributions of all the non- resonant TLSs using the probability distribution P/H20849/H90040x,/H90040z/H20850 =P0//H90040x.43,44For the case of weak coupling, /H9251b/H110211/2 and /H92750/H33356/H208492/H9251b/H208502/H9275co, which is the one found in experiments, one has Aofftot/H20849/H92750/H20850/H11011/H20885 /H6036/H20851/H92750+/H9003/H20849/H92750/H20850/H20852/H9255max d/H90040x/H20885 −/H90040x/H90040xd/H9004zP /H90040x/H6036/H9251b/H20881/H9275co/H20881/H92750 /H20849/H90040x/H208502 +/H20885 /H6036/H208492/H9251b/H208502/H9275co/H6036/H20851/H92750−/H9003/H20849/H92750/H20850/H20852 d/H90040x/H20885 −/H90040x/H90040xd/H9004zP /H90040x/H9251b/H20881/H9275co/H20849/H90040x/H208502 /H60363/H927507/2, obtaining the result Aoff-restot/H20849/H92750/H20850/H110152P/H9251b/H20881/H9275co//H92750. 2. Off-resonance contribution for T/H110220 Using the same scheme, the modifications due to the tem- perature will appear in the density of states of absorption andemission of energy corresponding to a dressed TLS,A/H20849/H9004, /H9275,T/H20850. Now that there will be a probability for the TLS to be initially in the excited antisymmetric state, /H20841a/H20856, propor- tional to exp /H20851−/H9004/kT/H20852, and to emit energy /H6036/H9275, giving it to our externally excited mode, /H20841k,n/H20856, thus compensating the ab- sorption of energy corresponding to the opposite case /H20849tran- sition from /H20841s/H20856/H20841k,n/H20856to/H20841a/H20856/H20841k,n−1/H20856/H20850, but contributing in an additive manner to the total amount of fluctuations, whichare the ones defining the linewidth of the vibrational modeobserved in experiments, fixing the value of Q −1/H20849/H9275,T/H20850. The expression for A/H20849/H9004,/H9275,T/H20850is given by A/H20849/H9275,T/H20850=1 Z/H20858 i/H20858 f/H20841/H20855i/H20841/H9268z/H20841f/H20856/H208412e−Ei/kT/H9254/H20851/H6036/H9275−/H20849Ef−Ei/H20850/H20852. /H20849B5/H20850 We consider a generic state /H20841ia/H20856=/H20841a/H20856/H20841k1n1,..., kjnj,... /H20856 or /H20841is/H20856=/H20841s/H20856/H20841k1n1,..., kjnj,... /H20856, and states that differ from it in /H6036/H9275j,/H20841fa/H11006/H20856=/H20841a/H20856/H20841k1n1,..., kjnj/H110061,... /H20856and /H20841fs/H11006/H20856/H20841s/H20856/H20841k1n1,..., kjnj/H110061,... /H20856. As for T=0, we will correct them to first order in the interaction Hamiltonian Hint =/H6036/H9261/H9268z/H20858k/H20881/H9275k/H20849ak†+ak/H20850, and then calculate the square of the matrix element of /H9268z,/H20841/H20855i/H20841/H9268z/H20841f/H20856/H208412. Elements /H20841/H20855ia,s/H20841/H9268z/H20841fa+,s+/H20856/H208412correspond to absorption by the dressed TLS of an energy /H6036/H9275jfrom the mode kj, while ele- ments /H20841/H20855ia,s/H20841/H9268z/H20841fa−,s−/H20856/H208412correspond to emission and “feeding” of the mode with a phonon /H6036/H9275j. The expressions for the initial states are /H20841is/H20856=/H20841s/H20856/H20841k1n1, ..., kjnj, ... /H20856→/H20841s/H20856/H20841k1n1, ..., kjnj, ... /H20856 +/H20858 ki/H33529ni/H110220/H6036/H9261/H20881ni/H9275i /H6036/H9275i+2/H9004/H20841a/H20856/H20841...kini−1... /H20856 −/H20858 ∀ki/H6036/H9261/H20881/H20849ni+1/H20850/H9275i /H6036/H9275i−2/H9004/H20841a/H20856/H20841...kini+1... /H20856,SURFACE DISSIPATION IN NANOELECTROMECHANICAL … PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-11/H20841ia/H20856=/H20841a/H20856/H20841k1n1, ..., kjnj, ... /H20856→/H20841a/H20856/H20841k1n1, ..., kjnj, ... /H20856 +/H20858 ki/H33529ni/H110220/H6036/H9261/H20881ni/H9275i /H6036/H9275i−2/H9004/H20841s/H20856/H20841...kini−1... /H20856 −/H20858 ∀ki/H6036/H9261/H20881/H20849ni+1/H20850/H9275i /H6036/H9275i+2/H9004/H20841s/H20856/H20841...kini+1... /H20856, /H20849B6/H20850 and, for example, the state /H20841fa+/H20856is given by /H20841fa+/H20856=/H20841a/H20856/H20841k1n1, ..., kjnj+1, ... /H20856→/H20841a/H20856/H20841k1n1, ..., kjnj +1, ... /H20856+/H20858 ki/H33529ni/H110220,i/HS11005j/H6036/H9261/H20881ni/H9275i /H6036/H9275i−2/H9004/H20841s/H20856/H20841...kini−1... kjnj +1... /H20856+/H6036/H9261/H20881/H20849nj+1/H20850/H9275j /H6036/H9275j−2/H9004/H20841s/H20856/H20841...kjnj.../H20856 −/H20858 ∀ki,i/HS11005j/H6036/H9261/H20881/H20849ni+1/H20850/H9275i /H6036/H9275i+2/H9004/H20841s/H20856/H20841...kini+1... kjnj+1... /H20856 −/H6036/H9261/H20881/H20849nj+2/H20850/H9275j /H6036/H9275j+2/H9004/H20841s/H20856/H20841...kjnj+2... /H20856/H20849 B7/H20850 with a similar expression for the rest of the states. The value of /H20841/H20855ia,s/H20841/H9268z/H20841fa+,s+/H20856/H20841 /H20849 absorption /H20850 is /H20841/H20855ia,s/H20841/H9268z/H20841fa+,s+/H20856/H20841 =/H20841/H9261/H20881/H20849nj+1/H20850/H9275j4/H9004 /H20849/H6036/H9275j/H208502−4/H90042/H20841, with nj=0,1,...., while for emission, the re- sult is /H20841/H20855ia,s/H20841/H9268z/H20841fa−,s−/H20856/H20841=/H20841/H9261/H20881nj/H9275j4/H9004 /H20849/H6036/H9275j/H208502−4/H90042/H20841, with nj=1,.... By taking the limits we considered at T=0 /H20849/H6036/H9275j/H11270/H9004 and/H6036/H9275j/H11271/H9004 /H20850, the results are the same as for T=0 for absorption, except for a factor /H20849nj+1/H20850, and we also have now the possibility of emis- sion, with the same matrix element but with the factor nj: Absorption/H20902/H11011/H20849nj+1/H20850/H6036/H9251b/H20881/H9275co/H20881/H9275 /H90042,/H6036/H9275j/H11270/H9004 /H11011/H20849nj+1/H20850/H9251b/H20881/H9275co/H90042 /H60363/H92757/2,/H6036/H9275j/H11271/H9004,/H20903 Emission/H20902/H11011nj/H6036/H9251b/H20881/H9275co/H20881/H9275 /H90042,/H6036/H9275j/H11270/H9004 /H11011nj/H9251b/H20881/H9275co/H90042 /H60363/H92757/2,/H6036/H9275j/H11271/H9004./H20903/H20849B8/H20850 Now we have to sum over all initial states, and by noting that the first order correction to the energy of any eigenstate is 0,the partition function Zis easy to calculate, everything fac- torizes, and the result is, for example, in the case /H6036 /H9275j/H11270/H9004,a s follows: Aabs/H20849/H9004,/H9275j,T/H20850=1 Z/H20858 i/HS11005j,ni=0/H11009 /H20858 nj=0/H11009/H6036/H9251b/H20881/H9275co/H20881/H9275j /H90042/H20849nj+1/H20850 /H11003exp/H20873−/H6036/H9275jnj kT/H20874exp/H20873−/H6036/H20858ini/H9275i kT/H20874 =/H6036/H9251b/H20881/H9275co /H90042/H20881/H9275j 1−e−/H6036/H9275j/kT,Aem/H20849/H9004,−/H9275j,T/H20850=1 Z/H20858 i/HS11005j,ni=0/H11009 /H20858 nj=1/H11009/H6036/H9251b/H20881/H9275co/H20881/H9275j /H90042nj /H11003exp/H20873−/H6036/H9275jnj kT/H20874exp/H20873−/H6036/H20858ini/H9275i kT/H20874 =/H6036/H9251b/H20881/H9275co /H90042/H20881/H9275je−/H6036/H9275j/kT 1−e−/H6036/H9275j/kT. /H20849B9/H20850 InAabs/H20849/H9004,/H9275j,T/H20850, we have added the contributions from the matrix elements /H20841/H20855is/H20841/H9268z/H20841fs+/H20856/H208412e/H9004/kT+/H20841/H20855ia/H20841/H9268z/H20841fa+/H20856/H208412e−/H9004/kT =/H20841/H20855is/H20841/H9268z/H20841fs+/H20856/H208412/H20851e/H9004/kT+e−/H9004/kT/H20852. The sum of exponentials cancels with the partition function of the TLS /H20849appearing as a factor in the total Z/H20850, leading in this way to the expression above /H20851the same applies for Aem/H20849/H9004,−/H9275j,T/H20850/H20852. The total fluctuations will be proportional to their sum, which thus turns out to be at this level of approximation /H11008cotanh /H20851/H6036/H9275j/kT/H20852as follows: Adiss/H20849/H9004,/H9275j,T/H20850=Aabs/H20849/H9004,/H9275j,T/H20850+Aem/H20849/H9004,−/H9275j,T/H20850=Adiss/H20849/H9004,/H9275j,T =0/H20850cotanh/H20875/H6036/H9275j kT/H20876. /H20849B10 /H20850 In fact, this result applies for any other type of modes, inde- pendent of its dispersion relation, provided the coupling Hamiltonian is linear in /H9268zand /H20849a−k†+ak/H20850and everything is treated at this level of perturbation theory. Moreover, it can be proven that if one has an externally excited mode with anaverage population /H20855n j/H20856, and fluctuations around that value are thermal-like, with a probability /H11008exp/H20851/H20849−/H20841nj−/H20855nj/H20856/H20841/kT/H20850/H20852, one recovers again the same temperature dependence, /H11008cotanh /H20851/H6036/H9275j/kT/H20852. APPENDIX C: DERIVATION OF Qrel−1„/H92750,T…[EQUATION (10)] As discussed after Eq. /H208499/H20850, we have to sum over under- damped TLSs, /H9255/H33356 /H2085130/H9251b/H20881/H9275coT/H208522/3, using the approximation for/H9003,/H9003/H20849/H9255,T/H20850/H1101130/H9251b/H20881/H9275coT//H20881/H9255. Moreover, if /H92750/H33356/H9003 /H20849/H9255 =/H2085130/H9251b/H20881/H9275coT/H208522/3,T/H20850, then in the whole integration range /H92750/H11271/H9003 /H20849/H9255,T/H20850⇔/H92750/H9270/H20849/H9255,T/H20850/H112711/H20849see Fig. 7/H20850, so that/H92750/H9270//H208511 +/H20849/H92750/H9270/H208502/H20852/H110151//H20849/H92750/H9270/H20850.Qrel−1/H20849/H92750,T/H20850follows Qrel−1/H20849/H92750,T/H20850/H11015P0/H92532 ET/H20885 /H2085130/H9251b/H20881/H9275coT/H208522/3T d/H9255/H20885 umin1 du/H208811−u2 u/H9003/H20849/H9255,T/H20850 /H92750. /H20849C1/H20850/CID6 /CID1/CID7/CID3/CID6/CID4 /CID6/CID5/CID1 FIG. 7. /H20849Color online /H20850Evolution with /H9004rof the different quan- tities determining the approximations to be taken in the integrand /H92750/H9270//H208511+/H20849/H92750/H9270/H208502/H20852of Eq. /H208499/H20850.SEOÁNEZ, GUINEA, AND CASTRO NETO PHYSICAL REVIEW B 77, 125107 /H208492008 /H20850 125107-12For temperatures T/H11271/H2085130/H9251b/H20881/H9275coT/H208522/3, which holds for rea- sonable Tand sizes, the integral, which renders a result of the kind Qrel−1/H20849/H92750,T/H20850/H11015/H208491//H92750/H20850/H20849A/H20881T−BT1/3/H20850, can be approxi- mated by just the first term, obtaining Eq. /H2084910/H20850. In any case, for completeness, we give the expression for Bas follows: B/H11015500P0/H925314/3 t2w4/3/H208491+/H9263/H208504/3/H208491−2/H9263/H208504/3 E11/3/H208493−5/H9263/H20850/H20873/H9267 E/H208741/3 . /H20849C2/H20850 Also, for completeness, we give the result for higher temperatures, although for current sizes the condition /H9003/H20849/H9255=/H2085130/H9251b/H20881/H9275coT/H208522/3,T/H20850/H11022/H92750implies values of Tabove the range of applicability of the standard tunneling model. Now for some range of energies, /H9003/H20849/H9255,T/H20850/H11022/H92750and the range of integration is divided into two regions, one where /H92750/H9270/H112711 and one where the opposite holds: Qrel−1/H20849/H92750,T/H20850=P0/H92532 ET/H20885 umin1 du/H208811−u2 u/H20877/H20885 /H2085130/H9251b/H20881/H9275coT/H208522/3/H2085116/H9251b/H20881/H9275co2T//H20881/H92750/H208522 /H11003dE/H92750/H9270/H20849/H9255,T/H20850+/H20885 /H2085116/H9251b/H20881/H9275co2T//H20881/H92750/H208522T dE1 /H92750/H9270/H20849/H9255,T/H20850/H20878. /H20849C3/H20850 The final result is Qrel−1/H20849/H92750,T/H20850/H11015−7P0/H92532/H92750/T+A/H20881T//H92750 −CT//H927502, with Adefined by Eq. /H2084910/H20850and Cby C/H110151500 P0/H92536 t3w2/H208491+/H9263/H208502/H208491−2/H9263/H208502 E3/H208493−5/H9263/H208502/H20873/H9267 E/H208741/2 . /H20849C4/H20850 All the results for Qrel−1/H20849/H92750,T/H20850have to be multiplied by the fraction of volume of the resonator presenting amorphous features, Vamorph /twL. APPENDIX D: DERIVATION OF EQUATIONS ( 14) and ( 15) To derive the interaction Hamiltonian /H20851Eq. /H2084914/H20850/H20852, note that in terms of the deformation gradient matrix /H11509iuj, it must be Hint=/H9253/H9268z/H11509iuj, where the deformations are caused in this case by the twisting of the resonator about its main axis. Thetwisting modes correspond 51to the rotation angle around the longest main axis /H9278obeying the wave equation C/H115092/H9278 /H11509z2=/H9267I/H115092/H9278 /H11509t2/H20849D1/H20850 so the variable /H9278can be expressed in terms of boson opera- tors/H9278/H208490/H20850=/H20858 k/H20881/H6036 2/H9267twL/H9275k/H20849ak†+ak/H20850, /H20841/H11509/H9278/H20841z=0=/H20881/H6036 2/H9267twL/H20858 kk /H20881/H9275k/H20849ak†+ak/H20850. /H20849D2/H20850 To obtain an approximate expression for /H11509iujin terms of the modes’ operators ak, we relate /H11509uto/H11509/H9278through the expres- sions for the free energy of the rod in terms of both variablesas follows: F rod=1 2/H20885dzC/H20873/H11509/H9278 /H11509z/H208742 =1 2/H20885dz/H20885dS4/H9262/H20875/H20873/H11509ux /H11509z/H208742 +/H20873/H11509uy /H11509z/H208742/H20876/H110151 2/H20885dz/H20885dS8/H9262/H20873/H11509ux /H11509z/H208742 /H20849D3/H20850 and the approximate relation /H20841/H11509u/H20841z=0=/H11509ux//H11509z =/H20881C/8/H9262tw/H20841/H11509/H9278/H20841z=0is found. This relation together with Eq. /H20849D2/H20850leads to the stated result /H20851Eq. /H2084914/H20850/H20852. 1. Calculation of E 0 To classically calculate the energy stored by a torsional mode/H9278j/H20849z,t/H20850=Asin/H20851/H208492j−1/H20850/H9266z/2L/H20852sin/H20849/H9275jt/H20850, we just calculate the kinetic energy in a moment where the elastic energy is zero, for example, at time t=0. If an element of mass is originally at position /H20849x,y,z/H20850/H20849x,ytransversal coordinates /H20850, with a torsion/H9278/H20849z,t/H20850it moves to r/H6023/H20849t/H20850=/H20873/H20881x2+y2cos/H20875arccosx /H20881x2+y2+/H9278/H20876, /H20881x2+y2sin/H20875arccosx /H20881x2+y2+/H9278/H20876,z/H20874. /H20849D4/H20850 The kinetic energy at time t=0 is E0=/H20885 0L dz/H20885 −t/2t/2 dx/H20885 −w/2w/2 dy·1 2/H9267/H20879dr/H6023 dt/H20879 t=02 . /H20849D5/H20850 Substituting the expression for r/H6023/H20849t/H20850in the integrand, one ar- rives at E0=/H20885 0L dz/H20885 −t/2t/2 dx/H20885 −w/2w/2 dy1 2/H9267A2/H9275j2sin2/H20875/H208492j−1/H20850/H9266 2Lz/H20876 /H11003/H20849x2+y2/H20850=1 48A2/H9275j2/H9267L/H20849t3w+w3t/H20850. /H20849D6/H20850 In terms of the creation and annihilation operators, /H9278j/H20849z,t/H20850=/H6036 2L/H9267I/H9275j/H20849aj†+aj/H20850ei/H20849kjz−/H9275t/H20850, so the mean square of its am- plitude is /H20855/H9278j2/H20856=A2/2=/H6036/H208492n+1/H20850/2L/H9267I/H9275j. 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PhysRevB.73.104505.pdf

Critical charge instability on the verge of the Mott transition and the origin of quantum protection in high- Tccuprates T. K. Kope ć Institute for Low Temperature and Structure Research, Polish Academy of Sciences, POB 1410, 50-950 Wroclaw 2, Poland /H20849Received 20 September 2005; revised manuscript received 24 January 2006; published 10 March 2006 /H20850 The concept of topological excitations and the related ground state degeneracy are employed to establish an effective theory of the superconducting state evolving from the Mott insulator for high- Tccuprates. The theory includes the effects of the relevant energy scales with the emphasis on the Coulomb interaction Ugoverned by the electromagnetic U /H208491/H20850compact group. The results are obtained for the layered t−t/H11032−t/H11036−U−Jsystem of strongly correlated electrons relevant for cuprates. Casting the Coulomb interaction in terms of composite-fermions via the gauge flux attachment facility, we show that instanton events in the Matsubara “imaginarytime,” labeled by topological winding numbers, are essential configurations of the phase field dual to thecharge. This provides a nonperturbative concept of the topological quantization and displays the significance ofdiscrete topological sectors in the theory governed by the global characteristics of the phase field. We show thatfor topologically ordered states these quantum numbers play the role of an order parameter in a way similar tothe Landau order parameter for conventionally ordered states. In analogy to the usual phase transition that ischaracterized by a sudden change of the symmetry, the topological phase transitions are governed by adiscontinuous change of the topological numbers signaled by the divergence of the zero-temperature topologi-cal susceptibility. This defines a quantum criticality ruled by topologically conserved numbers rather than theLandau principle of the symmetry breaking. We show that in the limit of strong correlations topological chargeis linked to the average electronic filling number and the topological susceptibility to the electronic compress-ibility of the system. We exploit the impact of these nontrivial U /H208491/H20850instanton phase field configurations for the cuprate phase diagram which displays the “hidden” quantum critical point covered by the superconducting lobein addition to a sharp crossover between a compressible normal “strange metal” state and a region character-ized by a vanishing compressibility, which marks the Mott insulator. Finally, we argue that the existence ofrobust quantum numbers explains the stability against small perturbation of the system and attributes to thetopological “quantum protectorate” as observed in strongly correlated systems. DOI: 10.1103/PhysRevB.73.104505 PACS number /H20849s/H20850: 74.20.Fg, 74.72. /H11002h, 71.10.Pm I. INTRODUCTION Superconductivity represents a remarkable phenomenon where quantum coherence effects appear at macroscopicscale. 1The superconducting /H20849SC/H20850properties, especially the perfect diamagnetism, are microscopic manifestations of thespontaneous breakdown of one of the fundamental symme-tries of matter, namely the U /H208491/H20850gauge symmetry. 2For a superconductor this introduces a state with no definite par-ticle number, but with a definite conjugate variable identifi-able with the phase . The discovery of the cuprate superconductors, 3which is believed to emerge due to the strong electron-electron interactions, has sparked a wide-spread interest in physics which goes beyond the traditionalFermi-liquid framework usually employed for understandingthe effect of interactions in metals. 4It has been recognized that the superconductivity occurs in the region of the dopedMott insulator near the Mott transition, so that the wholemicroscopic foundations on which BCS theory was built oncannot be accommodated to explain cuprate superconductiv-ity. The conventional BCS theory identifies SC order as aninstability of the Fermi sea. That assumes the existence of anormal Landau-Fermi-liquid 5which forms well above the critical temperature Tc, which eventually turns into supercon- ductors below Tcdue to residual attraction among low-lying quasiparticles. In cuprates, in turn, there is overwhelmingevidence that superconductivity does not appear as an insta- bility of a Landau-Fermi liquid. The reason is that the fer-mion quasiparticle does not seem to reflect the character ofthe measured low energy eigenstates. In this sense, the elec- tron /H20849or electronlike quasiparticle /H20850may no longer be the ap- propriate way to think of elementary excitations. It is intrigu-ing to conjecture that in the strongly correlated systems theubiquitous competitions between the variety of possibleground states govern the essential physics—the formation ofa highly degenerate state seems to open the way to transfor-mation into alternative stable electronic configurations.Strong correlations that suppress electron motion may trans-form the system into a kind of a unstable state which will bevery sensible to charge and/or spin ordering. In this critical-like state the superconductivity might emerge as a competi-tion between different ground states. Indeed, in cupratesthere is clear evidence for the existence of a special dopingpoint “hidden” near optimal doping below the SC domewhere superconductivity is most robust. 6,7Such a behavior suggests that this point could be a quantum critical point/H20849QCP /H20850while the associated critical fluctuations might be re- sponsible for the unconventional normal state properties. 8 However, it is unclear whether this QCP is “truly critical” inthe sense that it is characterized by universality and scaling.For example, if excitations at QCP also carry the electricalcurrent, then a resistivity linear in temperature, as is ob-PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 1098-0121/2006/73 /H2084910/H20850/104505 /H2084921/H20850/$23.00 ©2006 The American Physical Society 104505-1served in cuprates, is obtained only if the dynamical expo- nent is unphysically negative.9The resemblance to a conven- tional QCP is also hampered by the lack of any clearsignature of thermodynamic critical behavior. Usually, aQCP would be the end point of a critical line below which anordered phase takes place and it could be made manifestbelow the superconducting dome. Experiments appear to ex-clude any broken symmetry around this point although asharp change in transport properties is observed. 10Unfortu- nately, our understanding of the underlying orders in cu-prates is far from being satisfactory and identifying the na-ture of the putative QCP is an open question. A possible wayout from this difficulty would be if the degrees of freedomthat rule QCPs are different from the energy degrees of free-dom that govern the stable phases the critical point separates.Thus these “nonconventional” degrees of freedom could pro-vide critical fluctuations beyond those of the order parameterfluctuations usually included in the standard Ginsburg- Landau-Wilson /H20849LGW /H20850description. 11,12Within the LGW ap- proach the order parameter fluctuations are implemented byconstructing models which mimic the lowenergy properties of solids—a procedure which relies on the separability of thehigh and low energy scales. Under this proviso the high en-ergy variables can be removed out yielding an effectiveHamiltonian /H20849as exemplified, e.g., by the t−Jmodel 13/H20850which describes the relevant low energy and long-wavelength phys-ics. This procedure is often implemented via the projectivetransformation, which results in removing of high-energy de-grees of freedom and replacing them with kinematical con-straints. In such approaches, the high energy scale associatedwith the charge gap is argued to be irrelevant, hence thefocus exclusively on the spin sector to characterize the Mottinsulator. However, the charge transfer nature of the cupratesplays an essential role in the doped systems 14so that with discarding charge degrees of freedom an important part ofthe physics may be lost. However, there is also mountingexperimental evidence which put in question the validity ofthe various projection schemes. For example, it has beenfound that above any temperature associated with ordering inboth electron and hole-doped cuprates a charge gap of order2 eV is present in the optical conductivity and a rapid reshuf-fling of spectral weight with hole doping. 15,16Angle-resolved photoemission spectroscopy /H20849ARPES /H20850also reveals a similar charge gap.17Surprisingly, when superconductivity emerges, the low and high energy degrees of freedom are still coupled.It has been shown that changes in the optical conductivityoccur at energies 3 eV which exceed by two orders of mag-nitude the maximum of the superconducting gap. 18But the high energy scale involved is hard to understand unless it isassumed that the spectral weight is transferred from both thelower and the upper Hubbard bands, thus beyond the rangeof applicability of the pure t−Jmodel. The intimate connect- edness between the low and high energy degrees of freedomin doped Mott insulators was firmly appreciated and termedmottness . 19Due to the intrinsic mixing of high and low en- ergy degrees of freedom no low energy reduction is possiblein a conventional sense, so that the doped Mott insulators areinherently asymptotically slaved. 20In a similar spirit a detour from the strict projection program was recently proposed in aform of the “gossamer” superconductor 21recognizing therole of the expensive, double-occupancy charge configura- tions. While spontaneous symmetry breaking has become one of the main guiding principles in physics,22there are other sig- natures in a physical system that are associated with the to- pological effects. These are instrumental for a full under- standing of the physics and lead to a host of ratherunexpected and exotic phenomena, which are in general of anonperturbative nature. For example, the fundamental char-acter of a vector potential is evident in the Aharonov-Bohm/H20849AB /H20850effect, 23where the topology of the U /H208491/H20850group is essen- tial: when an electron is transported in a magnetic fieldaround a closed loop, it acquires a phase that is equal to themagnetic flux through the surface spanned by the electronpath. Strongly correlated electronic systems are no exceptionin this regard. In particular, the fractional quantum Hall/H20849FQH /H20850effect 24,25is the prominent representative. Here, the striking fact is that FQH systems may contain many differentphases at zero temperature which have the same symmetry. Thus different states cannot be distinguished by symmetriesand the Landau symmetry-breaking principle fails becausealso topological characteristics of the configurational spacecome into play. In the most interesting cases configurationalspaces are not simply connected. There are space-time con-figurations of quantum fields which cannot be continuouslydeformed one into another. Further, an adiabatic motionalong a noncontractible closed path in a configurationalspace leads to a geometric /H20849Berry /H20850phase 26acquired by the wave function. In condensed matter systems with large num-bers of mutually interacting particles, the subject of Berry’sphases becomes a key issue. For example, a Berry phasedistinguishes between integer and half-integer spin chainsand results in different ground states and excitations. 27The other examples are the possible geometric phase effects onstatistical transmutation 28that can be achieved by a “flux attachment” which now becomes a very powerful theoreticalmethod. 29,30In many cases the topological character of the quantum field is captured by a single integer, called the to-pological charge, or winding number of the field which clas-sifies topological excitations. These are found by integratingthe so-called Chern-Simons terms which enter theLagrangians of the theory. What is common to all the aboveissues is the appearance of gauge fields to characterize vari- ous interactions: field configurations which differ by a gaugetransformation are to be regarded as physically the same. In the present work we argue that the important properties of cuprates are controlled by the large Mott gap and considerthe representation of strongly correlated electrons as fermi-ons with attached “flux tubes.” This introduces a conjugateU/H208491/H20850phase variable, which acquires dynamic significance from the electron-electron interaction. This means that anelectron is not a quasiparticle /H20849in the Landau sense /H20850, but has a composite nature governed by the electromagnetic gaugegroup. Furthermore, we recognize the nontrivial topology ofthe electromagnetic U /H208491/H20850group by observing that the funda- mental group /H92661/H20851U/H208491/H20850/H20852=Zis given by a set of integers. Therefore the elementary excitations in a strongly correlated system always carry 2 /H9266-kinks of the phase field character- ized by the topological winding number31—a quantized U /H208491/H20850 topological charge. Due to the nontrivial first homotopyT. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-2group/H92661the configuration space of the quantum phase fields is multiply connected, so that inequivalent paths in the“imaginary-time” /H20849paths that are not deformable to one an- other /H20850naturally emerge. Hence, the U /H208491/H20850gauge field gives rise to a “topological interaction,” which is felt by the elec-trons and can be separated from ordinary dynamical oneslumping them into particle “statistics.” To facilitate this taskwe employ the functional integral formulation of the theorythat encompasses all of these topological possibilities: onehas to perform the functional integral over fields defined ondifferent topologically equivalent classes, i.e., with differentwinding numbers. From a canonical /H20849operator /H20850point of view, however, the different topological sectors seem to give rise tocompletely different Hilbert spaces and the resultant fieldoperators would satisfy quite complicated nonlocal commu-tation relations. The fact that a prospective theory of elec-tronic states in strongly correlated electron systems mustgive up on either standard fermion commutation relations or standard particle conservation laws has already been pointedout. 32Furthermore, we exploit the impact of these topologi- cal excitations for the phase diagram of cuprates /H20849with its various crossovers and transition lines /H20850and show that they can induce its unusual feature: a “hidden” quantum criticalpoint of the type that results from the topological ground-state degeneracy. It can be probed by the topological suscep-tibility as a robust, nonperturbative property that is related tothe physical quantity of interest, namely, the divergingcharge compressibility. It also provides a natural descriptionof the Mott state where the system is said to be incompress-ible when there is a gap in the chemical potential as a func-tion of the electron density. This topological underpinningestablishes the source of quantum protection as a collectivestate of the quantum field, whose excitations pertain to thewhole system. Therefore macroscopic behavior is mostly de-termined by topological conservation laws which do notarise just out of a symmetry of the theory /H20849as “conventional” conservation laws based on Noether’s theorem /H20850but it is a consequence of the connectedness, i.e., topology of the phasespace, related to the topological properties of the associatedgauge group manifold. The organization of this paper is as follows. In Sec. II we introduce the electronic model for cuprates which capturesthe layered structure and epitomizes the hierarchy of relevantenergy scales, with the largest set up by the Coulomb inter-action. In Sec. III we describe the details of the flux attach-ment transformation which results in the representation ofstrongly correlated electrons as fermions plus attached U /H208491/H20850 gauge “flux tubes” that leads to a composite particle picture.Section IV is devoted to the basic concepts of the algebraictopology /H20849homotopy groups /H20850that becomes instrumental for the Feynman’s path integral formulation of quantum statis-tics. This is followed by Secs. V and VI where the fermionicpart of the theory is elaborated in terms of the momentumdependent “ d-wave” spin-gap and microscopic phase stiff- nesses. Here, the most important results are summarized inthe effective bosonic model written with the help of the col-lective phase variables. This enables us to study the super-conductivity as the condensation of the “flux tubes” from theelectron composite which is presented in Sec. VII. We eluci-date there the role of doping for the superconducting order tooccur and the key role played by the topological degeneracy. In the subsequent section VIII the topological susceptibilityis used to probe the change of the topological order. Here, weshow that the topological susceptibility is related to thecharge compressibility that diverges at the degeneracy pointat zero temperature and defines a type of topological quan-tum criticality, beyond the paradigm of the symmetry break-ing. In Sec. IX we present calculated phase diagrams forcuprates displaying, beyond the conventional ordered states,regions that are related to the change of the topological order.Section X is dedicated to the discussion of the robustness ofthe ground states in cuprates and its source in the topologicalconservation laws. Finally, in Sec. XI we conclude with asummary of the results, while the Appendixes collect mate-rial that is related to the technical part of the work. II. ELECTRONIC MODEL FOR CUPRATES The Hubbard model is viewed as the generic model for interacting electrons in the narrow-band and strongly corre-lated systems 33that captures the physics of Mott transition.34 The existence of the Mott insulator in the cuprates’ parentcompounds implies that a viable theory of high-temperaturesuperconductivity must explicitly incorporate the Mott- Hubbard gap for charge transfer. 14While it is believed that the basic pairing mechanism in cuprates arises from the an-tiferromagnetic /H20849AF/H20850exchange correlations, 35it is apparent the charge fluctuations also play an essential role in dopedsystems. Hence the Coulomb charge fluctuations associatedwith double occupancy of a site are controlled by the param-eter Uin the Hubbard model, which also determines the strength of the AF exchange coupling J. Therefore energy scale of the charge fluctuation is characterized by the Mottgap, which is by far larger than the energy scale of magneticfluctuations. Although the t−Jmodel is usually viewed as theU→/H11009limit of the single band Hubbard model, the one- particle spectra of the two models differ considerably. 36As already mentioned, this limitation of the t−Jmodel comes from having projected out the doubly occupied states origi-nally contained in the Hubbard model. To explore the moreflexible arrangements between JandUthan encoded either in the Hubbard or t−Jmodels we employ in the present paper a generalized t−t /H11032−U−Jmodel for the CuO 2plane in high- Tcsuperconducting. In this way we retain basic features of both models: the charge fluctuations present in the Hubbard model /H20849but re- moved from the constrained t−Jmodel /H20850and the robust su- perconducting correlations described by the exchange inter-action J. Furthermore, we incorporate besides the direct nearest-neighbor hopping talso the next-nearest-neighbor t /H11032 hopping parameter, whose importance in cuprates has been emphasized.37A good deal of the existing literature on the cuprates invokes model Hamiltonians based only on theproperties of a single CuO layer. Obviously, the interlayerstructure cannot be ignored: 38the measured critical tempera- ture Tcis strongly dependent on the interlayer structure.39 Therefore three-dimensional /H208493D/H20850coupling of planes must play an important role in the onset of superconductivity,40,41 which have to be incorporated by means of the interlayerCRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-3couplings and the associated c-axis dispersion effects in modeling of the cuprates.42 Summing up, we consider an effective one-band elec- tronic Hamiltonian on a tetragonal lattice that emphasizesstrong anisotropy and the presence of a layered CuO 2stack- ing sequence in cuprates: H=Ht+HJ+HU, where Ht=/H20858 /H9251/H5129/H20875−/H20858 /H20855rr/H11032/H20856/H20849t+/H9254r,r/H11032/H9262/H20850c/H9251/H5129†/H20849r/H20850c/H9251/H5129/H20849r/H11032/H20850 +/H20858 /H20855/H20855rr/H11032/H20856/H20856t/H11032c/H9251/H5129†/H20849r/H20850c/H9251/H5129/H20849r/H11032/H20850−/H20858 rr/H11032t/H11036/H20849rr/H11032/H20850c/H9251/H5129†/H20849r/H20850c/H9251/H5129+1/H20849r/H11032/H20850/H20876, HJ=/H20858 /H5129/H20858 /H20855rr/H11032/H20856J/H20875S/H5129/H20849r/H20850·S/H5129/H20849r/H11032/H20850−1 4n/H5129/H20849r/H20850n/H5129/H20849r/H11032/H20850/H20876, HU=/H20858 /H5129rUn↑/H5129/H20849r/H20850n↓/H5129/H20849r/H20850. /H208491/H20850 Here /H20855r,r/H11032/H20856and /H20855/H20855r,r/H11032/H20856/H20856identify summations over the nearest-neighbor and next-nearest-neighbor sites labeled by 1/H33355r/H33355Nwithin the CuO plane, respectively, while 1 /H33355/H5129 /H33355N/H11036labels copper-oxide layers. Subsequently, tandt/H11032are thebare hopping integrals, while t/H11036stands for the interlayer coupling. The operator c/H9251/H5129†/H20849r/H20850/H20851c/H9251/H5129/H20849r/H20850/H20852creates /H20851annihilates /H20852 an electron with spin /H9251at the lattice site /H20849r,/H5129/H20850and n/H5129/H20849r/H20850 =n↑/H5129/H20849r/H20850+n↓/H5129/H20849r/H20850stand for number operators, where n/H9251/H5129/H20849r/H20850 =c/H9251/H5129†/H20849r/H20850c/H9251/H5129/H20849r/H20850and/H9262is the chemical potential. Furthermore S/H5129a/H20849r/H20850=/H20858 /H9251/H9252c/H9251/H5129†/H20849r/H20850/H9268/H9251/H9252ac/H9252/H5129/H20849r/H20850/H20849 2/H20850 denotes the vector spin operator /H20849a=x,y,z/H20850with/H9268/H9251/H9252abeing the Pauli matrices. Finally, Uis the on-site repulsion Cou- lomb energy and Jthe AF exchange. Owing the lattice ar- rangement the full electronic dispersion is given by /H9280/H20849k,kz/H20850=/H9280/H20648/H20849k/H20850+/H9280/H11036/H20849k,kz/H20850, /H208493/H20850 where the in-plane contribution reads /H9280/H20648/H20849k/H20850=−2 t/H20851cos /H20849akx/H20850+ cos /H20849aky/H20850/H20852+4t/H11032cos /H20849akx/H20850cos /H20849aky/H20850 /H208494/H20850 with t/H11032/H110220. Furthermore, the c-axis dispersion is given by /H9280/H11036/H20849k,kz/H20850=2t/H11036/H20849k/H20850cos /H20849ckz/H20850, t/H11036/H20849k/H20850=t/H11036/H20851cos /H20849akx/H20850− cos /H20849aky/H20850/H208522/H208495/H20850 as predicted on the basis of band calculations.43 III. ELECTRON AS A COMPOSITE OBJECT We now provide the representation of interacting elec- trons as fermions plus attached “flux tubes.”29This leads to a picture of composite particles which are void of the mutualinteractions among fermions: the electron-electron Coulombinteraction will be transformed into the action of U /H208491/H20850gauge /H20849phase /H20850fields governed by the effective kinetic term of “free” quantum rotors.A. Fermionic action The partition function for the system governed by the Hamiltonian in Eq. /H208491/H20850can be represented as a path integral using fermionic coherent states. Introducing Grassmann fields c/H9251/H5129/H20849r/H9270/H20850,c¯/H9251/H5129/H20849r/H9270/H20850that depend on the “imaginary time” 0/H33355/H9270/H33355/H9252/H110131/kBT, with Tbeing the temperature, we write the path integral for the statistical sum Zas Z=/H20885/H20851Dc¯Dc¯/H20852e−S/H20851c¯,c/H20852/H208496/H20850 with the fermionic action S/H20851c¯,c/H20852=/H20885 0/H9252 d/H9270/H20875/H20858 /H9251r/H5129c¯/H9251/H5129/H20849r/H9270/H20850/H11509/H9270c/H9251/H5129/H20849r/H9270/H20850+H/H20849/H9270/H20850/H20876. /H208497/H20850 The Hubbard term in Eq. /H208491/H20850we write in a SU /H208492/H20850invariant way as HU/H20849/H9270/H20850=U/H20858 r/H5129/H20853/H208491/4 /H20850n/H51292/H20849r/H9270/H20850−/H20851/H9024/H5129/H20849r/H9270/H20850·S/H5129/H20849r/H9270/H20850/H208522/H20854/H20849 8/H20850 singling out the charge-U /H208491/H20850and spin-SU /H208492/H20850/U/H208491/H20850sectors, where the unit vector /H9024/H5129/H20849r/H9270/H20850labels varying in space-time spin quantization axis.44The spin-rotation invariance one can make explicit by performing angular integration over /H9024/H20849r/H9270/H20850 at each site and time. In the following we fix our attention on the U /H208491/H20850invariant charge sector leaving aside possible mag- netic orderings such as antiferromagnetism. Although some-times concurrent magnetic transitions occur with the Motttransition, the mechanism of the Mott transition is primarilyindependent of the symmetry breaking of spins. Thus westress our primary interest in cuprates due to their supercon-ducting properties and the fact that the superconductivity /H20849re- sulting from condensation of charge /H20850should not be viewed as inextricably connected with the quantum antiferromag-netism. A clear support for this point comes from the obser-vation of the spectral weight transfer through the supercon-ducting transition in cuprates, 18which cannot be explained by invoking the antiferromagnetic order: the spectral weighttransfer persists well above the Neel temperature and at thedoping level where antiferromagnetism is absent. B. Gauge flux attachment transformation To proceed, we employ the Hubbard-Stratonovich trans- formation to decouple the Coulomb term in Eq. /H208498/H20850with the help of the fluctuating imaginary electrochemical potentialiV /H5129/H20849r/H9270/H20850conjugate to the charge number n/H5129/H20849r/H9270/H20850. Furthermore, we write the field V/H5129/H20849r/H9270/H20850as a sum of a static V0/H5129/H20849r/H20850and periodic function45 V˜/H5129/H20849r/H9270/H20850/H11013V˜/H5129/H20849r/H9270+/H9252/H20850, V/H20849r/H9270/H20850=V0/H20849r/H20850+V˜/H20849r/H9270/H20850, /H208499/H20850 where using Fourier series V˜/H20849r/H9270/H20850=/H208491//H9252/H20850/H20858 n=1/H11009 /H20851V˜/H20849r/H9275n/H20850ei/H9275n/H9270+c.c. /H20852, /H2084910/H20850 with/H9275n=2/H9266n//H9252/H20849n=0,±1,±2 /H20850being the /H20849Bose /H20850MatsubaraT. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-4frequencies. Now, we introduce the phase /H20849or “flux” /H20850field /H9278/H5129/H20849r/H9270/H20850via the Faraday-type relation /H9278˙/H5129/H20849r/H9270/H20850/H11013/H11509/H9278/H5129/H20849r/H9270/H20850 /H11509/H9270=V˜/H5129/H20849r/H9270/H20850, /H2084911/H20850 to remove the imaginary term i/H20885 0/H9252 d/H9270/H9278˙/H5129/H20849r/H9270/H20850n/H5129/H20849r/H9270/H20850/H11013i/H20885 0/H9252 d/H9270V˜/H5129/H20849r/H9270/H20850n/H5129/H20849r/H9270/H20850/H20849 12/H20850 for all the Fourier modes of the V/H5129/H20849r/H9270/H20850field, except for the zero frequency by performing the local gauge transformation to the new fermionic variables f/H9251/H5129/H20849r/H9270/H20850: c/H9251/H5129/H20849r/H9270/H20850= exp/H20875i/H20885 0/H9270 d/H9270/H11032V˜/H5129/H20849r/H9270/H11032/H20850/H20876f/H9251/H5129/H20849r/H9270/H20850. /H2084913/H20850 Thus as a result of Coulomb correlations the electron ac- quires a phase shift similar to that in the electric /H20849i.e., scalar /H20850 AB effect.23The expression in Eq. /H2084913/H20850means that an elec- tron has a composite nature made of the fermionic part f/H9251/H5129/H20849r/H9270/H20850with the attached “flux” /H20849or AB phase /H20850exp /H20851i/H9278/H5129/H20849r/H9270/H20850/H20852. Here, the quantity C0/H20849r/H9270/H20850defined by C/H51290/H20849r/H9270/H20850/H11013/H9278˙/H5129/H20849r/H9270/H20850=V˜/H5129/H20849r/H9270/H20850, /H2084914/H20850 is the one-dimensional /H20849temporal /H20850component Chern-Simons term46that makes the minimal coupling with the fermion density field. Since the Abelian Chern-Simons term is just asa total /H20849time /H20850derivative, the integral of it becomes simply converted into a “surface” integral, sensitive only to the glo-bal properties of the U /H208491/H20850gauge field along an “imaginary time” path that starts at imaginary time /H9270=0 and ends at /H9270 =/H9252. Thus the paths can be divided into topologically distinct classes, characterized by a winding number defined as the netnumber of times the world line wraps around the system inthe “imaginary time” direction. As we shall see in the nextsection, our considerations related to the “imaginary time”boundary conditions can be formalized using homotopicallynontrivial gauge transformations for which the strength ofthe phase shift must be quantized, so that the gauge changeof the Chern-Simons term will be an integral multiple of 2 /H9266. IV. QUANTUM STATES ON MULTIPLY CONNECTED SPACES A. Homotopy theory The algebraic topology and precisely the concept of ho- motopy groups47provides the necessary background by mak- ing reference to the topological structure of the group mani-fold, let us say M. The nth homotopy group /H9266n/H20849M/H20850is a group of equivalent classes of loops that can be smoothly deformed into each other without leaving M, where nrefers to the dimensionality of the loops in question. The already men-tioned AB effect is of topological nature, since mappingsfrom the electron’s configuration space to the gauge groupconstitute the nontrivial homotopy group /H92661/H20851U/H208491/H20850/H20852=Z, where the elements of Zare integers and represent winding numbers, i.e., U /H208491/H20850topological charges. We are precisely fac-ing a similar situation in a strongly correlated system since the electromagnetic U /H208491/H20850gauge group governs the charge sector. This becomes apparent in an electrodynamic dual de-scription of the charge in terms of the phase field, see Eq./H2084911/H20850, having its roots in the quantum mechanical complemen- tarity of phase and number. In the “imaginary time” evolu-tion of the phase, two field configurations lie in the sameconnected component of configurational space if they can becontinuously deformed into each other. There is a naturalequivalence relation between these paths called homotopy:two paths are equivalent /H20849i.e., belong to the same class /H20850if they can be “smoothly deformed” into each other. Theclasses are labeled by the winding numbers and are endowedwith a group structure by appropriately defining the compo-sition of two mappings. 47The rule of thumb is that if the homotopy group is trivial, then there cannot be any topologi-cal field configurations in the underlying theory. B. Homotopy classes and the path integral If we work in the Feynman’s path integral formulation of the quantum statistics48then the statistical sum Ztakes a form, in which homotopically distinct paths have to besummed according to various possibilities for inequivalentquantizations /H20849superselection sectors /H20850. Specializing to the case of the U /H208491/H20850group one obtains Z= /H20858 m/H33528/H92661/H20851U/H208491/H20850/H20852/H9267/H20849m/H20850Z/H20849m/H20850. /H2084915/H20850 Here, m/H33528Zlabels equivalence classes of homotopically con- nected paths and /H9267/H20849m/H20850marks the “statistical” weight which is related to a homotopy class. Furthermore, the partial sum Z/H20849m/H20850within the mth topological sector is given by the usual path integral Z/H20849m/H20850=/H20858 m/H20885/H20851D/H9278/H20852m/H20851Df¯Df/H20852e−S/H20851/H9278,f¯,f/H20852/H2084916/H20850 with the integration restricted to the mth homotopy class. Furthermore, the weight factors /H9267/H20849m/H20850form unitary irreduc- ible representations of the homotopy group /H92661/H20851U/H208491/H20850/H20852, so that the conditions for the weights are /H20841/H9267/H20849m/H20850/H20841=1 , /H9267/H20849m1/H20850/H9267/H20849m2/H20850=/H9267/H20849m1/L50195m2/H20850. /H2084917/H20850 In Eq. /H2084917/H20850m1andm2label the homotopy classes of the two paths with common end points, while m1/L50195m2label the ho- motopy class of the path obtained by joining the two. Inparticular, the weight factor, which furnishes the representa-tions of Ztakes the form /H9267/H20849m/H20850=ei/H9258m, where/H9258/H33528/H208510,2/H9266/H20850is the “statistical angle” parameter.49From the canonical point of view, if the configuration space of the system is not simplyconnected, as for the U /H208491/H20850group, then the quantization pre- scription becomes ambiguous since paths belonging to dif-ferent homotopy classes can get the extra relative Berryphase factor acquired by the wave function and the /H9258factor represents exactly this quantization ambiguity. Moreover, the /H9258term cannot be traced in a perturbation theory because itCRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-5has no effect upon the equations of motion. Therefore, in performing the integration in Eq. /H2084916/H20850one should take phase configurations that satisfy the boundary condition /H9278/H5129/H20849r/H9252/H20850 −/H9278/H5129/H20849r0/H20850=2/H9266m/H5129/H20849r/H20850andm/H5129/H20849r/H20850=0,±1,±2,.... For the sake of convenience it is desirable to “untwist” the boundary condi- tion by setting /H20849see Fig. 1 /H20850 /H9278/H5129/H20849r/H9270/H20850→/H9278/H5129/H20849r/H9270/H20850/H11013/H9272/H5129/H20849r/H9270/H20850+2/H9266/H9270 /H9252m/H5129/H20849r/H20850. /H2084918/H20850 Summing over all the phases /H9278/H5129/H20849r/H9270/H20850amounts to integrating over the /H9252-periodic field /H9272/H5129/H20849r/H9270/H20850/H20851/H9272/H5129/H20849r0/H20850=/H9272/H5129/H20849r/H9252/H20850/H20852and to sum over the set of integer winding numbers /H20853m/H5129/H20849r/H20850/H20854. The integral over the static /H20849zero frequency /H20850part of the fluctuating elec- trochemical potential /H20855V0/H20849r/H20850/H20856we calculate by the saddle point method to give 2 iU/H20855V0/H20849r/H20850/H20856=/H20883/H20858 /H9251f¯/H9251/H20849r/H9270/H20850f/H9251/H20849r/H9270/H20850/H20884+2/H9262 U. /H2084919/H20850 Explicitly, following the prescription given in Eqs. /H2084915/H20850and /H2084916/H20850we obtain for the statistical sum Z=/H20858 /H20853m/H5129/H20849r/H20850/H20854/H20885 02/H9266 /H20863 r/H5129d/H92720/H5129/H20849r/H20850 /H11003/H20885 /H9278/H5129/H20849r0/H20850=/H92720/H5129/H20849r/H20850/H9278/H5129/H20849r/H9252/H20850=/H9272/H51290/H20849r/H20850+2/H9266m/H5129/H20849r/H20850 /H20863 r/H5129/H9270d/H9272/H5129/H20849r/H9270/H20850/H20885/H20851Df¯Df/H20852e−S/H20851/H9272,m,f¯,f/H20852, /H2084920/H20850 with the action involving the topological Chern-Simons term and the statistical angle parameterS/H20851/H9272,m,f¯,f/H20852 =/H20858 /H5129/H20885 0/H9252 d/H9270/H208771 U/H20858 r/H20875/H11509/H9272/H5129/H20849r/H9270/H20850 /H11509/H9270+2/H9266 /H9252m/H5129/H20849r/H20850/H208762 +2/H9262 U/H20858 r1 i/H20875/H11509/H9272/H5129/H20849r/H9270/H20850 /H11509/H9270+2/H9266 /H9252m/H5129/H20849r/H20850/H20876+H/H20851/H9278,f¯,f/H20852/H20878. /H2084921/H20850 Here, H/H20851/H9278,f¯,f/H20852is the effective Hamiltonian that is void of the Coulomb interaction H/H20851/H9278,f¯,f/H20852 =/H20858 /H5129/H20885 0/H9252 d/H9270/H20877/H20858 r/H9251f¯/H9251/H5129/H20849r/H9270/H20850/H11509/H9270f/H9251/H5129/H20849r/H9270/H20850−/H9262¯/H20858 r/H9251f¯/H9251/H5129/H20849r/H9270/H20850f/H9251/H5129/H20849r/H9270/H20850 −/H20858 /H20855r,r/H11032/H20856te−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H20858 /H9251f¯/H9251/H5129/H20849r/H9270/H20850f/H9251/H5129/H20849r/H11032/H9270/H20850 +/H20858 /H20855/H20855r,r/H11032/H20856/H20856t/H11032e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H20858 /H9251f¯/H9251/H5129/H20849r/H9270/H20850f/H9251/H5129/H20849r/H11032/H9270/H20850 +/H20858 /H20855r,r/H11032/H20856t/H11036/H20849rr/H11032/H20850e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129+1/H20849r/H11032/H9270/H20850/H20852/H20858 /H9251f¯/H9251/H5129/H20849r/H9270/H20850f/H9251/H5129+1/H20849r/H11032/H9270/H20850 −J/H20858 /H20855rr/H11032/H20856B¯/H5129/H20849r/H9270,r/H11032/H9270/H20850B/H5129/H20849r/H9270,r/H11032/H9270/H20850/H20878, /H2084922/H20850 where/H9262¯=/H9262−nfU/2 is the shifted chemical potential, while nf=/H20858 /H9251/H20855f¯/H9251/H20849r/H9270/H20850f/H9251/H5129/H20849r/H9270/H20850/H20856 /H20849 23/H20850 is the occupation number for the fermionic part of the elec- tron composite. In Eq. /H2084922/H20850, while writing the term that gov- erns AF interaction we made use of the following represen-tation: B¯/H5129/H20849r/H9270,r/H11032/H9270/H20850=1 /H208812/H20851f¯↑/H5129/H20849r/H9270/H20850f¯↓/H5129/H20849r/H11032/H9270/H20850−f¯↓/H5129/H20849r/H9270/H20850f¯↑/H5129/H20849r/H11032/H9270/H20850/H20852 /H2084924/H20850 which is just the singlet-pair /H20849valence bond /H20850operator50 emerging from the decomposition J/H20858 /H5129/H20858 /H20855rr/H11032/H20856/H20875S/H5129/H20849r/H9270/H20850·S/H5129/H20849r/H11032/H9270/H20850−1 4n/H5129/H20849r/H9270/H20850n/H5129/H20849r/H11032/H9270/H20850/H20876 =−J/H20858 /H5129/H20858 /H20855rr/H11032/H20856B¯/H5129/H20849r/H9270,r/H11032/H9270/H20850B/H5129/H20849r/H9270,r/H11032/H9270/H20850. /H2084925/H20850 It is obvious that a quasiparticle description /H20849of any kind /H20850 makes sense only when the constituent objects are weaklyinteracting. The chief merit of the transformation in Eq. /H2084913/H20850 is that we have managed to cast the strongly correlated prob-lem into a system of weakly interacting f -fermions with re- sidual interaction given by J, submerged in the bath of strongly fluctuating U /H208491/H20850gauge potentials /H20849on the high en- ergy scale set by U/H20850minimally coupled to f-fermions via “dynamical Peierls” phase factors. It is clear that the action FIG. 1. Schematic representation of the mapping from the real lineR/H20851the covering group of U /H208491/H20850/H20852to the circle S1/H11011U/H208491/H20850which is locally invertible provided the topological sector with the windinginteger number mis chosen. The map is defined by a continuous function f/H20849 /H9258/H20850on /H208510,2/H9266/H20852, where f/H208490/H20850=0 and continuity of the map requires that f/H208492/H9266/H20850=2/H9266m, so that the homotopy classes of /H92661/H20849S1/H20850 are labeled by integers.T. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-6of these phase factors “frustrates” the motion of the fermi- onic subsystem. However, as we demonstrate in the follow-ing, it is only when charge fluctuations become phase coher- entthe frustration of the kinetic energy is released. From Eq. /H2084921/H20850we can read off the explicit expression for the statistical weight /H9267/H20849m/H20850= exp/H20875i·2/H9262 U·2/H9266m/H5129/H20849r/H20850/H20876 /H2084926/H20850 so that the statistical angle reads /H9258 2/H9266/H110132/H9262 U. /H2084927/H20850 The phase factor in Eq. /H2084926/H20850, being a topological quantity, is closely related to the concept of the geometric Berry phases.To explicate this we write explicitly the composite structureof the physical electron field using Eq. /H2084913/H20850: c /H9251/H5129/H20849r/H9270/H20850= exp/H20875i/H20885 0/H9270 d/H9270/H11032V˜D/H5129/H20849r/H9270/H11032/H20850/H20876ei/H9253/H5129B/H20849r/H9270/H20850f/H9251/H5129/H20849r/H9270/H20850./H2084928/H20850 The first term in the exponential in Eq. /H2084928/H20850is the usual dynamical phase factor where /H9278˙/H5129/H20849r/H9270/H20850=V˜D/H5129/H20849r/H9270/H20850and/H9278/H5129/H20849r/H9252/H20850 =/H9278/H5129/H20849r0/H20850. The second one, in turn, is the nonintegrable Berry phase factor:26/H9253/H5129B/H20849r/H9270/H20850=2/H9266/H9270m/H5129/H20849r/H20850//H9252, where m/H5129/H20849r/H20850marks the integer U /H208491/H20850topological number. C. Topological ground state degeneracy It is known that the ground state degeneracy may arise from broken symmetries. Here, we argue that the groundstate degeneracy of the charge states in strongly correlatedsystem states is a reflection of the topological properties ofthe system. The existence of topological features implies thatthe quantum eigenstates are not single values under the con-tinuation of the parameters in the Hamiltonian. Consider the“free” part of the action in Eq. /H2084921/H20850describing the dynamics of the U /H208491/H20850gauge field S 0/H20851/H9278/H20852=/H20858 /H5129/H20885 0/H9252 d/H9270/H208771 U/H20858 r/H20875/H11509/H9278/H5129/H20849r/H9270/H20850 /H11509/H9270/H208762 +2/H9262 U/H20858 r1 i/H11509/H9278/H5129/H20849r/H9270/H20850 /H11509/H9270/H20878. /H2084929/H20850 This is the action of a particle moving in a plane around “magnetic flux” /H9021//H90210/H110132/H9262/U. The Hamiltonian corre- sponding to this action is simply H0/H20851/H9278/H20852=U 4/H20858 r/H5129/H20875/H11509 /H11509/H9278/H5129/H20849r/H20850−2/H9262 U/H208762 , /H2084930/H20850 with the eigenenergies given by E0/H20849m/H20850=U 4/H20858 r/H5129/H20875m/H5129/H20849r/H20850−2/H9262 U/H208762 . /H2084931/H20850 The energy in Eq. /H2084931/H20850can be interpreted as the square of the kinetic angular momentum of a set of quantum rotors dividedby the “moment of inertia” I=2/Uand the allowed angular momenta are uniformly displaced from integers by 2 /H9262/U,which may be any real number. The energy spectrum is la- beled by integers m/H5129/H20849r/H20850and for integer 2 /H9262/Uthe ground state is nondegenerate. However, for the half-odd integer 2/H9262/Uthe ground state is doubly degenerate, see Fig. 2. In this case the destructive interference between even and oddtopological sectors is responsible for the ground state degen-eracy. As we shall see in the following, this degeneracy willbe crucial in explaining the occurrence of superconductivityand anomalous properties on the verge of Mott transition,where the correspondence between the filling factor and theground state degeneracy will also be established. V. BARE AND DRESSED BAND PARAMETERS With the help of the angle-resolved photoemission spec- troscopy one gets direct access to the density of low-energyelectronic excited states in the momentum-energy space ofCu-O planes in cuprates. Obviously, all the interactions ofthe electrons are encapsulated in ARPES data, but these arestill difficult to evaluate. Since the underlying band structureof the bare electrons is a priori unknown, one way to think about these interactions is to consider simply the electronicexcitations as quasiparticles which are characterized by ef- fective electronic parameters. Consequently, the tight-binding interpolation of the electronic structure is often used for fit-ting the experimental ARPES data for cuprates. In this typeof analysis a conceptually simpler band theory is used toreveal how the strongly correlated electronic effects can betaken into account via the influence of the electronic bandparameters. In the context of the present work in order toestablish a link between the “bare” band parameters of thehigh-energy model in Eq. /H208491/H20850and the “dressed” one of the low energy models we have to perform the averaging overthe fluctuation of the U /H208491/H20850gauge phase fields according to /H20885/H20851D/H9278/H20852/H20851Df¯Df/H20852e−S/H20851/H9278,f¯,f/H20852=/H20885/H20851Df¯Df/H20852e−SLE/H20851f¯,f/H20852, /H2084932/H20850 where the low energy action SLEis given by SLE/H20851f¯,f/H20852=−l n/H20885/H20851D/H9278/H20852e−S/H20851/H9278,f¯,f/H20852. /H2084933/H20850 Performing the cumulant expansion in Eq. /H2084933/H20850we can de- duce in the lowest order the corresponding low energy fer- FIG. 2. /H20849Color online /H20850Energy levels of the quantum Hamil- tonian, Eq. /H2084930/H20850. Half-odd integer values of the “statistical” param- eter 2/H9262/Uyield ground state degeneracy.CRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-7mionic Hamiltonian that has to be compared with the origi- nal one in Eq. /H208491/H20850: HLE=/H20858 /H5129/H20877−/H20858 /H20855r,r/H11032/H20856,/H9251/H20849t/L50195+/H9254r,r/H11032/H9262¯/H20850f/H9251/H5129†/H20849r/H20850f/H9251/H5129/H20849r/H11032/H20850 +/H20858 /H20855/H20855r,r/H11032/H20856/H20856,/H9251t/H11032/L50195f/H9251/H5129†/H20849r/H20850f/H9251/H5129/H20849r/H11032/H20850−J/H20858 /H20855rr/H11032/H20856B/H5129†/H20849r,r/H11032/H20850B/H5129/H20849r,r/H11032/H20850 +/H20858 /H20855r,r/H11032/H20856,/H9251t/H11036/L50195/H20849rr/H11032/H20850f/H9251/H5129†/H20849r/H20850f/H9251/H5129+1/H20849r/H11032/H20850/H20878, /H2084934/H20850 where the dressed parameters encapsulating the effect of Coulomb interaction are given by t/L50195=t/H20855e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H20856/H9254/H20841r−r/H11032/H20841,1st, t/H11032/L50195=t/H11032/H20855e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H20856/H9254/H20841r−r/H11032/H20841,2nd, t/H11036/L50195/H20849rr/H11032/H20850=t/H11036/H20849rr/H11032/H20850/H20855e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129+1/H20849r/H11032/H9270/H20850/H20852/H20856, /H2084935/H20850 and /H20855¯/H20856refers to the averaging over the U /H208491/H20850phase field. /H20855¯/H20856/H11013/H20885/H20851D/H9278/H20852¯e−S0/H20851/H9278/H20852 /H20885/H20851D/H9278/H20852e−S0/H20851/H9278/H20852/H2084936/H20850 with the strongly fluctuating kinetic part /H20849on the energy scale U/H20850S0/H20851/H9278/H20852given by Eq. /H2084929/H20850. On average, the effect of this renormalization due to the presence of the phase-phase cor- relation functions is the effective mass enhancement of thecarriers as a result of the band narrowing, so that the “dressed” band parameters t X/L50195/H20849where tX=t,t/H11032,t/H11036/H20850are used to match electronic spectra using the low-energy scale t−J model.51Typically, in cuprates t/L50195/H110110.5 eV, t/H11032/L50195/t/L50195 /H110110.15–0.35, and t/H11036/L50195is of order of magnitude smaller than the in-plane hopping parameters.43 VI. PSEUDOGAP AND PHASE STIFFNESSES A. RVB pairs: Single-particle “ d-wave” gap A routine Hubbard-Stratonovich decoupling is applied to the resonating valence bond /H20849RVB /H20850term in Eq. /H2084925/H20850to give exp/H20875−J/H20885 0/H9252 d/H9270/H20858 /H5129/H20858 /H20855rr/H11032/H20856B¯/H5129/H20849r/H9270,r/H11032/H9270/H20850B/H5129/H20849r/H9270,r/H11032/H9270/H20850/H20876 =/H20885/H20851D/H9004/L50195D/H9004/H20852/H20885e−S/H20851f¯,f,/H9004/L50195,/H9004/H20852, /H2084937/H20850 where/H9004/H20849r/H9270;r/H11032/H9270/H20850is the complex pair field that is nonlocal in space. Furthermore, the corresponding effective action is of the formS/H20851f¯,f,/H9004/L50195,/H9004/H20852=1 J/H20858 /H5129/H20858 /H20855rr/H11032/H20856/H20885 0/H9252 d/H9270/H20841/H9004/H5129/H20849r/H9270,r/H11032/H9270/H20850/H208412 −/H20858 /H5129/H20858 /H20855rr/H11032/H20856/H20885 0/H9252 d/H9270/H20877/H9004/H5129/H20849r/H9270,r/H11032/H9270/H20850 /H208812/H20851f↑/H20849r/H9270/H20850f↓/H20849r/H11032/H9270/H20850 −f↓/H20849r/H9270/H20850f↑/H20849r/H11032/H9270/H20850/H20852+/H9004/H5129/L50195/H20849r/H9270,r/H11032/H9270/H20850 /H208812/H20851f¯↑/H20849r/H9270/H20850f¯↓/H20849r/H11032/H9270/H20850 −f¯↓/H20849r/H9270/H20850f¯↑/H20849r/H11032/H9270/H20850/H20852/H20878. /H2084938/H20850 A saddle-point method can be applied to the action in Eq. /H2084938/H20850to give the self-consistency equation for the momentum- dependent gap parameter /H20841/H9004/H20849k/H20850/H20841belonging to the Cu-O plane. Assuming that /H9004/H20849k/H20850is not changing along the c-direction, we can drop the layer index for this quantity. Out of the many possible mean-field translationally invariant so-lutions in the RVB theory the “ /H9266-flux” phase is selected here because of its relation to the “ d-wave” symmetry of the pseudogap in cuprates. It is governed by the equation 1=J N/H20858 k/H92572/H20849k/H20850 2E/H20849k/H20850tanh/H20875/H9252E/H20849k/H20850 2/H20876, /H2084939/H20850 where/H9257/H20849k/H20850=cos /H20849kxa/H20850−cos /H20849kya/H20850with the quasiparticle spec- trum of the fermionic part of the electron composite, E2/H20849k/H20850=/H20851/H9280/H20648/L50195/H20849k/H20850−/H9262¯/H208522+/H20841/H9004/H20849k/H20850/H208412/H2084940/H20850 and /H9004/H20849k/H20850=/H9004/H20851cos /H20849akx/H20850− cos /H20849aky/H20850/H20852. /H2084941/H20850 The gap parameter is a quantity with the short-range prop- erty, lim /H20841r−r/H11032/H20841→/H11009/H9004/H20849r−r/H11032/H20850=0, essentially tied to local correla- tions on neighboring sites. As we see in the following, the presence of the “ d-wave” pair function /H9004/H20849k/H20850isnota signa- ture of the superconducting state—it merely marks the re- gion of nonvanishing phase stiffness. However, the presenceof/H9004/H20849k/H20850should be visible, e.g., in ARPES spectra that picture the momentum-space occupation and therefore can detect the dispersion E/H20849k/H20850with a gap for single particle excitations. 17 B. Microscopic phases stiffnesses We can take advantage of the effective fermionic action in Eq. /H2084938/H20850which is now quadratic in the Grassmann field vari- ables that can be integrated out without any difficulty yield-ing a fermionic determinant: Z=/H20885/H20851D/H9278/H20852e−S0/H20851/H9278/H20852+Tr ln Gˆ−1, /H2084942/H20850 where Gˆ−1=Gˆ o−1−T=/H208491−TGˆo/H20850Gˆ o−1/H2084943/H20850 is the effective propagator of the theory, whileT. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-8T/H11013/H20851T/H20852/H5129/H5129/H11032/H20849r/H9270,r/H11032/H9270/H11032/H20850 =/H20853te−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H92683ˆ/H9254/H5129/H5129/H11032/H9254/H20841r−r/H11032/H20841,1st/H9268ˆ3/H9254/H20849/H9270−/H9270/H11032/H20850 −t/H11032e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H92683ˆ/H9254/H5129/H5129/H11032/H9254/H20841r−r/H11032/H20841,2nd +t/H11036/H20849r−r/H11032/H20850e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H11032/H20849r/H11032/H9270/H20850/H20852/H92683ˆ/H9254/H20841/H5129−/H5129/H11032/H20841,1/H20854/H9268ˆ3/H9254/H20849/H9270−/H9270/H11032/H20850 /H11013Tˆ+Tˆ/H11032+Tˆ/H11036 /H2084944/H20850 is the matrix composed of the hopping integrals and phase factors. Furthermore, /H20851Gˆ o−1/H20852/H5129/H20849r/H9270,r/H11032/H9270/H11032/H20850=/H20875/H20873/H11509 /H11509/H9270−/H9268ˆ3/H9262/H20874/H9254r,r/H11032+/H9004/H5129/H20849r/H9270,r/H11032/H9270/H20850 /H208812/H9268ˆ+ +/H9004/H5129/L50195/H20849r/H9270,r/H11032/H9270/H20850 /H208812/H9268ˆ−/H20876/H9254/H5129/H5129/H11032/H9254/H20849/H9270−/H9270/H11032/H20850/H20849 45/H20850 stands for the inverse of the “free” fermion propagator con- taining the gap /H9004field. Here /H9268ˆ+=1 2/H20849/H9268ˆ1+i/H9268ˆ2/H20850, /H9268ˆ−=1 2/H20849/H9268ˆ1−i/H9268ˆ2/H20850, /H2084946/H20850 where/H9268ˆa/H20849a=1,2,3 /H20850are the Pauli matrices acting in the Nambu spinor space, so that Gˆo/H20849r/H9270r/H11032/H9270/H11032/H20850=/H20875G/H20849r/H9270r/H11032/H9270/H11032/H20850 F/H20849r/H9270r/H11032/H9270/H11032/H20850 F/L50195/H20849r/H9270r/H11032/H9270/H11032/H20850 −G/H20849r/H11032/H9270/H11032r/H9270/H20850/H20876. /H2084947/H20850 Using the self-consistency solution, Eq. /H2084939/H20850, and Fourier transforming to the frequency and momentum domain oneobtains Gˆ o/H20849k/H9263n/H20850=/H20898−i/H9263n+/H9262 /H9263n2+/H92622+/H20841/H9004/H20849k/H20850/H208412,/H20841/H9004/H20849k/H20850/H20841 /H9263n2+/H92622+/H20841/H9004/H20849k/H20850/H208412 /H20841/H9004/H20849k/H20850/H20841 /H9263n2+/H92622+/H20841/H9004/H20849k/H20850/H208412,−i/H9263n−/H9262 /H9263n2+/H92622+/H20841/H9004/H20849k/H20850/H208412/H20899, /H2084948/H20850 where/H9263n=/H208492n+1/H20850/H9266//H9252are the /H20849Fermi /H20850Matsubara frequen- cies, n=0,±1,±2,.... Now , expanding the trace of the loga- rithm in Eq. /H2084942/H20850we obtain up to the second order in the hopping matrix elements Tr ln Gˆ−1=T r Gˆ o−1−1 2Tr/H20849GˆoTˆ/H208502−T r /H20849GˆoTˆ/H11032/H20850 −1 2Tr/H20849GˆoTˆ/H11036/H208502+¯. /H2084949/H20850 Finally, by performing summations over frequencies and mo- menta that are implicitly assumed in the trace operation inEq. /H2084949/H20850we obtain an effective action expressed in the U /H208491/H20850 phase fieldsS ph/H20851/H9278/H20852=/H20858 /H5129/H20885 0/H9252 d/H9270/H20877/H20858 r/H208751 U/H9278˙/H51292/H20849r/H9270/H20850+2/H9262 U1 i/H9278˙/H5129/H20849r/H9270/H20850/H20876 −/H20858 /H20855rr/H11032/H20856J/H20648/H20849/H9004/H20850cos /H208512/H9278/H5129/H20849r/H9270/H20850−2/H9278/H5129/H20849r/H11032/H9270/H20850/H20852 −/H20858 /H20855/H20855rr/H11032/H20856/H20856J/H20648/H11032/H20849/H9004/H20850cos /H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852 −/H20858 rJ/H11036/H20849/H9004/H20850cos /H208512/H9278/H5129/H20849r/H9270/H20850−2/H9278/H5129+1/H20849r/H9270/H20850/H20852/H20878./H2084950/H20850 In our case U/H11271J,J/H11032,J/H11036so that the microscopic phase stiff- nesses can be regarded as residual interactions to the domi-nant kinetic term of the phase model in Eq. /H2084950/H20850. This justi- fies the retention of only the lowest order nonvanishing termsin the electron hopping tand t /H11032in Eq. /H2084949/H20850since for the ensuing microscopic phase stiffnesses one has, e.g.,J/H20849/H9004/H20850/U/H11011t/UandJ /H11032/H20849/H9004/H20850/U/H11011t/H11032/U, respectively. All the stiffnesses in Eq. /H2084950/H20850/H20851see also Eq. /H20849A2/H20850in Appendix A /H20852rest on the single-particle gap due to the in-plane momentum space pairing among fermionic parts of the electron compos- ites governed by the AF exchange J: when/H9004/H20849k/H20850=0 all the phase stiffnesses collapse. While J/H20648andJ/H11036depend on the square of the corresponding hopping elements, the stiffness J/H20648/H11032is different: it depends linearly ont/H11032and governs the process of correlated particle-hole motion. Collective pair transfer events are costly for large U, so that excitonic co- herent charge transfer dominates the in-plane chargemotion. 45The interplane stiffness J/H11036is essential, however, in establishing bulk superconductivity via the Josephson-likeinterplanar coupling. VII. FROM THE MOTT INSULATOR TO SUPERCONDUCTOR A. Off-diagonal long-range order (ODLRO) vs charge frustration Because of the composite nature of the electron field in a strongly correlated system the occurrence of superconductiv-ity requires both the condensation of the fermion pairs de- scribed by f¯↓/H20849r/H9270/H20850,f¯↑/H20849r/H9270/H20850as well as the phase coherence which follows from the condensation of “flux tubes” ei/H9278/H5129/H20849r/H9270/H20850 attached to the f-fermions. Thus nonvanishing of the pair- wave function /H9004isnota sufficient signature of ODLRO.52 We can deduce this relationship from the definition of the superconducting order parameter which implies /H9023/H5129/H5129/H11032/H20849r/H9270,r/H11032/H9270/H20850/H11013/H20855 c¯↓/H5129/H20849r/H9270/H20850c¯↑/H5129/H11032/H20849r/H11032/H9270/H11032/H20850/H20856 =/H9254/H5129/H5129/H11032/H20855f¯↓/H5129/H20849r/H9270/H20850f¯↑/H5129/H20849r/H11032/H9270/H11032/H20850e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H11032/H20850/H20852/H20856 →/H9254/H5129/H5129/H11032/H9004/H20849r/H9270,r/H11032/H9270/H20850/H927402, /H2084951/H20850 where/H92740=/H20855ei/H9278/H5129/H20849r/H9270/H20850/H20856. The condensation of the “flux tubes” from the electron composite has a transparent physical ex- planation. The phase factors which are introduced into thehopping elements by the gauge transformation in Eq. /H2084913/H20850 frustrate the motion in the fermionic subsystem. However,CRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-9when charge fluctuations become phase coherent , as signaled by /H20855ei/H9278/H5129/H20849r/H9270/H20850/H20856/HS110050, the frustration of the kinetic energy is re- leased. The role of the gap parameter /H9004also becomes appar- ent: pairing among fermionic parts of the electron compos-ites is a necessary precondition for the existence of themicroscopic phase stiffnesses /H20851cf. Eq. /H20849A2/H20850/H20852, and, thereby, for the whole superconducting order. The opposite is obviouslynot true: the pseudogap state with /H9004/HS110050 may be phase inco- herent. However, the appearance of bulk phase coherence inthe presence of large Coulomb interaction U, whose energy scale by far exceeds that of microscopic phase stiffnesses, isnot so obvious. Therefore in the following we elucidate theinstrumental role of doping, since for the superconductingorder to occur the system should be brought in the vicinity ofthe degeneracy point, that is on the brink of change of thetopological order. B. Effective nonlinear /H9268-model To proceed, we replace the phase degrees of freedom in Eq. /H2084950/H20850by the unimodular complex scalar z/H5129/H20849r/H9270/H20850=ei/H9278/H5129/H20849r/H9270/H20850 fields via suitable Fadeev-Popov resolution of unity 1/H11013/H20885/H20851D2z/H20852/H20863 r/H5129/H9254/H20849/H20841z/H5129/H20849r/H9270/H20850/H208412−1/H20850/H9254/H20853Re/H20851z/H5129/H20849r/H9270/H20850−ei/H9278/H5129/H20849r/H9270/H20850/H20852/H20854 /H11003/H9254/H20853Im/H20851z/H5129/H20849r/H9270/H20850−ei/H9278/H5129/H20849r/H9270/H20850/H20852/H20854, /H2084952/H20850 where the unimodularity constraint can be imposed using the Dirac/H9254-functional, thus bringing the partition function into the following form: Z=/H20885/H20851D2z/H20852/H20863 r/H5129/H9254/H20849/H20841z/H5129/H20849r/H9270/H20850/H208412−1/H20850e−S/H20851z,z/L50195/H20852. /H2084953/H20850 The /H20849nonlinear /H20850unimodularity constraint can be conve- niently resolved with the help of a real Lagrange multiplier /H9261, so that the phase action in Eq. /H2084950/H20850can be suitably ex- pressed by the effective nonlinear /H9268-model /H20849NL/H9268M/H20850repre- sented by the action S/H20851z,z/L50195/H20852=/H20858 /H5129/H20885 0/H9252 d/H9270/H20877−/H20858 r/H5129/H20885 0/H9252 d/H9270/H11032z/H5129/L50195/H20849r/H9270/H20850/H9253−1/H20849/H9270−/H9270/H11032/H20850z/H5129/H20849r/H9270/H20850 −2J/H20648/H20849/H9004/H20850/H20858 /H20855rr/H11032/H20856/H208751 2z/H5129/L50195/H20849r/H9270/H20850z/H5129/H20849r/H11032/H9270/H20850+c.c./H208762 −J/H20648/H11032/H20849/H9004/H20850/H20858 /H20855/H20855rr/H11032/H20856/H20856/H208751 2z/H5129/L50195/H20849r/H9270/H20850z/H5129/H20849r/H11032/H9270/H20850+c.c./H20876 −2J/H11036/H20849/H9004/H20850/H20858 r/H208751 2z/H5129/L50195/H20849r/H9270/H20850z/H5129+1/H20849r/H9270/H20850+c.c./H208762/H20878,/H2084954/H20850 where /H9253/H20849/H9270−/H9270/H11032/H20850=1 Z0/H20885/H20851D/H9278/H20852ei/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H9270/H11032/H20850/H20852e−Sph/H208490/H20850/H20851/H9278/H20852, Z0=/H20885/H20851D/H9278/H20852e−S/H208490/H20850/H20851/H9278/H20852/H2084955/H20850 is the phase-phase correlation function calculated with the action in Eq. /H2084929/H20850/H20849see Appendix B /H20850. Since a part of theaction in Eq. /H2084954/H20850is quartic in the unimodular z-fields we employ the mean-field-like decoupling /H208751 2z/H5129/L50195/H20849r/H9270/H20850z/H5129/H11032/H20849r/H11032/H9270/H20850+c.c./H208762 →/H20855z/H5129/L50195/H20849r/H9270/H20850z/H5129/H11032/H20849r/H11032/H9270/H20856/H20851z/H5129/L50195/H20849r/H9270/H20850z/H5129/H11032/H20849r/H11032/H9270/H20850+c.c. /H20852/H20849 56/H20850 to perform the closed-form integration over z-variables in Eq. /H2084953/H20850. Here, the average /H20855¯/H20856should be determined with the resulting effective action. To justify Eq. /H2084956/H20850, we observe that, formally, Eq. /H2084956/H20850follows from the decoupling of quar- tic terms in Eq. /H2084954/H20850with the help of suitable Hubbard- Stratonovich transformation and subsequent use of thesaddle-point method with respect to the emerging auxiliaryvariables. Next, we introduce the Fourier transformed vari-ables z /H5129/H20849r/H9270/H20850=1 /H9252NN/H11036/H20858 /H9275n/H20858 qz/H20849q,/H9275n/H20850e−i/H20849/H9275n/H9270−k·r−kzc/H5129/H20850, /H2084957/H20850 where q/H11013/H20849k,kz/H20850with q=/H20849kx,ky/H20850andkzlabeling “in-plane” momenta and the wave vectors associated with the third di- mension along the c-axis, respectively. With the help of Eq. /H2084957/H20850the resulting quadratic action in the z-variables becomes S/H20851z,z/L50195/H20852=1 /H9252NN/H11036/H20858 q/H9275nz/L50195/H20849q,/H9275n/H20850/H9003−1/H20849q,/H9275n/H20850z/H20849q,/H9275n/H20850,/H2084958/H20850 where /H9003−1/H20849q,/H9275n/H20850=/H9261−J/H20849q/H20850+/H9253−1/H20849/H9275n/H20850/H20849 59/H20850 is the inverse propagator for the z-fields and J/H20849q/H20850=J¯/H20849/H9004/H20850/H20851cos /H20849kxa/H20850+ cos /H20849kya/H20850/H20852+2J¯ /H20648/H11032/H20849/H9004/H20850cos /H20849kxa/H20850cos /H20849kya/H20850 +J¯/H11036/H20849/H9004/H20850cos /H20849kzc/H20850/H20849 60/H20850 is the dispersion associated with the microscopic phase stiff- nesses, where J¯ /H20648/H11032/H20849/H9004/H20850=J/H20648/H11032/H20849/H9004/H20850/H20855z/H5129/L50195/H20849r/H9270/H20850z/H5129/H20849r+1st,/H9270/H20850/H20856, J¯/H11036/H20849/H9004/H20850=J/H11036/H20849/H9004/H20850/H20855z/H5129/L50195/H20849r/H9270/H20850z/H5129+1/H20849r/H9270/H20850/H20856. /H2084961/H20850 Furthermore, /H92530/H20849/H9275n/H20850is the Fourier transform of the bare phase propagator in Eq. /H2084955/H20850, see Appendix B. C. SC critical boundary At the critical boundary demarcating the long-range or- dered phase-coherent true superconducting state the staticand uniform “order parameter” susceptibility diverges, sothat the suitable condition that can be read off from the ac-tion in Eq. /H2084958/H20850is /H9003 −1/H208490,0/H20850/H20841/H9261=/H9261c=0 , /H2084962/H20850 which fixes the Lagrange parameter /H9261at the transition boundary and within the ordered state. The parameter /H9261cis given by the solution of the unimodularity constraint equa-tion:T. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-101=1 /H9252NN/H11036/H20858 q/H9275n/H9003/H20849q,/H9275n/H20850. /H2084963/H20850 The emerging ground-state phase diagram is depicted in Fig. 3. It exhibits a periodic arrangement of phase incoherentMott-insulating lobes with the superconducting state aboveand between them. Apparently, at the degeneracy point /H9262c defined by 2/H9262c U=1 2/H2084964/H20850 the superconducting state is most robust. Clearly, the above picture resembles that of a system described by the well-known Bose-Hubbard Hamiltonian 53as a generic Hamil- tonian for strongly correlated bosons. It covers the physicsoriginating from the competition between the repulsive andkinetic term of the Hamiltonian, whose magnitudes are pro-portional, in the present setting, to the parameters Uand phase stiffnesses, respectively. Moreover, this similarity isnot accidental, since this is just an example of statisticaltransmutation where bosons emerge as fermions with at-tached “flux tubes” as a result of the gauge transformation. 30 Within the phase coherent superconducting state order pa- rameter is given by 1−/H92742=/H208791 /H9252NN/H11036/H20858 q/H9275n/H9003/H20849q,/H9275n/H20850/H20879 /H9261=/H9261c. /H2084965/H20850 With the aid of Eqs. /H20849B1/H20850and /H20849B3/H20850, by performing the sum- mation over Bose-Matsubara frequencies, we obtain 1−/H92742=1 4NN/H11036/H20858 q1 /H208812/H20851J/H208490/H20850−J/H20849q/H20850/H20852 U+h2/H208732/H9262 U/H20874/H20877coth/H20875/H9252U 4/H20873/H208812/H20851J/H208490/H20850−J/H20849q/H20850/H20852 U+h2/H208732/H9262 U/H20874+h/H208732/H9262 U/H20874/H20874/H20876 + coth/H20875/H9252U 4/H20873/H208812/H20851J/H208490/H20850−J/H20849q/H20850/H20852 U+h2/H208732/H9262 U/H20874−h/H208732/H9262 U/H20874/H20874/H20876/H20878, /H2084966/H20850 where h/H20849x−1/2 /H20850=x−/H20851x/H20852, while /H20851x/H20852is the greatest integer less than or equal to x. The remaining summation over the mo- menta can be efficiently performed by resorting to the latticedensity of states as explained in Appendix C. The parameter /H9274as a function of temperature and the chemical potential is depicted in Fig. 4, which shows substantial enhancement ofthis quantity near the degeneracy point. Note that the three-dimensional anisotropic lattice structure is essential, sinceeven a very small interplanar coupling renders the phasetransition in the 3D universality class as observed incuprates. 54Thus the absence of t/H11036will suppress the bulk critical temperature to zero because for an isolated stack oftwo-dimensional layers the NL /H9268M strictly predicts Tc=0, in agreement with the Mermin-Wagner theorem. VIII. TOPOLOGICAL CRITICALITY AT THE DEGENERACY POINT In the preceding paragraphs we have shown that a theory of strongly interacting electrons can be transformed to an FIG. 3. The ground state phase diagram resulting from the ef- fective action in Eq. /H2084950/H20850. Here, in the filling-control transition, the control parameter is the chemical potential, which is conjugate tothe carrier density. The picture shows the arrangement of Mott-insulating /H20849incompressible /H20850lobes MI· mwith topological order characterized but the winding number m=0 and m=1, respectively, with the phase coherent superconducting ground state between andabove them. For large Coulomb energy Uthe phase coherent state is only possible in the vicinity of the degeneracy point 2 /H9262/U =1/2. The curves are plotted for different ratios of the inter- to intralayer couplings as input parameters: J/H11036/J/H20648=0.001,0.01, and 0.1 /H20849from the top to the bottom /H20850and show the proliferation of the superconducting state as the stack of coupled two-dimensionalplanes system crosses from 2D to 3D behavior.CRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-11equivalent description of weakly interacting fermions which are coupled to the “fluxes” of the strongly fluctuating U /H208491/H20850 gauge field. In regard to the nonperturbative effects, we re-alized the presence of an additional parameter, the topologi-cal angle /H9258/2/H9266/H110132/H9262/U, which is related to the chemical potential. We argued that the configuration space for thephase field /H9278consists of distinct topological sectors, each characterized by an integer entering the weight factors in thefunctional integral and counts the topological excitations ofthe system. On the other hand, the ground state degeneracydepends also on the topology of the configurational spaceand the transition we encountered at /H9258/2/H9266=1/2 corresponds to an abrupt change of the ground state that is not related toany visible symmetry breaking. However, the existence ofdifferent ground states that are related to the topologicalproperties of the interacting electronic system is a hallmarkof the topological order. 55The latter is not associated with the symmetry breaking pattern, so it cannot be characterizedby conventional order parameters in the Landau sense. Weargue that the ground state degeneracy can be parametrizedby a topological order parameter being the average of thetopological charge, i.e., the elements of the homotopy groupof the U /H208491/H20850gauge group and this parameter has a direct physical significance: in the large- Ulimit the electron den- sity /H20849i.e., the filling factor /H20850is just given the mean topological charge rather than the number of fermionic oscillators. Fur-thermore, in analogy to the Landau theory where the diver-gence of the order parameter susceptibility signals the phasetransition between states with different symmetry, to indicatethe change between different topologically ordered states, weintroduce the topological susceptibility being a derivative ofthe mean topological charge with respect to the statisticalangle /H9258. Its divergence is related to the existence of distinct “vacua,” which cross in energy at the degeneracy point. Sub-sequently, we show that the topological susceptibility has a direct physical relevance, since it is related to the chargecompressibility. It diverges at the degeneracy point at T=0 and thus defines a type of topological quantum criticality,beyond the Landau paradigm of the symmetry breaking. A. Topological charge and the electron density In addition to the Coulomb energy Uand temperature, the chemical potential /H9262plays a crucial role in Mott transition, since it controls the electron filling ne. An immediate impli- cation of the composite nature of the electrons is that theelectron occupation number /H20849i.e., the average number of electrons per site in the Cu-O plane /H20850 n e=1 N/H20858 r/H9251/H5129/H20855c¯/H9251/H5129/H20849r/H9270/H20850c/H9251/H5129/H20849r/H9270/H20850/H20856 /H20849 67/H20850 consists of the fermion occupation coming from the fermi- onic part of the composite and a topological contributionresulting from the “flux-tube” attachment: /H20883/H20858 /H9251c¯/H9251/H5129/H20849r/H9270/H20850c/H9251/H5129/H20849r/H9270/H20850/H20884 =/H20883/H20858 /H9251f¯/H9251/H5129/H20849r/H9270/H20850f/H9251/H5129/H20849r/H9270/H20850/H20884+2 iU/H20883/H11509/H9278/H5129/H20849r/H9270/H20850 /H11509/H9270/H20884. /H2084968/H20850 The appearance of the topological contribution in Eq. /H2084968/H20850is not surprising given the fact that “statistical angle,” see Eq./H2084926/H20850, depends on the chemical potential and the occupation number is just its conjugate quantity. Owing that the U /H208491/H20850 topological charge /H20849the winding number /H20850is given by m /H5129/H20849r/H20850=1 2/H9266/H20885 0/H9252 d/H9270/H9278˙/H5129/H20849r/H9270/H20850=1 2/H9266/H20885 /H9278/H51290/H20849r/H20850/H9278/H51290/H20849r/H20850+2/H9266m/H5129/H20849r/H20850 d/H9278/H5129/H20849r/H9270/H20850 /H2084969/H20850 the mean value of the density of the topological charge can be written after performing the Legendre transformation as nb=2/H9262 U+2 U/H208831 i/H11509/H9278/H5129/H20849r/H9270/H20850 /H11509/H9270/H20884. /H2084970/H20850 Therefore the average electron occupation number neis given by ne=nf+nb−2/H9262 U. /H2084971/H20850 In the limit of strong /H20849weak /H20850correlations neinterpolates be- tween topological nb/H20849fermionic nf/H20850occupation numbers. Clearly, in the large Ulimit/H9262→nfU/2, so that ne→nband the system behaves as governed entirely by density of topo-logical charge. The latter behaves in the large Ulimit as the typical density of hard-core bosons showing characteristic“staircase” behavior, see Figs. 5–7. Indeed, in this limit thesystem is described by the quantum rotor action in Eq. /H2084929/H20850, in which the probability distribution function of the densityof topological charge is Gaussian and the problem has FIG. 4. /H20849Color online /H20850Superconducting order parameter /H9274that signals the global phase stiffness shown as a function of the tem-perature and chemical potential, for U=4 eV, J=0.15 eV, t /L50195 =0.5 eV, t/H11032/L50195/t/L50195=0.25, and t/H11036/L50195=0.01 eV.T. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-12single-site character that can be analytically solved in a closed form: n0b/H20849/H9262/H20850=2/H9262 U−1 /H9252/H11509/H9262/H92583/H20875/H9252/H9262 2/H9266i,e−/H9252U/4/H20876 /H92583/H20875/H9252/H9262 2/H9266i,e−/H9252U/4/H20876/H2084972/H20850 by making use of the Jacobi theta-function identity /H11509v/H92583/H20849iv,q/H20850 /H92583/H20849iv,q/H20850=/H20858 m=1/H11009/H20849−1/H20850m4/H9266iqm 1−q2msinh /H208492/H9266mv/H20850. /H2084973/H20850 The calculation of the mean topological density for the inter- acting problem, i.e., with the full phase action given byEq. /H2084950/H20850involving phase stiffnesses is a bit demanding since spatial correlations have to be included, as well. However,we can resort to the unimodular-field NL /H9268M description given in Eq. /H2084952/H20850. The result for nbboth within the Mott lobe and in the superconducting region is given by nb=/H20902nb/H20849/H9261/H20850, within MI nb/H20849/H92610/H20850−2/H92742h/H208732/H9262 U/H20874,within SC,/H2084974/H20850 where/H9274is the order parameter given by Eq. /H2084966/H20850, while nb/H20849/H9261/H20850=n0b/H20849/H9262/H20850−1 2NN/H11036/H20858 q/H20877coth/H20875/H9252U 4/H20873/H208812/H20851J/H208490/H20850−J/H20849q/H20850/H20852 U+/H9254/H9261+h2/H208732/H9262 U/H20874+h/H208732/H9262 U/H20874/H20874/H20876 − coth/H20875/H9252U 4/H20873/H208812/H20851J/H208490/H20850−J/H20849q/H20850/H20852 U+/H9254/H9261+h2/H208732/H9262 U/H20874−h/H208732/H9262 U/H20874/H20874/H20876/H20878, /H2084975/H20850 with/H9254/H9261=/H9261−/H92610. Here, the parameter /H9261is self-consistently determined via Eq. /H2084963/H20850whereas /H92610is given by the solution of Eq. /H2084962/H20850. The summation over the wave vectors can be conveniently performed with the help of the lattice density ofstates. B. Topological susceptibility and charge compressibility Mott insulators have a clear distinction from metals by vanishing of the charge compressibility at zero temperature,while this quantity has a finite value in metals. 57In physicalterms, the charge compressibility measures the stiffness to the twist of the phase of the wave function in the “imaginarytime” direction. As we have shown, in the limit of strongcorrelations the physical properties of the system are gov- erned by the fluctuations of the topological charge. Thus theeffects connected with the nontrivial topological configura-tions of the gauge fields can be tested by performing thesecond derivative of the free energy with respect to the sta-tistical parameter /H9258, see Eq. /H2084927/H20850, which gives the topological susceptibility,58i.e., the connected part of the two-point cor- relator of the topological charge densities at zero momentum: FIG. 5. Evolution of the fermionic occupation number nfwith increasing correlations at T=0 and for t/L50195=1, 0.5, 0.25, and 0.125 eV from the upper left to the lower right. For all plots U =4 eV, J=0.15 eV, and t/H11032/L50195/t/L50195=0.3. For large values of the Coulomb-to-band energy ratio the fermionic filling factor behavesasn f/H110112/H9262/U. FIG. 6. Evolution of the average topological number nbwith decreasing correlations at T=0:U=4, 1, 0.5, and 0.2 eV from the upper left to the lower right. For all plots J=0.15 eV, t/L50195=0.5 eV, t/H11032/L50195/t/L50195=0.25, and t/H11036/L50195=0.01 eV. For large values of the Coulomb-to- band energy ratio the electronic filling factor behaves as nc/H11011nb. The nearly linear dependence of nbas a function of the chemical potential near the degeneracy point 2 /H9262/U=0.5 signals the presence of the global phase stiffness, see Eq. /H2084974/H20850.CRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-13/H9273t=1 Z/H20858 m/H33528/H92661/H20851U/H208491/H20850/H20852/H115092/H9267/H20849m/H20850 /H11509/H92582Z/H20849m/H20850 −/H208751 Z/H20858 m/H33528/H92661/H20851U/H208491/H20850/H20852/H115092/H9267/H20849m/H20850 /H11509/H92582Z/H20849m/H20850/H208762 . /H2084976/H20850 In Eq. /H2084976/H20850, as a result of the nontrivial topology in the group manifold group caused by the nonsimply connected struc-ture, the partition functions Z/H20849m/H20850are given by the functional integrals taken over the field configurations in the topologi- cal class monly. The full partition function Z, as defined by Eq. /H2084915/H20850, involves all topological sectors. Since the statistical angle parameter /H9258/H20849and thereby the chemical potential /H9262/H20850 acts as a ground state selector, the topological susceptibilitycan be conveniently employed to detect transition betweendifferent topologically ordered states. Since these are labeledby the average topological charge, the abrupt change of thisquantity will be signaled by the divergence of /H9273tat the de- generacy point lim /H9262→/H9262c/H9273t/H20849T=0 ,/H9262/H20850=/H11009, /H2084977/H20850 where/H9262cis the value of the chemical potential at the degen- eracy point. The topological susceptibility in Eq. /H2084976/H20850can be directly linked with the physical quantities, namely thecharge compressibility /H9260=/H11509ne /H11509/H9262, /H2084978/H20850 which expresses the total density response of the system to a local change of the chemical potential. It is related to theshift of the electron chemical potential as a function of elec-tron density which can be measured, e.g., through the shiftsof spectral features in photoemission spectra. 56Taking the derivative of Eq. /H2084971/H20850with respect to the chemical potential we obtain U 2/H9260=U 2/H11509nb /H11509/H9262+U 2/H11509nf /H11509/H9262−1 . /H2084979/H20850 While the fermionic contribution /H11509nf//H11509/H9262is regular, for the bosonic part /H11509nb//H11509/H9262one gets 2/H9266/H9273t=U 2/H11509nb /H11509/H9262. /H2084980/H20850 Therefore in the large U-limit the charge compressibility is entirely governed by the topological susceptibility and servesto distinguish that the iis zero in a Mott insulating 34region while it remains a finite superfluid region and diverges at thedegeneracy point. C. Electron mass enhancement at the degeneracy point Another remarkable aspect of the transition from one to- pologically ordered state to another is the great enhancement of the effective mass me/L50195of the electrons due to the collapse of electron kinetic energies due to the formation of the de- generate state at 2 /H9262/U=0.5. To estimate the change of me/L50195 we calculate me/L50195 me=/H115092/H9280/H20648/H20849k/H20850//H11509kx2/H20841k=0 /H115092/H9280/H20648/L50195/H20849k/H20850//H11509kx2/H20841k=0=1 R, /H2084981/H20850 where R=/H20855e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H20856/H20841/H20841r−r/H20841=d /H2084982/H20850 is the band renormalization factor, see Eq. /H2084935/H20850, where d stands for the lattice vector connecting nearest-neighbor siteson a two-dimensional lattice. Figure 8 illustrates the evolu-tion of the effective mass as a function of temperature in thevicinity of the degeneracy point. Interestingly, at 2 /H9262/U =0.5 which marks the “topological quantum critical point”the electronic matter in its charge aspect is very “soft” /H20849see Fig. 9 /H20850making it very susceptible to transformation into al- ternative stable electronic configurations, namely to super-conductivity, which we are going to analyze. IX. PHASE DIAGRAM FOR CUPRATES There has been a considerable amount of controversy re- garding the observed pseudogap phenomena in cuprates.59 One general class of theories views the pseudogap phase asresulting from performed pairs. 60,61The cuprates, however, are not in the strict Bose condensation /H20849“local pair” /H20850limit, since the photoemission still reveals the presence of a largeFermi surface. Furthermore, in the Bose limit, the chemicalpotential would actually be located beneath the bottom of theenergy band, which is also not the case. The other class ofscenarios /H20849coined as “competing order” /H20850consider the pseudogap as not intrinsically related to superconductivity,but rather proclaim it as competitive with superconductivity.Most of these proposals involve either a charge density FIG. 7. /H20849Color online /H20850Average topological number nbas a func- tion of temperature and chemical potential for U=4 eV, J =0.15 eV, t/L50195=0.5 eV, t/H11032/L50195/t/L50195=0.3, and t/H11036/L50195=0.01 eV.T. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-14wave62or spin density wave,63usually without long range order. However, if there is a phase transition underlyingpseudogap formation, a direct thermodynamic evidence /H20849i.e., nonanalytic behavior of the specific heat, the susceptibility,or some other correlation function of the system /H20850must show up in existing experiments. Unfortunately, none of the spec-troscopic data support a picture where the pseudogap phase represents a phase with true long range order. For thetemperature-doping phase diagrams the two delineated abovescenarios generally predict that the pseudogap characteristictemperature T /L50195/H20849x/H20850merges with Tc/H20849x/H20850on the overdoped side60 /H20849“precursor” scenario /H20850orT/L50195/H20849x/H20850falls from a high value at low doping, comparable to the exchange energy to zero at a criti- cal doping point inside the superconducting dome8/H20849“compet- ing order” picture /H20850. Below we show that both scenarios are consistently accommodated within the presence of topologi-cal order, degeneracy point, and accompanying phase coher-ence around it, as shown in Fig. 10. We see the evolution ofthe charge compressibility /H9260as a function of the chemical potential from the Mott insulator34with/H9260=0 /H20849at 2/H9262/U=1/H20850to a point of degeneracy on the brink of the particle occupationchange at 2 /H9262/U=1/2 where/H9260=/H11009atT=0. This is also the point on the phase diagram from which the superconductinglobe emanates. It is clear that the nature of the divergence of /H9260here has nothing to do with singular fluctuations due to spontaneous symmetry breaking as in the “conventional”phase transition. Rather, this divergent response appears as akind of topological protection built in the system against thesmall changes of /H9262. Further, /H9260→/H11009implies that /H11509/H9262//H11509nebe- comes vanishingly small at T=0 which results in the chemi- cal potential pinning, as observed in high- Tccuprates.64,65 A. Low energy scale pseudogap temperature T/L50195 In the pseudogap state at high temperatures one thus finds the coexistence of two distinct components: a state withgapped fermionic excitations /H20849described by the fermionic part of the composite electron /H20850and incoherent charge exci- tations /H20849given by the attached “flux tube” /H20850, which, as the FIG. 8. The ratio of the effective-to-bare electron mass param- eterme/L50195/meas a function of the chemical potential /H9262in the vicinity of the degeneracy point for U=4 eV, J=0.15 eV, t/L50195=0.5 eV, t/H11032/L50195/t/L50195=0.25, t/H11036/L50195=0.01 eV, and different temperatures T=1, 50, 115, and 300 K from the top to the bottom. FIG. 9. /H20849Color online /H20850Contrasting behaviors at the degeneracy point T=0, 2/H9262/U=1 2, and around it. Right picture: the density plot of the electron effective mass parameter me/L50195/meas a function of temperature and the chemical potential showing the strong depression of electron kinetic energies. Left picture: The superconducting lobe with the density plot of the charge compressibility /H9260diverging at the degeneracy point. In the “V-shaped” region fanning out to finite temperatures the electronic matter in its charge aspect is very “soft” /H20849i.e., highly compressible as opposed to the Mott state /H20850. The values of the microscopic parameters for creation of the plots are the same as in Fig. 3.CRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-15temperature is lowered, enter the superconducting state. The pseudogapped state is largely unaffected by the supercon-ducting transition and does not participate directly in thesuperconducting behavior. As explained above the underly-ing mechanism for the appearance of a gapped state withnonvanishing /H9004is intimately connected with the antiferro- magnetic correlations represented by the AF exchange J.W e identify the temperature for which /H9004sets in with the pseudogap temperature T /L50195. This is also the temperature at which the microscopic phase stiffnesses in Eq. /H20849A2/H20850vanish, so that/H20855e−i/H20851/H9278/H5129/H20849r/H9270/H20850−/H9278/H5129/H20849r/H11032/H9270/H20850/H20852/H20856→0. /H2084983/H20850 Using results of Sec. IV A, the fermionic filling factor nf defined by Eq. /H2084923/H20850can be computed explicitly as nf−1=−1 N/H20858 k/H20875/H9280/L50195/H20849k/H20850−/H9262¯ E/H20849k/H20850/H20876tanh/H20875/H9252E/H20849k/H20850 2/H20876. /H2084984/H20850 With the help of Eq. /H2084983/H20850and using Eq. /H2084939/H20850we obtain that at T/L50195 1 J=1 2/H20841/H9262¯/H20841tanh/H20873/H9252/H20841/H9262¯/H20841 2/H20874, nf−1=/H9262¯ /H20841/H9262¯/H20841tanh/H20873/H9252/H20841/H9262¯/H20841 2/H20874. /H2084985/H20850 By eliminating the chemical potential from Eq. /H2084985/H20850we get kBT/L50195 J=/H20841nf−1/H20841 ln/H20875−2/H20841nf−1/H20841+/H20849nf−1/H208502+1 2/H20841nf−1/H20841−/H20849nf−1/H208502−1/H20876/H2084986/H20850 revealing that T/L50195is a universal function of the fermionic filling number nf. Approaching half-filling /H20849nf=1/H20850we can infer from Eq. /H2084986/H20850 lim nf→1kBT/L50195/H20849nf/H20850=J 4l n /H208493/H20850/H110150.228 J/H11013kBTmax/L50195, /H2084987/H20850 where Tmax/L50195is the maximum value of the pseudogap tempera- tureT/L50195. For J=0.15 eV we obtain Tmax/L50195=396 K, see Fig. 12. B. Effect of doping on AF exchange and x−Tphase diagram for cuprates It is well-known that the antiferromagnetic exchange J originates from the interplay between on-site repulsion /H20849U/H20850 and the delocalization energy /H20849t/H20850. The effect can be derived straightforwardly by expanding the energy to the second or- der in the hopping matrix element. This involves virtualdouble occupation and can be represented by an exchangeprocess taking place on neighboring lattice sites: /H2084988/H20850 which yields a contribution /H11011t2/U. There are also processes that are prohibited by the Pauli exclusion principle such as /H20841↑,↑/H20856→t 0. However, with increasing hole doping /H20849i.e., de- parting from the half-filling /H20850a given electron has fewer neighboring electrons to pair with, which results in degrada-tion of the exchange process described in Eq. /H2084988/H20850. Therefore the AF exchange is leading to an effective interaction, 66 which is a steadily decreasing function of x, vanishing at the critical doping xc: FIG. 10. /H20849Color online /H20850Finite temperature phase diagram: the superconducting lobe Tc/H20849/H9264/H20850translated from the chemical potential /H20849 /H9264/H11013/H9262, upper right panel /H20850to the particle occupation number /H20849/H9264/H11013nb, lower left panel /H20850. Shaded area: the density plot of the charge com- pressibility /H9260˜=U/H9260/2. The degeneracy point /H20849T=0,2/H9262 U=1 2/H20850, where /H9260 diverges transforms into the critical line on the nb−Tphase dia- gram. Upper left panel: /H9260andnbas a function of temperature for T=0.1 Ushowing the transition from the incompressible Mott state at 2/H9262/U=1 to a highly compressible region around the degeneracy point. The values of the parameters for creation of the plots are thesame as in Fig. 9.T. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-16Jeff/H20849x/H20850=J/H208491−Kx/H20850, /H2084989/H20850 where Kis treated as the lattice connectivity and the factor of K=4 is the coordination number on the 2D square lattice. Therefore the vanishing of Jeffdetermines the critical doping value xc=0.25. It is clear by inspecting Eq. /H2084939/H20850that dimin- ishing of the AF exchange with doping will spoil the RVBpairing of the fermionic part of the electron composite bysuppressing the d-wave gap function /H9004/H20849k/H20850. Since phase stiff- nesses in Eq. /H20849A2/H20850rest on/H9004/H20849k/H20850the doping dependence of J eff/H20849x/H20850can be directly translated into the calculation of the x−Tphase diagram that involves the doping effect on Jeff, see Fig. 12. By comparing Figs. 11 and 12we can clearly seethat the diminishing of the superconductivity in the over-doped region is just the result of the pair-breaking effecttriggered by the doping dependent AF exchange. To summa-rize: T /L50195/H20849x/H20850demarcates the region of nonzero microscopic phase stiffnesses which persist in the region characterized by the nonvanishing of the spin gap, as observed in high-frequency conductivity measurements. 67The origin of the spin gap is purely electronic and results from the restrictedspace of available states that strongly correlated excitationson neighboring sites encounter. It is described by the reso-nating valence bond singlet d-wave spin pairing of the fer- mionic part of the electron composite and is controlled bythe AF superexchange parameter J. The coincidence of the Fermi surface with the minimum gap locus as obtained fromARPES measurements 68also supports a pairing gap interpre- tation of the pseudogap.69 C. Crossover to the “strange metal” state at Tg: High energy scale feature The doping-dependent characteristic temperature T/L50195in Eq. /H2084986/H20850at which this pseudogap opens is in the underdoped region significantly larger than Tc. The physical reason for this is transparent: T/L50195marks the region of nonvanishing phase stiffness, albeit without global phase coherence /H20849that appears at much lower temperature Tc/H20850. However, when the copper oxide superconductor is driven in the normal state byapplying a high magnetic field, a clear pseudogap feature at asimilar energy scale to the superconducting gap is observedin the quasiparticle tunneling spectra and the pseudogap fea-ture persists up to the highest applied fields and does notdepend on the magnetic field. 70Surprisingly, the Hall coeffi- cient does not vary monotonically with doping but ratherexhibits a sharp change at the optimal doping level forsuperconductivity. 71This observation would support the idea that two competing ground states underlie the high-temperature superconducting phase. From this perspectiveone has to conclude that any prospective order consistentwith these observations implies that the pseudogap coexistswith superconductivity and is essentially unchanged by alarge applied external field. It is clear that sudden onset ofthe pseudogap at critical doping right at the point where therigidity of the condensate wave function is at its maximumwould be very difficult to reconcile with the precursor sce-nario, but is very consistent with the onset of correlationswhich compete with superconductivity. This is precisely the FIG. 11. /H20849Color online /H20850Temperature-doping /H20849x=1− nb/H20850phase diagram calculated for the doping independent AF exchange J/H20849for the values of the parameters, see Fig. 9 /H20850. Depicted is the supercon- ducting lobe /H20851bounded by the critical temperature Tc/H20849x/H20850/H20852and the pseudogap temperature T/L50195which marks the region of nonvanishing microscopic phase stiffnesses /H20851below T*/H20849x/H20850/H20852. Shaded area: intensity plot of the charge compressibility /H9260diverging along the critical T =0 line /H20849see previous figure /H20850. FIG. 12. /H20849Color online /H20850Temperature-doping phase diagram as in Fig. 11 but with doping dependent AF exchange parameter Jeff/H20849x/H20850 given by Eq. /H2084989/H20850with K=0.25. The temperature Tg/H20849x/H20850marks the sudden onset of greatly enhanced charge compressibility /H9260in the “strange metal” state as departing from the half-filling /H20849x=0/H20850. Note that the characteristic energy scale of these charge fluctuations /H20849con- trolled by the Mott energy U/H20850is much larger than that of spin gap, controlled by AF exchange J.CRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-17outcome of the topological order which differentiates the electronic ground state into two states labeled by the topo-logical winding number and with the degeneracy point sepa-rating them. It controls a remarkable concurrence betweennormal state properties and the ground-state superconductingorder setting up a unique critical doping point in the phasediagram where the transport properties change very suddenlyand where superconductivity is most robust. Therefore weidentify the crossover line T g/H20849x/H20850where the charge compress- ibility undergoes a sudden change as an additional boundary hidden in the cuprate phase diagram, see Fig. 12. It is impor-tant to realize that in contrast to T /L50195/H20849x/H20850the crossover line Tg/H20849x/H20850is controlled by the highest energy scale in the prob- lem, namely the Mott scale set up by U. This explains why the anomalous behavior still persists even the superconduct-ing order is suppressed, e.g., by the strong magnetic field.Considering the Hall effect one has to conclude that the mo-bile carrier density varies considerably with both doping andtemperature. On the other hand, the ARPES Fermi surfaceremains essentially unchanged in the whole of the metallicdoping range suggesting a constant density of states. Theseapparently contradictory results could be reconciled by ob-serving that ARPES is sensitive to the momentum-space oc-cupation and therefore detect the excitation described by thefermionic part of the electron composite, whereas the chargetransport properties are governed mainly by the “flux tubes”which constitute charge collective variables. Given fact thatthe inverse of the Hall coefficient is proportional to the num-ber of carriers 1/ R H/H11011neandneis governed by the topologi- cal charge nbit is apparent why 1/ RHjumps in the vicinity of the QCP that is of topological origin. X. QUANTUM PROTECTORATE It is often difficult to formulate a fully consistent and adequate microscopic theory of complex cooperative phe-nomena and great advances in the solid-state physics are to agreat extent due to the use of simplified and schematic modelrepresentations for the theoretical interpretation. In particu-lar, the method of model Hamiltonians has proved to be veryeffective. However, as it was recently argued, 72ab initio computations have failed completely to explain phenomenol-ogy of high T ccuprates: it would appear that not only the deduction from microscopics failed to explain the wealth ofcrossover behaviors, but as a matter of principle it probablycannot explain it. Therefore it was concluded that a moreappropriate starting point would be to focus on the results ofexperiments in the hope of identifying the corresponding“quantum protectorates”—stable states of matter whose ge-neric low energy properties, insensitive to microscopics, aredetermined by a “higher organizing principle and nothingelse.” From this perspective, each protectorate can be char-acterized by a small number of parameters, which can bedetermined experimentally, but which are, in general, impos- sible to calculate from first principles. 72However, as we saw, a system with many microscopic degrees of freedom canhave ground states whose degeneracy is determined by thetopology of the system. Prototypes of this kind of systemsare provided by fractional quantum Hall effect. For example,the ground state degeneracy in FQH liquids is not a conse- quence of symmetry of the Hamiltonian. It is robust againstarbitrary perturbations, even impurities that break the sym-metries in the Hamiltonian. Thus the topological ground statedegeneracy on nontrivial manifolds provides a precise theo-retical distinction between a topological and conventionalorder. The Hilbert space of quantum states decomposes intodistinct topological sectors, each sector remaining isolatedunder the action of local perturbations. This is a signature ofits topological nature. Choosing the states from ground statesin different sectors protects these states from unwanted mix-ing through the change of system parameters—protectionwithin the sector is secured through a gapped excitationspectrum. In particular, we found that for strong correlationsthe system is governed by the topological Chern numbers.However, the Chern number is a topologically conservedquantity and is “protected” against the small changes of sys-tem parameters. Being an integer it cannot change at all if ithas to change continuously. However, changing the interac-tion by a large amount may cause abrupt changes in groundstate properties described by a different topological quantumnumber, which leads to a change of topological order. Thiskind of stability might be generic for quantum systems gov-erned by topologically nontrivial group manifolds. Thereforeone is left not only with the low-energy principle /H20849the classic prototype being the Landau Fermi liquid /H20850, but the emergent physical phenomena are regulated also by topological prin-ciples that have a property of their insensitivity to micro-scopics and this quantum protectorate functions under certaintopological environments, through conserving of topologicalcharges. XI. SUMMARY AND DISCUSSION In the present work focusing on the t−t/H11032−t/H11036−U−Jmodel it is shown that the topological excitations of charge given bythe collective U /H208491/H20850phase field in a form of “flux tubes” at- tached to fermions can reproduce many robust featurespresent in the phase diagram of high- T ccuprates, thus sub- stantiating one of the emerging paradigms in the condensedmatter physics, namely the ubiquitous competitions instrongly correlated systems. The fundamental entities thatcarry charge /H20849and spin /H20850in the copper oxides are no longer the usual Landau quasiparticles but the “flux tube” fermion com-posites, so that the charge is no longer tied to the Fermistatistics. When charges are “liberated” then they can con-dense leading to superconductivity. This picture naturallyleads to the pseudogap physics that is observed in the under-doped cuprates, which originates from the momentum pair-ing /H20849in a d-wave pattern /H20850of the fermionic parts of the elec- tron composite controlled by the antiferromagneticsuperexchange J. This underlines the necessity of the funda- mental concept of fermion pairing in achieving the supercon-ductivity. In the mathematical structure of the theory thegauge field is governed by the U /H208491/H20850Chern-Simons term in the action of purely topological nature. From the canonicalpoint of view the Hilbert space of a quantum theory has anontrivial structure marked by the topological sectors whichcorresponds to a set of degenerate ground states. The topo-T. K. KOPE Ć PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-18logical ground state degeneracy provides a precise theoreti- cal distinction between a topological and conventional order:states with the same conventional broken symmetries maystill be distinguished from each other on the basis of whetherthey are characterized by different topological quantum num-bers captured by the homotopy theory gauge group manifold.In this paper, it has been shown that physics of the Motttransition is successfully covered in a topological frame-work. The natural order parameter for the Mott transition isthe topological charge related to electron concentration forthe filling-control scenario that selects topologically orderedstates. Expectation value of the density of topological chargedetermines also a dominant contribution to the topologicalsusceptibility which, in turn, is related to the charge com-pressibility of the system. It diverges at the degeneracy pointat zero temperature and defines a type of topological quan-tum criticality, beyond the Landau paradigm of the symmetrybreaking. Although the charge compressibility is completelysuppressed in the Mott insulator because of the Mott gap, thecriticality on the “strange metallic” side can be describedadequately by the divergence of the charge compressibility atzero temperature. This critical enhancement of the densityfluctuations extends to finite-temperature and is controlled bythe Mott energy scale. This gives rise to another crossoverline hidden in the cuprate phase diagram, where the chargecompressibility undergoes a sudden change to the “strangemetal” state. The crossover is governed by the Coulomb en-ergy U, so that the density fluctuations at the instability to- wards the superconductivity surpass the effects of the spinfluctuation mechanisms governed by the antiferromagneticexchange extensively studied for the cuprate superconduct-ors. This clearly demonstrates the inseparability of the high-energy Mott scale from the low-energy physics in the cuprateproblem and redefines the role of the chemical potential froma quantity that simply demarcates the boundary betweenfilled and empty states to a selector of topologically orderedelectronic ground state. APPENDIX A: MICROSCOPIC PHASE STIFFNESSES The microscopic phase stiffnesses to the lowest order in the hopping amplitudes are given by J/H20849/H9004/H20850=2t2 N/H9252/H20858 k/H9263nF/L50195/H20849k/H9263n/H20850F/H20849k/H9263n/H20850, J/H20648/H11032/H20849/H9004/H20850=t/H11032 /H9252N/H20858 k/H9263ncos /H20849akx/H20850cos /H20849aky/H20850G/H20849k/H9263n/H20850, J/H11036/H20849/H9004/H20850=1 N/H20858 qt/H110362/H20849q/H20850 /H9252N/H20858 k/H9263nF/L50195/H20849k/H9263n/H20850F/H20849k/H9263n/H20850. /H20849A1/H20850 Explicitly, after performing frequency and momentum sums in Eq. /H20849A1/H20850we obtainJ/H20849/H9004/H20850=t2 4/H20885 −22 dxdyx2y2 y2−x2/H9267/H20849x/H20850/H9267/H20849y/H20850 /H11003/H20902tanh/H208751 2/H9252/H9280/H20849x/H20850/H20876 /H9280/H20849x/H20850−tanh/H208751 2/H9252/H9280/H20849y/H20850/H20876 /H9280/H20849y/H20850/H20903, J/H20648/H11032/H20849/H9004/H20850=−t/H11032/H9262¯/H20885 −22 dx/H9267¯/H20849x/H20850 /H9280/H20849x/H20850tanh/H208751 2/H9252/H9280/H20849x/H20850/H20876, J/H11036/H20849/H9004/H20850=9t/H110362/H20841/H9004/H208412 16/H20885 −22 dxx2/H9267/H20849x/H20850 /H92803/H20849x/H20850 /H11003/H208772 tanh/H20875/H9252/H9280/H20849x/H20850 2/H20876−/H9252/H9280/H20849x/H20850sech2/H20875/H9252/H9280/H20849x/H20850 2/H20876/H20878. /H20849A2/H20850 Here,/H9280/H20849x/H20850=/H20881/H9262¯2+/H20841/H9004/H208412x2and /H9267/H20849x/H20850=/H208491//H92662/H20850K/H20851/H208811− /H20849x2/4/H20850/H20852, /H9267¯/H20849x/H20850=/H9267/H20849x/H20850−/H208492//H92662/H20850E/H20851/H208811− /H20849x2/4/H20850/H20852, /H20849A3/H20850 where K/H20849x/H20850andE/H20849x/H20850are the complete elliptic integrals of the first and second kind, respectively.73 APPENDIX B: PHASE-PHASE CORRELATION FUNCTION By performing the functional integration over the phase variables in Eq. /H2084955/H20850we obtain /H9253−1/H20849/H9270−/H9270/H11032/H20850=/H92773/H208732/H9266/H9262 U+/H9266/H9270−/H9270/H11032 /H9252,e−4/H92662//H9252U/H20874 /H92773/H208732/H9266/H9262 U,e−4/H92662//H9252U/H20874 /H11003exp/H20877−U 4/H20875/H20841/H9270−/H9270/H11032/H20841−/H20849/H9270−/H9270/H11032/H208502 /H9252/H20876/H20878,/H20849B1/H20850 where/H92773/H20849z,q/H20850is the Jacobi theta function,73which comes from the topological part of the functional integral over the phase variables. The function /H92773/H20849z,q/H20850is defined by /H92773/H20849z,q/H20850=1+2 /H20858 n=1/H11009 cos /H208492nz/H20850qn2/H20849B2/H20850 and is/H9252-periodic in the imaginary time /H9270as well as in the variable 2 /H9262/Uwith the period of unity. Fourier transforming one obtains /H9253/H20849/H9275n/H20850=1 Z0/H20858 m=−/H11009+/H110098 Uexp/H20875−/H9252U 4/H20873m−2/H9262 U/H208742/H20876 1−4/H20875/H20873m−2/H9262 U/H20874−2i/H9275n U/H208762, /H20849B3/H20850 whereCRITICAL CHARGE INSTABILITY ON THE VERGE OF ¼ PHYSICAL REVIEW B 73, 104505 /H208492006 /H20850 104505-19Z0= exp/H20873−/H9252/H92622 U/H20874/H92583/H20873/H9252/H9262 2/H9266i,e−/H9252U/4/H20874 /H20849B4/H20850 is the partition function for the “free” rotor Hamiltonian in Eq. /H2084930/H20850. APPENDIX C: LATTICE DENSITY OF STATES In this appendix we give the explicit formulas for the densities of states /H20849DOS /H20850for the anisotropic three- dimensional lattice that is helpful for evaluation of the sumsover the momenta that appear in Secs. VII and VIII of thepresent paper. Our starting point is the dispersion relevant forthe two-dimensional lattice with next-nearest interactions: E/H20849k/H20850=−2 tcos /H20849ak x/H20850−2tcos /H20849bky/H20850+4t/H11032cos /H20849akx/H20850cos /H20849aky/H20850 =2t/H20851− cos /H20849akx/H20850− cos /H20849aky/H20850+rcos /H20849akx/H20850cos /H20849aky/H20850/H20852. /H20849C1/H20850 The choice of such a dispersion is obviously motivated by its relevance as a simple means of modeling the quasiparticleband in the high- T ccuprates. The density of states reads /H9267/H20849E/H20850=/H20885 −/H9266/a/H9266/adkx /H208492/H9266/a/H20850/H20885 −/H9266/a/H9266/adkx /H208492/H9266/a/H20850/H9254/H20851E−E/H20849k/H20850/H20852 /H110131 2t/H9267˜/H20849/H9280/H20850 /H20849C2/H20850 with/H9267˜/H20849/H9280/H20850=K/H20875/H208814− /H20849/H9280−r/H208502 4/H208491+r/H9280/H20850/H20876 /H92662/H208811+r/H9280 /H11003/H20851/H9008/H208492+r−/H9280/H20850/H9008/H20849/H9280+r/H20850+/H9008/H20849−r−/H9280/H20850/H9008/H20849/H9280+2− r/H20850/H20852, /H20849C3/H20850 where/H9280=E/2tand r=2t/H11032/t. Here/H9008/H20849x/H20850is the Heavyside /H20849unit-step /H20850function. The expression in Eq. /H20849C3/H20850is valid for r/H333551. For r/H110221 one has to make the replacement /H9267˜/H20849/H9280/H20850 →/H20841Ree/H9267˜/H20849/H9280/H20850/H20841. The effect of the interplanar /H20849c-axis /H20850interaction can be incorporated as well via the following dispersion: E3D/H20849k/H20850=E/H20849k/H20850−2tzcos /H20849cqz/H20850. /H20849C4/H20850 In the presence of tzthe system is a three-dimensional aniso- tropic one for which the density of states becomes /H92673D/H20849E/H20850=/H20885 −/H9266/a/H9266/adkx /H208492/H9266/a/H20850/H20885 −/H9266/b/H9266/adkx /H208492/H9266/a/H20850/H20885 −/H9266/c/H9266/cdqz /H208492/H9266/c/H20850 /H11003/H9254/H20851E−E3D/H20849k/H20850/H20852 /H110131 2t/H9267˜3D/H20849/H9280/H20850. /H20849C5/H20850 Performing the integration over kzwe obtain /H92673D/H20849E/H20850=1 2/H9266t/H20885 −/H11009+/H11009 d/H9264/H9267˜/H20849/H9264/H20850/H9008/H20851rz2−/H20849/H9280−/H9264/H208502/H20852 /H20881rz2−/H20849/H9280−/H9264/H208502/H20849C6/H20850 in a form of the convolution involving previously calculated DOS in Eq. /H20849C3/H20850. 1J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. 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PhysRevB.96.064529.pdf

PHYSICAL REVIEW B 96, 064529 (2017) Odd-triplet superconductivity in single-level quantum dots Stephan Weiss and Jürgen König Theoretische Physik, Universität Duisburg-Essen and CENIDE, 47048 Duisburg, Germany (Received 6 July 2017; published 31 August 2017) We study the interplay of spin and charge coherence in a single-level quantum dot. A tunnel coupling to a superconducting lead induces superconducting correlations in the dot. With full spin symmetry retained, onlyeven-singlet superconducting correlations are generated. An applied magnetic field or attached ferromagneticleads partially or fully reduce the spin symmetry, and odd-triplet superconducting correlations are generated aswell. For single-level quantum dots, no other superconducting correlations are possible. We analyze, with the helpof a diagrammatic real-time technique, the interplay of spin symmetry and superconductivity and its signaturesin electronic transport, in particular current and shot noise. DOI: 10.1103/PhysRevB.96.064529 I. INTRODUCTION The formation of singlet Cooper pairs in conventional superconductors is a consequence of the broken U(1)-gauge symmetry in the electronic system below the critical tem-perature [ 1]. The SU(2)-spin symmetry, on the other hand, remains intact in a BCS superconductor. The combinationof superconductivity and magnetism, however, allows forother, unconventional types of pairing. Bulk rare-earth metalscan behave as ferromagnetic superconductors [ 2,3]. Heavy- fermion and organic superconductors in strong magneticfields can accommodate Fulde-Ferrell-Larkin-Ovchinnikovstates [ 4,5], superconducting states with oscillating order parameter. More recently, heterostructures built from conven-tional superconducting and ferromagnetic metals have becomean active research field. Theoretical investigations involvingwave-function [ 6–8] or semiclassical [ 6,9,10] methods have predicted that the proximity of conventional superconduc-tors leads to triplet pairing in the ferromagnets. This hasbeen experimentally confirmed in a variety of differentheterostructures [ 11,12], in part by observing 0- πtransitions in Josephson junctions [ 13–15]. To establish a long-range superconducting coherence in the ferromagnet, it is necessaryto break any residual uniaxial spin symmetry, e.g., by magneticinhomogeneities [ 9] or spin-orbit coupling [ 16]. Superconducting correlations can be classified by their symmetry in spin, space, and time. Since electrons arefermions, the correlation function has to be antisymmetricunder exchange of two particles. This allows for four dif-ferent classes of correlations. First, we distinguish betweencorrelators that are even or odd in time (or, equivalently, infrequency). Second, each of them can be characterized bytheir spin symmetry, which may be even for triplets or oddfor singlets. Then, the symmetry in space is fixed. We denotethe four classes of correlations as (i) even singlet, (ii) eventriplet, (iii) odd triplet, and (iv) odd singlet, where “even” and“odd” refer to the symmetry in time (frequency) and “singlet”and “triplet” refer to spin. (i) Even-singlet correlations appearin conventional s-wave superconductors and also in d-wave superconductors [ 17]. (ii) Even-triplet pairing is possible for p-wave superconductors, as suggested for Sr 2RuO 4[18]. (iii) Odd-triplet Cooper pairs have been demonstrated todeeply penetrate into ferromagnets. In such a way, it waspossible to control Josephson junctions [ 19,21], including0-πtransitions [ 20], via the magnetic configuration of the heterostructure, which gave rise to the term superconducting spintronics [22]. Therefore, not only ferromagnetic metals but also ferromagnetic insulators have been used [ 23]. Based on theoretical investigations it has been shown that odd-frequency triplet pairing also appears in diffusive normalmetals contacted by an even-frequency triplet superconductor[24]. (iv) Odd-singlet superconductivity has been predicted only theoretically [ 25] and has not been found experimentally yet. In bulk systems, the possibility to influence the pairing mechanism by tuning external parameters is rather limited.For the heterostructures studied in Refs. [ 19–21], external magnetic fields have been used to change the magnetic con-figurations, which allowed at least for some degree of external control. The possibility of manipulation is, however, much better in mesoscopic systems such as quantum dots. There,system parameters are widely tunable via applied gate and biasvoltages in addition to external fields. This is one motivationfor many theoretical and experimental studies of single anddouble quantum dots coupled to superconducting leads (seeRefs. [ 26,27] for an overview). For example, Cooper-pair splitting in double-quantum-dot (DQD) devices has attracted a lot of attention [ 28–30]. The possibility to control the nonlocal entanglement in a Cooper-pair-splitting device bymeans of spin-orbit interaction together with inter- and intradotCoulomb interactions has been proposed theoretically [ 31]. Recently, we proposed DQDs as a minimal model to gener- ate all four classes of superconducting pairing in a single device[32] and discussed how triplet pairing influences the electric current in various transport regimes. The discussed signatures, however, did not distinguish between even- and odd-triplet pairing and both are, in general, generated simultaneously. Tocircumvent this problem, we focus in this paper on a singlequantum dot hosting only one orbital level. In this case, thelack of orbital degrees of freedom implies that odd-tripletpairing is the only unconventional pairing mechanism that canoccur in addition to the conventional even-singlet one; even-triplet and odd-singlet pairings are ruled out by symmetry. Ithas been shown recently that odd-triplet correlations mightbe rigorously detected by a Majorana scanning tunnelingmicroscope [ 33]. For a full removal of spin symmetry, inhomogeneous magnetic fields, as proposed for DQDs [ 32], 2469-9950/2017/96(6)/064529(11) 064529-1 ©2017 American Physical SocietySTEPHAN WEISS AND JÜRGEN KÖNIG PHYSICAL REVIEW B 96, 064529 (2017) cannot be employed for a single-level quantum dot. Instead, we suggest coupling the quantum dot to two ferromagneticleads with noncollinear magnetization directions. The feasi-bility to achieve this has been experimentally demonstratedrecently [ 34]. A theoretical treatment of quantum dots coupled to su- perconductors in nonequilibrium situations is complicated bythe presence of Coulomb interaction in the quantum dot. Ifthe Coulomb interaction is weak, it can be treated perturba-tively [ 35–41]. To allow for an arbitrarily strong Coulomb interaction, one can perform a perturbation expansion in thetunnel coupling between the dot and leads [ 42,43]. In the limit of an infinitely large superconducting gap in the lead,an exact treatment of both the Coulomb interaction and thetunnel coupling between the quantum dot and superconductingleads is possible [ 43–46]. Here, we employ the diagrammatic real-time technique developed for single [ 43] and double [47] quantum dots attached to superconductors, which also allows for calculating shot noise [ 48]. While the diagrammatic technique is formulated (and has been applied [ 49]) for arbitrary values of the superconducting gap /Delta1in the lead, we assume, in the following, the limit of /Delta1→∞ . The structure of this paper is as follows. First, in Sec. II,w e define the model, introduce superconducting order parametersto describe conventional and unconventional correlations, andpresent the technique to calculate the order parameters aswell as the current and current noise. Then, in Sec. III A , we analyze the case of full spin symmetry. In contrast toour previous studies [ 43,48], we allow for an arbitrary ratio of tunnel-coupling strengths to the superconducting ( /Gamma1 S) and normal ( /Gamma1N) leads, instead of relying on either of the limits /Gamma1S//Gamma1N/greatermuch1o r/lessmuch1. Because of full spin symmetry, triplet pairing does not occur. In Sec. III B, we treat the case of partially reduced spin symmetry by an applied magnetic fieldor by involving one ferromagnetic lead. Finally, Sec. III C is devoted to the case of full removal of spin symmetry by meansof two noncollinearly magnetized ferromagnetic leads. II. MODEL AND METHOD In order to explore unconventional superconducting cor- relations in a single-level quantum dot, we use the modeldepicted in Fig. 1. A quantum dot containing a single (spinful) orbital is coupled to a conventional BCS superconductordenoted by the label S. Spin symmetry may be reduced by an external magnetic field and/or coupling the quantum dot totwo ferromagnetic leads ( LandR). A. Hamiltonian The Hamiltonian is H=Hdot+/summationdisplay α=L,R,S(Hα+Htun,α). (1) The quantum dot is described by the Hamiltonian Hdot=/summationdisplay σ/epsilon10d† σdσ+gμBB·S/¯h+Un↑n↓, (2) with fermionic operators d† σ(dσ) that create (annihilate) electrons on the quantum dot with spin σ. We denote byμS μR,nR,φR ε0,UΓS ΓLσ ΓRσμL,nL,φLnL nRx y zΔ FIG. 1. A single-level quantum dot is tunnel coupled to two ferromagnetic leads with coupling strength /Gamma1α,α=L/R , magnetized in directions nαand a conventional superconductor with gap /Delta1.T h e magnetization of each ferromagnet encloses an angle φαwith the x axis. Tunneling between the superconductor and quantum dot with coupling strength /Gamma1Sgives rise to a finite pairing amplitude on the quantum dot. The underlying coordinate system is shown as well. nσ=d† σdσthe particle-number operator. The single-particle energy /epsilon10can be tuned by a gate voltage. The parameter U models the charging energy for double occupation of the dot.An externally applied magnetic field is denoted B, and the dot spin is S=(¯h/2)/summationtext σσ/primed† σσσσ/primedσ/prime, with the vector of Pauli matrices σ=(σx,σy,σz). To treat the superconducting lead, we assume the limit of a large superconducting gap, /Delta1→∞ [43–46]. This allows us to integrate out the superconductor to arrive at an effectiveHamiltonian for the quantum dot, H dot+HS+Htun,S→Heff, with Heff=Hdot−/Gamma1S 2(d† ↑d† ↓+d↓d↑), (3) with/Gamma1S=2π|tS|2ρS, where ρSis the (spin-independent) density of states for electrons in the normal state at the Fermienergy in the superconductor. In the absence of a Zeemanfield, B=0, the spectrum of the effective Hamiltonian is given by /epsilon1 0for the single-electron states |↑/angbracketrightand|↓/angbracketrightand/epsilon1±= δ/2±/epsilon1A, with 2 /epsilon1A=/radicalBig δ2+/Gamma12 Sand detuning δ=2/epsilon10+U, for the coherent superposition of an empty dot and a doublyoccupied dot, |±/angbracketright =1 √ 2/parenleftbigg/radicalBigg 1∓δ 2/epsilon1A|0/angbracketright∓/radicalBigg 1±δ 2/epsilon1A|d/angbracketright/parenrightbigg , (4) referred to as Andreev bound states. The two ferromagnetic leads, α=L,R, are described by grand-canonical Hamiltonians Hα=/summationdisplay kτ(/epsilon1kατ−μα)c† kατckατ, (5) where μαis the electrochemical potential of lead α, which can be changed by applying bias voltages. We assume non-interacting electrons in the leads having energy /epsilon1 kατand use operators c† kατ(ckατ) to create (annihilate) an electron in lead αwith spin τand momentum k. The spin-quantization axis can be chosen differently for each lead. For the ferromagnets,it is convenient to choose the magnetization directions n Land nR, respectively, as natural quantization axes, such that τ=± describe majority and minority spins with density of statesρ ±at the Fermi energy. The degree of spin polarization at the Fermi energy is given by pα=(ρα+−ρα−)/(ρα++ρα−). 064529-2ODD-TRIPLET SUPERCONDUCTIVITY IN SINGLE-LEVEL . . . PHYSICAL REVIEW B 96, 064529 (2017) Paramagnetic leads are described by pα=0, whereas pα=1 corresponds to a half-metallic lead hosting only majority spins.The hybridization between the dot and lead αdue to a tunneling amplitude t αyields a finite linewidth /Gamma1α±=2π|tα|2ρα±, which, for pα/negationslash=0, is spin dependent. We define the average linewidth as /Gamma1α:=/summationtext σ=±/Gamma1ασ/2. Tunneling between the quantum dot and the leads with tunneling amplitude tαis modeled by Htun,α=/summationdisplay kτσ(tαc† kατuα τσdσ+H.c.), (6) where the unitary operator uα τσdescribes the rotation of the dot’s spin coordinate system to that of lead α. To be specific, we choose the quantization axis for the dot electrons along thez-axis, while n αlie in the x-yplane, enclosing an angle φα with the x-axis (see Fig. 1). This fixes the coordinate axes as ex=(nR+nL)/|nR+nL|,ey=(nR−nL)/|nR−nL|,ez= ex×ey[50,51]. As a result, we get explicitly Htun,α=tα√ 2/summationdisplay k/bracketleftbig c† kα+(eiφα/2d↑+e−iφα/2d↓) +c† kα−(−eiφα/2d↑+e−iφα/2d↓)/bracketrightbig +H.c. (7) B. Kinetic equation An applied bias voltage between any two leads αand βyields a difference eV=μα−μβin the electrochemical potentials. In such a nonequilibrium situation, a net chargecurrent can flow. The main idea of the diagrammatic techniqueused here is to analyze the system’s dynamics in terms ofthe reduced density matrix P≡(P χ2χ1) of the dot degrees of freedom that is obtained after integrating out the leads.Diagonal elements P χ χdescribe the nonequilibrium occupation probabilities of the respective states, whereas off-diagonalelements P χ/prime χdescribe coherent superpositions. The kinetic equation for the reduced density matrix is given by d dtPχ1 χ2(t)=−i/summationdisplay χ/parenleftbig hχ1χPχ χ2−hχχ2Pχ1 χ/parenrightbig (t) +/summationdisplay χ/prime 1χ/prime 2/integraldisplayt −∞dt/primeWχ1χ/prime 1 χ2χ/prime 2(t,t/prime)Pχ/prime 2 χ/prime 1(t/prime). (8) The first line describes the coherent evolution of the system due to the effective Hamiltonian Heffwith matrix elements hχχ/prime=/angbracketleftχ|Heff|χ/prime/angbracketright. The second line, involving the irreducible self-energy kernels W, describes both coherent and dissipative dynamics due to coupling to the ferromagnets. In the steady-state limit, the time derivatives of the reduced-density-matrixelements vanish. The stationary-density-matrix element is,then, obtained from 0=−i/summationdisplay χ/parenleftbig hχ1χPχ χ2−hχχ2Pχ1 χ/parenrightbig +/summationdisplay χ/prime 1χ/prime 2Wχ1χ/prime 1 χ2χ/prime 2Pχ/prime 2 χ/prime 1,(9) where we introduced the generalized transition rates Wχ1χ/prime 1 χ2χ/prime 2:= /integraltextt −∞dt/primeWχ1χ/prime 1 χ2χ/prime 2(t,t/prime). For the basis states χ, it is natural to use {|0/angbracketright,|↑/angbracketright,|↓/angbracketright,|d/angbracketright}. In addition to the kinetic equations for the diagonal reduced-density-matrix elements, we need to include the off-diagonalmatrix elements describing the coherence between |0/angbracketrightand|d/angbracketright. In the regime /Gamma1 S/greatermuch/Gamma1N, it is advantageous to use the basis {|↑/angbracketright,|↓/angbracketright,|+/angbracketright,|−/angbracketright} instead since, then, only diagonal matrix elements need to be taken into account and compact analyticformulas can be obtained for various transport quantities[48,52]. In the following, however, we will extend those previous studies by making no restriction on the ratio /Gamma1 S//Gamma1N. As a consequence, we need to keep all off-diagonal elementsthat are induced by the presence of the superconductor and/orthe ferromagnets. Throughout this work, however, we willassume that the tunnel coupling between the quantum dot andleadsα=L/R is weak, such that only generalized transition rates up to first order in /Gamma1 αwill be included. They are computed with the help of diagrammatic rules presented in Ref. [ 52]. C. Superconducting order parameters Superconducting correlations between two electrons of spin σandσ/primeare quantified by the Green’s function Fσσ/prime(t)=/angbracketleftTdσ(t)dσ/prime(0)/angbracketright, (10) where Tis the time-ordering operator. It is an anomalous (Gorkov) Green’s function as it involves two annihilationoperators and therefore probes coherences between particle-number states that differ by two electrons. The time argumentsof the two operators indicate that Eq. ( 10) measures correla- tions between electrons not only at equal but also at differenttimes. To construct out of Eq. ( 10) Green’s functions that possess definite symmetry in the spin and time arguments, weform proper linear combinations. For even-singlet correlations,we define F S e(t)=F↑↓(t)−F↓↑(t) 2, (11) which is a scalar under rotation in spin space. Furthermore, odd-triplet correlations are characterized by the three combi-nations F Tx o(t)=F↓↓(t)−F↑↑(t)√ 2, (12a) FTy o(t)=−iF↓↓(t)+F↑↑(t)√ 2, (12b) FTz o(t)=F↓↑(t)+F↑↓(t)√ 2, (12c) which transform under rotations like the Cartesian components of a vector F(t). Since there are only four possibilities to choose σandσ/primein Eq. ( 10), we cannot construct any other linear combination that is independent of Eqs. ( 11) and ( 12). For a single-level quantum dot, fixed symmetry in spin spaceimmediately determines the symmetry of the Gorkov Green’sfunction in frequency (time) [ 53]. The Gorkov Green’s functions contain the full information about the time dependence of the superconducting correla-tions. To define an order parameter, we want to characterize thecorrelations by a single number. A natural candidate for such anorder parameter is the equal-time correlator. This works wellfor even-singlet correlations, which yields the dimensionlessorder parameter /Delta1 S e:=FS e(0)=/angbracketleftd↓d↑/angbracketright=Pd 0, (13) 064529-3STEPHAN WEISS AND JÜRGEN KÖNIG PHYSICAL REVIEW B 96, 064529 (2017) which is nothing but the amplitude of the coherent superposi- tion of the dot being empty and doubly occupied. For odd-frequency correlations, however, FTio(0),i= x,y,z , vanishes due to symmetry, and another definition of the order parameter is required. Following Ref. [ 54], we define the order parameter as the derivative of the Gorkov Green’sfunction at equal time, /Delta1 T o:=¯hdFo(t) dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=0, (14) which has units of energy. Importantly enough, in the small quantum dot studied here, there is no room to construct Green’sfunctions describing even-triplet or odd-singlet correlations.Throughout this paper, we assume a weak tunnel couplingbetween the quantum dot and leads α=L/R , such that only first-order tunnel processes from and to the normal leads needto be taken into account. This is consistent with discussingthe superconducting order parameters to only the lowest,i.e., zeroth, order in /Gamma1 N. In this limit, we can completely express the odd-triplet order parameters in terms of reduced-density-matrix elements of the quantum dot. To do so, we firstevaluate ¯ h˙d σ(t)=i[Heff,dσ](t) and then plug the result into ¯h/angbracketleft˙dσ(0)dσ/prime(0)/angbracketright, which yields ¯h/angbracketleft˙d↑d↑/angbracketright=−i 2(Bx−iBy)/angbracketleftd↓d↑/angbracketright+i 2/Gamma1S/angbracketleftd† ↓d↑/angbracketright, (15a) ¯h/angbracketleft˙d↑d↓/angbracketright=− i/epsilon1/angbracketleftd↑d↓/angbracketright−i 2Bz/angbracketleftd↑d↓/angbracketright+i 2/Gamma1S/angbracketleftd† ↓d↓/angbracketright,(15b) ¯h/angbracketleft˙d↓d↑/angbracketright=− i/epsilon1/angbracketleftd↓d↑/angbracketright+i 2Bz/angbracketleftd↓d↑/angbracketright−i 2/Gamma1S/angbracketleftd† ↑d↑/angbracketright,(15c) ¯h/angbracketleft˙d↓d↓/angbracketright=−i 2(Bx+iBy)/angbracketleftd↑d↓/angbracketright−i 2¯h/Gamma1S/angbracketleftd† ↑d↓/angbracketright. (15d) Combining this results with Eqs. ( 12) and ( 14) leads to the very compact result /Delta1T o=i√ 2B/Delta1S e−i√ 2¯h/Gamma1SS. (16) It indicates that odd-triplet correlations in a single-level quan- tum dot are associated either with even-singlet correlations inthe presence of a Zeeman field or with a finite spin polarizationin the presence of a tunnel coupling to a superconducting lead.Therefore, it is irrelevant whether the finite spin polarizationis induced by a Zeeman field or by nonequilibrium spinaccumulation due to voltage-biased ferromagnetic leads. Notethat the Coulomb interaction strength Udoes not enter explicitly Eqs. ( 15). However, it is taken into account when calculating expectation values, e.g., /Delta1 S eandSin Eq. ( 16), with respect to the full Hamiltonian ( 1). We remark that Eqs. ( 13) and ( 16) are valid both in equilibrium and in nonequilibrium. To compute therefromorder parameters, one needs to find ρ d 0andSwith the help of the kinetic equations of the reduced density matrix. D. Charge current V oltages applied to the leads drive currents through the quantum dot. The current flowing from the normal leadα=L,R into the quantum dot can be calculated from Iα=e 2¯hTr(WIαP). (17) Here, the stationary density-matrix element Pχ2χ1has to be determined from the kinetic equation in the stationary limit,Eq. ( 9), and the matrix elements ( W Iα)χχ1χχ2ofWIαare the current rates. They are given by the generalized transitionrates multiplied by the number of electrons transferred fromleadαto the dot. Rules for an explicit calculation are given in Ref. [ 52]. The current I Sinto the superconductor, referred to as the Andreev current, follows from current conservation, IS=IL+IR. (18) E. Current noise and cross correlations In addition to current, we also study zero-frequency current fluctuations, given by Sαβ=lim ω→0/integraldisplay∞ −∞dt/angbracketleftδIα(t)δIβ(0)+δIβ(0)δIα(t)/angbracketrighte−iωt,(19) withδIα=Iα−/angbracketleftIα/angbracketright.F o rα=β, the correlations ( 19) de- scribe the shot-noise power; for α/negationslash=βthey describe cross correlations between leads αandβ. We have ensured that the definition, Eq. ( 19), reproduces the results of an equivalent study based on derivatives of the cumulant generating function[48]. Following Ref. [ 55], this can be expressed diagrammati- cally as S αβ=e2 2¯hlim ω→0Tr/parenleftbig/braceleftbig WIαIβ+WIβIα+WIα/bracketleftbig /Pi1−1 0(ω)−W/bracketrightbig−1WIβ +WIβ/bracketleftbig /Pi1−1 0(−ω)−W/bracketrightbig−1WIα/bracerightbig P/parenrightbig . (20) In this expression the kernels WIαIβare obtained from Wχ1χ/prime 1 χ2χ/prime 2in Eq. ( 8) by replacing two tunneling vertices with current vertices, the one earlier in time by Iβand the one later in time by Iα. To lowest order in /Gamma1N, this term is diagonal in the lead indices ,WIαIβ∼δαβ. The free propagation of the system between the successive current measurements, which enters inEq. ( 20), is given by (/Pi1 0)χ1,χ/prime 1 χ2,χ/prime 2(ω)=iδχ1,χ/prime 1δχ2,χ/prime 2 /epsilon1χ2−/epsilon1χ1−¯hω+i0+. (21) The term ¯ hωin the denominator prevents the appearance of divergencies for χ1=χ2for any finite ω. In the limit ω→0, i.e., the zero-frequency noise, the sum of the terms occurringin Eq. ( 20) is well defined. III. RESULTS The kinetic equation ( 8) is solved numerically taking into account matrix elements χ1,χ2∈{ |0/angbracketright,|↑/angbracketright,|↓/angbracketright,|d/angbracketright}; that is, we obtain all occupation probabilities together with theoverlaps between the respective states to first order in /Gamma1 N. Within this weak-tunneling approximation, i.e., /Gamma1N/lessmuchkBT, the current and the noise are calculated with the help ofEqs. ( 17) and ( 20). For the proximitized dot, /Gamma1 S/negationslash=0, in the presence of ferromagnetic (FM) leads with φα/negationslash=0, we have eight independent nonvanishing elements of Pχ/prime χ: four 064529-4ODD-TRIPLET SUPERCONDUCTIVITY IN SINGLE-LEVEL . . . PHYSICAL REVIEW B 96, 064529 (2017) occupation probabilities and four off-diagonal elements are different from zero due to the presence of the superconductor, Pd 0=(P0 d)∗/negationslash=0, as well as the FM leads, P↓ ↑=(P↑ ↓)∗/negationslash=0. The tensor of irreducible kernels Wis of dimension 8 ×8i n our case. We rearrange Eq. ( 8) in the stationary limit such that 0=˜WP and solve for Pby calculating the null space of ˜W, a matrix that combines both Wand the first line of Eq. ( 8) involving the effective Hamiltonian. A subtlety arises from the intrinsic inconsistency of the master equation when expanded to first order in /Gamma1N.I ft h e contribution of the dissipative kernels in Eq. ( 8) is much smaller than the line describing coherent evolution, entriesof the density matrix may not be of order /Gamma1 N, but higher-order contributions are mixed in. These next-to-leading-order ef-fects, for instance, the possibility to have finite currents withinthe Coulomb-blockaded region (e.g., due to cotunneling), scaleat least with /Gamma1 2 N. Within our approach we have carefully monitored that these artifacts do not have a visible effect in thenumerical results. Recently, several analytic approximations have been put forward. The regime /Gamma1 S,/Gamma1N/lessmuchkBTaround the resonance δ≈0, where only first-order tunneling processes to either the superconducting or the normal leads need to be takeninto account, was explored in Ref. [ 42]. Within the regime, /Gamma1 N/lessmuch/Gamma1S, where the dot spectrum is dominated by the formation of Andreev bound states, it is sufficient to takeinto account diagonal elements of the density matrix forthe calculation of the current [ 48,52]. They correspond to single occupancy or occupancy of the respective states ofEq. (4). This approximation is justified as long as the first line in Eq. ( 8)i so fo r d e r /Gamma1 S/greatermuch/Gamma1N. Consequently, the regime /Gamma1S/lessmuch /Gamma1Nor/Gamma1S∼/Gamma1Nneeds a different treatment. Overlap matrix elements P+ −become important when approaching /Gamma1S∼/Gamma1N. Within this work we focus on transport properties as current,noise, and cross correlators in the regime /Gamma1 N/lessmuch/Gamma1S. For this subgap regime it is possible to resolve the avoided crossingsbetween Andreev bound states most clearly. However, inthe opposite regime /Gamma1 S/lessmuch/Gamma1N, our results are not changed qualitatively. Three different scenarios are distinguished in the following: In scenario A the spin symmetry is intact. In scenario B weinclude a finite magnetic field along a specific axis that breaksSU(2) symmetry; however, a residual U(1) spin symmetry is still kept. In scenarios A and B bias voltages are chosensuch that μ S=0,μL=μR. Within scenario C, we study the regime with no spin symmetry at all, i.e., ( pα/negationslash=0,φL/negationslash=φR) in a nonequilibrium situation μL/negationslash=μR. We discuss the Andreev current as well as induced conventional and uncon-ventional order parameters together with shot-noise data. A. Full spin symmetry This section presents results for scenario A. In the absence of magnetic fields, i.e., pα=0 in each of the leads and B=0, theSU(2) symmetry is intact. The total spin of electrons on the dot and in the paramagnetic leads is conserved. The biasis chosen as μ L=μR=μfor the left and right lead, whereas μS=0. Units of energy are fixed by the Coulomb interaction U, and temperature is set to kBT=U/100. −3 −2 −1 0 1 2 3 δ/U−1.5−1.0−0.50.00.51.01.5μ/U −1.50.01.5 I[eΓN/¯h] FIG. 2. Andreev current as a function of gate voltage δand bias voltage μ. The coupling to the SC is /Gamma1S=U/2, the coupling to the paramagnetic leads is /Gamma1N=U/1000, and temperature is kBT=U/100. 1. Current In Fig. 2the Andreev current into the superconductor is shown as a function of gate and bias voltages ( δ,μ), respectively. We have chosen a strong coupling of the QD to thesuperconductor /Gamma1 S=U/2 to achieve a strong proximity effect. With respect to the applied bias voltage we distinguish threeregimes: (i) low bias, |μ|</epsilon1 −−/epsilon10; (ii) intermediate bias, /epsilon1−−/epsilon10<|μ|</epsilon1+−/epsilon10; and (iii) high bias, |μ|>/epsilon1+−/epsilon10. The boundaries between these regimes are given by theAndreev addition energies depicted as dashed lines in Fig. 2. In case (i) the dot is mostly occupied by a single electron,and around δ=0, a pronounced Coulomb blockade area is developed; hence, transport is blocked. Tuning μ> 0i n regime (ii) provides enough energy to allow for a finitecurrent flow into the superconductor (SC). In that case, asecond electron enters the dot, and an enhanced probability forAndreev reflection results in a finite current, as two electronsare able to enter the SC as a Cooper pair. Of course, theopposite process, for μ< 0, is possible as well: Cooper pairs are expelled from the superconductor, are split by thecharging energy on the dot, and leave the dot towards thesame lead or to different leads. In general, there is symmetryI(μ,δ)=−I(−μ,−δ). Within the high bias regime (iii), the Andreev current becomes insensitive to the sign of δ; that is, the stronger symmetry condition I(−δ)=I(δ) holds. 2. Shot noise and cross correlations In Fig. 3(a), we present data for the shot noise Sαα,α=L orR, of the Andreev current as a function of ( δ,μ). In the low-bias regime, the shot noise vanishes as the current does.Within regimes (ii) and (iii), finite and positive shot noise ispresent. Like for the current, there is a main peak of the shotnoise around the resonance δ=0. The main contribution to S ααstems from the first line in Eq. ( 20). In contrast to the current, however, there is an enhancement of the shot noise atthe Andreev addition energies in the top right and bottom leftof Fig. 3(a). The cross correlations S αβ,α/negationslash=βare depicted in Fig. 3(b). There are two notable features: first, the vanishing of the cross 064529-5STEPHAN WEISS AND JÜRGEN KÖNIG PHYSICAL REVIEW B 96, 064529 (2017) −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)Sαα[e2ΓN/¯h] −3−2−10123 δ/U(b)Sαβ[e2ΓN/¯h] −0.2−0.10.00.10.2 0.00.20.40.60.8 FIG. 3. Andreev shot noise for (a) the same lead (left or right) and (b) cross correlations between the left and right lead. Other parameters are as in Fig. 2. correlations for δ≈0 and, second, the appearance of negative cross correlations. Since WIαIβ=WIβIα=0f o rα/negationslash=βin the weak-coupling limit, the only contribution to Sαβis due to the second and third line in Eq. ( 20). For δ≈0 these are strongly suppressed, and cross correlations vanish. Along theAndreev addition-energy lines in the top right and bottom leftof Fig. 3(b), we obtain S αβ<0, indicating that the transport channels from the left and right lead block each other [ 56,57]. 3. Even-singlet order parameter A finite coupling of the SC to the quantum dot induces a finite amplitude /Delta1S efor even-singlet pairing. It is induced by the proximity effect, i.e., the coupling to the superconducting lead.In the Hamiltonian H Sfor the superconducting lead, Eq. ( 1), we chose a gauge in which the phase of the superconductingorder parameter is zero. This does not mean, however, thatthe induced order parameter /Delta1 S ehas to have the same phase. Since we are dealing with a nonequilibrium situation, /Delta1S e is complex, and its phase contains, in general, nontrivial information. Therefore, we show in Figs. 4(a) and4(b) the modulus and the phase of /Delta1S e, respectively. Areas with finite −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)|ΔS e| −3−2−10123 δ/U(b)arg(ΔS e) 0.00.10.20.3 −ππ/20π/2π FIG. 4. (a) Modulus and (b) phase of the induced even-singlet order parameter /Delta1S eas a function of gate and bias voltages. Other parameters are as in Fig. 2. −3 −2 −1 0 1 2 3 δ/U−1.5−1.0−0.50.00.51.01.5μ/U −0.750.000.75 I[eΓN/¯h] FIG. 5. Andreev current for scenario B1, i.e., the proximitized QD is tunnel coupled to ferromagnetic leads. Polarizations are chosensymmetrically p α=0.8,φα=0. Other parameters are as in Fig. 2. |/Delta1S e|are present in the intermediate-bias regime for δμ < 0. At high bias, the order parameter is suppressed around δ=0. This dip in the modulus of the order parameter as a functionofδis accompanied by a crossover of the phase from zero to π.I nF i g . 4, this crossover is rather sharp since we chose /Gamma1 N//Gamma1S/lessmuch1. With increasing /Gamma1N, the crossover becomes smoother. It has been shown in Refs. [ 42,43] that in the limit of a large superconducting gap in the lead, /Delta1→∞ ,t h e current into the superconductor is directly related to theeven-singlet order parameter. For real and positive /Delta1, we get I S=(e/¯h)/Gamma1SIm/Delta1S e; that is, the Andreev current in Fig. 2 displays, up to a constant prefactor, the imaginary part of theeven-singlet order parameter. The modulus of the even-singletorder parameter, shown in Fig. 4(a), looks different because it is, for the chosen parameters, dominated by its real part. We summarize our findings for scenario A as follows. In the presence of SU(2) spin symmetry, a conventional spin- singlet order parameter exists on the dot. In a nonequilibriumsituation a 0- πtransition can be observed. The noise looks qualitatively similar to the current, but the cross correlationsdisplay a suppression around zero detuning. B. Residual U(1) spin symmetry We now consider the reduction of the SU(2) spin symmetry toU(1), achieved in one of the two following ways. For scenario B1, the proximitized QD is coupled to FM leads withparallel magnetizations n L/bardblnRbetween the left and right side, accompanied by finite polarizations pα/negationslash=0( s e eF i g . 1). The second scenario B2 is realized when spin symmetry isbroken by a local static magnetic field B=Bˆxapplied to the QD. In this situation, the leads are chosen to be paramagneticmetals ( p α=0). 1. Scenario B1 In Fig. 5we depict IS(δ,μ) for B1 where the parameters for the FM leads are φα=0,pα=0.8,α=L/R . Temperature and tunnel couplings are chosen as in Fig. 2. As described above, the three regimes of low-, intermediate-, and large-biasvoltage show up. Note that the magnitude of the current 064529-6ODD-TRIPLET SUPERCONDUCTIVITY IN SINGLE-LEVEL . . . PHYSICAL REVIEW B 96, 064529 (2017) −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)Sαα[e2ΓN/¯h] −3−2−10123 δ/U(b)Sαβ[e2ΓN/¯h] −0.10−0.050.000.050.10 0.00.10.20.30.4 FIG. 6. Andreev shot noise of scenario B1 for (a) the same lead (left or right) and (b) cross correlations between the left and right lead. Other parameters are as in Fig. 5. is reduced due to the coupling of the QD to ferromagnets. Especially within the intermediate-bias regime, a clear asym-metry between positive and negative δis visible; however, the symmetry I(−δ,−μ)=I(δ,μ) is always intact. Shot noise S ααand cross correlations Sαβare presented in Figs. 6(a) and6(b), respectively. Like for the current, the finite spin polarization leads to a reduction of the shot noiseand cross correlations. Many features discussed for scenarioA hold qualitatively for scenario B1 as well. An importantdifference is that the cross correlations are now not suppressedanymore around zero detuning δ≈0. In Fig. 7, we show how the presence of the FM leads modify the even-singlet order parameter. The modulus of /Delta1 S eis, in general, suppressed in magnitude. In the regime where the QDis most likely singly occupied, however, |/Delta1 S e|is enhanced. The phase of /Delta1S eshows again a crossover from zero to πasδgoes through zero. The most important effect of the FM leads is that now odd-triplet order parameters are induced as well. Accordingto Eq. ( 16), the finite odd-triplet correlation is related to an accumulated dot spin S(since there is no magnetic field considered in scenario B1). From Eq. ( 16), we conclude that −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)|ΔS e| −3−2−10123 δ/U(b)arg(ΔS e) 0.00.10.2 −ππ/20π/2π FIG. 7. (a) Modulus and (b) phase of the induced even-singlet order in scenario B1 for the same parameters as in Fig. 5. −2 0 2 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)|ΔT o|/U −2 0 2 δ/Uμ/U(b)arg(ΔT o) −π−π/20π/2π 0.00.10.2 FIG. 8. Odd-triplet order parameter in scenario B1. The FM leads yield a finite spin accumulation on the QD, which results in finite /Delta1T oin the direction of the induced exchange magnetic field. Other parameters are as in Fig. 5. /Delta1T opoints along the axis of the residual U(1) symmetry. The modulus of this component is shown in Fig. 8(a). All other components vanish here. Since /Delta1T ois purely imaginary in scenario B1, the phase is either π/2o r−π/2[ s e eF i g . 8(b)]. Compared to the even-singlet order parameter, the phase of theodd-triplet one is shifted by π/2, in accordance with Eq. ( 16). 2. Scenario B2 In scenario B2, a local magnetic field applied to the QD is responsible for the reduction of the spin symmetry fromSU(2) to U(1), while the leads are paramagnetic, p α=0. In Fig. 9, we present the Andreev current IS(δ,μ)a saf u n c t i o n of detuning and bias voltage. The Zeeman field acting on thedot enters the Andreev addition energies. This generates anadditional substructure visible in Fig. 9. The shot noise S αα(δ,μ) is shown in Fig. 10(a) .A g a i n ,w e find a maximum around the resonance δ=0 and a substructure of the Andreev addition energies, like for the current. Thereis an enhancement of the shot noise along two of the Andreevaddition energies in the top right and bottom left. The cross −3 −2 −1 0 1 2 3 δ/U−1.5−1.0−0.50.00.51.01.5μ/U −1.5−1.0−0.50.00.51.01.5 I[eΓN/¯h] FIG. 9. Andreev current for scenario B2. The leads are assumed to be paramagnetic metals pα=0. A local magnetic field Bx=U/5 lifts the spin degeneracy on the QD. Other parameters are as in Fig. 2. 064529-7STEPHAN WEISS AND JÜRGEN KÖNIG PHYSICAL REVIEW B 96, 064529 (2017) −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)Sαα[e2ΓN/¯h] −3−2−10123 δ/U(b)Sαβ[e2ΓN/¯h] −0.2−0.10.00.10.2 0.00.20.40.60.8 FIG. 10. Andreev shot noise for scenario B2. (a) The shot noise and (b) the cross correlators. Parameters are as in Fig. 9. correlations [see Fig. 10(b) ] display a suppression around zero detuning, similar to scenario A and in contrast to B1. The main impact of the Zeeman field on the induced even- singlet order parameter is the additional substructure in theAndreev addition energies (see Fig. 11). As a consequence, there is now a finite region between the split Andreev additionin which the transition from phase 0 to phase πbecomes smoother [see top right and the bottom left of Fig. 11(b) ]. The suppression of the even-singlet order parameter around zerodetuning is, however, similar to that in cases A and B1. We now turn to the induced odd-triplet order parameter. It is finite due to the combination of a finite Zeeman field and thepresence of even-singlet correlations [see first term in Eq. ( 16)] and due to spin accumulation, according to the second term ofEq. ( 16). In Figs. 12(a) and12(b) modulus and phase of /Delta1 Txo, i.e., the component of the odd-triplet order parameter alongthe direction of the magnetic field, are shown as a functionof gate and bias voltages. Also, the phase of the odd-tripletorder parameter is shifted by π/2 [see above and Eq. ( 16)]. A comparison of Fig. 12with Fig. 11reveals the interesting result that, for the chosen parameters, odd-triplet correlationsmost prominently appear in regions in which even-singletcorrelations are suppressed: in the Coulomb-blockade region −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)|ΔS e| −3−2−10123 δ/U(b)arg(ΔS e) 0.00.10.20.3 −ππ/20π/2π FIG. 11. (a) Modulus and (b) phase of the induced even-singlet order parameter scenario B2 for parameters as in Fig. 9. −2 0 2 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)|ΔT o|/U −2 0 2 δ/Uμ/U(b)arg(ΔT o) −π−π/20π/2π 0.00.10.2 FIG. 12. Odd-triplet order parameter scenario B2. Parameters are as in Fig. 9. in the middle of Fig. 12and in the small stripes between the split Andreev addition energies in the top right and bottom left.These are regions in which spin accumulation is pronounced.The very fact that superconducting correlations also showup in the Coulomb-blockade regime underlines the notion ofsuperconducting pairing of electrons at different times sincethe probability to have both electrons in the quantum dot at thesame time is exponentially suppressed due to charging energy. To summarize the main feature of scenario B, a reduction of the spin symmetry, either by an external magnetic field orby FM leads, induces finite odd-frequency order parameters/Delta1 T oin the quantum dot. Triplet order parameters appear in the direction of the applied magnetic field or the magnetizationdirection. Other components are excluded by the residual spinsymmetry. C. Complete removal of rotational symmetry In scenario C, spin symmetry is fully broken by two FM leads, pα/negationslash=0, with noncollinear magnetization directions, φL/negationslash=φR, that are put at different chemical potentials. To be specific, we choose μL=−μR, relative to μS=0 and φL−φR=π/2. Choosing different chemical potentials for −3 −2 −1 0 1 2 3 δ/U−1.5−1.0−0.50.00.51.01.5μ/U −0.080.000.08 I[eΓN/¯h] FIG. 13. Andreev current for scenario C, here realized with FM leads, finite polarization p=0.8,φL−φR=π/2, and μL=−μR. Other parameters are as in Fig. 2. 064529-8ODD-TRIPLET SUPERCONDUCTIVITY IN SINGLE-LEVEL . . . PHYSICAL REVIEW B 96, 064529 (2017) −2 0 2 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)Sαα[e2ΓN/¯h] −2 0 2 δ/U(b)Sαβ[e2ΓN/¯h] −0.50−0.250.000.250.50 0.00.20.40.6 FIG. 14. (a) Shot noise and (b) cross correlations for scenario C for parameters as in Fig. 13. the two ferromagnets is important for a full removal of spin symmetry. Two FM leads with φL/negationslash=φRbutμL=μRcan be effectively described as one lead with some effectivemagnetization direction; that is, the induced superconductingcorrelations are in this case the same as in scenario B1. The Andreev current is shown in Fig. 13. It now looks, as a consequence of the biasing scheme μ L=−μR, very different from all the previously discusses cases. It is possible to switchbetween I S>0 andIS<0, i.e., electrons entering or leaving the SC lead, by changing the gate voltage. The vanishingAndreev current at δ=0 is a consequence of μ L=−μRand /Gamma1L=/Gamma1R. The corresponding shot noise and cross correlations are shown in Fig. 14. We find for all voltages negative cross correlations, Sαβ<0, while the shot noise is always positive. The even-singlet order parameter is shown in Fig. 15.T h e condition μL=−μRtogether with /Gamma1L=/Gamma1Rresults in /Delta1S e being real, positive, and symmetric with respect to δ→−δ. The imaginary part is suppressed due to the cancellation ofdifferent tunneling processes on the left and right side [ 52]. Finite Coulomb interaction at finite bias creates an ex- change field that affects the accumulated spin on the dot.The resulting spin precession yields a spin component not −3−2−10123 δ/U−1.5−1.0−0.50.00.51.01.5μ/U(a)|ΔS e| −3−2−10123 δ/U(b)arg(ΔS e) 0.00.10.20.3 −ππ/20π/2π FIG. 15. (a) Modulus and (b) phase of the even-singlet order parameter for scenario C for parameters as in Fig. 13. FIG. 16. (left column panels (a), (c), and (e)) give the Modulus and the (right column panels (b), (d), and (f)) the phase of the odd- triplet order parameter vector ( /Delta1Txo,/Delta1Ty o,/Delta1Tzo). only within but also out of the plane defined by the lead magnetization directions. Since the direction of the quantum-dot spin is, in general, not collinear with the exchange fieldgenerated by the FM leads, the direction of the odd-tripletorder parameter shows, according to Eq. ( 16), a nontrivial dependence of the gate and bias voltages. This is illustratedin Fig. 16, where we show the modulus and the phase of all components ( /Delta1 Txo,/Delta1Ty o,/Delta1Tzo) of the odd-triplet order parameter. Due to finite values of Sx,Sy,Sz, all three components of /Delta1oare finite [see Eq. ( 16)]. For the chosen parameters, the phase of all order parameters is ±π/2, indicating a purely 064529-9STEPHAN WEISS AND JÜRGEN KÖNIG PHYSICAL REVIEW B 96, 064529 (2017) imaginary odd-triplet order parameter. Sign changes in the order parameter are due to a sign change of the spin, whengoing, for instance, from negative to positive gate voltage.We mention that all components are of the same order ofmagnitude due to a large spin-valve effect. In comparison,forφ L=φRall order parameters vanish since the exchange field vanishes and spin symmetry is restored for the caseμ L=−μR. The positions of the resonances are determined by Andreev addition energies. To summarize scenario C, spin symmetry can be completely removed by using two noncollinearly magnetized FM leads.The combination of spin accumulation due to a voltage biasbetween the two ferromagnets and spin precession due toexchange fields yields finite values of all three componentsof the odd-triplet order parameter. When tuning gate voltages,it is possible to flip the spin and therefore change the sign ofthe imaginary part of the order parameter. IV . CONCLUSIONS In conclusion, we have studied the role of spin symmetry in the presence of superconducting correlations in a single-level quantum dot. Conventional even-singlet correlations areinduced by a tunnel coupling to a superconducting lead withordinary BCS pairing symmetry. In addition, superconductingodd-triplet correlations can be generated by breaking theSU(2) spin symmetry with the help of a Zeeman field or ferromagnetic leads. In a single-level quantum dot, noother pairing symmetries (even triplet or odd singlet) are possible. We have determined the superconducting order parameters by numerically solving the kinetic equations for the dot degreesof freedom. Therefore, we have taken all coherences betweendifferent charge states and between different spin states intoaccount. We have discussed three different scenarios in orderto investigate the influence of reduced spin symmetry on thesuperconducting correlations: (A) full SU(2) spin symmetry, (B) residual U(1) spin symmetry, and (C) full removal of spin symmetry. We have found that odd-triplet correlationsare present in the dot whenever the dot is coupled to asuperconductor and either a finite spin is accumulated ora magnetic field is applied. All induced order parametersdisplay a strong dependence on applied gate and bias voltages,magnetic fields, and the magnetization directions of attachedferromagnetic leads. This opens the possibility to selectivelytune the even-singlet and odd-triplet order parameters. Inparticular, we found regimes in which the unconventionalodd-triplet correlations are large while the conventional even-singlet correlations are suppressed. ACKNOWLEDGMENTS We acknowledge financial support of the Deutsche Forschungsgemeinschaft (DFG) under Projects No. WE5733/1 and No. KO 1987/5. We thank P. Stegmann and B.Sothmann for discussions. [1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev. 108,1175 (1957 ). [2] W. A. Fertig, D. C. Johnston, L. E. DeLong, R. W. McCallum, M. B. Maple, and B. T. Matthias, Destruction of Supercon-ductivity at the Onset of Long-Range Magnetic Order in theCompound ErRh 4B4,P h y s .R e v .L e t t . 38,987(1977 ). [3] M. Ishikawa and Ø. 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PhysRevB.74.214515.pdf

Optical determination of the superconducting energy gap in electron-doped Pr 1.85Ce0.15CuO 4 C. C. Homes,1,*R. P. S. M. Lobo,2P. Fournier,3A. Zimmers,4and R. L. Greene4 1Condensed Matter Physics & Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA 2Laboratoire de Physique du Solide (UPR 5 CNRS) ESPCI, 10 Rue Vauquelin 75231 Paris, France 3Département de Physique, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1 Canada 4Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, Maryland 20742, USA /H20849Received 26 June 2006; revised manuscript received 23 October 2006; published 22 December 2006 /H20850 The optical properties of single crystal Pr 1.85Ce0.15CuO 4have been measured over a wide frequency range above and below the critical temperature /H20849Tc/H1122920 K /H20850. In the normal state the coherent part of the conductivity is described by the Drude model, from which the scattering rate just above Tcis determined to be 1/ /H9270 /H1122980 cm−1. The condition that /H6036//H9270/H110152kBTnear Tcappears to be a general result in many of the cuprate superconductors. Below Tcthe formation of a superconducting energy gap is clearly visible in the reflectance, from which the gap maximum is estimated to be /H90040/H1122935 cm−1/H208494.3 meV /H20850. The ability to observe the super- conducting energy gap in the optical properties favors the nonmonotonic over the monotonic description of thed-wave gap. The penetration depth for T/H11270T cis/H9261/H112292000 Å, which when taken with the estimated value for the dc conductivity just above Tcof/H9268dc/H1122935/H11003103/H9024−1cm−1places this material on the general scaling line for the cuprates defined by 1/ /H92612/H11008/H9268dc/H20849T/H11229Tc/H20850·Tc. These results are consistent with the observation that 1/ /H9270/H110152/H90040, which implies that the material is not in the clean limit. DOI: 10.1103/PhysRevB.74.214515 PACS number /H20849s/H20850: 74.25.Gz, 74.72. /H11002h I. INTRODUCTION The high-temperature copper-oxide superconductors may be grouped into two broad categories; hole- and electron-doped materials. The hole-doped materials constitute the ma-jority of the cuprate superconductors, while electron dopingis observed in a relatively small number of materials, mainlythe /H20849Nd,Pr /H20850 2−xCexCuO 4systems1,2and the infinite-layer /H20849Sr,L/H20850CuO 2/H20849L=La,Sm,Nd,Gd /H20850materials.3,4The phase dia- grams of the hole- and electron-doped materials have some similarities,5with the parent materials being antiferromag- netic /H20849AFM /H20850insulators in both cases. However, the electron- doped materials are also noticeably different in that the AFMregion extends to a much higher doping with an almost non-existent pseudogap region. 6–8In addition, the superconduct- ing dome is quite small, with relatively low critical tempera-tures /H20849T c’s/H20850. These differences have prompted some debate as to whether or not the electron-doped materials were high-temperature superconductors at all, or if they resembledmore conventional superconductors. While some work indi-cated that the electron-doped materials possess an isotropics-wave superconducting energy gap, 9,10more recent studies have suggested that the energy gap has nodes and is d-wave in nature,11–21similar to the hole-doped materials.22,23How- ever, while the d-wave gap in the hole-doped materials may be described in a monotonic way, /H9004/H20849/H9278/H20850=/H90040cos/H208492/H9278/H20850, where /H90040is the gap maximum and /H9278is a Fermi surface angle, the energy gap in the electron-doped materials appearsto be nonmonotonic, i.e., /H9004/H20849 /H9278/H20850=/H90040/H20851cos/H208492/H9278/H20850−bcos/H208496/H9278/H20850 +ccos/H2084910/H9278/H20850/H20852.18–20 In electron-doped materials, a pseudogap exists in the un- derdoped regime and overlaps superconductivity over asmall portion of the phase diagram. 24,25In the normal state above the superconducting dome, the conductivity is reason-ably metallic. 24The formation of a superconducting energy gap and the commensurate change in the density of states/H20849DOS /H20850should lead to observable changes in the low-energy optical properties. In initial optical studies of the electron- doped /H20849Nd,Pr /H208502−xCexCuO 4materials,24,26–34there was no de- finitive signature in the reflectance of a gap opening. It was suggested that this was due to the fact that the high-temperature superconductors were in the clean limit /H20849i.e., the regime where the normal-state scattering rate 1/ /H9270/H11270/H9004 0,o r alternatively when the mean free path is much greater thanthe coherence length, l/H11271 /H92640/H20850, where the formation of a super- conducting gap is difficult to observe.35Interestingly, we have recently observed changes in the reflectance of thinfilms of Pr 2−xCexCuO 4above and below Tcwhich track with doping,36indicating that these features are associated with the superconducting energy gap. However, in any study ofthin films there is always the concern that substrate-inducedstrain may affect the structural and electronic properties ofthe film. In this work we examine the optical properties of an op- timally doped single crystal of Pr 1.85Ce0.15CuO 4/H20849Tc/H1122920 K /H20850 for light polarized in the copper-oxygen planes over a wide frequency range in the normal and superconducting states.Some aspects of this work have been previously reported. 37 The results show a clear signature of the formation of a su-perconducting energy gap in the reflectance and the opticalconductivity, validating the earlier thin-film work. 36The normal-state properties are well described by a simple two-component model /H20849coherent and incoherent components /H20850, which allows the plasma frequency of the coherent Drudecomponent /H9275pdand the scattering rate 1/ /H9270to be determined. At 30 K, 1/ /H9270/H1122980 cm−1or about 10 meV, which is consis- tent with the observation that /H6036//H9270/H110152kBTnear Tcin many of the cuprate superconductors. In the superconducting state,the real part of the dielectric function and optical conductiv-ity sum rules both yield estimates for the in-plane penetrationdepth of /H9261/H112292000 Å. From the structure observed in the re- flectance below T c, the gap maximum is estimated to be /H90040PHYSICAL REVIEW B 74, 214515 /H208492006 /H20850 1098-0121/2006/74 /H2084921/H20850/214515 /H208498/H20850 ©2006 The American Physical Society 214515-1/H1122935 cm−1/H20849about 4.3 meV /H20850. For values of 1/ /H9270determined just above Tc, the result that 1/ /H9270/H110152/H90040/H20849valid primarily along the antinodal direction /H20850indicates that the material is not in the clean limit, in agreement with recent scalingarguments. 38,39In addition, we speculate that the nonmono- tonic nature of the superconducting gap in this material re-sults in large changes to the joint density-of-states /H20849JDOS /H20850 below T crelative to the monotonic case, allowing the forma- tion of the gap to be observed more easily in the opticalresponse. II. EXPERIMENT Single crystals of Pr 1.85Ce0.15CuO 4were grown from a CuO-based flux using a directional solidification technique.40 A mixture of high-purity /H2084999.9% /H20850starting materials of Pr6O11, CeO 2, and CuO were heated rapidly to just above the melting point /H20849/H110111270 °C for this Ce concentration /H20850. After a soak of several hours at the maximum temperature, the ma-terials were cooled slowly to room temperature. To inducesuperconductivity, the crystals /H20849typical size of 2 /H110032m m 2and /H1101120/H9262m thickness /H20850were oxygen reduced by annealing in an inert gas atmosphere following a procedure similar to thatdescribed by Brinkmann et al. 41The superconducting transi- tion was characterized by a SQUID magnetometer fromQuantum Design in a field of 1 Oe /H20849ZFC /H20850, and the critical temperature determined to be T c/H1122920 K. The observed value ofTcfor this Ce concentration is somewhat less than ideal, suggesting that the sample may have been over-reduced. The reflectance of single-crystal Pr 1.85Ce0.15CuO 4has been measured at a near-normal angle of incidence for lightpolarized in the a-bplanes from /H1101518 to over 34 000 cm −1, above and below Tc, on Bruker IFS 66v/S and IFS 113v spectrometers using an in situ evaporation technique.42The noise in the far-infrared reflectance is less than 0.05%, result-ing in a signal-to-noise ratio of better than 2000:1. The op-tical properties are calculated from a Kramers-Kronig analy-sis of the reflectance, where extrapolations are supplied for /H9275→0,/H11009. At low frequency, a metallic Hagen-Rubens re- sponse is assumed in the normal state /H20849R/H110081−/H92751/2/H20850, and a two-fluid model was applied in the superconducting state /H20849R/H110081−/H92752/H20850. Above the highest-measured frequency in this experiment the reflectance of Pr 1.85Ce0.15CuO 4has been em- ployed to about 35 eV;29above that frequency a free- electron approximation has been assumed /H20849R/H110081//H92754/H20850. III. RESULTS AND DISCUSSION A. Optical and transport properties The ab-plane reflectance of Pr 1.85Ce0.15CuO 4/H20849Tc/H1122920 K /H20850 is shown in Fig. 1over a wide spectral range, at a variety of temperatures above and below Tc. The reflectance in the mid- infrared region and above is consistent with previous studiesof electron-doped cuprates; 24,26–34there is a plasma edge at /H110151.2 eV, and the reflectance throughout the mid-infrared re- gion increases with decreasing temperature. The inset in Fig.1shows the low-frequency reflectance at several tempera- tures above and below T c. The sharp structures in the reflec- tance that appear variously as resonant and antiresonant fea-tures are infrared-active lattice modes.43The reflectance increases with decreasing temperature, but there is littlechange between the 30 and 6 K spectra, with the exceptionof a kink at /H1122970 cm −1, below which the reflectance increases rapidly with decreasing frequency. This is the same feature inthe reflectance that was observed in thin-film studies, 36that signals the formation of a superconducting energy gap. The optical conductivity is shown over a wide frequency range in Fig. 2/H20849a/H20850, and at low frequency in Fig. 2/H20849b/H20850. The conductivity in the normal state can be described as a com-bination of a coherent Drude component that describes thefar-infrared response, and an incoherent component thatdominates the mid-infrared region. The “two-component”expression for the real part of the optical conductivity is /H92681/H20849/H9275/H20850=1 4/H9266/H9275pd2/H9003 /H92752+/H90032+/H9268MIR, /H208491/H20850 where /H9275pd2=4/H9266nde2/m*is the square of the Drude plasma frequency, ndis a carrier concentration associated with co- herent transport, m*is an effective mass, /H9003=1//H9270is the scat- tering rate, and /H9268MIRis the mid-infrared component. /H20851When /H9275pdand 1/ /H9270are in cm−1,/H92681also has the units cm−1;t o recover units for the conductivity of /H9024−1cm−1, the term 1/4/H9266in Eq. /H208491/H20850should be replaced with 2 /H9266/Z0, where Z0 =377/H9024is the characteristic impedance of free space. /H20852The Drude contribution has the form of a Lorentz oscillator cen-tered at zero frequency. The features in the reflectance attrib-uted to lattice modes appear as sharp resonances in the con-ductivity, shown in detail in Fig. 2/H20849b/H20850. To apply the two- component model to the data, it is necessary to specify thenature of /H9268MIR. The mid-infrared conductivity is often de- scribed by a series of overdamped Lorentzian oscillatorswhich yield a flat, incoherent response in this region. FIG. 1. /H20849Color online /H20850The temperature dependence of the re- flectance at a near-normal angle of incidence of single crystalPr 1.85Ce0.15CuO 4/H20849Tc/H1122920 K /H20850from /H1101518 to 34000 cm−1for light polarized in the a-bplane above and below Tc. Inset: The detailed temperature dependence of the far-infrared reflectance above andbelow T c. Note the kink and the sudden increase in the reflectance below /H1122970 cm−1forT/H11270Tc. The sharp features in the reflectance are infrared-active lattice modes. /H20849The resolution is 2 cm−1./H20850HOMES et al. PHYSICAL REVIEW B 74, 214515 /H208492006 /H20850 214515-2However, this approach can be somewhat arbitrary. To simplify the fitting a constant background /H20849/H9268MIR /H112291000/H9024−1cm−1/H20850has been used at low temperature. The fitted results show that while the scattering rate decreases from 1/ /H9270=130±3 to 80±2 cm−1when the temperature is reduced from 100 K to just above Tcat 30 K, the Drude plasma frequency remains constant at /H9275pd =13 000±200 cm−1. Transport measurements in the electron- doped cuprates typically show a quadratic form of theresistivity, 17,44–47/H9267=/H92670+aT2, with a weak temperature de- pendence near Tc, indicating that at low temperatures 1/ /H9270is dominated by /H92670. Interestingly, the result that /H6036//H9270/H110152kBT close to Tcseems to be true for many of the cuprates. Note that /H9275pd2is a reflection of only those carriers that participate in coherent transport, rather than the total numberof doped carriers determined from the classical plasma fre- quency /H9275p2=4/H9266ne2/m. The value for /H9275pmay be estimated using several different techniques. The zero-crossing of the real part of the dielectric function /H92801/H20849/H9275/H20850occurs at /H9275p//H20881/H9280/H11009. However, the presence of interband absorptions that overlap with the coherent component, as well as the difficulty inchoosing the correct value of /H9280/H11009, makes this approach unre- liable. Another method is the finite-energy f-sum rule48/H20885 0/H9275c /H92681/H20849/H9275/H20850d/H9275/H11015/H9275p2/8, /H208492/H20850 where /H9275cis a cutoff frequency. In the absence of other exci- tations, this sum rule is exact in the /H9275c→/H11009limit. While this condition is difficult to achieve in the cuprates, modificationsto this sum rule based on an analysis of the absorption coef-ficient /H9251/H20849/H9275/H20850have been suggested by Hwang et al.49Adopting this approach yields /H9275p/H1122919 300 cm−1, suggesting that only about half of the doped carriers participate in coherent trans-port /H20849assuming the masses do not change /H20850, similar to the hole-doped materials. 50 While the majority of this paper is concerned with the far-infrared optical properties, it is worth commenting brieflyon the behavior of the optical conductivity throughout therest of the infrared frequency range. As Fig. 2/H20849a/H20850indicates, at room temperature the conductivity is quite broad, with littlefrequency dependence at low energies; however, as the tem-perature decreases the Drude component narrows rapidly.The reduction in 1/ /H9270leads to changes in the conductivity over much of the far infrared; however, with the exception ofa weak feature at /H1122918 000 cm −1, the high-frequency conduc- tivity displays little structure. While this result is consistentwith an earlier study of single crystal Nd 1.85Ce0.15CuO 4 grown by a flux technique,30other works on crystals grown using the traveling-solvent floating-zone method have re-vealed some unusual structure in the 300–400 cm −1 /H2084940–50 meV /H20850region.24,32,33The as-grown electron-doped materials are not superconducting; they must be oxygen- reduced to induce a Tc. Interestingly, the as-grown /H20849or oxy- genated /H20850samples show prominent structure in the reflectance in the 300–400 cm−1region that manifests itself as a sup- pression of the conductivity, that is almost completely re-moved upon oxygen reduction. 31,32We speculate that the dif- ferences observed in the various works may be related todifferent levels of oxygen reduction. As previously men-tioned, the conductivity is reasonably well described in thefar-infrared region by a Drude response. However, it hasbeen noted in the hole-doped cuprates that the modulus ofthe conductivity obeys a power law over much of the mid- infrared region, 51,52/H20841/H9268˜/H20849/H9275/H20850/H20841/H11008/H9275−0.65. The log-log plot of the temperature dependence of the modulus of the optical con- ductivity vs frequency is shown in Fig. 3for a variety of temperatures over a wide frequency range. Throughout muchof the mid infrared, the modulus of the conductivity follows the power law /H20841 /H9268˜/H20849/H9275/H20850/H20841/H11008/H9275−0.69, in good agreement with the behavior observed in the hole-doped cuprates.51–56Below roughly 1000 cm−1there is a deviation from this power-law behavior at high temperature. As the temperature decreases alinear behavior is once again recovered; however, the expo- nent is now larger /H20841 /H9268˜/H20849/H9275/H20850/H20841/H11008/H9275−0.81, suggesting that the charac- ter of the conductivity is different in these two regions. Well below Tc, there is a substantial reduction in the low- frequency conductivity /H20851indicated by the arrow in Fig. 2/H20849b/H20850/H20852. The Ferrell-Glover-Tinkham /H20849FGT /H20850sum rule57,58states that the difference between the conductivity curves at T/H11229Tcand T/H11270Tc/H20849the so-called “missing area” /H20850is related to the forma- tion of a superconducting condensate FIG. 2. /H20849Color online /H20850/H20849a/H20850The temperature dependence of the real part of the optical conductivity over a wide frequency range forPr 1.85Ce0.15CuO 4above and below Tcfor light polarized in the a -bplanes. /H20849b/H20850The low-frequency optical conductivity. The conduc- tivity in the normal state is described by a Drude component thatnarrows rapidly with decreasing temperature. Below T c, there is a significant loss of spectral weight at low frequency /H20849arrow /H20850due to the formation of a condensate. The sharp features superimposedupon the electronic background at /H11229306, 338, and 433 cm −1/H20849aster- isks/H20850are infrared-active phonon modes /H20849Ref. 43/H20850.OPTICAL DETERMINATION OF THE SUPERCONDUCTING … PHYSICAL REVIEW B 74, 214515 /H208492006 /H20850 214515-3/H20885 0+/H9275c /H20851/H92681/H20849/H9275,T/H11229Tc/H20850−/H92681/H20849/H9275,T/H11270Tc/H20850/H20852 /H11015/H9275ps2/8, /H208493/H20850 where /H9275ps2=4/H9266nse2/m*is the square of the superconducting plasma frequency, nsis the superconducting carrier concen- tration, and /H9275cis chosen such that /H9275ps2converges smoothly. The strength of the condensate is simply /H9267s=/H9275ps2, which is related to the penetration depth by /H9267s=c2//H92612. The value of /H9267s may also be estimated from the response of the dielectric function in the zero-frequency limit to the formation of acondensate, which is expressed purely by the real part /H92801/H20849/H9275 →0/H20850=/H9280/H11009−/H9275ps2//H92752forT/H11270Tc. This allows the strength of the condensate to be calculated from /H9275ps2/H11229−/H92752/H92801/H20849/H9275/H20850as/H9275→0. The frequency dependence of /H20881−/H92752/H92801/H20849/H9275/H20850is shown in Fig. 4for Pr 1.85Ce0.15CuO 4at 30 and 6 K. The low-frequency ex- trapolations employed in the Kramers-Kronig analysis of thereflectance are included to allow the /H9275→0 values to be de- termined more easily. In the normal state the function goessmoothly to zero, indicating the absence of a condensate.Well below T c, the estimate for the superconducting plasma frequency is /H9275ps/H112297800 cm−1. This is consistent with values of/H9275ps/H112298000 cm−1determined from the FGT sum rule. From 1/ /H9261=2/H9266/H9275 psthe penetration depth is determined to be /H9261=2000±100 Å, similar to results obtained from thin films with similar Tc’s.36,59A comparison of /H9275psto/H9275pdindicates that less than half of the carriers involved in the coherentDrude component have collapsed into the condensate /H20849as- suming similar effective masses /H20850, a result that is consistent with the larger body of work on the hole-doped materials. 50 The relatively low values for Tcand/H9261for this material, and the electron-doped materials in general, have always pre-sented a challenge for the Uemura plot, 60which relates the density of carriers in the superfluid to the transition tempera-ture, /H9267s/H11008Tc. While the Uemura plot works well for the hole- doped cuprates in the underdoped region, the electron-dopedmaterials have typically fallen well off of the Uemuraplot. 30,61,62Recently, a more general scaling relation /H9267s /H11008/H9268dcTc/H20849where /H9268dcis determined just above Tc/H20850which al- lows the points for the electron- and hole-doped cuprates tobe scaled onto the same universal line. 38,39The value of /H9268dc close to Tcis taken from both /H92681/H20849/H9275→0/H20850as well as Drude fits to the lineshape of the conductivity, resulting in the estimate /H9268dc/H20849T/H11015Tc/H20850=35 000±3000 /H9024−1cm−1; this places the mate- rial almost directly on the /H9267s/H11008/H9268dcTcscaling line. An impli- cation of this result is the suggestion that this material is notin the clean limit; 39this controversial point will be examined in more detail in a subsequent section. B. Determination of the gap maximum A reasonable estimate for the gap maximum may be taken from a comparison of the reflectance at T/H11229TcandT/H11270Tc, shown in Fig. 5/H20849a/H20850. Above about 70 cm−1the two curves are nearly identical, but below this value the 6 K reflectancedisplays a kink followed by an abrupt increase; this featurehas also been observed in our measurements of the reflec-tance of thin films of this material. 36The optical conductivity /H20849and by extension, the reflectance /H20850of a material may be de- scribed by the Kubo-Greenwood formula, which considersall the single-electron transitions across the Fermi surface, ameasure of the JDOS. 63It is not possible to reproduce the structure in the reflectance simply by considering the re-sponse of the dielectric function to the formation of a con-densate; instead, the kink is associated with a DOS effect dueto the formation of a superconducting gap. From the JDOS,the position of the kink should correspond to twice the gapmaximum. A reasonable estimate for twice the gap maximumis therefore 2 /H9004 0/H1122970 cm−1,o r/H90040/H112294.3 meV; this yields a ratio of 2 /H90040/kBTc/H112295, in good agreement with previous thin film36and single crystal results.64Furthermore, the value for /H90040is in excellent agreement with the values determined from Raman studies.18,19 The nature of the superconducting energy gap also plays a critical role in our ability to observe it. Gaps with sharp FIG. 3. /H20849Color online /H20850The log-log plot of the temperature de- pendence of the modulus of the optical conductivity over a widefrequency range above and below T c. Throughout the mid-infrared region /H20849between the square and the diamond /H20850, the modulus obeys a power-law behavior, /H20841/H9268˜/H20849/H9275/H20850/H20841/H11008/H9275−0.7. At low temperature the power- law behavior is recovered again at low frequency /H20849between the square and the dot /H20850,/H20841/H9268˜/H20849/H9275/H20850/H20841/H11008/H9275−0.8./H20851From the relation /H9268˜/H20849/H9275/H20850 =/H92681/H20849/H9275/H20850+i/H92682/H20849/H9275/H20850=−i/H9275/H20851/H9280˜/H20849/H9275/H20850−/H9280/H11009/H20852/4/H9266, a value of /H9280/H11009=3.6 was used to determine /H92682/H20849/H9275/H20850./H20852 FIG. 4. /H20849Color online /H20850The temperature dependence of/H20881−/H92752/H92801/H20849/H9275/H20850of Pr 1.85Ce0.15CuO 4at 30 and 6 K. The results of the low-frequency extrapolations employed in the Kramers-Kroniganalysis below /H1122918 cm −1/H20849dashed line /H20850are included as a guide to the eye.HOMES et al. PHYSICAL REVIEW B 74, 214515 /H208492006 /H20850 214515-4features in the DOS, and therefore the JDOS, should have features that are more easily observed in the optical proper-ties. To illustrate this point, gap functions for isotropics-wave, monotonic and nonmonotonic d-wave materials are shown in Fig. 6/H20849a/H20850and/H9004/H20849 /H9278/H20850//H90040is shown over the first quad- rant in Fig. 6/H20849b/H20850. The nonmonotonic d-wave gap will have a gap maximum much closer to the nodes, resulting in a largerpart of the Fermi surface that is effectively gapped. In com-parison, the monotonic d-wave gap associated with the hole- doped cuprates has a gap maximum far from the nodal re-gions and the resulting JDOS is rather smeared out. Wepropose that the nonmonotonic nature of the d-wave gap in the electron-doped material makes it possible to identify /H9004 0 from the optical properties. However, this is by no means restricted to the electron-doped materials. There are largechanges observed in the reflectance of the hole-doped cu-prates below T c, the so-called “knee” in the reflectance65 located at roughly 2 /H90040, that may be due to DOS effects re- lated to the formation of a superconducting energy gap.66In those cases where this feature was observed above Tc,i tw a s argued that it was not related to the superconductivity. How-ever, many of the cuprate superconductors initially studiedwere naturally underdoped; these materials display apseudogap that develops in the normal state. 7Angle-resolved photoemission spectroscopy has demonstrated that thepseudogap entails a partial gapping of the Fermi surface in amanner similar to that of the superconducting energy gap. 67 In addition, the calculations of the optical conductivity basedon a monotonic d-wave gap are in excellent agreement with the experiment. 68Thus, the appearance of the knee in the reflectance above Tcin the underdoped materials does not rule out the association of this feature with the superconduct-ing energy gap for T/H11270T c. While it is therefore possible to observe the DOS effects of the superconducting gap in materials with a simple mono-tonic d-wave gap, we argue that this task is simplified con- siderably if the gap is nonmonotonic. To elaborate on thispoint, we calculate the temperature dependence of the reflec-tance of a material using the BCS model with an isotropics-wave gap for an arbitrary purity level. 69The normal state is described using the Drude model with a plasma frequency of /H9275pd=13 000 cm−1and scattering rate 1/ /H9270=80 cm−1, while below Tcthe optical properties have been calculated with a gap of 2 /H9004=70 cm−1. The calculated reflectance curves are shown in Fig. 5/H20849b/H20850. The normal-state reflectance at Tcis re- produced quite well, while below Tcthe formation of an isotropic s-wave gap produces a region of steadily increasing reflectance for /H9275/H113512/H90040; for T/H11270Tcthe gap is fully formed and the reflectance is unity below 2 /H90040. While it is clear that the gap in Pr 1.85Ce0.15CuO 4isdwave, the nonmonotonic nature of the gap results in more of the Fermi surface beingmore effectively gapped than the monotonic case /H20849reminis- cent of an isotropic gap /H20850, resulting in a JDOS which allows for the unambiguous determination of 2 /H9004 0. C. Anisotropy and the effects of disorder In terms of the optically determined values for 1/ /H9270and 2/H90040, the clean and dirty limits are defined as 1/ /H9270/H112702/H90040and 1//H9270/H114072/H90040, respectively /H20849where it is understood that “dirty” refers to the effects of disorder and electronic correlationsrather than impurity effects /H20850. Although the cuprates are gen- erally considered to be in the clean limit, we now are facedwith the condition that 1/ /H9270/H110152/H90040, which places the material close to the dirty limit. The widely accepted statement thatthis class of materials is in the clean limit is based on theincorrect comparison of the small value for the quasiparticle FIG. 5. /H20849Color online /H20850/H20849a/H20850The far-infrared reflectance of Pr1.85Ce0.15CuO 4above and below Tc. The kink in the reflectance below Tcsignals the formation of a superconducting energy gap and denotes the value of 2 /H90040. The estimated noise in the reflectance is indicated by the thick line at 50 cm−1; the signal-to-noise ratio is in excess of 2000:1, and no smoothing has been applied to the data./H20849b/H20850The calculated reflectance of a BCS superconductor with an isotropic s-wave gap for a series of temperatures at and below T c. FIG. 6. /H20849Color online /H20850/H20849a/H20850A radial plot of the amplitude of an isotropic s-wave gap /H20849dotted line /H20850, a monotonic d-wave gap, /H9004/H20849/H9278/H20850 =/H90040cos/H208492/H9278/H20850/H20849dashed line /H20850, and a nonmonotonic d-wave gap, /H9004/H20849/H9278/H20850=/H90040/H20851cos/H208492/H9278/H20850−0.42 cos /H208496/H9278/H20850+0.17 cos /H2084910/H9278/H20850/H20852. Note that the gap functions are rotated by 45° with respect to the hole-dopedcuprates. /H20849b/H20850A linear plot of the same gap functions over the first quadrant.OPTICAL DETERMINATION OF THE SUPERCONDUCTING … PHYSICAL REVIEW B 74, 214515 /H208492006 /H20850 214515-5scattering along the nodal direction and the gap value along the antinodal direction, two different directions in momen-tum space. This statement does not take into considerationthe anisotropic nature of the Fermi surface of these materialsin which both 1/ /H9270and the superconducting energy gap vary significantly, depending on whether the nodal or antinodaldirections are being considered. For instance, the quasiparti-cle scattering rate is observed to drop abruptly in the cupratesbelow T cafter the antinodal region of the Fermi surface is gapped, suggesting that the scattering rate in the nodal direc-tion is much smaller than the antinodal direction. 70,71/H20849In the underdoped cuprates the formation of a pseudogap leads tomuch the same behavior in the normal state; the gapping ofthe antinodal regions restricts the quasiparticles to the nodalpart of the Fermi surface, where they display a metalliccharacter, 66,72i.e., a “nodal metal.” /H20850In the absence of a pseudogap, Matthiesen’s rule implies that the 1/ /H9270observed in the normal state therefore arises from scattering mainly inthe antinodal direction, and that as a consequence 1/ /H9270is anisotropic. In addition, the superconducting energy gap ishighly anisotropic due to its d-wave nature, with /H9004 0in this work estimated to be /H112294.3 meV. The comparison of the normal-state scattering rate with the superconducting energygap maximum correctly compares two quantities associatedwith the antinodal direction. In terms of the mean free path land the superconducting coherence length /H92640, the clean and dirty limits are expressed asl/H11271/H92640andl/H33355/H92640, respectively. Allowing that the mean free path is simply the Fermi velocity times the scattering time,l= vF/H9270, and that the coherence length for an isotropic gap in the weak-coupling regime is /H92640/H11229vF//H20849/H2088112/H9004/H20850,73then the statement that 1/ /H9270/H110152/H90040andl/H11015/H92640are roughly equivalent. In a previous study of Nd 1.85Ce0.15CuO 4, it was determined that 1//H9270/H11015100 cm−1just above Tc.30Using an average value for the Fermi velocity of vF=2.2/H11003107cm/s /H20849Ref. 74/H20850yields l /H11015730 Å. Given that /H92640/H1122980–90 Å in the electron-doped materials,75,76this appeared to justify the statement that this material was in the clean limit, contradicting the present re-sult. However, in addition to 1/ /H9270, the anisotropy of the Fermi velocity is also well documented in the high-temperature su-perconductors, and is found to vary from vF/H110152.5–2.7 /H11003107cm/s along the nodal direction,77,78tovF/H110150.5 /H11003107cm/s along the antinodal direction;79while these val- ues appear to be remarkably universal, some sample anddoping dependence is expected. Thus, the mean free pathdetermined from 1/ /H9270should be based on the antinodal vF; this yields a significantly smaller value of l/H11229250 Å. Based on the anisotropy of the Fermi velocity alone, the mean freepath along the nodal direction will be considerably larger.The estimated value /H9004 0in the present work is /H112294.3 meV; using the value for the antinodal Fermi velocity yields /H92640,calc/H11229165 Å, which is about twice as large as the com- monly quoted experimental values of /H92640=80–90 Å; the lack of perfect agreement may be partially due to the uncertaintyin vF, but it is more likely a result of the naïve approach taken to calculate /H92640. In the nodal direction, the gap vanishes and the coherence length diverges. These results are summa-rized in Table I.What these calculations indicate is that the assertion that l/H11271 /H92640arises only if the mean free path along the nodal direc- tion is compared with the coherence length in the antinodaldirection; if the nodal and antinodal directions are consideredas separate cases, then it is indeed the case that l/H11015 /H92640. Thus, the result that 1/ /H9270/H110152/H90040is in fact consistent with the state- ment that l/H11015/H92640, where it is understood that we are referring to the antinodal direction. In fact, the statement 1/ /H9270/H110152/H90040 should be considered more robust because it does not rely on vF. This implies that the material is not in the clean limit. Note that this statement should be qualitatively correct alongthe nodal direction as well, but because precise values of l and /H92640/H20851or 1//H9270and/H9004/H20849/H9278/H20850/H20852are difficult to determine, this statement is somewhat speculative. IV. CONCLUSIONS The ab-plane optical properties of single crystal Pr1.85Ce0.15CuO 4/H20849Tc/H1122920 K /H20850have been examined above and below Tc. In the normal state just above Tc, the coherent part of the optical conductivity may be described by a simpleDrude component with /H9275pd/H1122913 000 cm−1and 1/ /H9270 /H1122980 cm−1. It is noted that the condition /H6036//H9270/H110152kBTnear Tc observed in this material is generally true for many other cuprate superconductors. Below Tc, the superconducting plasma frequency is estimated to be /H9275ps/H112297800 cm−1, yield- ing a penetration depth of /H9261/H112292000 Å; when combined with the optical estimate for the dc resistivity /H9268dcjust above Tc, this material falls on the scaling relation /H9267s/H11008/H9268dcTcrecently proposed for the cuprate superconductors.38,39The estimate for the superconducting gap maximum 2 /H90040/H1122970 cm−1is in good agreement with previous results,18,36and is consistent with the view that the superconducting energy gap is mostlikely nonmonotonic dwave. The result that 1/ /H9270/H110152/H90040im- plies that this material is not in the clean limit, a self-consistent result that by its own nature allows for the directobservation of the superconducting energy gap. ACKNOWLEDGMENTS We would like to acknowledge helpful discussions with D. N. Basov, A. V . Chubukov, A. J. Millis, T. Valla, and N. L.Wang. This work was supported by the Office of Science,U.S. Department of Energy, under Contract No. DE-AC02-98CH10886; work in Maryland is supported by the NSF con-tract DMR-0352735.TABLE I. The estimated values at the nodal and antinodal points of the Fermi surface for the Fermi velocity, scattering rateand mean-free path /H20849T/H11407T c/H20850; the magnitude of the superconducting energy gap and the coherence length /H20849T/H11270Tc/H20850. RegionvFa /H20849107cm/s /H208501//H9270 /H20849cm−1/H20850l /H20849Å/H20850/H9004/H20849/H9278/H20850 /H20849meV /H20850/H92640,calcb /H20849Å/H20850 Nodal 2.5 /H1102180 /H110221000 →0→/H11009 Antinodal 0.5 80 250 4.3 165 aThe experimental error is /H1101120–30%. bThe coherence length is calculated from /H92640=vF//H20849/H2088112/H9004/H20850.HOMES et al. 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PhysRevB.70.045312.pdf

Anomalous temperature dependence of electrical transport in quantum Hall multilayers H. A. Walling,1D. P. Dougherty,1D. P. Druist,1E. G. Gwinn,1K. D. Maranowski,2and A. C. Gossard2 1Physics Department, University of California, Santa Barbara, California 93106, USA 2Materials Department, University of California, Santa Barbara, California 93106, USA (Received 16 December 2003; revised manuscript received 19 March 2004; published 20 July 2004 ) We study the temperature dependence of vertical transport through the chiral sheath of surface states that exists near the sidewalls of GaAs/Al 0.01Ga0.09As multilayer structures in the regime of the integer quantum Hall effect. Because variable-range hopping through the bulk provides a parallel conduction channel, wedesign our experiment to extend the temperature range of sheath-dominated transport. To do so, we increasedevice perimeter by using fractal-perimeter mesas. We report on the nearly linear increase of the sheathconductivity with temperature, a result not predicted by existing theories for the edge state sheath. DOI: 10.1103/PhysRevB.70.045312 PACS number (s): 73.21.Ac, 73.43. 2f A two-dimensional (2D), chiral sheath of surface states exists near the sidewalls of multilayer semiconductor mesas intheregimeoftheintegerquantumHalleffect.1Fermi-level states within the bulk of the multilayer are localized at lowtemperatures, so transport perpendicular to the plane of thelayers occurs primarily via the edge states of the quantumwells. As electrons tunnel between layers, the edge statescouple weakly to form the surface sheath [Fig. 1 (a)]. Trans- port on the sheath is ballistic in the plane of the quantumwells, diffusive in the perpendicular direction, 2and chiral because electrons circle the mesa in one direction only. Anelectron circling the mesa cannot backscatter, so if the mesaperimeter is large enough to preclude coherent transportaround the perimeter, the system will not localize. 3This sup- pression of localization makes the chiral sheath an interest-ing experimental test bed for understanding the different fac-tors that influence electrical transport in low-dimensionalsystems. Here, we use an MBE-grown GaAs/AlGaAs multilayer structure that we patterned into optimized geometries tostudy the temperature sTddependence of the vertical conduc- tivity, ssheath, of this unusual 2D system.4Because the mesas we study have large perimeters, the observed temperaturedependence should not exhibit localization physics, but in-stead will reflect disorder, inelastic, and interaction effects inthis chiral tunneling system. 5,6Although edge states in the integer quantum Hall effect are well-studied and are thoughtto have simple physics, 7–10the behavior we observe in ssheathsTdis unexpected and points to inelastic or interaction effects that have not been considered previously. Our results are of general interest as a complementary probe of transportphysics to the more heavily studied isotropic 2D systems. Early work on GaAs/A1GaAs multilayers established that in-plane transport measurements in the quantum Hall(QH)regime yield results qualitatively similar to the QH effect in a single 2D system: plateaus in the Hall resistancesR Hdaccompany vanishing longitudinal resistance.11Quan- tum Hall states in vertical transport measurements are char- acterized by minima in the vertical conductance, Gzz, that correspond to the in-plane plateaus in RH.1Size scaling ex- periments on vertical transport mesas showed that at low Tin QH states, Gzzis proportional to the mesa perimeter, P.A thigher temperatures, transport through the bulk dominates andGzzis proportional to the sample area, A.1 At any nonzero temperature, parallel transport through the bulk contributes to the total vertical conductance. We there-fore model the measured conductance, G zz, as the sum of a sheath conductance, Gsheath=sP/Hdssheath, and a bulk con- ductance, Gbulk=sA/Hdsbulk, whereHis the height of the multilayer. Because the sheath conductivity, ssheath, has a weak temperature dependence compared to the bulk conduc-tivity, sbulk, we must take care to distinguish the temperature dependence of ssheathfrom that of sbulk. We design our ex- periment to maximize the temperature range of sheath-dominated transport, while remaining in the incoherent limitso that localization effects are negligible. FIG. 1. (a)Schematic of samples used to study sheath transport. (b)–(f)Top-view micrographs of samples show roughly 1/4 of the sample area. Light areas are the gold-covered mesas. Dark areas arethe etched wafer. In sample 3, light-colored features are fractal-shaped holes where the contact metal did not lift off before themesa was etched. These areas are therefore not etched and do notaffect the total sample perimeter or area. Dark, square-shaped fea-tures in samples 2 and 4 are holes etched into the fractal-shapedmesas. These regions decrease overall sample area and increaseperimeter. In all samples, there are areas (darker )where the mesa etch removed some of the contact gold near the sample edges. (g) All samples have 1- mm minimum features.PHYSICAL REVIEW B 70, 045312 (2004 ) 0163-1829/2004/70 (4)/045312 (5)/$22.50 ©2004 The American Physical Society 70045312-1Earlier work used a different approach to estimate ssheathsTdin a similar multilayer structure, and did not ana- lyze the low- Tlimit that is the focus of this work.4This group fabricated two mesas with the same area, but differentperimeters. They assumed the high- Tbulk contributions would be identical, and thus interpreted the difference of thetwo data sets as ssheathsTd. This work assumed negligible variation in materials properties across the semiconductor wafers. We do not make this assumption, and instead exploitthe geometry of our mesas to maximize sensitivity to ssheath, for temperatures between 50 and 300 mK. This method al-lows us to characterize G zzin the sheath-dominated transport regime without complicating the analysis with a subtractionprocedure. A preliminary report on part of the data has ap-peared elsewhere. 12 Our strategy for optimizing sensitivity to sheath proper- ties is to increase Pby using mesas with fractal perimeters. Figures 1 (b)–1(f)show a top-view photograph of one quad- rant of each of the five mesas studied. Fractals 1, 2, and 3have identical outer perimeters of fractal dimension d=1.5. Samples 2 and 3 have holes removed from their interiors,resulting in larger total perimeters and smaller areas thansample 1. Samples 4 and 5 have identical outer perimeterswithd=1.67, but different total PandA.Table I gives total P andAfor the five samples. The minimum feature size for all of the fractals is 1 mm[Fig. 1 (g), a blow-up of one region, displays the quality of the lithography ]. A different way to maximize Prelative to Ais to make samples with small P.The problem with this approach is that localization effects are expected in samples small enoughthat electrons can circumnavigate the perimeter phasecoherently. 13,14Our fractals have large enough P(to ,2m m )that such circumnavigation is highly unlikely. Since we observe similar behavior in samples with Pspanning a factor of 4, localization effects from phase-coherent wrap-ping paths appear negligible at the size scales studied. We fabricated all samples from a multilayer structure with 160 periods of 150 Å GaAs quantum wells alternating with150 Å Al 0.1Ga0.9As barriers. Thus the multilayer height, H, of all mesas is 4.79 mm. The barriers are Si-doped at their centers to give the quantum wells a sheet density of 3.4310 11cm−2, as extracted from in-plane transport experi- ments on a companion structure. In the vertical transportsamples, there is a layer of degenerately doped n+GaAs above and below the top and bottom Al 0.1Ga0.9As layers, to which we made Ohmic electrical contacts using alloyedNiAuGe. We defined the fractal mesas with e-beam lithogra- phy and deposited the contact metal to act as both an etchmask and Ohmic top contacts. To give the mesas verticalwalls we dry etched the sample in a reactive ion etcher, usingSiCl 4. We used photolithography to define the bottom con- tacts for lift-off, and deposited NiAuGe. We alloyed the topand bottom contacts for 1 min at 430°C in a rapid thermalannealer. Finally, we deposited a thick layer of Ti/Au toallow wire bonding. We relied on the Schottky barrier inGaAs not to short the layers together. We conducted our experiment twice, thermally cycling the samples between runs. We measured samples 1, 3, and 5over a temperature range from 50 mK to 2 K during the firstrun, and 2, 4, and 5 from 50 mK to 1.3 K during the second.We repeated measurements of sample 5 to observe the effectsof thermal cycling on our results. We used small excitationcurrents to measure the vertical conductance, G zz, of the me- sas at dilution refrigerator temperatures, taking care to ensurelinear IV characteristics at 50 mK. Measurement signals cor-responded to ,10 −15W at 50 mK. We applied a magnetic field perpendicular to the layers and swept it slowly from0–17.9 T to locate the quantum Hall states.At low tempera-tures, we found G zz~Pwithin QH states; at high tempera- tures,Gzz~Afor all fields. The temperature scale for the crossover from bulk-dominated sGzz~Adto sheath- dominated sGzz~Pdtransport depends on the sample geom- etry, but is on the order of 500 mK at the center of the v =2 per layer quantum Hall state, which was well-defined for all of the mesas. We set the magnet to the center of this state,B=6.75 T, and proceeded to sweep the temperature from 50 mK to 2 K. See Fig. 1 of Ref. 12 for a magnetic fieldsweep between 0 and 17 T that shows the center of the v =2 QH state. Here, we use vto denote the number of filled Landau bands below the Fermi energy rather than the fillingfactor. To display the quality of size scaling at low temperatures, Fig. 2 (a)shows a log-log plot of G zzversus sample perim- eter,P, for samples 1–5 at T=100 mK. The line with slope 1 on this log-log plot shows good overall agreement withG zz~P. For comparison, the inset in Fig. 2 (a)showsGzz versus sample area, A, at 100 mK. Comparison to the line with slope 1 in the inset shows that Gzzdoes not scale with A in the limit of low temperatures. To display the quality ofsize scaling at high temperatures, Fig. 2 (b)shows a log-log plot ofG zzversus sample area, A, for the five samples at T =1.34 K. The line with slope 1 on this log-log plot showsgood overall agreement with G zz~Aat 1.34 K. The inset in Fig. 2 (b)shows a log-log plot of GzzversusPat 1.34 K. The line with slope 1 in the inset shows that at this high tempera-ture,G zzis not proportional to P. Figures 3 (a)and 3 (b)showGzzas a function of tempera- ture for samples 1, 3 and 5 (2 and 4 ). For all five samples Gzz is weakly temperature dependent below ,500 mK and rises rapidly at higher temperatures, where bulk transport domi-nates. Figure 4 plots the low- Tsheath conductivity, ssheath, estimated from the low- TGzzusing ssheath=HGzz/P, where His the sample height and Pis the perimeter. As shown in Fig. 4, for all samples ssheathrises roughly linearly at low T, with similar slopes. The standard deviation of the intercept is ,4%. We believe that this spread is due toTABLE I. Total perimeters, areas, and afor the samples studies. Sample number Perimeter smmdArea smm2d a 1 16 384±650 65 536 1.21±0.14 2 18 000±720 40 000 0.89±0.423 22 272±890 59 648 1.26±0.324 4 146±170 4 046 0.99±0.105 4 096±170 4 096 1.01±0.385(second time )4 096±170 4 096 1.31±0.21WALLING et al. PHYSICAL REVIEW B 70, 045312 (2004 ) 045312-2a combination of factors, including an approximately 4% uncertainty in the sample perimeters, based upon high-magnification photographs of our samples. These imagesshow slight rounding at the sample corners that accounts for,1% uncertainty. In addition, the long mesa etch removed the contact metal from some regions of the sample edges, asshown by the small, darker gray areas at the mesa edges inFig. 1 (b). This likely caused some degree of sample erosion in the affected regions. We estimate that this effect adds amaximum of 4% uncertainty in P. Uncertainty in the sample area due to such perimeter imperfections or to errors in fieldstitching during electron-beam lithography is negligible.Gradients over the wafer during MBE growth that slightlychange barrier and well thicknesses provide an additionalsource of variability between samples. Because the align-ment of edge states between layers affects the strength oftunneling, mesa sidewalls that are not perfectly flat will pro-duce stronger tunneling at points where the edge states over-lap. Thus, variations in flatness of the sidewall profile be-tween mesas may also contribute to the spread in ssheath. To be sure that the observed weak temperature depen- dence in Gzzbelow ,500 mK was not the result of poor thermal contact between the sample and the mixing chamberof the dilution refrigerator, we used the mesa with the largestareaAtoPratio sA/P=4 mmd, sample 1, as a thermometer. Because the n=4 QH state becomes developed at lower tem- peratures than for n=2, we expected transport to be domi- nated by the bulk in the n=4 quantum Hall state s3.55 T dat FIG. 2. (a)Log-log plot of Gzzversus sample perimeter sPdfor samples 1–5 at T=100 mK. The line with slope=1 indicates that Gzz~Pat low temperatures. The inset shows a log-log plot of Gzzversus sample area at T=100 mK. The solid line with slope=1 shows that at low temperatures, Gzzis not proportional to A.(b) Log-log plot of Gzzversus sample area sAdfor samples 1–5 at T =1.34 K. The solid line with slope=1 shows that at high tempera-tures,G zz~A. The inset shows a log-log plot of Gzzversus sample perimeter at T=1.34 K. The solid line with slope=1 shows that at high temperatures, Gzzis not proportional to P. FIG. 3. (a)Gzzversus temperature at 6.75 T for fractals 1, 3, and 5. (b)Gzzversus temperature at 6.75 T for fractals 2 and 4. Gzzis weakly T dependent below ,500 mK and then rises rap- idly with temperature. FIG. 4. Vertical sheath conductivity, ssheath=HGzz/P, for all samples, in units of e2/h. The sheath conductivity initially rises approximately linearly with temperature. The solid lines are fits toG zz=G0+K1Tabetween 50–200 mK. The avalues are given in Table I and the text.ANOMALOUS TEMPERATURE DEPENDENCE OF PHYSICAL REVIEW B 70, 045312 (2004 ) 045312-3all but the lowest temperatures, with corresponding strong temperature dependence in Gzzdue to the dominance of hop- ping transport through the bulk. This was the case from 2 Kto,100 mK: G zzfollowed a variable-range hopping (VRH ) temperature dependence.At lower temperatures Gzzfell more slowly, and showed a crossover in size scaling from Gzz~A toGzz~P, indicating that the bulk contribution had become negligible. To illustrate this bulk-to-sheath crossover, Fig. 5shows the ratio of G zzfor sample 1 to Gzzfor sample 3 in the center of the n=4 QH state s3.55 T d. We expected this ratio to equal the ratio of the two samples’perimeters only at very low temperatures.The solid horizontal line shows the ratio ofthe area of sample 1 to the area of sample 3, and the dashedhorizontal line shows the ratio of the perimeter of sample 1to the perimeter of sample 3.The data approach the solid lineat high temperatures and the dashed line at low temperatures,indicating a crossover to sheath-dominated transport at tem-peratures less than 100 mK. We therefore concluded that theelectrons reached temperatures below 100 mK. Thus, we areconfident that the data’s behavior at low temperatures is nota result of poor equilibration with the mixing chamber. As an initial investigation of the temperature dependence of sheath conduction, we fit the data to G zz=Gsheath+Gbulk withGsheath=G0+K1Ta, whereG0is the zero-temperature sheath conductance.To account for parallel transport throughthe bulk, we used a VRH form that fits the bulk well at highT,G bulk=K2Tbexpf−sT0/Tdgg, with g=1/2forthe bulk hop- ping exponent. Figure 6 shows that the observed weakly temperature- dependent behavior that we find at low temperatures is char-acteristic of the edge state sheath and not a remnant contri-bution from bulk hopping. The low- Tdata points are for sample 1 and the dashed line is G 0+Gbulk, withGbulkthe VRH form fitted to the high- Tdata. As shown, the variationinGbulkis negligible relative to the observed temperature dependence for temperatures below ,300 mK. We have tested that the value of the bulk hopping exponent gdoes not strongly affect fits to the bulk conductance: values of g =1,1/3,and1/4all give a fitted Gbulkthat makes negligible contribution to Gzzbelow ,300 mK. After we characterized the bulk and determined that the data’s weak temperature dependence at low temperatureswas not simply a remnant bulk contribution, we assumedG zz>Gsheathand fit the low- Tdata for all five fractals to Gzz=G0+K1Ta. Fits from 50–200 mK, a range over which the bulk contribution is negligible, give a=1.11±0.17 as the average and standard deviation over all samples. Table I liststhe values of afor each sample and the corresponding fit uncertainties. The fits to the low- Tdata show that ssheathrises roughly linearly with temperature. Because the fitted exponent ais slightly greater than 1, and because the data have an upwardcurvature at temperatures at which we expect negligible bulkcontribution, the sheath conductance is perhaps a power se-ries with the linear term dominating at the lowest tempera-tures accessed by our experiment. To observe the effects of thermal cycling on our measure- ments, we compared G zzsTdfor fractal 5 for both data sets. We found good agreement at low temperatures. Fitting the low-Tdata (to 300 mK )to a straight line yields slopes and 0-K intercepts that agree within 2%, indicating a high de-gree of reproducibility of sheath conductivity with thermalcycling. High- Ttransport exhibits larger changes with ther- mal cycling, with bulk hopping slightly stronger for the sec-ond data set than the first. The boundaries of the QH stateschange slightly between thermal cycles, so this sensitivity ofbulk transport to thermal cycling may reflect correspondingchanges in the bulk localization length, 15to which hopping transport is quite sensitive. Although we concentrated our studies on the center of the n=2 QH state s6.75 T d, we also studied ssheathsTdat off- center magnetic field values, at B=6.25 T and B=7.0 T. At 7.0 T,GzzsTdclosely resembles GzzsTdat 6.75 T, but has a slightly smaller low- Tslope. The 6.25-T data show that at this field the bulk contribution grows much faster than at FIG. 5. Symbols show the ratio of Gzzfor sample 1 to Gzzfor sample 3.At high temperatures this ratio approaches the solid hori-zontal line, which shows the ratio of the area of sample 1 to the areaof sample 3.At temperatures below 100 mK, the data approach thedashed horizontal line, which shows the ratio of the perimeter ofsample 1 to the perimeter of sample 3.This crossover in size scalingindicates a crossover to sheath-dominated transport at lowtemperatures. FIG. 6. Variable-range hopping form as a function of tempera- ture compared to experimental data (symbols ). The dashed line has g=1/2.TheVRH form has a much weaker temperature dependence at low temperatures than the data.WALLING et al. PHYSICAL REVIEW B 70, 045312 (2004 ) 045312-46.75 and 7 T, presumably reflecting an enhancement of the bulk localization length at fields closer to the transition be-tween QH states. We note that other groups 16,17have studied the breakdown of the QHE in the regime of nonlinear, in-plane transport in single 2-dimensional electron gas. Becausewe take measurements well within the regime of linear re-sponse, and because our vertical transport experiments donot give rise to a transverse Hall voltage, we believe that themore rapid rise in conductance that we observe away frominteger vis not related to the breakdown of the QHE that these groups studied. The nearly linear increase in ssheathwith temperature is surprising. Barely metallic, three-dimensional systems haveshown similar temperature dependence in the vicinity of ametal-insulator transition; 18however, the nature of electron trajectories in such systems is quite different from chiral flowon the sheath, and we see no reason to expect similar depen- dence on temperature. Theory for the chiral sheath predictsthat interactions in weakly coupled quantum wells give ssheath=s0+KT2, withKnegative.14Apparently factors not included in present theory are important in transport on thesurface sheath. To attempt to understand our results, we used a simple, noninteracting model to investigate the temperature depen-dence of the tunneling conductance, G, through a barrier with energy-dependent transmission TsEd. In such a system G~eTsEdgsEds− ]f/]EddE, wherefis the Fermi-Dirac func- tion. We assumed gsEd, the density of states, to be energy independent, and approximated TsEdusing the WKB method for our quantum-well barriers to calculate the integral. The result was a calculated increase in Gzzof 7310−4% between 50 and 300 mK. Our experimentally observed Gzzincreases by,10% over the same temperature range. We conclude that thermal broadening effects on tunneling are too weak toexplain our much stronger observed temperature depen-dence. We also considered inelastic effects that disorder could produce in interlayer tunneling in our system. Impurity po-tentials will cause the edge states to meander relative to oneanother, producing maxima in the tunneling rate in regionswhere edge states overlie. This meandering of edges pro- duces a shift in the dispersion relations in adjacent layers. Ifwe consider two straight sections where the edges of adja-cent layers are separated by s, their dispersion relations ac- quire a relative momentum shift qssd=s/,m 2, where ,mis the magnetic length.As a result, momentum and energy can- not be conserved simultaneously. To tunnel between layersand conserve momentum, electrons must gain energy fromsome source: scattering from phonons or from other elec-trons. Qualitatively, we expect this inelastic scattering wouldcause ssheathto rise with increasing temperature, though we do not know its functional form. We note that electron-phonon interactions are predicted to destroy the Fermi-liquidstate at integer filling in single 2DEG systems, 19and thus may have important effects here. Alternatively, electron-electron interaction effects, such as scattering from collective excitations or energy-dependent density of states (DOS )atEF, could perhaps result in a linear increase in ssheathwith temperature over the range observed in the data. Energy-dependent DOS commonly arises in dif-fusive systems, though no estimates of similar effects on theconductivity of chiral systems are available. In summary, low-temperature transport perpendicular to the layers of a multilayer quantum Hall system showed anunexpected temperature dependence. In the absence of in-elastic and interaction effects, the vertical conductivity of thesheath, ssheath, would be independent of temperature. Theory that assumes a constant density of states predicts interactioneffects to produce a quadratic fall in ssheathwith temperature. Instead, we observe a linear increase in ssheathat low tem- peratures, with a slope that is too large to explain by thermalbroadening. We thank R. Sedgewick for writing code to generate the fractal images that we used to fabricate our mesas. We thankJohn Chalker, Matthew Fisher, and Leon Balents for helpfulconversations, and Ernie Caine for help with the electron-beam lithography. This work was supported by NSF-DMR9700767 and NSF-DMR 0071956. 1D. P. Druist, P. J. Turley, E. G. Gwinn, K. D. Maranowski, and A. C. Gossard, Phys. Rev. Lett. 80, 365 (1998 ). 2L. Balents and M. P.A. Fisher, Phys. Rev. Lett. 76, 2782 (1996 ). 3J. T. Chalker and A. Dohmen, Phys. Rev. Lett. 75, 4496 (1995 ). 4M. Kuraguchi and T. Osada, Physica E (Amsterdam )6, 594 (2000 ). 5J. T. Chalker and S. L. Sondhi, Phys. Rev. B 59, 4999 (1999 ). 6J. J. Betouras and J. T. Chalker, Phys. Rev. B 62, 10931 (2000 ). 7B. I. Halperin, Phys. Rev. B 25, 2185 (1982 ). 8A. H. Macdonald and P. Streda, Phys. Rev. B 29, 1616 (1984 ). 9B. E. Kane, D. C. Tsui, and G. Weimann, Phys. Rev. Lett. 59, 1353 (1987 ). 10R. J. Haug, Semicond. Sci. Technol. 8, 131 (1993 ). 11H. L. Stormer, J. P. Eisenstein, A. C. Gossard, W. Wiegmann, and K. Baldwin, Phys. Rev. Lett. 56,8 5 (1986 ).12H. A. Walling, D. P. Dougherty, D. P. Druist, E. G. Gwinn, K. D. Maranowski, and A. C. Gossard, Physica E (Amsterdam )12, 132(2002 ). 13I. A. Gruzberg, N. Read, and S. Sachdev, Phys. Rev. B 56, 13218 (1997 ). 14S. Cho, L. Balents, and M. P. A. Fisher, Phys. Rev. B 56, 15814 (1997 ). 15M. Furlan, Phys. Rev. B 57, 14818 (1998 ), and references therein. 16G. Nachtwei, Physica E (Amsterdam )4,7 9 (1999 ). 17L. B. Rigal, D. K. Maude, M. Potemski, J. C. Portal, L. Eaves. Z. R. Wasilewski, G. Hill, and M. A. Pate, Phys. Rev. Lett. 82, 1249 (1999 ). 18M. A. Dubson and D. F. Holcomb, Phys. Rev. B 32, 1955 (1985 ). 19O. Heinonen and S. Eggert, Phys. Rev. Lett. 77, 358 (1996 ).ANOMALOUS TEMPERATURE DEPENDENCE OF PHYSICAL REVIEW B 70, 045312 (2004 ) 045312-5

PhysRevB.86.245112.pdf

PHYSICAL REVIEW B 86, 245112 (2012) Angle-dependent spectral weight transfer and evidence of a symmetry-broken in-plane charge response in Ca 1.9Na0.1CuO 2Cl2 R. Schuster,1S. Pyon,2,*M. Knupfer,1M. Azuma,3M. Takano,4H. Takagi,2,5,6and B. B ¨uchner1,7 1IFW Dresden, Institute for Solid State Research, P .O. Box 270116, D-01171 Dresden, Germany 2Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan 3Materials and Structures Lab., Tokyo Institute of Technology, Yokohama, Kanagawa 226-8503, Japan 4Institute for Integrated Cell-Material Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan 5Magnetic Materials Laboratory, RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan 6Inorganic Complex Electron Systems Research Team, RIKEN Advanced Science Institute, Wako, Saitama 351-0198, Japan 7Institute for Solid-State Physics, Department of Physics, TU Dresden, D-01062 Dresden, Germany (Received 21 September 2012; revised manuscript received 7 November 2012; published 12 December 2012) We report about the energy and momentum dependent charge response in Ca 1.9Na0.1CuO 2Cl2employing electron energy-loss spectroscopy. Along the diagonal of the Brillouin zone (BZ) we find a plasmonpeak—indicating the presence of metallic states in this momentum region—which emerges as a consequence ofsubstantial spectral-weight transfer from excitations across the charge-transfer (CT) gap and is the two-particlemanifestation of the small Fermi pocket or arc observed with photoemission in this part of the BZ. In contrast, thespectrum along the [100]direction is almost entirely dominated by CT excitations, reminiscent of the insulating parent compound. We argue that the observed polarization dependent shape of the spectrum is suggestive of abreaking of the underlying tetragonal lattice symmetry, possibly due to fluctuating nematic order in the chargechannel. In addition we find the plasmon bandwidth to be suppressed compared to optimally doped cuprates. DOI: 10.1103/PhysRevB.86.245112 PACS number(s): 71 .30.+h, 73.20.Mf, 79 .20.Uv I. INTRODUCTION Cuprates are still the paradigm for many topics in condensed matter physics that cannot be reconciled within the conven- tional Fermi liquid approach. At the heart of the researchon their electronic structure lies the question of how the insulating parent compounds develop into high-temperature superconductors and finally into ordinary metals upon doping. It is well established that, upon introducing charges into the CuO 2plane, cuprates undergo a metal-insulator transition (MIT) which is seen in various transport and spectroscopic experiments.1This MIT, however, appears to be rather peculiar because additional charges are believed to aggregate in partic- ular patterns termed stripes or checkerboards,2a phenomenon that is also observed in other transition-metal systems such asthe nickelates. 3In general, the charge order is accompanied by a corresponding structure in the spin sector.4In this respect the very recent reports about the existence of charge-only orderinginRBa 2Cu3O6+x(R=Y, N d )5,6are remarkable. The Ca 2−xNaxCuO 2Cl2system exhibits charge order in the form of bond-centered stripes or nematic behavior as revealedby scanning-tunneling microscopy (STM) experiments. 7–9 Naturally, the role of those inhomogeneities for the super- conductivity or more generally for the electronic properties ofunderdoped cuprates is under strong debate. In particular itis known that in the La-based families there is a significantsuppression of the superconducting transition temperature forx=0.125 where the stripe order is most robust. 4,10While the spin dynamics in this region of the phase diagram iswell established by inelastic neutron scattering (see, e.g.,Ref. 11) the corresponding behavior for excitations in the charge channel as a function of energy and momentum is, though of urgent interest for a complete characterization ofthe collective-mode spectrum, only scarcely explored so far.Inelastic electron scattering, also known as electron energy- loss spectroscopy (EELS) in transmission, is a well estab-lished and bulk-sensitive tool to investigate the energy andmomentum dependence of the charge response in solids, asits cross-section is directly proportional to Im[ −1//epsilon1(ω,q)]— with/epsilon1(ω,q) being the complex dielectric function of the sample—and therefore allows us to investigate the dynamicsof collective charge excitations. 12 Here we employ EELS to study the charge dynamics in the system Ca 1.9Na0.1CuO 2Cl2. We find an angular dependent transfer of spectral weight between the excitations acrossthe charge-transfer (CT) gap and those associated with themetallic state. The result of this is a “nodal metal” that isobserved as a truncated Fermi surface (FS) arc or small pocketin angle-resolved photoemission spectroscopy (ARPES) onunderdoped cuprates. As the observed intensity modulationsof the spectrum are not compatible with the underlyingtetragonal lattice symmetry, we argue that this spectral-weighttransfer appears to be accompanied by nematic order in thecharge channel. II. EXPERIMENTS AND RESULTS The single crystals of Ca 1.9Na0.1CuO 2Cl2obtained by a flux method under high pressure13were cut into thin ( d∼100 nm) films using an ultramicrotome under nitrogen atmosphere toavoid sample damage due to air exposure. In the spectrometerthe films were aligned in situ with electron diffraction showing the high quality of our samples and allowingfor polarization dependent investigations along well defineddirections within the CuO 2plane. The measurements were carried out using a dedicated transmission electron energy-lossspectrometer 14equipped with a helium flow cryostat employ- ing a primary electron energy of 172 keV and energy and 245112-1 1098-0121/2012/86(24)/245112(6) ©2012 American Physical SocietyR. SCHUSTER et al. PHYSICAL REVIEW B 86, 245112 (2012) FIG. 1. (Color online) The EELS intensity in Ca 1.9Na0.1CuO 2Cl2for momentum transfers parallel to [100](left panel) and [110](right panel) measured at room temperature. Note the intensity enhancement around 1 eV parallel to the [110]direction. All spectra have been normalized on the high-energy side between 3.5 and 4 eV. momentum resolutions of /Delta1E=80 meV and /Delta1q= 0.035 ˚A−1, respectively. In Fig. 1we present the behavior of the EELS intensity for Ca1.9Na0.1CuO 2Cl2in two high-symmetry directions within the CuO 2plane. The momentum evolution in the region of the CT peak around 2 eV was investigated in a previousreport 15and we therefore focus on the energy range around 1 eV in the following. As can be seen for momentum transfersparallel to the [110]direction there is a well pronounced peak in the tail of the zero-loss line which disperses tohigher energies upon leaving the center of the Brillouin zone(BZ). This peak is, however, strongly suppressed for q/bardbl[100] (along the copper-oxygen bonds). Note that this anisotropy is absent in the insulating Sr 2CuO 2Cl2,16in the strongly underdoped Ca 1.95Na0.05CuO 2Cl2,15and also in the optimally doped Bi 2Sr2CaCu 2O8+δ.17 To further quantify the 1 eV feature we present its momen- tum evolution in Fig. 2for both lattice directions. In order to obtain an unbiased estimate for the energetic position wedid not remove the quasielastic line. While the energy valuesforq/bardbl[110]shown in Fig. 2track the position of the peak maximum around 1 eV, this procedure could not be applied FIG. 2. (Color online) The momentum dependence of the 1 eV feature seen in Fig. 1. See text for details.to the spectra in the [100]direction as there the peak is hardly observable. Therefore we took the zero crossing of the secondderivative between 1 eV and 1 .5 eV as the characteristic feature for the spectra, and we present its momentum dependencein Fig. 2. Consequently the different onset energies may be considered as an artifact of the data evaluation. As can be seenfrom this analysis, in both directions the dispersion is positiveand has a bandwidth of about 200 meV. To gain a deeper understanding we measured the EELS intensity for different temperatures and angles within the CuO 2 plane. A summary of these results is presented in Fig. 3.T h e left panel shows an angular map measured over one quadrantof the BZ at room temperature with a constant momentum transfer of q=0.1˚A−1. Obviously, there is a significant angular range where the 1 eV feature gains substantialweight at the expense of the CT peak. Though not perfectlysymmetric, 18the portion of the BZ where the low-energy peak is most pronounced clearly corresponds to the so-called nodaldirection: the momentum region where the superconductinggap and also the pseudogap (PG) are known to approachzero from ARPES. 19The temperature (in)dependence of the discussed effect is summarized in the right panel of Fig. 3. Those curves have been obtained by measuring angular cutsat constant energy losses of 1 and 2 .4 eV. Subsequently, these two spectra were normalized by a similar one measured at 4 eVenergy loss. The angular independence of the EELS signal onthe high-energy side allows us to consider this third spectrumas a suitable normalization background. Note, however, thatthe observed asymmetric signal does not depend on the energyrange taken for the normalization. Clearly, the obtained patternshows a fourfold symmetry over the entire angular range ofthe BZ, with the maximum of the 1 eV peak always locatedaround the nodal directions (indicated by the vertical dashedbars), in agreement with the left panel of Fig. 3.F r o mt h i sw e conclude that the observed effect is intrinsic and is not causedby the sample preparation. Were this the case the preparationprocedure would produce a distinguished axis parallel to thecutting direction of the sample, the result being a pattern witha periodicity of πinstead of the π/2 period we observe. In addition, the intensity redistribution between the featuresaround 1 and 2 eV is surprisingly robust against temperaturevariations, as we do not observe any changes of the periodicity 245112-2ANGLE-DEPENDENT SPECTRAL WEIGHT TRANSFER AND ... PHYSICAL REVIEW B 86, 245112 (2012) FIG. 3. (Color online) Left panel: Angular (polarization) dependence of the EELS intensity for q=0.1˚A−1measured at room temperature normalized on the high-energy side between 4 and 5 eV. Right panel: Constant energy cuts for q=0.1˚A−1around the full BZ (see text for details). Note the fourfold symmetry. The dashed bars correspond to the nodal directions as derived from the crystal structure measured withelastic scattering. In both panels the angle is measured relative to the [100]direction. or the amplitude between the highest and lowest measured temperatures shown in Fig. 3(at least with the resolution accessible to us). The same holds true for all intermediatetemperature steps investigated. We emphasize that we do not find evidence for superstruc- ture reflections in the elastic scattering either at high or atlow temperatures. This agrees with earlier x-ray scatteringexperiments on this system. 20In addition the pattern seen in Fig.3changes quantitatively (see below) but not qualitatively when changing the momentum transfer between q=0.08˚A−1 andq∼0.4˚A−1and we do not see any indication for a resonance-like enhancement of the effect at the momentumtransfer of the checkerboard q=2π/4a 0≈0.41˚A−1(see Fig.4)7with the lattice constant a0=3.85˚A. In order to track the momentum evolution of the intensity anisotropy we plot the spectra along the [100]and [110] directions for a series of qvalues in Fig. 4. Already by visual inspection it is clear that the anisotropy persists but is notmonotonic as a function of q. As a more quantitative measure of the anisotropic EELS intensity we define the following ratio: R SW q=SW[110] q SW[100] q, (1) where the “spectral weights” SWi qare determined by SWi q=/integraltextω2 ω1dωωI (ω,q) with the normalized intensities I(ω,q)s h o w n in Fig. 4and the momentum transfer qparallel to the directions indicated by the superscript i∈{[100],[110]}. Note that if one converted the intensities I(ω,q) to the absolute value of Im[ −1//epsilon1(ω,q)] by means of a Kramers-Kronig (KK) transform, the integrals SWi qwould indeed correspond to the spectral weight (and also be constrained by the f-sum rule12) which motivates our nomenclature. We did not, however,perform the KK calculation, because, in particular for the[100]direction, the subtraction of the quasielastic line is highly ambiguous. The boundaries of the integration ω 1andω2 correspond to the dips in the EELS intensity below and above the 1 eV feature along [110], respectively (see the vertical arrows in Fig. 4). The evolution of the ratio defined in Eq. (1) as a function of momentum transfer is visualized in Fig. 5.A s can be seen, there is a rather well defined maximum around the incommensurate qc∼0.175–0 .2˚A−1. III. DISCUSSION Although there is experimental evidence that the infrared optical response in underdoped cuprates contains more thana simple Drude term, 21,22we attribute the intensity increase around 1 eV and in particular the well pronounced peak inthe[110]direction to the Drude plasmon caused by the doped charge carriers. This is motivated by the metallic resistivityand the clearly visible plasma edge seen in the reflectivity ofCa 1.9Na0.1CuO 2Cl2at this energy.21In addition earlier EELS reports on optimally doped cuprates, in particular from theBi and Y families, found similar features that have beeninterpreted analogously. 17,23–26In contrast to our observation from Fig. 2, however, all these experiments show a plasmon dispersion with a bandwidth that is larger than the one reportedhere by at least a factor of 2. As the slope of the plasmondispersion in a simple metal is proportional to the squared Fermi velocity v F∼n1 3with the charge-carrier density n,27 the smaller dispersion in Ca 1.9Na0.1CuO 2Cl2might simply reflect the smaller (compared to the optimally doped systems)FS seen in ARPES. 28This is also in line with the systematic blueshift of the plasma edge as a function of doping inCa 2−xNaxCuO 2Cl2.21 We now turn our attention to the obvious anisotropy in the charge response for different directions within the CuO 2plane. 245112-3R. SCHUSTER et al. PHYSICAL REVIEW B 86, 245112 (2012) FIG. 4. (Color online) The asymmetry of the EELS intensity for the two in-plane directions as a function of momentum. For each momentum value, the curves are normalized between 3.5 and 4 eV and the arrows indicate the integration ranges for the evaluation of Eq. (1). According to earlier reports13,21,29,30but also from our elastic scattering data (not shown) the crystal structure ofthe Ca 2−xNaxCuO 2Cl2system is perfectly tetragonal without any signs of buckling or orthorhombicity often found in othercuprate families. This implies that for q=0 the dielectric function—which is in the general case of anisotropic systemsa tensor—is diagonal and contains only one independent in-plane component and consequently should be isotropic throughout the entire CuO 2plane.31Therefore also 1 //epsilon1(ω,q), which we probe in EELS, should be describable by a singleisotropic component. Although we cannot measure at q≡0 due to the increasing influence of the zero-loss peak and theenhancement of surface scattering, 27in general the momentum transfer of q=0.1˚A−1, which corresponds to only about 6% of the BZ size, is small enough to be considered as theoptical limit ( q=0) (see also the isotropy for this values of momentum transfer in the response of other cuprates 15–17). Consequently, we attribute the behavior described above toan intrinsic (electronic) breaking of the underlying tetragonallattice symmetry, as our data imply strong polarization depen-dent changes of the charge response, equivalent to two in-planecomponents of the dielectric tensor even in the limit q→0. This is compatible with the STM data 8,9where evidence is found for a local breaking of the symmetry from C4(tetragonal) toC2(orthorhombic), which can be ascribed to the presence of a nematic order parameter.2,32Note, however, that while the STM data implies a higher metallic character parallel to thebonds, our data seem to indicate an enhanced density of states along the diagonals of the unit cell (BZ). Our data thereforealso appears at odds with results based on Raman 33,34and neutron scattering,35where stripes are along the diagonal only for a doping too low to induce superconductivity. The observedπ/2-periodic signal (see Fig. 3) might be explained either by an intrinsically 2D charge modulation or the existence oforthogonal domains within the CuO 2plane. In any case, from our data we conclude that metallic character—indicated bythe presence of the plasmon peak—is preferably present alongthe diagonals of the unit cell whereas the spectra along theCu-O bonds are dominated by CT excitations reminiscent ofthe insulator. These domains are highly fluctuating but thetimescale of the electron-scattering process is short enoughthat EELS measures a snapshot of this dynamic behavior.At low enough temperatures these charge fluctuations maylock in to form a well ordered pattern, but due to thetransmission geometry of the experiment we are limited to T/greaterorequalslant 20 K. From the absence of superstructure reflections and therobustness of the tetragonal lattice symmetry, we conjecturethat Ca 1.9Na0.1CuO 2Cl2tends to show nematic order. Our data therefore also indicate that the regular checkerboardpattern seen in the STM 7may be pinned by the surface, in agreement with the findings of Ref. 36.F r o mF i g . 5we identify a characteristic momentum qc∼0.175–0 .2˚A−1of the nematic fluctuations which translates to a length scale ofl c=2π qc∼30–35 ˚A∼8a0–9a0. 245112-4ANGLE-DEPENDENT SPECTRAL WEIGHT TRANSFER AND ... PHYSICAL REVIEW B 86, 245112 (2012) FIG. 5. The momentum dependence of the ratio defined in Eq. (1). From the ARPES literature on underdoped Ca2−xNaxCuO 2Cl228,37and also other cuprate systems19 it is known that the large FS observed in the optimally and overdoped compounds is confined to a small angularrange—either in the form of an arc or closed pocket 38—around the nodal direction which increases with doping. From ourdata, in particular from Fig. 3, it is clear where these metallic states derive from. They are created by a transfer of spectralweight from the upper Hubbard band (the excitations around2.5e Vi nF i g . 3) down to low energies in the vicinity of the Fermi level. In EELS these low-lying states show up atfinite energies (around 1 eV) because for a metal a peak inIm(−1//epsilon1)=/epsilon1 2/(/epsilon12 1+/epsilon12 2) occurs at the plasma frequency, which is determined by the density of free charges andtherefore is finite. Note that such a spectral-weight transfer isconsidered to be a hallmark of strongly correlated systems. 39 Angle-resolved inverse photoemission could reveal the corresponding behavior in the single-particle channel.Interestingly, a similar anisotropic appearance of metallic states upon doping is also predicted theoretically for theHubbard model by cluster extensions of dynamical mean-fieldtheory 40and variational cluster approaches.41 As the spectral-weight transfer, however, is expected to occur symmetrically for all four nodal points of the BZand therefore preserves the underlying tetragonal symmetryof the lattice, our observations seem to require the collabora-tive action of (fluctuating) charge order andspectral-weight transfer. IV . SUMMARY To summarize, we investigated the energy and momentum dependent charge response in Ca 1.9Na0.1CuO 2Cl2by means of electron energy-loss spectroscopy. We find a polariza-tion dependent transfer of spectral weight between the CTexcitations—dominating the spectrum around ( π,0)—and the metallic plasmon [around ( π,π)] which can be understood as an angular dependent metal-insulator transition within theBZ. The observed pattern of the charge response indicates thebreaking of the underlying tetragonal lattice symmetry whichwe take to be indicative for a nematic order parameter in theCuO 2plane. In addition, the plasmon is found to disperse with a bandwidth smaller than in optimally doped cuprates.All effects are robust against temperature variation, arguing infavor of a highly fluctuating order. ACKNOWLEDGMENTS We appreciate experimental support by R. H ¨ubel, R. Sch¨onfelder, and S. Leger and stimulating discussions with J. v. Wezel and J. v. d. Brink. 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PhysRevB.100.125201.pdf

PHYSICAL REVIEW B 100, 125201 (2019) Theoretical study of fluorine doping in layered LaOBiS 2-type compounds Naomi Hirayama,*Masayuki Ochi, and Kazuhiko Kuroki Osaka University, 1-1 Machikaneyama-cho, Toyonaka-shi, Osaka 560-0043, Japan (Received 23 March 2019; revised manuscript received 3 July 2019; published 5 September 2019) We theoretically investigate the fluorine doping in LaOBiS 2-type quaternary compounds (LaOBiS 2,N d O B i S 2, LaOBiSe 2, and LaOSbSe 2), which are promising candidates for thermoelectric and superconducting materials. These compounds possess a layered structure comprising blocking LnO ( Ln=La, Nd, etc.) layers and conduct- ingPnCh 2(Pn=Bi, Sb; Ch=S, Se) layers. Their carrier concentration is generally tuned via substitutional doping of F atoms in the O site for improving the thermoelectric performance or the superconductivity;however, the tunability of the electrical properties via F doping strongly depends on constituent elements.In order to elucidate the difference, we theoretically examine the electronic and structural properties of theseF-doped systems using first-principles calculation. Our results show that the monoclinic distortion of the mothercompound, which is closely related to the Pnelement, can drastically decrease the capability of F doping. Replacement of the Lna t o mf r o mL at oN di n LnOBiS 2makes F doping difficult, which is consistent with experimental observation. We also find that the tetragonal structure is gradually stabilized by F doping for allthe systems investigated in this study. Our results will be important knowledge for controlling the electricalproperties of LaOBiS 2-type compounds both as thermoelectric and superconducting materials. DOI: 10.1103/PhysRevB.100.125201 I. INTRODUCTION Materials science for solving the energy problem is cur- rently of crucial importance. To this end, several technologieshave been actively developed such as the thermoelectric ef-fects, the direct conversion between thermal and electric en-ergy, and the superconductivity, which enables efficient powertransmission. For these technologies, efficient thermoelectricconversion enabled by a high value of the dimensionless figureof merit ( ZT) and high superconducting transition temperature are the central objectives, respectively, for which materialssearch has been conducted by many researchers. In the searchfor such materials, the controllability of the carrier concen-tration is often an essential aspect of candidate materials tooptimize their functionalities. LaOBiS 2[1], which comprises blocking LaO layers and conducting BiS 2layers, has attracted much attention as a superconducting [ 2–10] and thermoelectric [ 11–18] material. One of the remarkable features of this material is a relativelylow thermal conductivity, ∼2Wm −1K−1at room tempera- ture [ 19], which is advantageous for increasing ZT. Another important characteristic is a rich variety of constituent ele-ments. For example, a capability of substitution of F atomswith O atoms in LaO layers offers a high controllability ofcarrier concentration. Since LaOBiS 2has a large band gap without doping carriers, [ 20,21] the capability of the carrier doping is indispensable for employing it as a superconductingor thermoelectric material. For instance, a high superconduct-ing transition temperature was observed in LaO 1-xFxBiS 2with x=0.5[1]. Also, the thermoelectric performance of LaOBiS 2 can be improved by utilizing the large degrees of freedom of *hirayama@presto.phys.sci.osaka-u.ac.jpconstituent elements not limited to O atoms. An experimental study revealed that the thermoelectric performance can beimproved by partial replacement of S atoms with Se atoms,where ZT=0.17 at 723 K is realized in LaOBiS 2−xSexwith x=0.8[19]. A further high ZTof approximately 0.36 with low thermal conductivity of 0 .8−1.2Wm−1K−1at around 650 K was achieved in a densified sample of LaOBiSSe [ 22]. It was also shown that the Se substitution reduces the latticethermal conductivity by softening the rattling phonon modesin LaOBiS 2-xSex[15]. In addition, a theoretical study [ 23]p r e - dicted that replacing Bi with Sb or As together with replacingS with Se can considerably improve the power factor of thiscompound. From this viewpoint, a recent experimental studythat reported a successful synthesis of LnOSbSe 2(Ln=La, Ce) is intriguing [ 24]. While low thermal conductivity of 1.5 and 0.8Wm−1K−1at room temperature were found for Ln= La and Ce, respectively, there remains a difficulty in loweringtheir high electrical resistivity, which was also reported inCe(O,F)SbS 2[25]. Although a recent experimental study on NdO 0.8F0.2Sb1-xBixSe2(x/lessorequalslant0.4) revealed that the electrical conductivity of the SbSe 2-based compound can be improved by Bi doping, [ 26] the microscopic origin of the dependence of the transport properties on constituent elements such aspnictogen has been unclear. It is of great importance toinvestigate what determines the controllability of the carrierconcentration and the electrical conductivity in various kindsof the pnictogen dichalcogenide layered compounds. In this study, we theoretically investigate the electronic and structural properties of several LaOBiS 2-type quaternary compounds (LaOBiS 2,N d O B i S 2,L a O B i S e 2, and LaOSbSe 2) for undoped and F-doped states using first-principles calcula-tion. Our results suggest that the monoclinic distortion, whichis closely related to the Pn(Pn=Bi, Sb) element, drastically decreases the F-doping capacity. The change in the Lnatom 2469-9950/2019/100(12)/125201(8) 125201-1 ©2019 American Physical SocietyHIRAYAMA, OCHI, AND KUROKI PHYSICAL REVIEW B 100, 125201 (2019) f r o mL at oN di n LnOBiS 2makes F doping difficult, which may have some relevance to experimental findings. We alsofind that the tetragonal structure is gradually stabilized byF doping for all the compounds investigated in this study.The present calculation provides important information on theF-doping capability in the LaOBiS 2-type compounds. This paper is organized as follows. In Sec. II, we outline some computational conditions for crystal structure optimiza-tion and calculation of the formation energy of F-dopedsystems. In Sec. III A , we demonstrate the electronic and structural properties of compounds that have the same layeredstructure as LaOBiS 2and different compositions, such as NdOBiS 2and LaOBiSe 2, using the first-principles calculation with the generalized gradient approximation (GGA). Subse-quently, the feasibility of F doping in the systems is examinedin terms of the formation energy. In Sec. III B, we demon- strate the electronic properties of LaOBiS 2and LaOSbSe 2 obtained using the first-principles calculation with a hybridexchange-correlation functional. Their stable structures withand without F doping are also examined. Finally, we discussthe origin of the different doping abilities of these systemsin view of their structural properties. Section IVpresents the conclusions. II. CALCULATION METHOD A. Structural description and computational conditions for supercell calculations The LaOBiS 2crystal is composed of alternate stacking of LaO blocking and BiS 2conducting layers as shown in Fig. 1. Because LaOBiS 2and related compounds were reported to be the tetragonal crystal structure with the P4/nmm space group (No. 129) or the monoclinic structure with the P21/mspace group (No. 11) [ 28], we performed the cell-relaxation calcula- tion considering monoclinic as well as tetragonal symmetriesin order to elucidate the effect of the crystal symmetry ontothe capability of the F doping. We utilized the 2 ×2×1, 2×2×2, 3×3×1, 4×4 ×1, and/221a2×/221a2×1 supercells to deal with systems doped with F at the O sites with the doping concentration x, e.g., LaO 1-xFxBiS 2. When one of O atoms in these supercells is replaced by F, xis given as 0.125, 0.0625, 0.0556, 0.03125, and 0.25, respectively. In Sec. III A , we performed cell re- laxation and band structure calculations using the Perdew-Burke-Ernzerhof parametrization of the GGA functional [ 29] for LaOBiS 2,N d O B i S 2, and LaOBiSe 2. In Sec. III B,w e performed these calculations using Heyd, Scuseria, and Ernz-erhof (HSE06) hybrid functional [ 30] for LaOSbSe 2and LaOBiS 2, because we found that LaOSbSe 2becomes metallic in the GGA calculation by a well-known underestimation ofthe band gap. Hybrid-functional calculation of LaOBiS 2was performed to compare it with LaOSbSe 2. Owing to the high computational cost of calculations using the hybrid functional,we used only the unit cell and the /221a2×/221a2×1 and 2 ×2× 1 supercells in the analysis of LaOSbSe 2and LaOBiS 2shown in Sec. III B. For all the calculations in this paper, we used the pro- jector augmented wave method as implemented in Viennaab initio simulation package [ 31–34]. In crystal structure FIG. 1. (a) Crystallographic structure of LnOPnCh 2, which con- sists of alternate stacking of LnO blocking and PnCh 2conduction layers. Here, the Ln(green), O (red), Pn(purple), and Ch(yellow) atoms are displayed. (b) Unit cell shown with the lattice parameters. The lattice parameters follow the conditions a=b/negationslash=candα= β=γ=90◦for the tetragonal ( P4/nmm space group) structure and a/negationslash=b/negationslash=candβ/negationslash=90◦for the monoclinic ( P21/mspace group) structure. Depicted using the VESTA software [ 27]. optimization, both the atomic coordinates and the cell shapes were optimized by keeping the crystal symmetry. The plane-wave cutoff energy was 550 eV . Brillouin zone sampling wasperformed by the k-points grid with the following meshes: (12 12 3) for the unit cell; (6 6 3) for the 2 ×2×1 supercell; (6 6 2) for the 2 ×2×2 supercell; (4 4 3) for the 3 ×3×1 super- cell; (3 3 3) for the 4 ×4×1 supercell; and (8 8 3) for the /221a2 ×/221a2×1 supercells. For self-consistent field calculations, the convergence threshold of the total energy was 10 −7eV. The ionic relaxation was performed until the Hellmann-Feynmanforce acting on each atom becomes less than 10 −2eV/Å. To represent the strongly localized 4 forbitals of Nd atoms, we adopted an open-core treatment where 4 f3are included in the core. Spin-orbit coupling was not included in this study, theeffect of which on the doping capability will be an importantfuture issue. 125201-2THEORETICAL STUDY OF FLUORINE DOPING IN … PHYSICAL REVIEW B 100, 125201 (2019) B. Formation energy of F-doped system We will discuss how the structural properties influence the doping ability by comparing the formation energy of all theexamined compounds. In order to identify the energy stabilityof F-doped systems, we evaluated the following formationenergy: /Delta1E form=E(LnO1-xFxPnCh 2)+xμ(O) −E(LnOPnCh 2)−xμ(F), (1) where Ln=La, Nd, Pn=Bi, Sb, and Ch=S, Se in this study. Here, E(A) is the total energy that was calculated by assuming a single-crystalline structure of the system. μ(O) andμ(F) denote chemical potentials of O and F, respectively. For simplicity, we assumed μ(O)=E(O2)/2 and μ(F)=E(F2)/2, which corresponds to the O-rich and F-rich conditions. The way of estimating μ(O) and μ(F) does not af- fect the following discussion because, in the present study, we aim to examine differences of /Delta1Eformamong the compounds rather than the values of /Delta1Eformthemselves. In order to obtain E(O2) and E(F2), we calculated the total energy of the O 2(F2) single molecule in the 10 Å ×10 Å×10 Å unit cell, where the interatom distance was optimized in the calculation. We considered a spin-polarized state for the O 2molecule. For calculations of these molecules, we only took the /Gamma1point in the reciprocal space. We used the same exchange-correlationfunctional for evaluating all the total energies E(A)i nE q .( 1), i.e., GGA in Sec. III A -3 and HSE06 in Sec. III B 3 . Other computational conditions are the same as those shown in Sec. II A. We note that Eq. ( 1) estimates the difference in total energy between the system before and after the F doping, without considering the charged states. Although the estimation of the defect formation energy generally requires consideration of the charged states of the defect (see Ref. [ 35], for instance), we used Eq. ( 1) for simplicity. This can be reasonable for a system where carriers are doped with high concentrations because, in such a case, the doped carriers can be regarded as being within the unit cell. Several experimental studies [ 2,36,37] suggested that substantial electron carriers generated by par- tial substitution of F for O were doped in the conduction bandin La(O ,F)BiS 2. Although Eq. ( 1) is quite simplistic, these experimental results may verify the validity of using it in the present study because we are interested in the stability of systems with relatively high doping concentrations. Furthermore, although the chemical reaction presented with Eq. ( 1) is not very realistic, it suffices for our aim of comparing the doping abilities of the compounds, where only the relative values of /Delta1Eare needed. For LaOSbSe 2,w ea l s o evaluated the energy difference between the nondoped and F-doped systems using more realistic chemical reaction asshown in Sec. III B 3 . III. RESULTS AND DISCUSSION A. Comparison of LaOBiS 2,N d O B i S 2, and LaOBiSe 2using the GGA functional 1. Electronic band structure of mother compounds First, we have examined the electronic band structure of LaOBiS 2,N d O B i S 2, and LaOBiSe 2using first-principles FIG. 2. Electronic band structures of tetragonal (a) LaOBiS 2, (b) NdOBiS 2, and (c) LaOBiSe 2crystals obtained using the first- principles calculation with the GGA functional. calculation with the GGA functional. The electronic band structures of the undoped systems with the tetragonal structureare shown in Fig. 2. Since all the systems shown here are gapped, we can safely investigate the F-doping effects in thefollowing analysis using the GGA. The electronic band structures of doped unit cells— LaO 1-xFxBiS 2,N d O 1-xFxBiS 2, and LaO 1-xFxBiSe 2with x= 0.5—are plotted in Fig. 3. By comparing Figs. 2and3,t h e addition of F atoms results in n-type conductivity without sub- stantially altering the conduction band dispersion around theFermi level of pure systems even at a high doping level suchasx=0.5. Moreover, from local density of states calculation, it has been revealed that the F atoms form deep impuritystates over 6 eV lower than the Fermi level. It suggests thatthe thermoelectric transport calculation under the rigid-bandassumption, where the Fermi level is shifted according to thecarrier concentration based on the band structure of a puresystem, is valid for these compounds. We note that, whilethe validity of this approximation was already checked forLaOBiS 2[38], the present calculation shows that it is valid also for other compounds (NdOBiS 2and LaOBiSe 2). Several theoretical studies adopted the rigid band approach to analyzethermoelectric properties (see Refs. [ 23] and [ 39], for in- stance); the above result assures us that the good performanceof LaOBiS 2-type materials that was predicted by the previous studies is convincing. 125201-3HIRAYAMA, OCHI, AND KUROKI PHYSICAL REVIEW B 100, 125201 (2019) FIG. 3. Electronic band structures of tetragonal (a) LaO 1-xFxBiS 2,( b )N d O 1-xFxBiS 2,a n d( c )L a O 1-xFxBiSe 2crystals with x=0.5, obtained from first-principles calculation with the GGA functional. The Fermi level is set to the origin of the energyaxis. 2. Structural properties of the optimized structure with F doping Figure 4presents the total energy difference between the monoclinic and tetragonal structures for LaO 1-xFxBiS 2, FIG. 4. Total energy difference between the monoclinic and tetragonal structures for LaO 1-xFxBiS 2(circles), NdO 1-xFxBiSe 2(tri- angles), and LaO 1-xFxBiSe 2(squares), calculated using GGA.NdO 1-xFxBiS 2, and LaO 1-xFxBiSe 2with several values of the doping concentration x. It is noteworthy that the total energy of pure LaOBiS 2is lower for the monoclinic phase with β=91.02◦than for the tetragonal phase, which indicates that the stable structure is monoclinic rather than tetragonal.The obtained lattice constants of pure LaOBiS 2(a=4.073 Å, b=4.050 Å, and c=14.295 Å) well reproduce the experi- mental results [ a=4.0769(4) Å, b=4.0618(3) Å, and c= 13.885(2) Å; b =90.12(2)◦][28]. This result indicates the va- lidity of the present calculation for examining these systems. On the other hand, NdOBiS 2and LaOBiSe 2exhibit smaller energy differences between the tetragonal and monoclinicphases than LaOBiS 2. For example, the energy difference between the two phases for NdOBiS 2is around 1.2 meV , which is negligibly small, i.e., can be regarded as a numer-ical error, for the present computational conditions, such asthe number of kpoints. According to experimental studies, LaOBiSe 2crystallizes in the tetragonal phase [ 40]. Further- more, NdOBiS 2was also reported to be the tetragonal struc- ture [ 41–44]. The lattice parameters were obtained as a=b= 4.009 Å and c=14.244 Å for NdOBiS 2;a=b=4.146 Å andc=14.96 Å for LaOBiSe 2through the cell optimization calculation. The calculation results well reproduced the ex-perimental results: a=b=3.98 Å and c=13.56 Å [ 41]f o r NdOBiS 2and a=b=4.1565(1) Å and c=14.1074(3) Å [40]f o rL a O B i S e 2. Our results are also consistent with the previous theoretical calculation showing that the P21/m (monoclinic) and P4/nmm (tetragonal) phases are the most stable for LaOBiS 2and LaOBiSe 2, respectively [ 45]. However, we note here that the calculated values of cwere approximately 3–6% overestimated when compared with theexperimental values, in contrast to aandb, where the discrep- ancies between the calculated and experimental values werewithin 1%. This overestimation of cmight be attributed to the van der Waals interaction between the stacking layers,which is not easy to describe accurately by popular energyfunctionals. The dependence of the lattice constants of LaOBiS 2on the doping concentration xis shown in Fig. 5. Here, the crystal symmetry was assumed to be P21/m(monoclinic). Our results are consistent with an experimental report [ 2], which revealed that the value of csizably decreases with increasing x. Similarly, F doping for the other two compounds (NdOBiS 2and LaOBiSe 2) induced the shrinkage of these cells along the caxis, which also replicates experimental data [40,44]. Although the magnitudes of ctend to be overesti- mated, as described above, the good agreement between thetheoretical and experimental behaviors after F doping assuresus that our analysis is suitable for comparing the stability ofF-doped compounds. More interestingly, our results indicatethat LaOBiS 2gradually falls into the tetragonal phase by the F doping, that is, the lattice constants aandbbecome closer and the angle βapproaches 90°. This tendency is also inferred from Fig. 5, where the energy difference between the tetragonal and monoclinic structures vanishes by increasing x. 3. Formation energy of F-doped systems The formation energies /Delta1Eformof the systems doped with F are presented in Fig. 6. The figure contains the calculation 125201-4THEORETICAL STUDY OF FLUORINE DOPING IN … PHYSICAL REVIEW B 100, 125201 (2019) FIG. 5. Dependences of the lattice constants (a) a,b,c,a n d( b ) βon the F-doping concentration xfor the optimized structures of LaO 1-xFxBiS 2. results obtained for LaOBiS 2,N d O B i S 2, and LaOBiSe 2for two different cases: structures optimized assuming the tetrag-onal and monoclinic phases. For example, a plot for “Tetrago-nal LaOBiS 2” means that we assumed the tetragonal structure when evaluating E(LaOBiS 2) and E(LaO 1-xFxBiS 2)i nE q .( 1). Among these compounds, LaOBiS 2has the lowest values of /Delta1Eform, as shown in Fig. 6. This good doping ability agrees with experimental facts [ 2], which indicates that the electrical properties are easily controlled by doping with F for LaOBiS 2. The next lowest /Delta1Eformwas obtained for NdOBiS 2and the largest was for LaOBiSe 2. That is, the change in chalcogens from S to Se led to a larger increase in /Delta1Eformthan the change in lanthanoids from La to Nd. We note that, on the basis ofexperimental studies that reported successful F doping intoNdOBiS 2[41–44] and LaOBiSe 2[40,46], the F doping into all these systems are still possible while the solubility can bedifferent among them. As shown in Fig. 6,t h e /Delta1E form values of monoclinic LaOBiS 2are slightly higher than those of the tetragonal one. This can be understood in terms of structural changesinduced by F doping. As shown in Fig. 6, F doping decreases FIG. 6. Dependency of the formation energy /Delta1Eform on the F-doping concentration xfor LaO 1-xFxBiS 2,N d O 1-xFxBiS 2,a n d LaO 1-xFxBiSe 2, calculated using GGA. The plot shows each com- pound with the tetragonal (solid symbols) and monoclinic crystallo- graphic (open symbols) structures. the monoclinic distortion of the system; i.e., the system ap- proaches the tetragonal structure. Therefore, in Eq. ( 1), al- lowing the monoclinic distortion in calculation mainly lowersE(LaOBiS 2), while a change in E(LaO 1-xFxBiS 2)i ss m a l l e r . As a result, a higher formation energy is obtained whenone allows the monoclinic distortion in calculation. NdOBiS 2 and LaOBiSe 2exhibit negligibly small differences between the/Delta1Eformvalues of the tetragonal and monoclinic phases because the structures of these optimized cells are almost thesame as discussed in Sec. III A 2 . We note that, for these systems, the effect of the monoclinic distortion onto the F-doping capability seems to be smaller than that introduced bychanging the constituent elements of LnandCh. Before proceeding to the next section, we point out that thex−/Delta1E formplots in Fig. 6exhibit good linearity up to xof 12.5%. In contrast, the plots show nonlinear relationships forlarger x. Because calculation of heavily doped systems with sizable interaction among dopant atoms is challenging owingto the necessity of trying several dopant configurations, weshall focus on xup to 25% in the remainder of this paper. B. Comparison of LaOBiS 2and LaOSbSe 2 using hybrid functional 1. Electronic band structure of mother compounds The band structures of the undoped LaOSbSe 2and LaOBiS 2with the tetragonal structure calculated using the HSE06 hybrid functional are shown in Fig. 7. The band dispersion of LaOBiS 2in Fig. 7(a) closely resembles that obtained using GGA shown in Fig. 2(a). The band dispersion of LaOSbSe 2shown in Fig. 7(b) is also similar to them, while the band gap is much narrower. In fact, this small band gap∼0.23 eV obtained by HSE06 disappears for GGA because of its well-known underestimation of the band gap, whilethe experimental study reported the insulating behavior for 125201-5HIRAYAMA, OCHI, AND KUROKI PHYSICAL REVIEW B 100, 125201 (2019) FIG. 7. Electronic band structures of (a) LaOBiS 2and (b) LaOSbSe 2for the tetragonal structure obtained using the HSE06 hybrid functional. LaOSbSe 2[24]. This is the reason why we adopted HSE06 here rather than GGA. 2. Structural properties of the optimized structure with F doping The lattice constants of the pure LaOBiS 2and LaOSbSe 2 optimized using HSE06 are listed in Table I. The results show that the monoclinic instability is enhanced in LaOSbSe 2than LaOBiS 2, as seen from the large energy difference between the tetragonal and monoclinic phases, the large differencebetween the lattice constants aand b, and a larger βin monoclinic LaOSbSe 2. The energy difference between the tetragonal and monoclinic phases in LaOBiS 2is of the same order of magnitude as that obtained by GGA, which is alsolisted in Table I. Table Ialso demonstrates that F doping reduces the monoclinic distortion for both systems, which isconsistent with GGA analysis performed for LaOBiS 2shown in the previous section. We note that, although we obtain apositive total energy of the monoclinic LaO 1-xFxSbSe 2(x= 0.25) relative to that for the tetragonal phase, this might be due to a numerical error since P21/mis a subgroup of P4/nmm . Nevertheless, we can safely say that the tetragonal structure isactually stabilized by F doping also for LaOSbSe 2because of the large total energy difference between the two phases forx=0 and the change in the lattice parameters from x=0t o 0.25, which suggests that the system falls into (at least closelyapproaches) the tetragonal phase by F doping. According to an experimental work [ 24], the x-ray diffraction analysis revealed that LaOSbSe 2has a tetragonal (P4/nmm space group) structure with a=b=4.14340(3) Å and c=14.34480(14) Å. The calculated lattice constants listed in Table Iagree with these experimental values within an error of 3%. We note that, however, the lattice structure de-termined in this experimental study differs from that predictedin our calculation, to say, the monoclinic lattice. Nevertheless,on the basis of our calculation results, which offer reasonableagreement with experimental observation for Pn=Bi sys- tems as we have seen in Sec. III A , we can speculate that a strong instability toward the monoclinic structure should bepresent in LaOSbSe 2. As a matter of fact, some experimental studies reported that Ce(O ,F)Sb(S ,Se) 2, where S : Se =1:0 or around 1:1, has the monoclinic structure [ 25,47]. Also in the previous theoretical calculation, replacement of Pnatom from Bi to Sb or As tends to stabilize the P21/m(monoclinic) phase [ 45]. Further discussion will be presented in the follow- ing section. 3. Formation energy of F-doped systems The formation energies /Delta1Eformof F-doped LaOSbSe 2and LaOBiS 2are shown in Fig. 8. For calculating the formation energies, we assumed the tetragonal or monoclinic structuresjust as in Sec. III A 3 . We can see two important features: mon- oclinic distortion remarkably increases the formation energyfor LaOSbSe 2, and the tetragonal structures of LaOSbSe 2and LaOBiS 2have rather close values of /Delta1Eform. Since the exper- imental study reported the tetragonal structure for LaOSbSe 2, our calculation results suggest that the capability of F dopingis not so different between LaOBiS 2and LaOSbSe 2. In fact, the experimental observation that the F doping induced somechange of the transport properties in LaOSbSe 2[24] suggests that F atoms are indeed doped in the system although thesystem remains insulating. Our calculation suggests that thestructural instability in tetragonal LaOSbSe 2can play some role in its insulating behavior through the local structuraldistortion leading to the difficulty in F doping. It is also TABLE I. Calculated total energies and lattice constants of LaO 1-xFxBiS 2and LaO 1-xFxSbSe 2forx=0 and 0.25. Total energy relative to that for the tetragonal structure is shown for the monoclinic structure. Material (functional) Doping concentration x Structure Total energy (eV/cell) a(Å) b(Å) c(Å) β x=0 Tetragonal − 4.026 4.026 14.15 90° Monoclinic −0.03 4.065 4.025 14.19 91.2°LaOBiS 2(hybrid)x=0.25 Tetragonal − 4.038 4.038 13.74 90° Monoclinic −0.01 4.046 4.046 13.69 90.3° x=0 Tetragonal − 4.050 4.050 14.25 90° Monoclinic −0.01 4.073 4.050 14.30 91.0°LaOBiS 2(GGA)x=0.25 Tetragonal − 4.066 4.066 13.81 90° Monoclinic 0.00 4.066 4.066 13.81 90.0° x=0 Tetragonal − 4.071 4.071 14.59 90° Monoclinic −0.18 4.157 4.062 14.76 92.0°LaOSbSe 2(hybrid)x=0.25 Tetragonal − 4.118 4.118 13.85 90° Monoclinic 0.04 4.117 4.117 14.09 90.2° 125201-6THEORETICAL STUDY OF FLUORINE DOPING IN … PHYSICAL REVIEW B 100, 125201 (2019) FIG. 8. Dependency of the formation energy /Delta1Eform on the F-doping concentration xfor LaO 1-xFxBiS 2and LaO 1-xFxSbSe 2, calculated using the HSE06 hybrid functional. The graph contains different crystallographic structures for each compound, the tetrag-onal (solid symbols) and monoclinic (open symbols) structures. For the monoclinic structure, only the case of x=0.25 is shown. noteworthy that the monoclinic distortion in LnOPnCh 2re- sults in the deformation of the conducting layer from thetwo-dimensional square lattice to the (quasi-)one-dimensionalchains [ 28] (Fig. 9), where the transport could be more easily disturbed by several scattering processes due to its lowdimensionality. While the formation energies calculated so far in this paper are all negative, this does not necessarily indicate thatthis impurity doping was achieved without any hindrancebecause of our simple treatment of F doping. For example,the chemical reaction used in Eq. ( 1) is clearly different from the actual synthesis. Whereas our analysis should be suffi-cient to roughly see the material dependence of the F-dopingcapability and the strength of the monoclinic distortion, wehere evaluated /Delta1E form based on a more practical chemical reaction for LaOSbSe 2. In its experimental synthesis, La 2O3, FIG. 9. The SbSe 2conductive layer in LaOSbSe 2crystal struc- tures optimized assuming (left) the tetragonal and (right) the mono- clinic phases. The bonds are displayed between the nearest Sb and Se atoms that are separated by a distance of 3 Å or less.Sb2Se3, LaSe, LaSe 2, and LaF 3were used [ 24]. Using these precursors, we evaluated the total energies before and aftersynthesizing LaO 1-xFxSbSe 2as follows: 1-x 3E(La2O3)+1 8E(Sb2Se3)+1+4x 24E(LaSe ) +1−2x 24E(LaSe 2)+x 18E(LaF 3) →E(LaO 1-xFxSbSe 2). (2) Thus, by taking the difference between the left-hand side with a finite xand that for x=0, the capability of the F doping into LaOSbSe 2can be evaluated by the following energy difference: E(LaO 1-xFxSbSe 2)−/bracketleftbigg E(LaOSbSe 2)−x 3E(La2O3) +x 6E(LaSe )−x 12E(LaSe 2)+x 18E(LaF 3)/bracketrightbigg . (3) We computed Eq. ( 3) and obtained the values of 0.42 and 0.64 eV for tetragonal and monoclinic LaO 0.75F0.25SbSe 2, respectively. This suggests that the F doping into LaOSbSe 2 seems difficult, at least for relatively large x=0.25. More careful analysis on a dilute F-doped system would be animportant future study. IV . CONCLUSIONS We have theoretically investigated the structural proper- ties of LnOPnCh 2-type quaternary compounds (LaOBiS 2, NdOBiS 2,L a O B i S e 2, and LaOSbSe 2) and examined their influences on F doping. Our results show that replacementof atoms from La to Nd and S to Se in LaOBiS 2crystal makes F doping difficult, which may have some relevance toexperimental observation. We also find that replacement of thePnelement from Bi to Sb increases the monoclinic distortion of the mother compound and, in consequence, decreases thecapability of F doping. That is, large formation energy isnecessary for doping the monoclinic LaOSbSe 2with F owing to the monoclinic-to-tetragonal transformation induced by Fdoping. Therefore, it is suggested that the difficulty in control-ling the electrical properties of LaOSbSe 2by F doping, which was experimentally observed, is partly attributed to its lowdoping ability related to the predisposition towards distortion.Our theoretical work can provide important information oncarrier density control in LnOPnCh 2systems. Although the present paper focuses on the substitutional doping of F in the O site as the primary structure of F-doped LaOBiS 2-type compounds, it would be important to discuss the possibility of occupying the interstitial site infuture studies. If this occurs, the interstitial F atoms can becharged as F −, which results in unintentional hole doping. In addition, intrinsic charged defects of the parent compoundscan be formed and affect the carrier concentration. Thus, thesedefects could be associated with the low electrical conductiv-ity of LaOSbSe 2. Further studies are necessary to clarify this point. 125201-7HIRAYAMA, OCHI, AND KUROKI PHYSICAL REVIEW B 100, 125201 (2019) ACKNOWLEDGMENTS We appreciate the fruitful discussions with Prof. Y . Mizuguchi and Prof. Y . Goto. This work was supported byJSPS KAKENHI (Grant No. JP17H05481 and Grant No.JP19H04697) and JST CREST (Grant No. JPMJCR16Q6), Japan. 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PhysRevB.96.144502.pdf

PHYSICAL REVIEW B 96, 144502 (2017) Multigap superconductivity in ThAsFeN investigated using μSR measurements Devashibhai Adroja,1,2,*Amitava Bhattacharyya,1,3,†Pabitra Kumar Biswas,1Michael Smidman,4Adrian D. Hillier,1 Huican Mao,5Huiqian Luo,5Guang-Han Cao,6Zhicheng Wang,6and Cao Wang7 1ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot Oxon OX11 0QX, United Kingdom 2Highly Correlated Matter Research Group, Physics Department, University of Johannesburg, P .O. Box 524, Auckland Park 2006, South Africa 3Department of Physics, Ramakrishna Mission Vivekananda University, Howrah 711202, India 4Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China 5Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 6Department of Physics, State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310027, China 7Department of Physics, Shandong University of Technology, Zibo 255049, China (Received 9 June 2017; revised manuscript received 11 September 2017; published 3 October 2017) We have investigated the superconducting ground state of the newly discovered superconductor ThFeAsN with a tetragonal layered crystal structure using resistivity, magnetization, heat capacity, and transverse-fieldmuon-spin rotation (TF- μSR) measurements. Our magnetization and heat-capacity measurements reveal an onset of bulk superconductivity with T c∼30 K. A nonlinear magnetic-field dependence of the specific heat coefficient γ(H) has been found in the low-temperature limit, which indicates that there is a nodal energy gap. O u ra n a l y s i so ft h eT F - μSR results shows that the temperature dependence of the superfluid density is better described by a two-gap model either isotropic s+swave or s+dwave than a single-gap isotropic s-wave model for the superconducting gap, consistent with other Fe-based superconductors. The combination of γ(H) and TF- μSR results suggest that the ( s+d)-wave model is the most consistent candidate for the gap structure of ThFeAsN. The observation of two gaps in ThFeAsN suggests a multiband nature of the superconductivitypossibly arising from the dbands of Fe ions. Furthermore, from our TF- μSR study we have estimated the magnetic penetration depth in the polycrystalline sample of λ L(0)=375 nm, superconducting carrier density ns=4.97×1027m−3, and carrier’s effective-mass m∗=2.48me. We compare the results of our present paper with those reported for the Fe-pnictide families of superconductors. DOI: 10.1103/PhysRevB.96.144502 I. INTRODUCTION In a conventional superconductor, the binding of electrons into the paired states, known as Cooper pairs, is responsiblefor superconductivity as described in the Bardeen-Cooper-Schrieffer (BCS) theory in 1957 [ 1]. However, the BCS theory often fails to describe the superconductivity (SC)observed in strongly correlated materials. Several stronglycorrelated superconducting materials, having magnetic f-o r d-electron elements, exhibit unconventional SC, and various theoretical models based on magnetic interactions (magneticglue) and spin fluctuations have been proposed to understandthese superconductors [ 2,3]. Gauge symmetry is broken in the case of conventional BCS superconductors, and forthis, other symmetries of the Hamiltonian are broken forunconventional superconductors in the superconducting state.BCS superconductors also can show gap anisotropy, althoughthey remain nodeless and the gap does not change sign overthe Fermi surface, whereas unconventional superconductorsmay have nodes (zeros) in the gap function along certaindirections, and the location of the nodes is closely associatedwith the pairing symmetry. Therefore investigation of thesuperconducting gap structure of strongly correlated f- and d-electron superconductors is very important for understand- *devashibhai.adroja@stfc.ac.uk †amitava.bhattacharyya@rkmvu.ac.ining the physics of unconventional pairing mechanisms in these classes of materials. Unconventional superconductivity has been observed in high-temperature cuprates [ 4], iron pnictides [ 5], and heavy fermion materials [ 6], which have strong electronic cor- relations and quasi-two dimensionality. Interestingly super-conductivity in the iron-based materials emerges after dop-ing electrons/holes into an antiferromagnetic parent com-pound [ 5], for example, LaFeAsO 1−xFx(1111 family) [ 7,8], BaFe 2−xCoxAs2(122 family) [ 9], NaFe 1−xCoxAs (111 fam- ily) [ 10], FeTe 1−xSex(11 family) [ 11,12], and Ca 1−xLaxFeAs 2 (112 family) [ 13,14], etc. Some special systems are self-doped by the ion deficiency, such as LaFeAsO 1−δ[15] and Li 1−δFeAs [16]. It is interesting that, in the 1111 family of Fe-based materials, superconductivity can be induced by chemicalsubstitution (i.e., electron and hole doping) on any atomic site, for example, an antiferromagnetically ordered ground state in LaFeAsO is transformed into a superconducting groundstate with fluorine and hydride doping on the oxygen site(e.g., LaFeAsO 1−xFx,LaFeAsO 1−xHx)[17–20]. It is of great interest to explore possible unconventional su- perconductivity in stoichiometric Fe-based layered materials,having tetragonal crystal structures with significant electroncorrelations. Recently, the first nitride iron pnictide supercon-ductor ThFeAsN, containing layers with nominal composi-tions [Th 2N2] and [Fe 2As2] (the inset of Fig. 1), has been dis- covered with Tc=30 K for the nominally undoped compound [21]. The transition temperature of this newly discovered 2469-9950/2017/96(14)/144502(7) 144502-1 ©2017 American Physical SocietyDEV ASHIBHAI ADROJA et al. PHYSICAL REVIEW B 96, 144502 (2017) FIG. 1. X-ray powder-diffraction pattern (at 300 K) with the Rietveld refinement fit of the data of ThFeAsN. The line drawn through the data points corresponds to the calculated pattern, andthe cross symbols represent observed data. The vertical bars show the Bragg peaks’ positions, and the blue line at the bottom shows the difference plot. The inset shows the tetragonal crystal structure [ 26]. material is as high as the electron-doped 1111-based super- conductors and another family of newly discovered stoichio- metric superconductors ACa2Fe4As4F2(A=K,Rb,and Cs , Tc∼30 K) [ 22,23]. Although the first-principles calculations of ThFeAsN indicate that the lowest-energy magnetic groundstate is the stripe-type antiferromagnetic state [ 24,25], the normal-state resistivity shows no obvious magnetic anomalybut only metallic behavior down to 30 K [ 21]. The elec- tron doping by substituting N with O or hole doping bysubstituting Th with Y only suppresses the superconductingT c. The density functional theory calculations of ThFeAsN show approximately nested hole and electron Fermi surfacesof Fe dcharacter involving the xz, yz , and xyorbitals, indicating strong similarity to the other Fe-pnictide families ofsuperconductors [ 24,25]. ThFeAsN shares similar electronic structures and magnetic properties with those of LaOFeAs[25]. The calculated bare susceptibility χ 0(q) of ThFeAsN peaks at the Mpoint, suggesting perfect nesting between the holelike and the electronlike Fermi surfaces with vectorq=(π,π, 0), similar to other FeAs-based superconductors [25]. Furthermore, the nonmagnetic ground state of ThFeAsN down to 2 K has been confirmed by powder neutron-diffractionmeasurements [ 26] and a 57Fe Mössbauer spectroscopy study on polycrystalline samples [ 27]. In addition to all the existing information, it is important to understand the superconducting and magnetic propertiesof ThFeAsN on a microscopic level. Transverse-field muon-spin rotation (TF- μSR) and relaxation measurements provide direct information on the nature of the superconducting gapsymmetry and absolute value of the magnetic penetrationdepth. We therefore have investigated the superconductingproperties of ThFeAsN using the bulk properties and TF- μSR measurements. Our study of the TF- μSR shows that the temperature dependence of the superfluid density is betterdescribed by a two-gap ( s+s)- or (s+d)-wave model than a single-gap isotropic s-wave model.II. EXPERIMENTAL DETAILS A polycrystalline sample of ThFeAsN was synthesized by the solid-state reaction method as described by Wanget al. [21]. The sample was characterized using powder x-ray diffraction, electrical resistivity, magnetic susceptibility, andheat-capacity measurements. The resistivity and heat capacitywere measured using a Quantum Design physical propertymeasurement system between 1.5 and 300 K. Temperature-dependent resistivity from 2 to 300 K was measured by astandard four-probe method. The heat capacity was measuredusing a standard thermal relaxation method with a sampleofm=18 mg. The dc magnetization of the same sample was measured on a Quantum Design magnetic propertymeasurement system. μSR experiments were carried out in the MUSR spectrometer at the ISIS pulsed muon source ofthe Rutherford Appleton Laboratory, United Kingdom [ 28]. TheμSR measurements were performed in TF mode. A pellet (12-mm diameter) of polycrystalline ThFeAsN was mountedon a silver (99.999%) sample holder. Hematite ( α-Fe 2O3) slabs were placed just after the sample to reduce the backgroundsignal. The sample was cooled under He-exchange gas in aHe-4 cryostat operating in the temperature range of 1.5–300 K.TF-μSR experiments were performed in the superconducting mixed state in an applied field of 400 G, well above the lowercritical field of H c1∼30 G of this material. Data were col- lected in the field-cooled (FC) mode where the magnetic fieldwas applied above the superconducting transition temperatureand the sample was then cooled down to base temperature.Muon-spin rotation and relaxation is a dynamic method thatallows one to study the nature of the pairing symmetry insuperconductors [ 29]. The vortex state in the case of type-II superconductors gives rise to a spatial distribution of localmagnetic fields; which demonstrates itself in the μSR signal through a relaxation of the muon polarization. The data wereanalyzed using the free software package WIMDA [30]. III. RESULTS AND DISCUSSIONS The analysis of the powder x-ray diffraction at 300 K reveals that the sample is single phase and crystallizes in theZrCuSiAs-type tetragonal crystal structure with space-groupP4/nmm (No. 129, Z=2) as shown in the inset of Fig. 1. The refined values of the lattice parameters are a=4.0367(2) andc=8.5262(2) ˚A. The layered structure of ThFeAsN is shown in the inset of Fig. 1perpendicular to the caxis where separated layers of Th and N ions at the bottom and top of theunit cell (along the caxis) can be seen. The As and Fe layers are halfway along the caxis. The Fe and As ions form tetrahedrons with two As-Fe-As bond angles α∼107.0 ◦andβ∼114.5◦ at 300 K. The layered structure of ThFeAsN is very similar to others in the 1111 family of iron pnictide superconductors [ 15]. The electrical resistivity reveals a sharp drop below 30 K followed by zero resistivity indicating the onset of supercon-ductivity with T c=30 K [Fig. 2(a)]. In the zero field, the temperature-dependent resistivity of ThFeAsN is metallic andexhibits a power-law behavior ρ=ρ 0+aTnwithn∼1.3 between Tcand 150 K, indicating non-Fermi-liquid behavior [26]. The low-field magnetic susceptibility measured in an applied field of 5 G shows an onset of diamagnetism below 144502-2MULTIGAP SUPERCONDUCTIVITY IN ThAsFeN . . . PHYSICAL REVIEW B 96, 144502 (2017) FIG. 2. (a) Temperature dependence of electrical resistivity, (b) Low-field dc-magnetic susceptibility measured in zero-field cooled (ZFC) and FC modes in an applied field of 5 G. (c) The isothermal field dependence of magnetization at 5 K. (d) The isothermal field dependenceof magnetization at low field at 2, 5, and 10 K. (e) Temperature dependence of the heat capacity divided by temperature in a zero field and in an applied field of 16 T, and the inset shows the difference of the heat-capacity data 0–16 T plotted as C/T vsT. The blue vertical arrow shows the jump in C/T atT c. (f) The magnetic-field dependence of the electronic specific heat coefficient /Delta1γ[=γ(H)−γ(0)] extrapolated toT∼0 and 7 K. The solid lines show a power-law fit γ(H)∼Hn. 30 K indicating that the superconductivity occurs at 30 K and the superconducting volume fraction is close to 100% at 5.0 K[Fig. 2(b)]. This result confirms the bulk nature of supercon- ductivity with T c=30 K in ThFeAsN, which is comparable toTc=26 K observed in fluorine-doped LaFeAsO [ 31]. Very similar behavior of the resistivity and magnetic susceptibilityhas been reported for ThFeAsN by Mao et al. [26]. The magnetization isotherm M(H) curve at 5 K [Fig. 2(c)] shows typical behavior for type-II superconductivity. The lower critical field H c1obtained from the MvsHplot at 5 K is 30 G [Fig. 2(d)]. The upper critical field Hc2=80 kG at 26 K (with a slope of dH/dT ∼− 2.4T/K) has been estimated using field-dependent resistivity measurements [ 32] compared to the Pauli limit of μ0HP=18.4Tc=552 kG (55 .2T )[ 33]. The specific heat as (C /T) is displayed in Fig. 2(e) for zero field and an applied field of 16 T. A clear anomaly isobserved in a zero field corresponding to the superconducting transition at around 30 K, which is suppressed in the 16 T field. The jump in (C /T) atT cwas estimated by subtractingthe 16 T data from that at the zero field, yielding a jump of /Delta1C/T c=25 (mJ mol−1K−2), which is a factor of 2.78 larger than 9 (mJ mol−1K−2) observed in LaFeAsO and SmFeAsO polycrystalline samples [ 34]. To shed light on the nature of the gap symmetry, we also performed field-dependent heat-capacity measurements up to a field of 16 T. The fielddependence of the specific heat coefficient γ(H) was estimated by plotting C/T vsT 2and extrapolating to T∼0 K and is displayed in Fig. 2(f). The estimated γ(H)a tT∼0K exhibits a nonlinear magnetic-field dependence, but at 7 K it shows linear field dependence. The nonlinear behavior γ(H)∼H0.65, found in the low-temperature limit indicates the presence of nodal gap behavior. A very similar behavior of γ(H)∼H0.5has been observed in LaFeAsO 0.9F0.1by Gang et al. [34] that has been attributed to the nodal gap structure. Figures 3(a) and3(b) show the TF- μSR precession signals above and below Tcobtained in the FC mode with an applied field of 400 G (well above Hc1∼30 G but below Hc2∼80 kG at 26 K). The observed decay of the μSR 144502-3DEV ASHIBHAI ADROJA et al. PHYSICAL REVIEW B 96, 144502 (2017) FIG. 3. Transverse-field μSR asymmetry spectra for ThFeAsN collected (a) at T=1.5 K and (b) at T=32.5 K (i.e., below and above Tc) in an applied magnetic field of H=400 G. For the sake of clarity we present here the time-dependent asymmetry in the short- time region. The solid line shows a fit using Eq. ( 1). signal with time below Tcis due to the inhomogeneous field distribution of the flux-line lattice. We have used an oscillatorydecaying Gaussian function to fit the TF- μSR time-dependent asymmetry spectra, which is given below, G x(t)=A1cos(2πν1t+φ1)e x p/parenleftbigg−σ2t2 2/parenrightbigg , (1) where A1is the muon initial asymmetry, ν1is the frequency of the muon precession signal associated with the full volumeof the sample, and φ 1is the initial phase offset. The frequency associated with muon precession on hematite (on whichthe sample pellet was mounted) is very high (209 MHz or15.48 kG) and is out of the time window of the MUSRspectrometer at the ISIS facility [ 35]. Furthermore, both the frequency and the relaxation rate of hematite are temperatureindependent below 100 K [ 35]. Equation ( 1) contains the total relaxation rate σfrom the superconducting fraction of the sample; there are contributions from the vortex lattice(σ sc) and nuclear dipole moments ( σnm) where the latter is assumed to be constant over the entire temperature range[where σ=/radicalbig (σ2sc+σ2nm)]. The contribution from the vortex lattice σscwas determined by quadratically subtracting the background nuclear dipolar relaxation rate obtained from thespectra measured above T c.A sσscis directly related to the superfluid density, it can be modeled by [ 36–38] σsc(T) σsc(0)=1+2/angbracketleftBigg/integraldisplay∞ /Delta1k∂f ∂EEdE/radicalBig E2−/Delta12 k/angbracketrightBigg FS, (2)FIG. 4. (a) Temperature dependence of the muon depolarization rateσsc(T) of ThFeAsN collected in an applied magnetic field of 400 G in a FC mode. σsc(T) of the FC mode (symbols) where the lines are the fits to the data using Eq. ( 2) for various gap models. The dotted magenta line shows the fit using an isotropic single-gap s-wave model with/Delta1(0)=5.1±0.1 meV , the dashed red line and blue solid line show the fit to a two-gap model, the s+swave and s+dwave, re- spectively, with /Delta11(0)=5.2±0.1 meV and /Delta12(0)=0.3±0.1m e V (for both models). The green long-dashed line shows the fit using an anisotropic s-wave model, and the solid purple line shows the fit using thed-wave model. (b) Temperature dependence of the internal field. where f=[1+exp(−E/k BT)]−1is the Fermi function and the brackets correspond to an average over the Fermi surface.The gap is given by /Delta1(T,ϕ)=/Delta1 0δ(T/T c)g(ϕ), whereas g(ϕ)’s refer to the angular dependence of the superconducting gap function, and ϕis the azimuthal angle along the Fermi surface. We have used the BCS formula for the tempera-ture dependence of the gap, which is given by δ(T/T c)= tanh{(1.82)[1.018(Tc/T−1)]0.51}[39].g(ϕ)[40,41]i sg i v e n by (a) 1 for the s-wave gap [also for the ( s+s)-wave gap], (b) |cos(2ϕ)|for the d-wave gap with line nodes, and (c) for the anisotropic s-wave model|1+cos(4ϕ)| 2[36,39,42,43]. Figure 4(a) shows the temperature dependence of σsc, measured in an applied field of 400 G collected in theFC mode. The FC mode is thermodynamically stable andprovides direct information on the nature of the flux-linelattice. The temperature dependence of σ scincreases with decreasing temperature confirming the presence of a flux-linelattice and indicates a decrease in the magnetic penetrationdepth ( λ 2∼1 σsc) with decreasing temperature. The onset of diamagnetism below the superconducting transition can beseen through the decrease in the internal field below T cas shown in Fig. 4(b). A small anomaly can be seen in the internal 144502-4MULTIGAP SUPERCONDUCTIVITY IN ThAsFeN . . . PHYSICAL REVIEW B 96, 144502 (2017) TABLE I. Fitted parameters obtained from the fit to the σsc(T) data of ThFeAsN using different gap models. The Tc=28.1±1K was estimated from the single isotropic s-wave gap fit and was kept fixed for fitting all the models. Model g(φ) Gap value Gap ratio χ2 /Delta1(0) (meV) 2 /Delta1(0)/kBTc swave 1 5.1(1) 4.21 1.6 s+swave 1 5.2(1), 0.3(1) 4.29, 0.25 1.4 Anisotropy gap|1+cos(4φ)| 26.29 5.2 1.63 dwave cos(2 φ) 7.75 6.40 4.3 s+dwave 1 ,cos(2φ) 5.2(1), 0.3(1) 4.29, 0.25 1.4 field below 10 K, and the origin of this is not clear at present. To find out whether this anomaly is due to any real phasetransition in the vortex lattice below 10 K one needs detailedμSR measurements on single crystals of ThFeAsN for the muon beam both parallel and perpendicular to the caxis. From the analysis of the observed temperature dependenceofσ scusing different models for the gap, the nature of the superconducting gap can be probed. We have analyzed thetemperature dependence of σ scbased on five different models, the single-gap isotropic s-wave, the anisotropic s-wave, the line nodal d-wave models, the isotropic ( s+s)-wave, and the (s+d)-wave two-gap models. The fits to the σsc(T) data of ThFeAsN with various gap models using Eq. ( 2)a r e shown by lines (dashed, dotted, and solid) in Fig. 4(a), and the estimated fit parameters are given in Table I. It is clear from Fig. 4(a) that the d-wave model does not fit the data. On the other hand, the isotropic s-wave, the ( s+s)-wave, the (s+d)-wave, and the anisotropic s-wave models show good fits to the σsc(T) data. However, upon examining the agreement with the low-temperature upturn in the data, itis clear that only two models which explain this feature arethe isotropic ( s+s)-wave and ( s+d)-wave two-gap models. Further support for the ( s+s)- and ( s+d)-wave models can be seen through the goodness of the fit χ 2given in Table I. The value of χ2=1.4 for these models is the lowest. The estimated parameters for the ( s+s)- and ( s+d)-wave models show one larger gap of /Delta11(0)=5.2±1 (meV) and another much smaller gap of /Delta12(0)=0.3±1 (meV). The smaller gap is a nodal gap for the ( s+d)-wave model. Our μSR analysis alone cannot distinguish between the ( s+s)- and the (s+d)-wave models, but from combining the results with the field-dependent heat capacity, we conclude that the ( s+d)- wave model is the best to explain the observed behavior ofσ sc(T) andγ(H). The value of σsc(0)=0.7637±3μs−1was estimated from the ( s+d)-wave fit while keeping Tc=28.1K fixed from the single isotropic s-wave gap fit. The estimated value of 2 /Delta11(0)/kBTc=4.29±0.08 from the ( s+s)- and (s+d)-wave fits is comparable to that of the s-wave model (4.21) but larger than the value of 3.53 expected for BCSsuperconductors. On the other hand, for the smaller gap thevalue of 2 /Delta1 1(0)/kBTc=0.25±0.08 is much smaller than the BCS value. The two-gap nature, one larger and another smallerthan the BCS value, are commonly observed in Fe-based super-conductors [ 44]a sw e l la si nB i 4O4S3[45]. The multigap andd-wave order parameters are universal and intrinsic to cuprate superconductors [ 46,47], whereas Cr-based superconductors A2Cr3As3(A=K and Cs) exhibit a nodal gap [ 48,49]. Fur- thermore, the large value of 2 /Delta10/kBTc=4.29±0.08 indi- cates the presence of strong coupling and unconventionalsuperconductivity in ThFeAsN. The two superconductinggaps (one larger and another smaller) also were observed inSrFe 1.85Co0.15As2withTc=19.2 K in the scanning tunneling microscope study [ 50]. Moreover combined angle-resolved photoemission spectroscopy (ARPES) and μSR studies on Ba1−xKxFe2As2withTc=32.0 K also reveal the presence of two gaps ( /Delta11=9.1 and/Delta12=1.5m e V )[ 51]. The muon-spin depolarization rate ( σsc) below Tcis related to the magnetic penetration depth ( λ). For a triangular lattice [29,42,52],σsc(T)2 γ2μ=0.00371φ2 0 λ4(T), where γμ/2π=135.5M H z /T is the muon gyromagnetic ratio and φ0=2.07×10−15Tm2 is the flux quantum. This relation between σscandλis valid for 0.13/κ2/lessmuch(H/H c2)/lessmuch1, where κ=λ/ξ/greatermuch70 [53]. As with other phenomenological parameters characterizing asuperconducting state, the penetration depth can also be relatedto microscopic quantities. Using London theory [ 29],λ 2 L= m∗c2/4πnse2, where m∗=(1+λe-ph)meis the effective mass andnsis the density of superconducting carriers. Within this simple picture, λLis independent of the magnetic field. λe-phis the electron-phonon coupling constant, which can be estimated from /Theta1DandTcusing McMillan’s relation [ 54] λe-ph=1.04+μ∗ln(/Theta1D/1.45Tc) (1−0.62μ∗) ln(/Theta1D/1.45Tc)+1.04, where μ∗is the repulsive screened Coulomb parameter and usually assigned as μ∗= 0.13. For ThFeAsN, we have used Tc=28.1 K and /Theta1D= 332 K [ 27], which together with μ∗=0.13, we have estimated λe-ph=1.48. Furthermore, assuming that roughly all the normal state carriers ( ne) contribute to the superconductivity (i.e.,ns≈ne), we have estimated the magnetic penetration depthλ, superconducting carrier density ns, and effective-mass enhancement m∗to be λL(0)=375 nm [from the ( s+d)- wave fit], ns=4.97×1027carriers /m3, and m∗=2.48me, respectively. The correlation between Tcandσscobserved in μSR studies has suggested a new empirical framework for classifyingsuperconducting materials [ 55]. Here we explore the role of muon-spin relaxation rate/penetration depth in the super-conducting state for the characterization and classification ofsuperconducting materials as first proposed by Uemura et al. [55]. In particular we focus upon the classification scheme of Uemura et al. [55] which considers the correlation between the superconducting transition temperature T cand the effective Fermi temperature TFdetermined from μSR measurements of the penetration depth [ 56]. Within this scheme strongly correlated exotic superconductors, i.e., high- Tccuprates, heavy fermions, Chevrel phases, and the organic superconductorsform a common but distinct group, characterized by a universalscaling of T cwithTFsuch that 1 /10>(TC/TF)>1/100 (Fig. 5). For conventional BCS superconductors, 1 /1000> (Tc/TF). Considering the value of Tc/TF=30/4969.4= 0.006 for ThFeAsN (see Fig. 5), this material can be classified as not an exotic superconductor but very closeto this limit according to the classification of Uemuraet al. [55]. 144502-5DEV ASHIBHAI ADROJA et al. PHYSICAL REVIEW B 96, 144502 (2017) FIG. 5. A schematic of the plot of Uemura et al. [55]o f superconducting transition temperature Tcagainst effective Fermi temperature TF. The big solid red square and small blue square (on top of the red square) show the points calculated using the (s+d)- and ( s+s)-wave models, respectively, for ThFeAsN. The “exotic” superconductors fall within a common band for which1/100<T c/TF<1/10, indicated by the region between two red color dashed lines in the figure. The solid black line corresponds to the Bose-Einstein condensation temperature ( TB)[56]. IV . CONCLUSIONS In conclusion, we have presented the resistivity, magne- tization, heat capacity, and TF- μSR measurements in the normal and superconducting states of ThFeAsN, which havetetragonal layered crystal structures. Our magnetization andheat-capacity measurements confirmed the bulk superconduc-tivity with T c=30 K. From the TF- μSR we have determined the muon depolarization rate in the FC mode associated withthe vortex lattice. The temperature dependence of σscfits better to a two-gap model with either an isotropic s+swave or a s+dwave than a single-gap isotropic s-wave, anisotropic s-wave, or d-wave models. Our μSR analysis alone cannot distinguish between the ( s+s)- and the ( s+d)-wave models, but combining the results with field-dependent heat capacity,we conclude that the ( s+d)-wave model is the best to explain the observed behavior of σ sc(T) and γ(H). Furthermore, the value (for the larger gap) of 2 /Delta11(0)/kBTc=4.29±0.08 obtained from the ( s+s)- and ( s+d)-wave gap models fit is larger than 3.53, expected for BCS superconductors,indicating the presence of strong-coupling superconductivityin ThFeAsN. Moreover, two superconducting gaps have alsobeen observed in the Fe-based families of superconductors,and hence our observation of two gaps is in agreement with thegeneral trend observed in Fe-based superconductors. Furtherconfirmation of the presence of two gaps in ThFeAsN wouldrequire ARPES studies on single crystals of ThFeAsN. Thepresent results will help to develop a realistic theoretical modelto understand the origin of superconductivity in ThFeAsN. ACKNOWLEDGMENTS D.A. and H.L. would like to thank the Royal Society of London for the U.K.-China Newton funding. D.A. and A.H.would like to thank CMPC-STFC, Grant No. CMPC-09108,for financial support. A.B. would like to acknowledge DSTIndia for an Inspire Faculty Research Grant and ISIS-STFCfor funding support. The work at IOP, CAS was supportedby the National Natural Science Foundation of China (GrantsNo. NSFC-11374011 and No. 11611130165) and the StrategicPriority Research Program (B) of the Chinese Academy ofSciences (Grants No. XDB07020300 and No. XDPB01). H.L.would like to acknowledge support from the Youth Innova-tion Promotion Association of CAS (Grant No. 2016004).We would like to thank Dr A. Hannon for help with the CRYSTALMAKER software. [1] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 106, 162(1957 ). [2] J. Paglione and R. L. Greene, Nat. Phys. 6,642(2010 ). [ 3 ] G .R .S t e w a r t , Adv. Phys. 66,75(2017 ). [4] P. Dai, H. A. Mook, S. M. Hayden, G. 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PhysRevB.89.014205.pdf

PHYSICAL REVIEW B 89, 014205 (2014) Diffusive transport in Weyl semimetals Rudro R. Biswas,*and Shinsei Ryu Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA (Received 6 November 2013; revised manuscript received 31 December 2013; published 27 January 2014) Diffusion, a ubiquitous phenomenon in nature, is a consequence of particle number conservation and locality in systems with sufficient damping. In this paper, we consider diffusive processes in the bulk of Weyl semimetals,which are exotic quantum materials, recently of considerable interest. In order to do this, we first explicitlyimplement the analytical scheme by which disorder with anisotropic scattering amplitude is incorporated into thediagrammatic response-function formalism for calculating the “diffuson”. The result, thus, obtained is consistentwith transport coefficients evaluated from the Boltzmann transport equation or the renormalized uniform currentvertex calculation as it should be. We, thus, demonstrate that the computation of the diffusion coefficient shouldinvolve the transport lifetime and not the quasiparticle lifetime. Using this method, we then calculate the densityresponse function in Weyl semimetals and discover an unconventional diffusion process that is significantly slowerthan conventional diffusion. This gives rise to relaxation processes that exhibit stretched exponential decay insteadof the usual exponential diffusive relaxation. This result is then explained using a model of thermally excitedquasiparticles diffusing with diffusion coefficients which are strongly dependent on their energies. We elucidatethe roles of the various energy and time scales involved in this novel process and propose an experiment by whichthis process may be observed. DOI: 10.1103/PhysRevB.89.014205 PACS number(s): 72 .10.Bg,72.10.Fk,05.60.Gg,03.65.Vf I. INTRODUCTION The last decade has witnessed an enormous surge of interest in two-dimensional (2D) semimetals, materials with a nodein the electronic density of states (DOS) where the valenceand conduction bands touch. Stoichiometrically, this band-touching point is also the position of the chemical potentialin the undoped state. This undoped state is challenging toimplement experimentally because of the two-dimensionalnature of these materials, which leads to oversize effects dueto the environment (e.g., the substrate). Spectacular examplesof such 2D semimetals are provided by graphene [ 1] and the surface states of strong topological insulators (STIs) [ 2] where the quasiparticles (QPs) obey the (2 +1)-dimensional [(2 +1)- D] two-component Dirac equation, thus, exhibiting lineardispersion. Since the Dirac equation can have two oppositechiralities and the full lattice band structure should containquasiparticles with the net chirality of all branches equalto zero, these quasiparticles are required to always come inpairs [ 3]. In the cases of graphene and STIs, these pairs are well separated in momentum space and real space, respectively.Thus, under appropriate circumstances, their behavior may beunderstood as arising from the behavior of independent copiesof (2+1)-D two-component Dirac fermions. An additional characteristic of these (2 +1)-D quasiparticles is that there is a relatively easy route for converting them into massivegapped bands and so, destroying the critical scale-free natureof the massless theory. This is accomplished by breaking,spontaneously or explicitly, the symmetry that protects themassless character. In graphene, this is a combination of timereversal and inversion symmetries [ 4,5]; it is the time-reversal symmetry in STIs [ 6,7]. *rrbiswas@illinois.eduIn recent years, there have been exciting analogous pro- posals for observing similar phenomena in three-dimensional(3D) semimetals [ 8,9]. In this case, the two-component theory near the touching point of the conduction and valence bandsis that of Weyl fermions. The theory of these fermions maybe obtained by setting the mass term for the relativisticfour-component Dirac equation to zero and then using one ofthe pair of decoupled two-component fermions with oppositechiralities that form the four-component Dirac spinor. Afterappropriate rotations and rescalings, the Hamiltonian of thetheory can always be recast as H W=±vσ·k, where vis the velocity, σ·k=σxkx+σyky+σzkz(σx,y,z are the Pauli spin matrices), kis the deviation from the band touching point in momentum space, and the ±sign denotes the two possible chiralities allowed. In this case, since the Hamiltonian is aHermitian 2 ×2 matrix parametrized by three real numbers (apart from an irrelevant shift in the zero energy), the threecomponents of the momentum quench all degrees of freedom,and the spectrum cannot be gapped continuously at a singleWeyl point. Also, since the Weyl nodes occur in the bulk of a3D material, they may be less susceptible to being doped by theenvironment, unlike the 2D Dirac case where such an undesiredoccurrence of (inhomogeneous) doping needs to be activelyeliminated by challenging methods, such as the isolation ofthe material from its environment (e.g., suspended graphene)or by sensitive chemical control (e.g., in STIs). However,Weyl points must come in pairs of opposite handedness, andscattering between different Weyl points can destroy the Weylpoints, which can happen because of atomic scale disorder. Inthe absence of such atomic scale disorder, a Weyl point is stableand may be more accessible in the laboratory than a 2D gaplessDirac theory if an appropriate material is found. Anotherreason why these materials have garnered much interest isthat they are expected to have topologically protected gaplesssurface states which are novel because they have discontinuousFermi surfaces (Fermi “arcs”) [ 8]. These are related to the fact 1098-0121/2014/89(1)/014205(12) 014205-1 ©2014 American Physical SocietyRUDRO R. BISW AS AND SHINSEI RYU PHYSICAL REVIEW B 89, 014205 (2014) that, depending on chirality, the node of the theory acts like a source or sink for the Berry curvature flux [ 10]. This leads the fermions to exhibit exotic behaviors, such as the chiralanomaly because of which, in the presence of parallel electricand magnetic fields, particles are lost or are created for Weylfermions of opposite chiralities [ 11]. As in the case of the 2D undoped Dirac theory, the gapless critical nature of the Weyl theory and the chiral nature ofthe quasiparticles give rise to unconventional features in theirtransport characteristics, and this, in particular, has drawnmuch research interest [ 12–14]. One aspect has been the effects due to the marginal nature of Coulomb interaction in thesetheories [ 15]. Another aspect that attracted much attention in the context of 2D Dirac materials was the possible absenceof the Anderson localization transition [ 16] and the nature of associated phenomena, such as weak localization effects [ 17]. This discussion originated a couple of decades ago [ 18,19] when the effects of quenched disorder in undoped 2D Diracand 3D Weyl theories were investigated for the cases wherethe symmetries protecting gaplessness were not broken. Theeffective coupling induced between the fermions in the replicapicture was shown to be marginal in 2D but irrelevant in 3D inthe sense of the renormalization group. Thus, the two theories(2D and 3D) are affected very differently by disorder. In the2D case, the presence of a small amount of disorder generates,in a nonperturbative way, a finite DOS at the Dirac node;consequently, the material becomes metallic. In the 3D Weylcase, however, the nodal nature of the Weyl point is preservedunless the disorder strength is larger than a critical value, andso, the semimetal phase is stable. In this paper, we investigate the existence and idiosyncrasies of diffusive processes in the Weyl semimetal, in the “quantumcritical” regime where the chemical potential is much smallerthan the temperature. We do this for weak disorder, assumedto be of the potential type and slowly varying so that there isno scattering between different Weyl nodes which may gapthe spectrum. Choosing uncorrelated “vector” disorder (i.e.,terms proportional to the Pauli matrices in the Hamiltonian)is found to not change the underlying physics. Its effect ondiffusion in the isotropic random case will be shown to be amodification of the ratio of the transport to the quasiparticlelifetimes. Unlike the case of the 2D Dirac fermions, for whichthe effect of disorder on the nodal states is nonperturbative,in the case of the 3D Weyl semimetal, disorder is irrelevant,and thus, well-known perturbative techniques may be used[20–22]. However, there are a handful of nuanced differences from the use of these methods for a conventional Fermi gas. First, the scattering from potential disorder is highly anisotropic here and is characterized by a complete suppres-sion of exact backscattering [ 23,24]. This is dramatically dif- ferent from the textbook example of a simple Fermi gas and isknown from the Boltzmann approach, or the calculation of therenormalized uniform current vertex, to result in the transportlifetime replacing the state lifetime in the Drude expression ofthe conductivity [ 25,26]. Here we explicitly derive the density response function in the diagrammatic approach, which isused to calculate quantum effects beyond the Boltzmannapproach (e.g., weak localization). We will reconcile thesetwo approaches, demonstrating how to calculate the chargevertex renormalization in the “conserving approximation” [ 20]for disorder (the “diffuson”) with an arbitrary anisotropic scattering amplitude. We, thus, show that the result obtainedis consistent with the Boltzmann picture. We find that, fordisorder with a potential character, the transport time is 3 /2 times the state lifetime [Eq. ( 48)]. The inclusion of isotropic vector disorder reduces this ratio to a number between 3 /2 and 9/10 [Eq. ( 88)]. The second characteristic of the diffusion process in Weyl semimetals is that there is a node in the DOS at zero energyand the DOS increases linearly with the energy. Due to thenode in the DOS, the quasiparticle lifetime becomes infiniteat the Fermi level (zero energy), and one expects that, atzero temperature, transport will be ballistic (however, we cannaively expect it to be zero since the DOS is zero at the node).The situation is quite different at a finite temperature Tfor a sufficiently large sample. We argue that, when the disorderis weak, the transport process may be modeled as diffusionby particle-hole pairs with an energy-dependent diffusioncoefficient, which becomes smaller at higher energies. Thecontribution of these pairs to transport is, however, not uniformbut proportional to the product of the DOS and the slope ofthe Fermi distribution function, which is greatly suppressed atenergies less than T(see Fig. 5). As a result, the contribution of the ballistic zero-energy modes may be neglected self-consistently. The bulk of diffusive transport occurs via modeswith excitation energies that are on the order of or greater thanT. Since the quasiparticle lifetime is a decreasing function of energy, this means that the diffusive process kicks in fortime scales larger than the state lifetime for quasiparticlesat energy T. Also, since higher-energy quasiparticles have a smaller diffusion coefficient and diffuse slowly, this combineddiffusive process is shown to result in diffusive relaxationof particle density, which is qualitatively different from andslower than what we obtain for a diffusive process with asingle diffusion coefficient. While the latter (conventional)process exhibits relaxation that decays exponentially in timet, in a Weyl semimetal, the relaxation process is exponential int 1/3—a stretched exponential in t. Hence, it is much slower [Eqs. ( 68) and ( 73); Figs. 7and6]. A process such as this is described by diffusion with a memory function [ 27], which decays slowly. There does not exist a well-defined time scalebeyond which it may be neglected. The energy and time scalesinvolved in Weyl diffusion are summarized in Fig. 1. A consequence of the finite-temperature diffusion is that the Weyl semimetal has a finite dc conductance, which atzero temperature, may naively be expected to be zero sincethe DOS is zero at the Fermi point. This finite-temperaturedc conductance, obtained when only impurity scattering ispresent, has been derived previously, except for the crucialsubstitution of the transport lifetime by the quasiparticlelifetime. This requires the gauge-invariance abiding currentvertex renormalization, which was ignored in those papers. The remainder of this paper is organized as follows. We first introduce the phenomenon of diffusion in the presenceof a general memory function in Sec. II. We then explain our technique for treating the case of quenched disorder withanisotropic scattering when calculating the density vertexrenormalization in the diffusion approximation in Sec. III. Proceeding to the case of Weyl semimetals in Sec. IV,w e first calculate the electron self-energy in the self-consistent 014205-2DIFFUSIVE TRANSPORT IN WEYL SEMIMETALS PHYSICAL REVIEW B 89, 014205 (2014) No diffusionE∝disorder−1 Temperature ( T)States involved in diffusion τ(T)∼E/T2ln (time scale)Diffusive transportTimescales involved in diffusion over length scale 1/ qDiffusion timescale[v2q2D()]−1ln|| Theory cutoff ( vΛ)Quasiparticle lifetime τ() FIG. 1. (Color online) Illustration of the energy and time scales involved in Weyl diffusion. The vertical/horizontal axes are en-ergy/time scales, plotted on a logarithmic scale. QP states with energies |/epsilon1|<T play a negligible role in diffusion. QPs with energy /epsilon1diffuse on time scales much larger than the state lifetime τ(/epsilon1)∝ /epsilon1 −2, and their participation is given roughly by the shading (more accurately by the function plotted in Fig. 5). Thus, diffusion occurs on time scales larger than the lifetime τ(T) of the lowest-energy QPs involved. Also, the process, for a given length scale q−1,i n v o l v e s a range of diffusing time scales [ v2q2D(/epsilon1)]∝/epsilon12attributed to the diffusing QPs at the various energies. Diffusion processes with thiswide range of time scales superpose to give an unconventional slow diffusion process shown in Figs. 7and6. Born approximation (SCBA) and show that it predicts a semimetal-to-metal phase transition as disorder strength isincreased [ 18,19]. We subsequently only consider the case of weak disorder when the Born approximation is suffi-cient. The density response function then is calculated usingthe techniques from the previous section, following which,we elaborate on the idiosyncrasies arising from the nodalnature and the strong variation in the DOS with energy.We conclude this section with a phenomenological modelof diffusion that independently diffuses thermally excitedquasiparticles with an energy-dependent diffusion coefficientand show that it is equivalent to the results obtained for theWeyl semimetal, thus, clarifying the physical picture thatis presented in this paper. In Sec. V, we comment on the dc conductivity in Weyl semimetals and the roles of vectordisorder and Coulomb interactions. In conclusion, we proposean experiment (see Fig. 8) using currently available techniques to observe this novel slow diffusive relaxation when a Weylsemimetal is found or is fabricated in the laboratory. II. DIFFUSION AND THE DENSITY RESPONSE FUNCTION Diffusion is a consequence of particle number conservation. The rate of change in the number density is, to the firstapproximation, related linearly to its history via a diffusion memory function [ 27]M(assuming space and time-translation invariance on the average; we will always work in thereciprocal qspace), ∂ tnq(t)+/integraldisplay∞ 0dt/primeMq(t/prime)nq(t−t/prime)=0. (1) The local nature of the underlying microscopic theory, in combination with the conserved nature of the particle number,ensures that ˆM q(/Omega1)→0a s q→0. For long enough time scales when all low-frequency “reactive” modes are dampedout and the system enters the diffusive regime, ˆM q(/Omega1)∝q2as q→0. The generalized Laplace transform [ 28] (denoted by a tilde over the function name), providing the t>0 evolution of this generalized diffusion equation [thus, Im( z)>0 below and throughout this paper] in terms of the initial density nq(0) att=0, is given by ˜nq(z)=nq(0) z−˜Mq(z),Im(z)>0. (2) If there is a time scale τM, decided by microscopic processes, beyond which Mis negligible as the system loses its memory, then on time scales much longer than τM,E q s .( 1) and above reduce to the well-known “Markovian” diffusion equation, (∂t+Dq2)nq(t)=0, (3) and so, ˜nq(z)=nq(0) z+iDq2,|z|τM/lessmuch1, (4a) ⇒nq(t)=nq(0)e−Dq2t,t/greatermuchτM. (4b) The diffusion coefficient Dis defined through D=/integraldisplay≈τM 0dt/primeMq(t/prime) q2/similarequali˜Mq(z) q2/vextendsingle/vextendsingle/vextendsingle/vextendsingle Im(z)>0,|z|τM/lessmuch1.(5) However, in the absence of such a well-defined time scale τM, as will be the case with the Weyl quasiparticles below, we needto use the general form Eq. ( 1)o r( 2). Given this linear relation between ˜n q(z) andnq(0), we can deduce the low-frequency behavior of the density linear-response function ˜ χq(z). This response function χqquantifies the density response to a chemical potential wave μq(t) according to nq(t)=−i/integraldisplayt −∞dt/primeχq(t−t/prime)μq(t/prime) (6) ⇒ ˆnq(/Omega1)=˜χq(/Omega1+i0) ˆμq(/Omega1), where ˆfdenotes the usual temporal Fourier transform of f(t). At low frequencies zwith Im( z)>0, we can show that the frequency dependence of ˜ χq(z) is determined completely by the memory function ˜Mq(z) via the relation [ 27], ˜χq(z)=/parenleftbigg−˜Mq(z) z−˜Mq(z)/parenrightbigg ˜χq(i0+) ≈/parenleftbiggiDq2 z+iDq2/parenrightbigg ˜χq(i0+),|z|τM/lessmuch1. (7) 014205-3RUDRO R. BISW AS AND SHINSEI RYU PHYSICAL REVIEW B 89, 014205 (2014) III. ELECTRON DIFFUSION WHEN THE IMPURITY SCATTERING AMPLITUDE IS ANISOTROPIC Before embarking on finding the diffusion mechanism in Weyl semimetals, we will take a detour and calculate thecharge-density response in an electron gas in the presenceof random potential disorder when the scattering amplitudeisanisotropic , i.e., varies with the angle between incoming and outgoing momenta. It is well known, from the Boltzmanntransport equation approach or a diagrammatic calculation ofthe renormalized uniform current vertex, that this changesthe appropriate microscopic time scale entering the diffusionprocess from the quantum state lifetime τto the transport lifetime τ tr, both of which are defined in Eqs. ( 10) and ( 18) below. Since the diagrammatic method is used for derivingquantum corrections to the Boltzmann result (such as weaklocalization), it also is useful to treat the case of anisotropicscattering in this framework. Even though this seems to befundamental enough to have been derived before, we foundthe research literature to be lacking in this respect and willderive the procedure in this section. A. Electron self-energy /Sigma1 We first recount the well-known procedure to obtain the single-particle Green’s function in the presence of a staticdisorder potential U, which has zero mean [ /angbracketleftU(r)/angbracketright=0] and satisfies the “white-noise” criterion at long enough lengthscales, /angbracketleftU(r)U(r /prime)/angbracketright=ζδ(r−r/prime). (8) Here, /angbracketleft•/angbracketrightdenotes averaging over different realizations of disorder. As an example, if we consider a density nimpof random short-range potential wells Vwith/angbracketleftV(r)/angbracketright=0 and /angbracketleft(/integraltext V(r)ddr)2/angbracketright=U2 0, where dis the spatial dimension, then we can show that ζ=nimpU2 0. We will assume that the electron wave functions are labeled by their momentum kand that the dispersion is isotropic (for convenience of calculation)and is given by ε k≡εkand that the scattering amplitude /angbracketleftk|U|q/angbracketright≡Ukqis allowed to depend sensitively on angle θkq between the incoming and the outgoing momenta. This is true, for example, in the cases of Dirac or Weyl fermions, the latter ofwhich, we will consider in the next section. In that case, we canassume the following general form for the disorder-averagedsquared magnitude of the scattering amplitude involving statesat similar energies [ 29], 1 V/angbracketleft|Ukq|2/angbracketright=U(/epsilon1,θ kq),/epsilon1=εk≈εq, (9) where Vis the total volume [ 30]. We will assume, for our purposes, that the dependence on /epsilon1is mild [defined by a relation analogous to ( 22)] and the above form will be substituted in place of the impurity-correlation potentialdenoted by the starred vertex dashed line in the Feynmandiagram. The disorder-averaged retarded/advanced electronic self- energy /Sigma1 R/A(ω,k) is calculated in the Born approximation by the Feynman diagram shown in Fig. 2. This yields the effectivek,ω k,ωp,ω FIG. 2. The self-energy /Sigma1(ω,k) in the Born approximation. state lifetime τ(ω)t ob ef o r ω≈εk, Im[/Sigma1R/A(ω,k)]=Im/bracketleftbigg/integraldisplayddp (2π)dV−1/angbracketleft|Ukp|2/angbracketright ω−εp±i0/bracketrightbigg =∓πg(ω)/angbracketleftU(ω,θ kp)/angbracketrightˆp≡∓1 2τ(ω).(10) Here,g(/epsilon1) is the electronic DOS at energy /epsilon1, and/angbracketleft•/angbracketright ˆpaverages the enclosed expression over directions of p. For example, /angbracketleftU(ω,θ kp)/angbracketrightˆpd=2=/integraldisplay2π 0dθkp 2πU(ω,θ kp) d=3=/integraldisplayπ 0sinθkpdθkp 2U(ω,θ kp). (11) The real part of /Sigma1renormalizes the spectrum and quasiparticle weight, both of which are not going to affect the followingdiscussion and will be neglected in this paper. The quasiparticlepeak of the single-particle Green’s function is, thus, going tobe given by G R/A(ω,k)/similarequal1 ω−εk±i 2τ(ω). (12) B. The density vertex correction The diffusion pole that is expected to arise in the density response function Eq. ( 7) is obtained by correctly calculating the density vertex, maintaining number conservation [ 20]. For this, we need to solve the Bethe-Salpeter equations for thedensity vertex [ 22,26,31,32] corresponding to ˆ ρ q(/Omega1)( a si s sketched schematically in Fig. 3), /Gamma1(k,ω|k+q,ω+/Omega1) =1+/integraldisplayddp (2π)d/angbracketleft|Ukp|2/angbracketright V/Gamma1(p,ω|p+q,ω+/Omega1) ×GA(ω,p)GR(ω+/Omega1,p+q). (13) k,ωk,ω k+q,ω+Ωk+q,ω+Ωp,ω p+q,ω+Ω=+ Γ(k,ω|k+q,ω+Ω ) Γ(p,ω|p+q,ω+Ω ) FIG. 3. Bethe-Salpeter equation for the vertex ˆ ρq(/Omega1). 014205-4DIFFUSIVE TRANSPORT IN WEYL SEMIMETALS PHYSICAL REVIEW B 89, 014205 (2014) There also are contributions corresponding to the products GRGRandGAGAwhich we will neglect in this paper as they do not influence the low-frequency diffusive contributionto transport [ 25] and only serve to cancel out unphysical contributions from the high-energy states. We will also onlybe interested in values of ω≈ε k, which tells us that the values of p, that contribute substantially to the integral on the right, are those for which εp≈ω≈εk(“classical” ballistic transport in between collisions). Thus, the kdependence of /Gamma1is reduced to those through εkand the dependence on the orientation of kwith respect to q, i.e., via angle θkq.I nt h e following, we will, thus, write the vertex renormalization as/Gamma1(ε k,θkq,q,/Omega1 )≡/Gamma1(k,ω|k+q,ω+/Omega1) when ω≈εk. Equa- tion ( 13) then becomes /Gamma1(εk,θkq,q,/Omega1 )=1+/integraldisplayddp (2π)dU(εk,θkp)/Gamma1(εp,θpq,q,/Omega1 ) ×GA(εk,p)GR(εk+/Omega1,p+q). (14) As far as we can tell, this explicit dependence on θkqhas been neglected in previous published research. This neglect is justified only when the impurity scattering is independent of the angle between the incoming and the outgoing momenta.Because of this, past calculations involving the charge vertexrenormalization have failed to distinguish between the trans-port lifetime and the state lifetime and cannot be applied tothe cases of Dirac and Weyl fermions without invoking theBoltzmann transport equation. In order to solve Eq. ( 14), we decompose the angular functions into a complete basis of angular modes for θ kq,t h e specifics of which depend on the dimension. For d=2, these would be the cosine cos( nθ), whereas, in d=3, which we will concentrate on, we will use the Legendre polynomialsP /lscript(cosθ), which are the m=0 components of the spherical harmonics Y/lscriptm(θ,φ) and are independent of the azimuthal angleφ. Thus, we define /Gamma1(εk,θkq,q,/Omega1 )=∞/summationdisplay n=0γn(εk,q,/Omega1 )Pn(cosθkq), (15) and the angular averages, /angbracketleftU(/epsilon1,θ)Pn(cosθ)/angbracketrightang=αn 2πg(/epsilon1)τ(/epsilon1). (16) In particular, using Eq. ( 10), we see that α0=1. (17) Also, we can define the transport lifetime via 1 τtr(εk)=2π/integraldisplayddp (2π)dU(εk,θkp)(1−cosθkp)δ(εp−εk) =1−α1 τ(εk). (18) Using the shorthand notations θa≡θakfor any vec- tora,εk=/epsilon1, τ≡τ(εk),/Delta1=/epsilon1−εp+i 2τ, andδ=vp·q− /Omega1=vqcosθpq−/Omega1, we can manipulate the Bethe-Salpeter equation ( 14)a sf o l l o w s : /Gamma1(θq,q,/Omega1 )−1=/integraldisplayddp (2π)dU(/epsilon1,θ p)/Gamma1(/epsilon1,θ qp,q,/Omega1 )×GA(/epsilon1,p)GR(/epsilon1+/Omega1,p+q) =/integraldisplayddp (2π)d/Gamma1(/epsilon1,θ qp,q,/Omega1 )U(/epsilon1,θ p) /Delta1∗(/Delta1−δ) =/integraldisplay dεpg(εp)/angbracketleftbigg /Gamma1(/epsilon1,θ qp,q,/Omega1 )U(/epsilon1,θ p) |/Delta1|2 ×/parenleftbigg 1+/Delta1∗δ |/Delta1|2+δ2(/Delta1∗)2 |/Delta1|4+···/parenrightbigg/angbracketrightbigg ˆp ≡T1+T2+T3+··· . (19) This expansion is justified when |δ|/lessmuch|/Delta1|, i.e., |/Omega1|,vq/lessmuch1 τ(εp)≈1 τ. (20) We have expanded until the third term to obtain the lowest- order contributions from both /Omega1andqas will be shown below. These terms are evaluated below. 1. Evaluating T1 Using the decomposition in Eq. ( 15) and suppressing the arguments of γn, we obtain T1=/summationdisplay nγn/integraldisplay dεpg(εp) |/Delta1|2/angbracketleftU(/epsilon1,θ pk)Pn(cosθqp)/angbracketrightˆp.(21) Here, we will make the assumption that the DOS g(/epsilon1) andU are slow functions of /epsilon1over the scale of τ−1, 1 τ(/epsilon1)g/prime(/epsilon1) g(/epsilon1)/lessmuch1, (22) which also allows us to replace g(εp)→g(/epsilon1) in the energy integral in Eq. ( 21) and to perform the remaining Lorentzian integral, /integraldisplaydεp |/Delta1|2=2πτ(/epsilon1) (23) to obtain T1=2πg(/epsilon1)τ(/epsilon1)/summationdisplay nγn/angbracketleftU(/epsilon1,θ pk)Pn(cosθqp)/angbracketrightˆp. (24) Now, we can show [ 33] that angular averaging /angbracketleft•/angbracketright p⊥kover the azimuthal angle of p, when kprovides the zenith direction, yields /angbracketleftPn(cosθqp)/angbracketrightp⊥k=Pn(cosθq)Pn(cosθp), (25) which helps us simplify further and use Eq. ( 16) to obtain the result, T1=/summationdisplay nγnαnPn(cosθq). (26) 2. Evaluating T2 Using the decomposition in Eq. ( 15), we obtain T2=/summationdisplay nγn/integraldisplay dεpg(εp)/Delta1∗ |/Delta1|4 ×/angbracketleftU(/epsilon1,θ pk)Pn(cosθqp)(vqcosθqp−/Omega1)/angbracketrightˆp.(27) 014205-5RUDRO R. BISW AS AND SHINSEI RYU PHYSICAL REVIEW B 89, 014205 (2014) As before, the DOS may be replaced by g(/epsilon1), and the energy integral evaluates to /integraldisplay dεp/Delta1∗ |/Delta1|4=− 2iπτ2. (28) In order to evaluate the angular integral, we need to use the relation, xPn(x)=(n+1)Pn+1(x)+nPn−1(x) 2n+1 ≡dnPn+1+fnPn−1, (29) and the angular averaging property Eq. ( 25) to obtain T2=−iτ/summationdisplay nγn{/Omega1Pn(θq)αn−vq[dnPn+1(θq)αn+1 +fnPn−1(θq)αn−1]}. (30) 3. Evaluating T3 Using the decomposition in Eq. ( 15), we obtain T3=/summationdisplay nγn/integraldisplay dεpg(εp)(/Delta1∗)2 |/Delta1|6 ×/angbracketleftU(/epsilon1,θ pk)Pn(cosθqp)(vqcosθqp−/Omega1)2/angbracketrightˆp.(31) Extracting the DOS from the energy integral, the remaining energy integral reduces to /integraldisplay dεp(/Delta1∗)2 |/Delta1|6=− 2πτ3. (32) In order to evaluate the angular integral in this case, we need to use the relation, x2Pn=anPn+2+bnPn+cnPn−2, (33) where an=(n+1)(n+2) (2n+1)(2n+3), (34a) bn=4n3+6n2−1 (2n+3)(2n+1)(2n−1), (34b) cn=n(n−1) (2n+1)(2n−1). (34c) Using these, Eq. ( 29) and the angular averaging property Eq. ( 25)i nE q .( 31), we find T3=−τ2/summationdisplay nγn{/Omega12Pn(θq)αn+v2q2 ×[anPn+2(θq)αn+2+bnPn(θq)αn+cnPn−2(θq)αn−2] −2/Omega1vq [dnPn+1(θq)αn+1+fnPn−1(θq)αn−1]}.(35) 4. Recursion relations for γn Substituting Eqs. ( 26), (30), and ( 35)i nE q .( 19), using the decomposition ( 15) and comparing the coefficients of Pn(θq) on either side, we obtain a set of recursion relations, γn−δn0=αn{γn[1+iτ/Omega1−τ2(bnv2q2+/Omega12)] −τ2v2q2(γn−2an−2+γn+2cn+2)+(−iτvq+2/Omega1vqτ2)(γn+1fn+1+γn−1dn−1)}. (36) This tells us that, if αn=0f o rn/greaterorequalslantm, then γn/greaterorequalslantm=0. Let us note that /Omega1τandvqτ are the small parameters [see Eq. ( 20)] by which we can perturbatively solve these recursion relations.Expressing the mth recursion relation as 2/summationdisplay n=−2Cm,nγm+n=δm,0, (37) we find that Cm,nis, in general, of the |n|th order in smallness. The only exception is C00, which is of the second order in smallness because α0=1 leads to an exact cancellation, a con- sequence of particle number conservation/gauge invariance.Because of this, to the lowest order, we find γ 0=C11 C00C11−C1,−1C01, (38a) γ1=−C1,−1 C00C11−C1,−1C01,etc. (38b) In general, all γn’s are going to have the quantity C00C11− C1,−1C01in their denominators, which, we will show now, yields the diffusive pole for the density vertex. Indeed, C00C11−C1,−1C01 ≈(−iτ/Omega1+b0v2q2τ2)(1−α1)−(−iτvq )2d0f1α1 =(1−α1)/parenleftbigg −iτ/Omega1+v2q2τ2 3(1−α1)/parenrightbigg , (39) whose zero at /Omega1≡−iD(/epsilon1)q2yields the diffusion coefficient, D(/epsilon1)=v2τ(/epsilon1) 3(1−α1)=v2τtr(/epsilon1) 3, (40) in terms of the transport lifetime τtr[Eq. ( 18)], consistent with the Boltzmann approach. To complete the current discussion, we will write down the density vertex up to the Legendre function of order 1, whichis the relevant case for the Dirac/Weyl fermions, /Gamma1(ε k,θkq,q,/Omega1 )=i τ(εk)+α1 1−α1vqcosθkq /Omega1+iD(εk)q2. (41) IV . DIFFUSION IN WEYL SEMIMETALS Semimetals have band structures where the valence and conduction bands touch at isolated point(s) in momentumspace. Near these touching points, the effective band theory interms of k, the deviation in momentum space from the touching point, is most generally given by the Weyl theory, H W=vσ·k. (42) In general, the velocities in different directions are different, but we can always use appropriate anisotropic scaling transfor-mations to obtain the isotropic form Eq. ( 42). A slowly varying background disorder potential in this theory has the form ˆV=U(r)1. (43) 014205-6DIFFUSIVE TRANSPORT IN WEYL SEMIMETALS PHYSICAL REVIEW B 89, 014205 (2014) The conduction ( +)/valence ( −) bands, respectively, of the Weyl theory Eq. ( 42), with dispersions ε±,k=±vk, possess the following wave functions: ψk,+=1√ 2/parenleftbigge−iφkcosθk 2 sinθk 2/parenrightbigg , (44a) ψk,−=1√ 2/parenleftbigge−iφksinθk 2 −cosθk 2/parenrightbigg , (44b) where ( θk,φk) are the colatitude and azimuthal angles for kin the polar coordinates. Using this, we can show that |/angbracketleftk/prime,s/prime|k,s/angbracketright|2=1+ss/primecosθkk/prime 2, (45) and hence, the disorder potential ( 43) satisfying the white- noise criterion ( 8) has the property, 1 V/angbracketleft|Usk,s/primeq|2/angbracketright=ζ 2(1+ss/primecosθkq). (46)When q≈kands=s/primein the above equation, it also is equal toU(/epsilon1s,k,θkq) by definition [Eq. ( 16)], and so, we obtain α1=1/3,α n>1=0, (47) which implies, using ( 18), τtr(/epsilon1)=3 2τ(/epsilon1). (48) A. Self-energy in the SCBA Although our derivation of the diffusion law in Weyl semimetals will involve the self-energy evaluated in theBorn approximation (valid for very weak disorder ζ→0), it will be instructive to evaluate it in the SCBA to exposeinteresting physics suggested previously [ 18,19]. In the SCBA, the self-energy again is given by the diagram in Fig. 2 where we interpret the electron propagator as also includingthe self-energy correction. The self-consistent equation thenbecomes, with the momentum independence of the self-energymatrix ˚leading to the simple diagonal form ˚∝/Sigma11, /Sigma1R/A(ω,k)=ζ4π 8π3/integraldisplay/Lambda1 0ω−/Sigma1R/A (ω−/Sigma1R/A)2−v2p2p2dp=ζ 2π2v3(ω−/Sigma1R/A)/bracketleftbigg −v/Lambda1+(ω−/Sigma1R/A)tanh−1v/Lambda1 ω−/Sigma1R/A/bracketrightbigg /similarequalζ 2π2v3(ω−/Sigma1R/A)/bracketleftBig −v/Lambda1∓i(ω−/Sigma1R/A)π 2/bracketrightBig +O/parenleftBigω /Lambda1/parenrightBig . (49) Forω/lessmuch/Lambda1andζ/Lambda1< 2π2v2, the above equations can be solved to yield /Sigma1R/A(ω,k)=ζ 2π2v3⎡ ⎣−ω 1−ζ/Lambda1 2π2v2∓iπ 2/parenleftBigg ω 1−ζ/Lambda1 2π2v2/parenrightBigg2⎤ ⎦ +O/parenleftBigω /Lambda1/parenrightBig . (50) This tells us that disorder introduces a field renormalization Z=1−ζ/Lambda1 2π2v2as well as renormalizes the fermionic velocity v→vZ. The latter is consistent with a QP tunneling at ζ≈v2//Lambda1when v→0 (the bandwidth collapses), and the DOS at zero energy becomes nonzero [ 18,19,34]. Taken literally, however, since Z→0+as one approaches this transition, the quasiparticles are destroyed both because of theloss of quasiparticle weight and because of a divergence in thequasiparticle damping (since |Im(/Sigma1)|∝Z −2), and it is clear that better approximation schemes are necessary to calculatethe quasiparticle behavior near the transition. In this paper, we will be interested in the limit of weak disorder when the Born approximation is valid. Introducing anew energy scale defined using the disorder strength, E=4πv 3 ζ, (51) the Born approximation is valid when E/greatermuch/Lambda1/greatermuchω. (52)In this approximation, the fermion propagator assumes the simple form GR/A(ω,k)=/parenleftbigg ω−vσ·k±i 2τ(ω)/parenrightbigg−1 , (53) with τ(ω)=2E ω2. (54) Upon rotation to the conduction-/valence-band basis ( s=± ), we recover the diagonal form ( 12) of the Green’s function, GR/A s=±(ω,k)=1 ω−εs,k±i 2τ(ω),ε s,k=svk. (55) B. The density vertex correction We can now use the results from Sec. IIIif condition ( 22) is satisfied. Indeed, for the Weyl semimetal, the DOS is g(ω)=ω2 2π2v3, (56) and condition ( 22) becomes equivalent to ω/lessmuchE, (57) which already is included in the condition ( 52). Thus, we can use Eq. ( 41) and can obtain /Gamma1(εs,k,θkq,q,/Omega1 )=i τ(εs,k)+vq 2cosθkq /Omega1+iD(εs,k)q2. (58) 014205-7RUDRO R. BISW AS AND SHINSEI RYU PHYSICAL REVIEW B 89, 014205 (2014) The diffusion coefficient is given by [using Eq. ( 40)] D(ω)=v2E ω2, (59) and is a very sensitive function of energy, in contrast to cases where the Fermi surface is not at/close to a node inthe DOS. Following the inequality ( 20), we find that the last two expressions are valid for |/Omega1|,vq/lessmuchω 2 E. (60) C. The density response function The density response function ( 6)i sg i v e nb y[ 25] ˜χq(/Omega1+i0) =/summationdisplay s=±/integraldisplaydω 2πi[nF(ω+/Omega1)−nF(ω)] ×/integraldisplayd3k (2π)3/Gamma1(εs,k,θkq,q,/Omega1 )GA s(ω,k)GR s(ω+/Omega1,k+q) −(value at q=0). (61) The subtraction at q=0 is required because the bare density operator used is not “normal ordered” while we are findingthe response function for the change in the density, which is a normal-ordered quantity. When condition ( 60) is satisfied for ag i v e n ω, using Eqs. ( 58) and ( 23), the product of the Green’s functions above can be evaluated as follows: /summationdisplay s/integraldisplayd3k (2π)3/Gamma1(εs,k,θkq,q,/Omega1 )GA s(ω,k)GR s(ω+/Omega1,k+q) /similarequali τ(ω) /Omega1+iD(ω)q2/integraldisplay∞ −∞d/epsilon1g(/epsilon1) /vextendsingle/vextendsingleω−/epsilon1+i 2τ(ω)/vextendsingle/vextendsingle2 /similarequal2πig(ω) /Omega1+iD(ω)q2, (62) which, in combination with ( 61), yields (see Fig. 4) ˜χq(/Omega1+i0)=/integraldisplay∞ −∞dω/parenleftbigg∂nF(ω) ∂ω/parenrightbigg g(ω)/Omega1 /Omega1+iD(ω)q2 −(value at q=0). (63) Of course, this evaluation required the condition ( 60)t o be satisfied for all ωin the above integral, which would k+q,, ω+Ω FIG. 4. The charge response function in the ladder approxima- tion. The renormalized vertex (shaded region) satisfies the Bethe-Salpeter equation in Fig. 3.10 5 5 10ΩTgΩΩnFΩ FIG. 5. (Color online) The contribution of particle-hole pairs at various energies to the diffusion process. The shaded region showsthe negligible ∼4% contribution from pairs with energies |ω|<T. naively require /Omega1→0. This is because the states close to the Weyl/Dirac point have lifetimes τ(ω→0)→∞ and, hence, exhibit ballistic transport. However, these states contributenegligibly to the full charge response since the weighingfunction |g(ω)∂n F(ω)/∂ω|in the ωintegral is peaked around ω/similarequal± 2.4Tand has negligible weight ( ∼4%) for |ω|<T as shown in Fig. 5. Thus, the condition for observing diffusive transport is actually |/Omega1|,vq/lessmuch1 τ(T)∼T2 E. (64) Thus, to conclude, for /Omega1,vq /lessmuchT2/E/lessmuchT, ˜χq(/Omega1+i0)=/integraldisplay /Omega1τ(ω)<1dω/parenleftbigg −∂nF(ω) ∂ω/parenrightbigg g(ω)iD(ω)q2 /Omega1+iD(ω)q2. (65) The lower limit in the ωintegral is decided by the condition for deriving the diffusive form of the density vertex, which translates to the condition ω>√ /Omega1E. The charge response is diffusive but with a continuum of diffusive poles D(ω)q2 ranging from D(ω→∞ )q2=0 until approximately D(T)q2. Thus, in the time domain, the charge response should be much slower than the usual exponential decay with a single time scale ( Dq2)−1that is present for usual diffusion with a fixed diffusion coefficient D:χq(t)∼e−Dq2t. Indeed, transforming Eq. ( 65) to the time domain and using Eqs. ( 59) and ( 56), we find χq(t)≈/Theta1(t)q2/integraldisplay |ω|>Tdω/parenleftbigg −∂nF(ω) ∂ω/parenrightbigg g(ω)D(ω)e−D(ω)q2t =/Theta1(t)Eq2 2π2v/integraldisplay∞ 1dxex (ex+1)2e−D(T)q2t/x2/parenleftBig x=/epsilon1 T/parenrightBig . (66) This integral can be calculated approximately using the saddle- point approximation for long times D(T)q2t/greatermuch1. The saddle point of the exponent of the integrand occurs at xs≈[2D(T)q2t]1/3+O(e−[D(T)q2t]1/3). (67) 014205-8DIFFUSIVE TRANSPORT IN WEYL SEMIMETALS PHYSICAL REVIEW B 89, 014205 (2014) Utilizing the usual saddle-point evaluation technique, we obtain the long-time behavior of the density response function, χq(t)≈/Theta1(t)Eq2 v√ 3π3/parenleftbiggD(T)q2t 4/parenrightbigg1/6 ×exp/bracketleftBigg −3/parenleftbiggD(T)q2t 4/parenrightbigg1/3/bracketrightBigg forD(T)q2t/greatermuch1. (68) Figure 5shows that this analytical long-time expression matches the solution from ( 66) very well. This stretched exponential time dependence, a new physical result, is qual-itatively different and is slower than the usual exponential inthe time decay mentioned previously. This behavior also isinsensitive to how the cutoff near |ω|=Tis handled. This is because the long-time behavior arises from the contributionof quasiparticle states at higher energies that have smallerdiffusion coefficients and, hence, slower diffusion time scales.Another point to note is that the integral expression in Eq. ( 66) is valid for time scales tlonger than τ(T)[τ(ω) is a decreasing function of ω], t/greatermuchτ(T)=2E T2⇒D(T)q2t/greatermuch[vqτ(T)]2. (69) Sincevqτ(T)/lessmuch1, we can trust the answer from the aforemen- tioned integral for almost the full range of D(T)q2t>0. This is plotted in Fig. 6in the form of χq(t)/χq(0) and is compared to the exponential decays e−Dq2tthat would have resulted if all particles diffused with a diffusion coefficient D=D(T)o r D=D(2.4T), where 2 .4Tis the typical energy of the Weyl quasiparticles taking part in the diffusion (see Fig. 5and the associated discussion). D. The diffusion memory function for Weyl fermions We are now in a position to figure out the diffusion equation obeyed by the Weyl fermions. Comparing Eqs. ( 7) and ( 65), we can solve for the memory function ˜M(z), −˜Mq(/Omega1+i0) /Omega1−˜Mq(/Omega1+i0)=˜χq(/Omega1+i0) ˜χq(i0+) =/integraldisplay dωP(ω)iD(ω)q2 /Omega1+iD(ω)q2, where the normalized weighing function is P(ω)∝/Theta1[/Omega1τ(ω)−1]/parenleftbigg −∂nF(ω) ∂ω/parenrightbigg g(ω), /integraldisplay dωP(ω)=1. (70) The Heaviside step function /Theta1explicitly removes the nondiffu- sive contribution from energies with /Omega1τ(ω)<1. Manipulating the previous equations, we find ˜Mq(/Omega1+i0)=−iq2/integraldisplay dωP(ω)D(ω) /Omega1+iD(ω)q2 /integraldisplay dωP(ω) /Omega1+iD(ω)q2. (71)0 2 4 6 8 10 12 14DTq2t0.150.30.51ΧqtΧq0 DT D2.4T Weyl Fit 0. 0.5 1. 1.5 2. 2.5DTq2t130.150.30.51ΧqtΧq0 DT D2.4T Weyl Fit(a) (b)[[ FIG. 6. (Color online) Comparing the slow diffusive decay of χq(t) in Weyl semimetals with that of particles diffusing with diffusion coefficient D=D(T)o rD=D(2.4T), where 2 .4Tis the typical energy of the quasiparticles taking part in the diffusion. The plots are log-linear, so exponential decays in the quantity on the horizontal axis [ ∝tin (a) and ∝t1/3in (b)] show up as straight lines. The analytical approximation (“fit”) for long times, as derived in Eq. ( 68), also is shown on these plots for comparison. Since D(ω) decreases without bound as ω→∞ , there is no small time scale beyond which this memory function can takeon the simple form in Eq. ( 5), and so, there is no time scale beyond which the Weyl fermions will follow a simple diffusionequation. This physical statement can be visualized by following the time evolution of a particle density wave n q(t). For particles that satisfy a diffusion equation with diffusion coefficient D, Eq. ( 3) tells us that nq(t)/nq(0)∼e−Dq2t. However, for the Weyl case, using Eqs. ( 2) and ( 7), we find ˜nq(/Omega1+i0) nq(0)=1 /Omega1/parenleftbigg 1−˜χq(/Omega1+i0) ˜χq(i0+)/parenrightbigg ∝/integraldisplay dω/parenleftbigg −∂nF(ω) ∂ω/parenrightbiggg(ω) /Omega1+iD(ω)q2.(72) This means that, for long times t/greatermuchτ(T), nq(t) nq(0)∝/integraldisplay dω/parenleftbigg −∂nF(ω) ∂ω/parenrightbigg g(ω)e−D(ω)q2t ∝/integraldisplay∞ 0dxx2ex (ex+1)2e−D(T)q2t/x2/parenleftBig x=/epsilon1 T/parenrightBig 014205-9RUDRO R. BISW AS AND SHINSEI RYU PHYSICAL REVIEW B 89, 014205 (2014) 0 2 4 6 8 10 12 14D/LParen1T/RParen1q2t0.40.71nq/LParen1t/RParen1/Slash1nq/LParen10/RParen1 D/LParen1T/RParen1 D/LParen12.4T /RParen1 Weyl 0. 0.5 1. 1.5 2. 2.50.40.71nq/LParen1t/RParen1/Slash1nq/LParen10/RParen1 D/LParen1T/RParen1 D/LParen12.4T /RParen1 Weyl(a) (b)DTq2t13[[ FIG. 7. (Color online) Comparing the slow diffusive decay of a density perturbation nq(t) in Weyl semimetals with that of conventional particles diffusing with diffusion coefficient D=D(T) orD=D(2.4T) according to Eq. ( 4), where 2 .4Tis the typical energy of the quasiparticles taking part in the diffusion. The plots are log-linear, so exponential decays in the quantity on the horizontal axis [∝tin (a) and ∝t1/3in (b)] show up as straight lines. ≈4/radicalbiggπ 3/parenleftbiggD(T)q2t 4/parenrightbigg5/6 ×exp/bracketleftBigg −3/parenleftbiggD(T)q2t 4/parenrightbigg1/3/bracketrightBigg forD(T)q2t/greatermuch1. (73) This stretched exponential decay, found using a saddle-point analysis, is similar to that for the density response function(Fig. 6). The numerically evaluated integral is compared against the exponential decays ( 4) of a density perturbation for conventional particles diffusing with the diffusion coefficientD(T)o rD(2.4T)i nF i g . 7[35]. E. A phenomenological picture of Weyl diffusion Equation ( 65) will now be derived from a simple phe- nomenological model of Weyl fermion quasiparticles withparticles at different energies diffusing independently with anenergy-dependent diffusion coefficient D(/epsilon1). This clarifies the physical picture of the unconventional behavior uncovered inthe past few subsections. Denoting the density of particles atenergy /epsilon1by the quantity n q(t|/epsilon1), we should have nq(t)=/integraldisplay d/epsilon1n q(t|/epsilon1). (74) Att=0, we will assume thermal equilibrium in the presence of a frozen fluctuation μqin the chemical potential. In that case, nq(t=0)=/integraldisplay d/epsilon1g(/epsilon1)/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbigg μq ⇒nq(t=0|/epsilon1)=g(/epsilon1)/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbigg μq, (75) and ˜χq(i0+)=nq(t=0) μq=/integraldisplay d/epsilon1g(/epsilon1)/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbigg .(76) Following the discussion surrounding Fig. 5, we see that nq(t=0|/epsilon1) is negligible for /epsilon1<T , and thus, we should disregard it in the discussion below. Now, we will relax thechemical potential to the constant equilibrium value and willcalculate the diffusive process by which the numbers relax totheir equilibrium (zero) values. Using Eq. ( 3), ˜n q(z|/epsilon1)=nq(t=0|/epsilon1) z+iD(/epsilon1)q2,|z|τ(/epsilon1)/lessmuch1 =/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbiggg(/epsilon1) z+iD(/epsilon1)q2μq. (77) Thus, using Eq. ( 74), the total density response is ˜nq(z)=/integraldisplay |z|τ(/epsilon1)<1d/epsilon1/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbiggg(/epsilon1) z+iD(/epsilon1)q2μq ≡nq(0) z−˜Mq(z)[using (2)] , (78) and so, ˜χq(i0+) z−˜Mq(z)=/integraldisplay |z|τ(/epsilon1)<1d/epsilon1/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbiggg(/epsilon1) z+iD(/epsilon1)q2.(79) Finally, using Eqs. ( 7) and ( 76), ˜χq(z)=/parenleftbigg 1−z z−˜Mq(z)/parenrightbigg ˜χq(i0+) =/integraldisplay /Omega1τ(ω)<1d/epsilon1/parenleftbigg −∂nF(/epsilon1) ∂/epsilon1/parenrightbigg g(/epsilon1)iD(/epsilon1)q2 z+iD(/epsilon1)q2,(80) which is identical to Eq. ( 65) and proves the applicability of this simple picture to the previous results of this section. V . DIFFUSIVE CONDUCTIVITY, VECTOR DISORDER, AND COULOMB INTERACTIONS We conclude the main body of this paper with comments about the conductivity of disordered Weyl semimetals at finitetemperatures and the effects of including isotropic vectordisorder, i.e., disorder terms proportional to the Pauli matricesin (42), assuming rotational isotropy. 014205-10DIFFUSIVE TRANSPORT IN WEYL SEMIMETALS PHYSICAL REVIEW B 89, 014205 (2014) A. Conductivity Using the continuity equation −e∂tρ+∇·J=0, the (longitudinal) conductivity σ=σxx=σyy=σzzmay be re- lated to the density response function χ, σq(/Omega1)=−ie2/Omega1 q2χq(/Omega1). (81) In the diffusing limit /Omega1,vq /lessmuchT2/E/lessmuchT, we can use ( 65)t o obtain σq(/Omega1)/similarequal/integraldisplay /Omega1τ(ω)<1dω/parenleftbigg −∂nF(ω) ∂ω/parenrightbigg g(ω)e2/Omega1D(ω) /Omega1+iD(ω)q2. (82) Taking the limit q→0 and then /Omega1→0, we arrive at the dc conductivity, σDC/similarequale2/integraldisplay dω/parenleftbigg −∂nF(ω) ∂ω/parenrightbigg g(ω)D(ω)=e2E 2π2v ≡4e2v2 ζh, (83) where Planck’s constant hhas been restored and the result is reexpressed in terms of ζdefined in ( 8). This answer is equivalent to earlier results [ 13,14], but with the correct numerical factors [ 13] and the renormalization of the current vertex [ 25], equal to the ratio τtr/τ=3/2. In addition, for small /Omega1, the diffusive component of the conductivity decreases from its dc value by a term proportionalto√ /Omega1as has been obtained previously [ 14] (but without using the conserving approximation or taking the diffusive-ballistic crossover into account). This arises due to the ∼√ /Omega1Ebehavior of the lower limit of the ωintegral in ( 82), below which ballistic transport occurs. However, in this energy range,the calculations need to incorporate the diffusive-ballistictransition accurately as well as interband scattering processesin order to correctly obtain the coefficient of this√ /Omega1term. B. Vector disorder In addition to the usual potential disorder term U(r)1,w e can also include vector disorder in the Weyl theory ( 42), HV=V(r)·σ. (84) We assume white-noise disorder with rotational symmetry restored on the average, which means that /angbracketleftVa(r)Vb(r/prime)/angbracketright=δabζVδ(r−r/prime), (85) and that this vector disorder is uncorrelated with the potential disorder on the average /angbracketleftVaU/angbracketright=0. With these assumptions, we can show that the net effect is to change the effectiveinteraction induced by disorder, Eq. ( 86), to the form /angbracketleftbig/vextendsingle/vextendsingleU total kq/vextendsingle/vextendsingle2/angbracketrightbig V=ζ 2(1+ss/primecosθkq)+3ζV 2/parenleftbigg 1−ss/prime 3cosθkq/parenrightbigg . (86) This means that depending on the relative strengths of the two kinds of disorder, we have −1 9/lessorequalslantα1/lessorequalslant1 3, (87)and so, 9 10/lessorequalslantτtr τ/lessorequalslant3 2. (88) Apart from this modification, the rest of the physics is the same as when only potential disorder is present. C. Coulomb interactions Unlike short-range quenched disorder, which is an irrele- vant perturbation to the Weyl theory, long-range instantaneousCoulomb interaction is a marginal perturbation and resultsin strong electron-electron scattering (including inelasticprocesses) as the temperature is decreased. This phenomenoncan approximately be treated using the quantum Boltzmannequation and shows that the conductivity goes to zero asT→0[12,13]. Thus, in the presence of Coulomb interactions, our results for quenched disorder-limited transport hold onlyfor high enough temperatures when the Coulomb scatteringmay be neglected in comparison to that due to quencheddisorder. VI. DISCUSSION In this paper, we have initially introduced the formalism required to correctly compute the renormalized charge vertexin the diffusion approximation in the presence of disorderwith an anisotropic scattering amplitude. We have then usedthis to study the case of Weyl fermions in the presence ofquenched disorder in the quantum critical regime when the Weyl semimetalSET charge probe Backgates -++-- Length scale λ of induced charge modulation FIG. 8. (Color online) Experimental setup for measuring charge diffusion in a Weyl semimetal. The backgate arrangement induces a charge-density modulation in the Weyl semimetal with a charac- teristic wavelength λ=2π/q that is equal to the distance between two successive gates with the same polarity. At t=0, the gates are switched off and the charge modulation relaxes to the equilibrium uniform value, and this decay is tracked using, say, a single electron transistor (SET) based charge-sensitive probe [ 36]. We predict that the charge decay will follow the form predicted in Fig. 7. 014205-11RUDRO R. BISW AS AND SHINSEI RYU PHYSICAL REVIEW B 89, 014205 (2014) chemical potential is small compared to the temperature. These quasiparticles exhibit anisotropic scattering, and theirDOS varies sharply with energy near the Fermi point. Theanisotropic scattering results in the transport time scale beingthe relevant quantity in the diffusion parameter, the quantumcritical nature of the problem results in temperature beingthe only low-energy scale in the theory, and the sharp linearincrease in the DOS leads to a very slow unconventionaldiffusion process, characterized by a memory function whichdoes not have a sharp upper limit on the time scalesinvolved. When a Weyl semimetal is found in the laboratory, this novel slow diffusive relaxation may be observed as a test ofour theory. The experimental setup would involve preparing asample of the Weyl semimetal with a frozen long-wavelengthexcess charge modulation at t=0 induced by, say, a backgate arrangement that is electrically insulated from the sample asshown schematically in Fig. 8. The backgate is then grounded, and the induced charge is allowed to relax to the steady-stateuniform distribution. The amount of charge present at a given time may be sensed using, say, a sensitive SET chargeprobe [ 36]. The relaxation of charge then is predicted to follow the slow process shown in Fig. 7with the value of qbeing provided by the inverse wavelength of the induced chargemodulation, which may be tuned by changing the separationbetween the backgates. The initial decay may, however, turnout to be much faster in the experiment due to charge diffusionalong the surface Fermi arcs. ACKNOWLEDGMENTS This research was supported by the Institute of Con- densed Matter Theory at UIUC. We would like to acknowl-edge stimulating discussions with E. Fradkin, I. Gruzberg,L. Kadanoff and T. Witten. We are grateful to T. Witten for hissuggestions regarding calculating the integral in Eq. ( 66)b y using the saddle-point approximation. [ 1 ] A .K .G e i m , Rev. Mod. Phys. 83,851(2011 ). [2] M. Z. Hasan and J. E. Moore, Annu. Rev. Condens. Matter Phys. 2,55(2011 ). [3] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 185,20(1981 ). [4] B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors (Princeton University Press, Princeton, 2013). [5] J. E. Drut and T. A. L ¨ahde, P h y s .R e v .L e t t . 102,026802 (2009 ). [6] R. R. Biswas and A. V . Balatsky, Phys. Rev. B 81,233405 (2010 ). [7] L. A. Wray, S.-Y . Xu, Y . Xia, D. Hsieh, A. V . Fedorov, Y . S. Hor, R. J. Cava, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Phys. 7,32(2011 ). [8] X. Wan, A. M. Turner, A. Vishwanath, and S. Y . Savrasov, Phys. Rev. B 83,205101 (2011 ). [9] A. A. Burkov and L. Balents, P h y s .R e v .L e t t . 107,127205 (2011 ). [10] L. Balents, Phys. 4,36(2011 ). [11] D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109,181602 (2012 ). [12] L. Fritz, J. Schmalian, M. M ¨uller, and S. Sachdev, Phys. Rev. B 78,085416 (2008 ). [13] P. Hosur, S. A. Parameswaran, and A. Vishwanath, Phys. Rev. Lett.108,046602 (2012 ). [14] A. A. Burkov, M. D. Hook, and L. Balents, P h y s .R e v .B 84, 235126 (2011 ). [15] D. E. Sheehy and J. Schmalian, Phys. Rev. Lett. 99,226803 (2007 ). [16] K. Nomura, M. Koshino, and S. Ryu, P h y s .R e v .L e t t . 99,146806 (2007 ). [17] E. McCann, K. Kechedzhi, V . I. Fal’ko, H. Suzuura, T. Ando, a n dB .L .A l t s h u l e r , P h y s .R e v .L e t t . 97,146805 (2006 ). [18] E. Fradkin, P h y s .R e v .B 33,3257 (1986 ). [19] E. Fradkin, P h y s .R e v .B 33,3263 (1986 ). [20] G. Baym and L. P. Kadanoff, Phys. Rev. 124,287(1961 ). [21] P. A. Lee and T. V . Ramakrishnan, Rev. Mod. Phys. 57,287 (1985 ).[22] A. Altland and B. D. Simons, Condensed Matter Field The- ory, 2nd ed. (Cambridge University Press, Cambridge, UK, 2010). [23] T. Ando and T. Nakanishi, J. Phys. Soc. Jpn. 67,1704 (1998 ). [24] T. Ando, T. Nakanishi, and R. Saito, J. Phys. Soc. Jpn. 67,2857 (1998 ). [25] H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics: An Introduction , 1st ed. (Oxford University Press, Oxford, 2004). [26] S. F. Edwards, Philos. Mag. 3,1020 (1958 ). [27] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry, And Correlation Functions (Westview, Boulder, CO, 1990). [28] The generalized Laplace transform of a function f(t)i sd e fi n e d via ˜f(z)=/integraldisplay∞ −∞dω 2πˆf(ω) z−ω, where ˆf(ω) is the usual Fourier transform of f(t). With this def- inition, ˜f(ω±i0) is the Fourier transform of ∓i/Theta1(±t)f(t),/Theta1 denoting the Heaviside step function. [29] Scattering processes effectively mix states whose energies are separated by the inverse state lifetime, i.e., the energy“linewidth.” [30] This arises from the δfunction in momentum space /angbracketleftU kqUpl/angbracketright∝ δ(k+p−q−l). [31] D. V ollhardt and P. W ¨olfle, P h y s .R e v .B 22,4666 (1980 ). [32] A. A. Abrikosov, L. P. Gorkov, and I. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963). [33] Using the formula http://dlmf.nist.gov/18.18#E9 . [34] K. Kobayashi, T. Ohtsuki, K.-I. Imura, and I. F. Herbut, Phys. Rev. Lett. 112,016402 (2014 ). [35] The convergence of the saddle-point estimate to the actual value of the integral is very slow as t→∞ and has not been included in Fig. 7. [36] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and A. Yacoby, Nat. Phys. 4,144(2007 ). 014205-12

PhysRevB.101.144401.pdf

PHYSICAL REVIEW B 101, 144401 (2020) Density functional theory of magnetic dipolar interactions Camilla Pellegrini ,1Tristan Müller,2John K. Dewhurst,2Sangeeta Sharma,3Antonio Sanna ,2and Eberhard K. U. Gross1 1Institute of Chemistry, Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel 2Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany 3Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489 Berlin, Germany (Received 15 January 2020; accepted 10 March 2020; published 2 April 2020) We propose a way to include magnetic dipole-dipole interactions in density functional theory calculations. To this end, we derive an approximation to the exchange-correlation energy functional associated with the spin-spincorrection to the Coulomb force in the Breit-Pauli Hamiltonian. The local spin-density approximation is shownto be identically zero. First order nonlocal corrections are evaluated analytically within linear response to anoncollinear external magnetic field. The functional obtained is based on the exact-exchange energy of themagnetic electron gas with dipolar interactions and is estimated to be relevant at interatomic distances, or in thelow electron density limit, where it amounts to one quarter of the magnetostatic energy. We expect our functionalto improve over the current description of ground-state properties of inhomogeneous magnetic structures at thenanoscale and dipolar spin systems. DOI: 10.1103/PhysRevB.101.144401 I. INTRODUCTION The need for higher density data storage and practical schemes to implement quantum information processing [ 1–4] has led, in the last decades, to thriving research on non- conventional magnetic systems [ 5]. These include ultracold dipolar gases in optical lattices [ 6,7], low-dimensional or frus- trated magnets [ 8], nanostructured magnetic materials [ 9], and molecular magnets [ 10]. All these systems show a complex magnetic behavior at the atomic scale, which results from thedelicate interplay between Heisenberg exchange interactionsand (tunable) spin-orbit coupling and dipolar interactions.In particular, new effects from the last, due to their long-range and anisotropic nature, are attracting great interest.While the picture of magnetic dipole-dipole interactions as asmall classical perturbation might be enough to understandtraditional magnetism, it seems to be no longer sufficient forthe description of spin systems with strong magnetic momentsand length scales approaching the nanometer range. Currently, the most common and feasible approach to describe the magnetic behavior of a given material is themicromagnetic approach [ 11], which ignores the atomic struc- ture of matter, neglects quantum effects, and uses classicalphysics in a continuum description of the system. Essentially,the atomic magnetic moments are assumed to vary slowlywithin a mesoscopic volume of the sample, so to define amesoscopic average magnetization M(r). The magnetic Gibbs energy, which is the sum of four major contributions, i.e.,exchange, magnetocrystalline anisotropy, Zeeman and dipolarenergy, is formulated in terms of the continuous magnetizationvector field and minimized to determine static magnetizationstructures. Specifically, the dipolar energy is derived fromsimplified magnetostatic Maxwell’s equations and computedin terms of effective magnetic volume charges ρ(r)=− ∇· M(r) and effective magnetic surface charges σ(r)=M(r)·nas E magstat =μ0 2/integraldisplay Vρ(r)U(r)d3r+μ0 2/integraldisplay Sσ(r)U(r)dS.(1) Here, the magnetic scalar potential U(r) is the solution of the Poisson’s equation /Delta1U(r)=−ρ(r), i.e., has the general form U(r)=1 4π/integraldisplay Vρ(r/prime) |r−r/prime|d3r/prime+1 4π/integraldisplay Sσ(r/prime) |r−r/prime|dS/prime.(2) As is clear from Eqs. ( 1) and ( 2), the magnetostatic energy arises from inhomogeneities of the average magnetizationM(r) on a mesoscopic scale. Micromagnetic simulations have proven to be generally a very reliable tool for investigatingthe properties of ferromagnetic nanostructures. However, dueto the rapid advances in the synthesis of nanostructured ma-terials, the continuum assumption behind these algorithmsmight be unjustified. E.g., highly inhomogeneous magneticstructures, laser-induced magnetization dynamics, and sam-ple sizes approaching the atomic scale are a few cases thatchallenge the limits of micromagnetic theory and demandextensions to the standard approach. On the other hand, the calculation of atomic proper- ties and interactions at the quantum mechanical level isthe realm of ab initio methods, such as density functional theory [ 12–14]. Density functional calculations of magnetic systems are mostly based on (nonrelativistic) spin-densityfunctional theory (SDFT) [ 15]. Within SDFT, the dominant source of magnetic coupling is exchange, which originatesfrom the Pauli exclusion principle and favors spin alignment(ferromagnetism). By minimizing the total energy functional,single-particle Kohn-Sham (KS) equations [ 16] are derived and solved self-consistently to determine the values of thecharge density and the magnetization density. The applica-bility of the method depends crucially on physically sound 2469-9950/2020/101(14)/144401(10) 144401-1 ©2020 American Physical SocietyCAMILLA PELLEGRINI et al. PHYSICAL REVIEW B 101, 144401 (2020) and numerically feasible approximations to the exchange- correlation (xc) part of the energy functional, which includesall quantum and many-body effects. The rotational invari-ance with respect to the spin quantization axis is broken byrelativistic corrections to the Hamiltonian, i.e., dipole-dipoleinteraction and spin-orbit coupling. Both are of the same order(1/c 2) in the weakly relativistic expansion of the full La- grangian of quantum electrodynamics [ 17]. Spin-orbit effects, which are responsible for magnetocrystalline anisotropy, areoften taken into account in practice by using nonrelativis-tic SDFT functionals together with Dirac- or Pauli-type KSequations. However, magnetic dipole-dipole interactions arecurrently not included in the formalism. SDFT calculationsin the local spin-density approximation (LSDA) and beyond,have proven over the years to yield reliable results for largeclasses of magnetic materials, with collinear and noncollinearspin alignment. Nevertheless, based on the recent experimen-tal advances, we propose in this paper a density functionaltreatment of the dipolar interaction as a pairwise interaction,with associated quantum effects. We start with the weakly relativistic Hamiltonian ˆH=ˆT+/integraldisplay d 3r{ˆn(r)vext(r)+μBˆm(r)·Bext(r)} +e2 2/integraldisplay d3r/integraldisplay d3r/primeˆn(r)ˆn(r/prime) |r−r/prime|+ˆHSS, (3) which includes, beyond the usual Hamiltonian of spin-density functional theory, the mutual 1 /c2interaction ˆHSSbetween the spin magnetic moments of the electrons [ 18]. This term has the form [ 17,19]: ˆHSS=−2πμ2 B/integraldisplay d3r/integraldisplay d3r/primeˆmi(r)δ⊥ ij(r−r/prime)ˆmj(r/prime), (4a) δ⊥ ij(r−r/prime)=2 3δijδ(r−r/prime)+dij(r−r/prime), (4b) representing the interaction −μ2 B 2/integraltext d3rˆm(r)·ˆB(r)o ft h e magnetization density ˆm(r) with the magnetic induction ˆBi(r)=4π/integraltext d3r/primeδ⊥ ij(r−r/prime)ˆmj(r/prime) generated by the magne- tization distribution within the sample. Here, the magneti-zation density operator is defined as ˆm(r)=ˆψ †(r)σˆψ(r), where ˆψ(r),ˆψ†(r) are the usual Pauli spinor field operators andσis the vector of Pauli matrices. δ⊥denotes the transverse delta function [ 20,21],μB=e¯h/(2mec) is the Bohr magneton. Repeated indices are assumed to be summed over. As onecan see, Eq. ( 4a) is the sum of two contributions. The first contribution, coming from the first term of Eq. ( 4b), ˆH SC=−4πμ2 B 3/integraldisplay d3rˆm(r)·ˆm(r), (5) is a contact interaction, which depends on the magnetization density at the same point. This is the counterpart of the Fermicontact interaction between an electron and a nucleus. Thesecond contribution, coming from the second term of Eq. ( 4b), ˆH dip=−2πμ2 B/integraldisplay d3r/integraldisplay d3r/primeˆmi(r)dij(r−r/prime)ˆmj(r/prime),(6)represents the dipolar interaction between two spin densities. Here, the tensor dijis defined as follows dij(r−r/prime)≡−1 4π∇i∇/prime j1 |r−r/prime|+1 3δijδ(r−r/prime) =1 4πR3(3¯Ri¯Rj−δij), (7) where R=r−r/primeis the relative position of the electrons, and ¯Rdenotes the unit vector along R. The expression for dij is understood to be regularized at R=0[19,21]. Physically, Eq. ( 6) together with Eq. ( 7) describe the interaction between the magnetization density at rand the dipolar field created by the magnetization distribution at all the other points r/prime/negationslash=r. Nevertheless, the contact term δijin Eq. ( 7) is included to ensure that the diagonal elements of dijsatisfy the Laplace equation /Delta1(1/|r−r/prime|)=−4πδ(r−r/prime) for the scalar poten- tial generated by the magnetic charge density in the system.The dipolar tensor, as defined by Eq. ( 7), is both traceless and symmetric. The present paper is organized as follows: In Sec. II Awe point out the equivalence between the magnetostatic energycontribution implemented in the micromagnetic approach andthe classical Hartree term of SDFT for the dipolar interaction.The SDFT framework then naturally leads to the inclusionof a dipolar xc functional to account for quantum effects. InSec. II Bwe address the adequacy of a LSDA for the dipolar xc energy. In Sec. II Cwe derive a nonlocal and noncollinear exchange energy functional as leading order quantum correc-tion to the magnetostatic energy. In Sec. IIIwe briefly consider the density functional treatment of the spin contact interactionof Eq. ( 4b). II. DIPOLE-DIPOLE INTERACTION FUNCTIONAL A. Hartree energy functional The Hartree term for the magnetic dipole-dipole interaction has been derived by Jansen [ 22] in a broader analysis of magnetic anisotropy contributions within the framework ofdensity functional theory. The dipolar energy in the Hartreeapproximation is obtained by simply replacing the magne-tization density operator ˆm(r)i nE q s .( 6) and ( 7)b yt h e expectation value m(r): E dip H[m]=−μ2 B 2/integraldisplay d3r/integraldisplay d3r/prime ×3/parenleftbig m(r)·¯R/parenrightbig/parenleftbig m(r/prime)·¯R/parenrightbig −m(r)·m(r/prime) R3.(8) The Hartree method, in which the dipolar interaction is taken into account by a mean field type of approximation,is qualitatively equivalent to the micromagnetic approach.[Formally, this can be seen by rewriting Eq. ( 1)a sE magstat = −μ0 2/integraltext d3r/integraltext d3r/primeM(r)·N(r−r/prime)·M(r/prime), where the demag- netizing tensor for a ferromagnetic body of arbitrary shapeis given by N(r−r /prime)=−1 4π∇∇/prime(1/|r−r/prime|).] Differences in the dipolar energy calculated from Eq. ( 8) and from the micromagnetic formula ( 1) are due to deviations of the ac- tual atomic distribution m(r) from its average M(r) over a mesoscopic cell of atomic volumes. We emphasize that, sofar, only this mean field contribution to the dipolar energy 144401-2DENSITY FUNCTIONAL THEORY OF MAGNETIC DIPOLAR … PHYSICAL REVIEW B 101, 144401 (2020) FIG. 1. Diagrammatic expansion of the exchange-correlation en- ergy of the homogeneous electron gas with dipolar spin-spin interac-tions, up to second order. (a) Exchange energy diagram. (b),(d) Direct and (c),(e) exchange diagrams contributing in second order to the correlation energy. Dashed lines indicate dipolar interactions; wigglylines indicate Coulomb interactions. has been implemented in actual calculations of magnetic structures. However, the Hartree treatment of a pairwise inter- action is usually a crude approximation (see, e.g., the case of the Coulomb interaction), as it completely neglects quantummany-body effects and is affected by a self-interaction error.In the next sections we discuss how to derive an improvedestimate of the real dipolar energy by adding, as a natural stepof the SDFT formalism, an approximate exchange-correlationterm. B. LSDA The approximation to the exchange ( x) energy functional most widely used in SDFT is the local spin-density approxi-mation [ 15]. In LSDA, the xenergy of a nonuniform magnetic system is approximated at each point by the xenergy of the homogeneous electron gas (HEG) with the same spindensity as the local density. Choosing a coordinate systemwith the zaxis along the direction of the local spin, the xenergy density of the spin polarized nonrelativistic HEG with dipolar interactions [see diagram (a) of Fig. 1] can be evaluated as e dip x(r)=μ2 B 2/integraldisplay d3r/primeραβ(R)σi ναdij(R)σj βμρμν(−R),(9) where ραβ(R)=/integraltext d3kψ† kσ(rα)ψkσ(r/primeβ) is the one-body den- sity matrix with spin orbitals ψkσ(rα)=1/√ Veik·rδσα. Since the HEG is spherically symmetric, the density matrix dependsonly on the modulus of the distance, i.e., ρ(R)=ρ(R). It is then easy to show that Eq. ( 9) gives a null contribution, as one essentially integrates a spherical harmonic with l=2 and m=0 over all angles. We thereby conclude that for the HEG, regardless of the spin polarization, the leadingrelativistic correction to the energy due to the dipole-dipoleinteraction vanishes. We point out that this is a generalproperty of the interaction, and the obtained result is notaffected by employing Dirac spinors for the electron fieldoperators.(a) (b) (c) FIG. 2. First order Feynman diagrams for the spin density re- sponse function with magnetic dipole-dipole interaction (dashed line). The gray dots represent the external magnetic fields. Nevertheless, the correlation energy of the dipolar HEG turns out to be finite. In second order it arises entirely fromdiagram (b) of Fig. 1, which readily translates into the Møller- Plesset (MP2) correlation energy per electron E (MP2)dip c =−e4¯h2k4 F 2m3ec4π243−46 ln 2 525. (10) For the same reason discussed above, in fact, diagram (c) vanishes, and it can be proven, in general, that all the diagramsinvolving one single dipole interaction line [thus includingalso diagrams (d) and (e) of Fig. 1], do not contribute to the energy. C. Nonlocal exchange energy functional We proceed to derive nonlocal corrections to the LSDA for the dipolar xenergy functional. Corrections to the stan- dard LSDA in SDFT are systematically constructed from theweakly inhomogeneous electron gas via the gradient expan-sion and the linear response. Here, we follow the second strat-egy, which allows one to account for small variations of m(r) at any r. We thus consider the (spin-unpolarized) dipolar HEG subject to a weak external magnetic field δV j q(r)∝eiq·rσj, which perturbs the magnetization density from the averagevalue m jtomj+δmj(r). The wave vector qis assumed to be arbitrary, so that the approach is fully noncollinear.By expanding the dipolar xenergy to second order in the deviation from the homogeneous limit, we have [ 14,23–25] E dip x[m]=μ2 B 2/integraldisplayd3q (2π)3Kij x(q)δmi(q)δmj(−q), (11) where δmi(q) is the induced magnetization density variation (to be obtained from an actual calculation), and the xkernel is given by Kij x(q)≡∂2Edip x ∂mi(q)∂mj(−q)=gkl(q) χik 0(q)χjl 0(q). (12) Here, Kij xis expressed in terms of the Lindhard response function of the HEG χik 0=∂mi/∂Vkand the linear response contribution to the dipolar xenergy functional gkl(q)≡ ∂2Edip x/∂Vk q∂Vl −q. The latter is represented diagrammatically in Fig. 2. Here, the vertex correction diagram (a) corresponds 144401-3CAMILLA PELLEGRINI et al. PHYSICAL REVIEW B 101, 144401 (2020) to the analytic expression glk(q,0)=2π β2/summationdisplay n,m/integraldisplayd3k (2π)3/integraldisplayd3k/prime (2π)3vij k−k/primeσi αδG0δ/epsilon1(k,i/epsilon1n)σl /epsilon1ηG0ηγ(|k+q|,i/epsilon1n)σj γβG0 βζ(|k/prime+q|,i/epsilon1/prime m)σk ζθG0θα(k/prime,i/epsilon1/prime m), (13) where vij k=(δij−3¯ki¯kj)/3 is the Fourier transform [ 19] of the dipolar tensor of Eq. ( 7), and G0 αβ(k,iωn)=δαβ/(iωn−εk)i s the unperturbed Matsubara Green’s function for the paramagnetic electron gas. Summing over the spin indices in Eq. ( 13)g i v e s Tr{σiσlσjσk}=4δilδjk. (14) From Eq. ( 14), since the HEG is isotropic, we observe that Eq. ( 13) can be written in the following form gij(q,0)=f(q)(δij−3¯qi¯qj), (15) where f(q) denotes a function of the modulus of qonly, and the angular dependence of gon the indices of qis through the traceless and symmetric interaction tensor vij k−k/prime. By assuming qin the direction of the zaxis, i.e., q=q¯z, we evaluate the function f(q)=−gzz(q¯z)/2 (note that one also has gxx=gyy=−1/2gzz). Summing over the Matsubara frequencies gives f(q)=8π 3/integraldisplayd3k (2π)3/parenleftbiggnk−nk+q εk−εk+q/parenrightbigg/integraldisplayd3k/prime (2π)3/parenleftbiggnk/prime−nk/prime+q εk/prime−εk/prime+q/parenrightbigg P2(cosθk−k/prime), (16) where nkis the Fermi distribution function and P2(cosθk)=1/2(3 cos2θk−1) is the Legendre polynomial of second order with cosθk=¯k·¯z. The main result of this paper is the exact evaluation of Eq. ( 16) in terms of one quadrature. We notice that by means of the transformations k(/prime)→± k(/prime)−q/2, it is possible to recast the ∼cos2θk−k/primeterm of Eq. ( 16) in the form I(q)=m2 e 8π5¯h4q2/integraldisplay d3k/integraldisplay d3k/primenk−q/2nk/prime−q/2 (k·q)(k/prime·q)/braceleftBigg/bracketleftbiggq·(k+k/prime) |k+k/prime|/bracketrightbigg2 +/bracketleftbiggq·(k−k/prime) |k−k/prime|/bracketrightbigg2/bracerightBigg , (17) which looks structurally similar to the response function of the electron gas with Coulomb interaction [ 26,27]. In evaluating Eq. ( 17) we generalize the analytic derivation presented in Ref. [ 27] (see Appendix Afor details). The additional term in Eq. ( 16) simply amounts to the square of the Lindhard function χ0, so that Eq. ( 16) reads as f(q)=I(q)−π 3χ2 0(q). In the zero temperature limit we obtain for f(q) the following expression: f(q)=m2 ek2 F 16π3¯h4q2/braceleftbigg2 45q(7q5−15q4+30q3−20q2−144) ln |a|+2 45q(7q5+15q4+30q3+20q2+144) ln b +4 45q2(7q2+60) ln2 q+16 45(11q2−18)−2 3q/bracketleftbigg (2b)3lnb/parenleftbigg lnb+ln2 q/parenrightbigg −(2a)3ln|a|/parenleftbigg ln|a|+ln2 q/parenrightbigg/bracketrightbigg +8/integraldisplayb −adzzln|z|[(a+z)(b−z)W1(z)−(b+z)(z−a)W2(z)]−4 3/parenleftbigg q+abln/vextendsingle/vextendsingle/vextendsingleb a/vextendsingle/vextendsingle/vextendsingle/parenrightbigg 2/bracerightbigg , (18) where W1(z)=ln|z+a z−b|andW2(z)=ln|z−a z+b|, with a=1−q/2 and b=1+q/2 in units of the Fermi vector kF. The remaining self-energy diagrams (b) and (c) of Fig. 2give a null contribution to the dipolar xenergy (as it can be easily checked by evaluating the trace over the spin indices). The physical explanation for this result is that diagram (a), including twotriplet Green’s functions, correspond to the Fock ( x) energy diagram for a magnetic HEG, whereas both diagrams (b) and (c) contain one singlet Green’s function [ 28]. For completeness we show here the expansions of f(q) at small and large q: f(q)=⎧ ⎨ ⎩m2 ek2 F 1080π3¯h4/bracketleftbig(127+60 log 2 −60 log q)q2 5−97q4 70−53q6 392+.../bracketrightbig ,q→0 16m2 ek2 F 675π3¯h4/parenleftbig25 q4+11 q6+.../parenrightbig ,q→∞.(19) From Eq. ( 19) we observe that the second derivative of the functional has a logarithmic divergence in the low wave-vector limit q=0. This nonanalyticity implies the nonexis- tence of standard semilocal gradient approximations and canbe traced back to the nonlocal character of the interaction. In Fig. 3we show the function f(q) and the xkernel K zz x(q) computed from Eq. ( 12)a trs=1. At large q,Kzz x(q) tends to a constant value. In this limit the dipolar xenergy amounts to one quarter of the magnetostatic energy. In Figs. 3(b)–3(d) we compare the Fourier transform of Kzz x(q) (see Appendix C)to the interaction between classical magnetic dipoles, 4 πdzz, for different values of the charge density. In the short-distancelimit ( R→0) both K zz x(R) and the classical dipole interaction increase as 1 /R3[Fig. 3(b)]. At distances of few atomic units thexkernel decays, faster at higher charge density, while developing an oscillatory behavior [Fig. 3(c)]. The oscilla- tions are readily seen in the ratio between Kzz x(R) and 4 πdzz [Fig. 3(d)], with a period that depends strongly on the value of the density. At high density ( rs=1) the oscillations are fast, with a period of few atomic units, whereas at low density 144401-4DENSITY FUNCTIONAL THEORY OF MAGNETIC DIPOLAR … PHYSICAL REVIEW B 101, 144401 (2020) 0 0.5 1 R (a.u)-400 -200 0 200 400Kzzx,4πdzz(a.u.)Hartree 1 5 10-0.02 -0.01 0 0.01 0.02rs=1 rs=3 rs=5 rs=10 0 5 10 q/kF 0 0.01 0.02 0.03f (a.u) 0 5 10 q/kF-3 -2 -1 0Kzzx (a.u) 0 1 02 03 04 0 50 60-0.2 -0.1 0 0.1Kzzx/4πdzz (a) (b) (c) (d)Exchange FIG. 3. Real and reciprocal space behavior of the dipolar exchange kernel. qandRare taken along the zaxis; qis measured in units of the Fermi vector kF,Ris given in atomic units (Bohr). (a) Dipolar exchange kernel Kzz x(q) (black line) and energy functional f(q)( d a s h e d line) at rs=1. (b) Short-distance Kzz x(R)a trs=1 (short-dashed line) compared to the Hartree (magnetostatic) contribution (green line). (c) Mid-distance Kzz x(R)a trs=1 (short-dashed line), 3 (long-dashed line), 5 (dot-dashed line), and 10 (red line) compared to the Hartree (magnetostatic) contribution (green line). (d) Ratio between dipolar exchange and Hartree at rs=1 (dashed line) and rs=10 (red line). (rs=10) they lie in the long period range and eventually disappear in the limit rs→∞ . The oscillation amplitude (of the ratio) decreases slowly with Rand, for realistic values of rs, remains sizable ( ≈5%) even at large interatomic distances. We expect that in conventional magnets, with mostly simplemagnetic patterns and domain wall geometries, these oscil-latory corrections to the magnetostatic energy will averageto zero, especially in the high density limit. However, inmore complex magnetic configurations, such as layered andfrustrated systems [ 8,29,30], the oscillatory xenergy may sum up leading to sizable and measurable effects. III. SPIN-SPIN CONTACT FUNCTIONAL As mentioned in Sec. I, in addition to the dipolar term, the magnetic interaction between two electrons includes alsoa spin-spin contact term [Eq. ( 5)]. This contact interaction has the same form of the Coulomb exchange interaction butis rescaled by the smaller factor μ 2 B. The associated Hartree functional is easily obtained as ESC H=−4πμ2 B 3/integraldisplay d3rm2(r), (20) while the LSDA for the xenergy is given by ESC x=2πμ2 B/integraldisplay d3r/bracketleftbigg n2(r)−1 3m2(r)/bracketrightbigg , (21) where nis the total electron density. IV . CONCLUSIONS We have proposed a density functional treatment of the dipolar interaction between electronic spin magnetic mo-ments. Within this approach, the dipolar Hartree term isgiven by the classical magnetostatic energy, currently imple-mented in magnetic structure codes. In addition, we havederived quantum corrections by evaluating analytically theexact xenergy (Fock term) for the magnetic electron gas with spin-spin interactions. The dipolar xenergy thereby obtained amounts to one quarter of the magnetostatic energy at shortinteraction distance, or in the limit of low electronic density.At long range, the dipolar xkernel displays an oscillatory behavior, while decaying in amplitude slightly faster than theclassical contribution [see Fig. 3(d)]. This quantum correction is expected to have negligible effects in most conventionalmagnetic materials, where it likely averages to zero. However,it might become significant in complex magnetic structures,especially in specific geometries where lattice and dipolexoscillations are commensurate, or in the presence of a delicate magnetic balance, like in frustrated systems. Furtherprogress in the functional approximation might be achievedby carrying out the Levy constrained search [ 31,32]f o rt h e exact functional F[n,m] =min /Psi1→n,m/angbracketleftbigg /Psi1|ˆT+e2 2/integraldisplay d3r/integraldisplay d3r/primeˆn(r)ˆn(r/prime) |r−r/prime|+ˆHSS|/Psi1/angbracketrightbigg , (22) via a stochastic minimization [ 33] over the many-body wave functions /Psi1that are eigenstates of the total (spin +orbital) an- gular momentum ˆJ2. Upcoming work comprises implement- ing and testing the new functional against experimental data.Applications include the study of crystalline layered magnetsand magnetic atoms on surfaces, as well as the dynamics ofdomain walls and skyrmions. Of particular interest is also theapplication of our functional to the physics of dipolar quantumgases, where it might serve as an exchange partner for the“strictly correlated particles” functional of Ref. [ 34]. ACKNOWLEDGMENTS We acknowledge financial support by the European Re- search Council Advanced Grant No. FACT (ERC-2017-AdG-788890) and German Research Foundation (DFG) throughSPP 1840 QUTIF, Grant No. 498/3-1. 144401-5CAMILLA PELLEGRINI et al. PHYSICAL REVIEW B 101, 144401 (2020) APPENDIX A: EVALUATION OF I(q) For convenience we evaluate Eq. ( 17) in cylindrical coordinates with the polar axis along q, where all the wave vectors are measured in units of kF. The integrations over the azimuthal and radial coordinates of kandk/primeare readily carried through obtaining I(q)=m2 ek2 F 16π3¯h4q23/summationdisplay i=0Ji, (A1) where J0=− 2/integraldisplay/integraldisplayb −adzdz/prime zz/prime[(z2+z/prime2)(λ+λ/prime)(2 ln 2 +1)+z4+6z2z/prime2+z/prime4], (A2) J1=2/integraldisplay/integraldisplayb −adzdz/prime zz/prime[α2/radicalbig R(z,z/prime)+β2|β|], (A3) J2=4/integraldisplay/integraldisplayb −adzdz/prime zz/primeλ[α2ln|2/radicalbig R(z,z/prime)+λ/prime−λ+α2|+β2ln|2|β|+λ/prime−λ+β2|], (A4) J3=−4/integraldisplay/integraldisplayb −adzdz/prime zz/primeλ[β2ln|β2|+α2ln|α2|]. (A5) We adopt the same notation as in Ref. [ 27]. Here, a=1−q/2,b=1+q/2,α=z+z/prime,β=z−z/prime, andλ(/prime)=(a+z(/prime))(b− z(/prime)). The function R is defined as R(z,z/prime)=C0(z)z/prime2+B0(z)z/prime+A0(z), where A0=z2,B0=(2+2qz−q2)z, and C0=1+ 2qz. Evaluating J0is straightforward and the resulting expression is J0=−(2+ln 2)8 q2−2q/bracketleftbigg q2ln 2−4 3(4+5l n2 )/bracketrightbigg ln/vextendsingle/vextendsingle/vextendsinglea b/vextendsingle/vextendsingle/vextendsingle. (A6) J 1can be rewritten in the following form J1=4/integraldisplay/integraldisplayb −adzdz/prime/parenleftbiggα z/prime/radicalbig R(z,z/prime)+β z/prime|β|/parenrightbigg =4/integraldisplayb −adz/parenleftbig¯JA 1(z)+¯JB 1(z)/parenrightbig , (A7) where ¯JA 1(z) and ¯JB 1(z) are evaluated to be [ 27] ¯JA 1(z)=1+1 4q2+5 2qz+/parenleftbigg 2−ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−4 q2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg z2+B0 4C0(2z+q)+1 4C3/2 0z2[8−q4+4qz(6−q2)+12q2z2]Y(z), ¯JB 1(z)=2qz−1−q2 4−z2(3−2l n|z|+ln|ab|), with Y(z)=ln|√C0+1√C0−1|. The remaining integration in Eq. ( A7) can also be carried through obtaining J1=−1 q2−1 9+44 3q2+4/parenleftbigg4 3+q2/parenrightbigg lnq 2+1 3[(q−2)3lnb−(q+2)3ln|a|]+1 2q3(q2−1)2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleq+1 q−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle +3 4q3η5−1 2qη3−/parenleftbigg5 2q3−3 2q+q 4/parenrightbigg η1−/parenleftbigg3 2q−2 q3−q 2/parenrightbigg η−1+/parenleftbigg1 2q−1 4q3−q 4/parenrightbigg η−3, (A8) where ηn=q/integraltextb −adzCn/2 0Y(z). The explicit expressions for η±1,−3are given in Ref. [ 27], forη3,5in Appendix B. Next, we evaluate J23=J2+J3. This term is conveniently rewritten as J23=4/integraldisplay/integraldisplayb −adzdz/primeλ zz/prime/parenleftbig α2+β2/parenrightbig ln|4λ|−4/integraldisplayb −adzλ z[¯N1(z)+¯N2(z)], (A9) where ¯N1(z) and ¯N2(z) are defined as follows: ¯N1(z)=/integraldisplayb −adz/primeα2 z/primeln|α2+λ/prime−λ−2/radicalbig R(z,z/prime)|, (A10) ¯N2(z)=/integraldisplayz −adz/primeβ2 z/primeln|2β(z−b)|+/integraldisplayb zdz/primeβ2 z/primeln|2β(z+a)|. (A11) Equations ( A10) and ( A11) can be integrated by parts obtaining ¯N1(z)=/integraldisplayb −adz/prime α/parenleftbigg z2ln|z/prime|+1 2z/prime2+2zz/prime/parenrightbigg/parenleftbiggqz√R(z,z/prime)−1/parenrightbigg +/parenleftbigg z2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleb a/vextendsingle/vextendsingle/vextendsingle/vextendsingle+q+4z/parenrightbigg ln|2λ| (A12) ¯N2(z)=/parenleftbigg z2ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleb a/vextendsingle/vextendsingle/vextendsingle/vextendsingle+q−4z/parenrightbigg ln|2λ|+/parenleftbigg3 2z2−ln|z|z2/parenrightbigg W1(z)+/integraldisplayb −adz/prime/parenleftbigg z2ln|z/prime|+1 2z/prime2−2zz/prime/parenrightbigg1 β, (A13) 144401-6DENSITY FUNCTIONAL THEORY OF MAGNETIC DIPOLAR … PHYSICAL REVIEW B 101, 144401 (2020) where we have used the notation W1(z)=ln|z+a z−b|. Subsequent substitution of Eqs. ( A12) and ( A13)i nE q .( A9)g i v e s J23=4q/bracketleftbigg q+/parenleftbigg ab+2 3/parenrightbigg ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleb a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg (2l n2+1)−8 3qln/vextendsingle/vextendsingle/vextendsingle/vextendsingleb a/vextendsingle/vextendsingle/vextendsingle/vextendsingle+6/integraldisplay b −adzλzW2(z)−4(q/Phi11+2/Phi12+q/Phi13−/Phi14). (A14) Here, we have defined W2(z)=ln|z−a z+b|, /Phi11=/integraldisplayb −adzλ/integraldisplayb −adz/prime/parenleftbigg1 2z/prime2+2zz/prime/parenrightbigg1 α√R(z,z/prime), (A15) /Phi12=/integraldisplayb −adzλz/integraldisplayb −adz/primez/prime αβln|z/prime|, (A16) /Phi13=/integraldisplayb −adzλ/integraldisplayb −adz/primez2ln|z/prime|1 α√R(z,z/prime), (A17) /Phi14=/integraldisplayb −adzλzW1(z)l n|z|. (A18) By writing /Phi11as /Phi11=1 2/integraldisplayb −adzλ/integraldisplayb −adz/prime 1√R(z,z/prime)/bracketleftBig z/prime+3z/parenleftBig 1−z α/parenrightBig/bracketrightBig , (A19) and performing the integrations over z/prime /integraldisplayb −adz/primez/prime √R(z,z/prime)=1 C0(2z+q)−B0 C3/2 0Y(z), (A20) /integraldisplayb −adz/prime 1√R(z,z/prime)=2√C0Y(z), (A21) /integraldisplayb −adz/prime 1 α√R(z,z/prime)=−1 qzW2(z), (A22) we get /Phi11=1 2/integraldisplayb −adzλ√C0/bracketleftbigg2z+q√C0+/parenleftbigg 6z−B0 C0/parenrightbigg Y(z)/bracketrightbigg +3 2q/integraldisplayb −adzλzW2(z). (A23) The last term in Eq. ( A23) cancels with the same contribution of opposite sign in Eq. ( A14). The remaining integrals can be carried out as follows 1 2/integraldisplayb −adzλ2z+q C0=1 24q4/bracketleftbigg −6q+16q3+6q5−3(q2−1)3ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleq+1 q−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketrightbigg , (A24) −1 2/integraldisplayb −adzλY(z) C3/2 0[B0−6zC0]=−5 16q4η5+/parenleftbigg9 16q2+1 q4/parenrightbigg η3−/parenleftbigg3 16−1 16q2+9 8q4/parenrightbigg η1 −/parenleftbiggq2 16−3 8+13 16q2−1 2q4/parenrightbigg η−1+/parenleftbigg3 16q2−1 16q4−3 16+q2 16/parenrightbigg η−3. (A25) We then write Eq. ( A16)a s /Phi12z↔z/prime =−/integraldisplayb −adzzln|z|/integraldisplayb −adz/primeλ/primez/prime αβ(A26) =−/integraldisplayb −adzzln|z|/bracketleftbigg (b+z)(a−z)/integraldisplayb −adz/primez/prime αβ+(q+z)/integraldisplayb −adz/primez/prime β−/integraldisplayb −adz/primez/prime2 β/bracketrightbigg , (A27) where each of the integrations in z/primecan be performed /integraldisplayb −adz/primez/prime αβ=1 2(W1(z)+W2(z)), (A28) /integraldisplayb −adz/primez/prime β=−2+zW1(z), (A29) /integraldisplayb −adz/primez/prime2 β=1 2[a(a−2z)−b(b+2z)]+z2W1(z). (A30) 144401-7CAMILLA PELLEGRINI et al. PHYSICAL REVIEW B 101, 144401 (2020) Substituting Eqs. ( A28)–(A30)i nE q .( A27), and carrying through the elementary integrations over z, we obtain the following result for /Phi12in terms of one quadrature /Phi12=−1 2/integraldisplayb −adzz[λW1(z)−(b+z)(z−a)W2(z)]ln|z|−1 2q(q+a2ln|a|−b2ln|b|). (A31) We follow the same procedure for /Phi13given in Eq. ( A17) /Phi13z↔z/prime =/integraldisplayb −adzln|z|/integraldisplayb −adz/primeλ/primez/prime2 α√R(z,z/prime) =/integraldisplayb −adzln|z|/bracketleftbigg (b+z)(a−z)/integraldisplayb −adz/prime z/prime2 α√R(z,z/prime)+(q+z)/integraldisplayb −adz/primez/prime2 √R(z,z/prime)−/integraldisplayb −adz/primez/prime3 √R(z,z/prime)/bracketrightbigg . (A32) Here, we have /integraldisplayb −adz/prime z/prime2 α√R(z,z/prime)=1 C0(2z+q)−1 C3/2 0(B0+2zC0)Y(z)−z qW2(z), (A33) /integraldisplayb −adz/primez/prime2 √R(z,z/prime)=/parenleftbiggb 2C0−3B0 4C2 0/parenrightbigg/radicalbig R(z,b)+/parenleftbigga 2C0+3B0 4C2 0/parenrightbigg/radicalbig R(z,−a)+2√C0/parenleftbigg3B2 0 8C2 0−A0 2C0/parenrightbigg Y(z), (A34) /integraldisplayb −adz/primez/prime3 √R(z,z/prime)=/parenleftbiggb2 3C0−5B0b 12C2 0+5B2 0 8C3 0−2A0 3C2 0/parenrightbigg/radicalbig R(z,b)−/parenleftbigga2 3C0+5B0a 12C2 0+5B2 0 8C3 0−2A0 3C2 0/parenrightbigg/radicalbig R(z,−a) −/parenleftbigg5B3 0 16C3 0−3A0B0 4C2 0/parenrightbigg2√C0Y(z). (A35) Substituting Eqs. ( A33)–(A35)i nE q .( A32), we obtain with some algebra /Phi13=¯/Phi11+¯/Phi12+¯/Phi13, (A36) where ¯/Phi11=−1 q/integraldisplayb −adzzln|z|(b+z)(a−z)W2(z), (A37) ¯/Phi12=/integraldisplayb −adz/bracketleftbigg −19 32q3C2 0+/parenleftbigg139 96q3−9 32q/parenrightbigg C0−15 16q3+25 16q−5 32q+/parenleftbigg1 16q3−3 4q+17 32q+q3 32/parenrightbigg C−1 0 +/parenleftbigg −1 16q−13 96q3+5q 32+q3 24/parenrightbigg C−2 0+/parenleftbigg5 32q3−15 32q+15 32q−5 32q3/parenrightbigg C−3 0/bracketrightbigg ln|z|, (A38) ¯/Phi13=/integraldisplayb −adzC−7/2 0/bracketleftbigg/parenleftbigg −4+2q2−q4 4/parenrightbigg z+/parenleftbigg −16q+11 2q3−q5 4/parenrightbigg z2+/parenleftbigg 8−20q2+11 2q4−q6 8/parenrightbigg z3 +/parenleftbigg 36q+2q3+5 4q5/parenrightbigg z4+/parenleftbigg 60q2+25 2q4/parenrightbigg z5+35q3z6/bracketrightbigg ln|z|Y(z). (A39) Evaluating ¯/Phi12is elementary. Moreover, it can be shown [ 27] that ¯/Phi13is equivalent to ¯/Phi13=1 8q3/summationdisplay n=−3γn1 2n+1/bracketleftbigg (1+q)2n+1lnbln/vextendsingle/vextendsingle/vextendsingle/vextendsingle2b q/vextendsingle/vextendsingle/vextendsingle/vextendsingle−˜q2n+1ln|a|ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜q+1 ˜q−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/Omega1 n/bracketrightbigg , (A40) where γ3=35 8q3,γ2=−45 4q3+25 8q,γ1=69 8q3−117 8q+5 8q,γ0=−3 2q3+29 4q+3q−q3 8, γ−1=−3 8q3+11 4q−7 4q−q3 8,γ−2=3 4q3−3 8q−3q3 8,γ−3=−5 8q3+15 8q−15 8q+5q3 8. Here, ˜ q=|1−q|and the explicit expressions for /Omega10,±1are given in Ref. [ 27], while for /Omega1±2,±3in Appendix B. APPENDIX B η3=1 5[4q(2+q2)−2q(5+10q2+q4)l nq−2(1−2q+4q2−3q3+q4)aln|2a|+2(1+2q+4q2+3q3+q4)bln 2b], (B1) 144401-8DENSITY FUNCTIONAL THEORY OF MAGNETIC DIPOLAR … PHYSICAL REVIEW B 101, 144401 (2020) η5=1 7/bracketleftbigg 4q/parenleftbigg 3+13 3q2+q4/parenrightbigg −2q(7+35q2+21q4+q6)l nq−2(1−3q+9q2−13q3+11q4−5q5+q6)aln|2a| +2(1+3q+9q2+13q3+11q4+5q5+q6)bln 2b/bracketrightbigg , (B2) η−5=1 3/bracketleftbigg4q (q2−1)2+2l n/vextendsingle/vextendsingle/vextendsingle/vextendsingleq+1 q−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/parenleftbigg 1+1 (1+q)3/parenrightbigg ln 2b+/parenleftbigg 1+1 (1−q)3/parenrightbigg ln|2a|+2q(q2+3) (q2−1)3lnq/bracketrightbigg . (B3) /Omega1−3=h0(q)−2q/bracketleftbigg/integraldisplayb −adz/parenleftbig C−3 0+C−2 0+C−1 0/parenrightbig ln|z|/bracketrightbigg +2(η−3+η−5), (B4) where h0(q)i sg i v e ni nR e f .[ 27], /Omega1−2=h0(q)−2q/bracketleftbigg/integraldisplayb −adz/parenleftbig C−2 0+C−1 0/parenrightbig ln|z|/bracketrightbigg +2η−3, (B5) /Omega12=h0(q)−1 30[q(416+108q2)+q(240+120q2)l n2−q(60+300q+80q2+75q3+12q4)l nq +(2b)(92−16q+38q2+21q3+12q4)l n2 b−(˜q+1)(137 −77 ˜q+47 ˜q2−27 ˜q3+12 ˜q4)l n|˜q+1| +(˜q−1)(137 +77 ˜q+47 ˜q2+27 ˜q3+12 ˜q4)l n|˜q−1|], (B6) /Omega13=h0(q)+1 210/bracketleftBig −q/parenleftbigg 4472+9028 3q2+520q4/parenrightbigg −q(2520 +3640 q2+840q4)l n2+q(420+4410 q+1260 q2 +3675 q3+924q4+490q5+60q6)l nq−2b(704−142q+386q2+437q3+464q4+230q5+60q6)l n2 b +(˜q+1)(1089 −669 ˜q+459 ˜q2−319 ˜q3+214 ˜q4−130 ˜q5+60 ˜q6)l n|˜q+1|−(˜q−1)(1089 +669 ˜q +459 ˜q2+319 ˜q3+214 ˜q4+130 ˜q5+60 ˜q6)l n|˜q−1|. (B7) APPENDIX C: FOURIER TRANSFORM OF Kij x(q) The real-space representation of Eq. ( 11) is given by Edip x[m]=μ2 B 2/integraldisplay d3rd3r/primeKij x(R)δmi(r)δmj(r/prime), (C1) where Kij x(R)=gkl(R) χik 0(R)χjl 0(R). (C2) Here, gkl(R) is evaluated as follows. We write the Fourier transform of gkl(q) in spherical coordinates as: gkl(R)=1 (2π)3/integraldisplay∞ −∞gkl(q)eiq·Rd3q=1 (2π)3/integraldisplay∞ 0dq/integraldisplay2π 0dφ/integraldisplayπ 2 −π 2dθf(q)2P2(cosθ)eiq·Rq2sinθ. (C3) Then, by expressing the exponential as a Rayleigh expansion, and using the addition theorem for spherical harmonics Ylm(θ,φ), we obtain: gkl(R)=2 (2π)2/integraldisplay2π 0/integraldisplayπ 2 −π 2∞/summationdisplay l=0l/summationdisplay m=−lY20(θ,φ)Ylm(θ,φ)sinθdφdθ/integraldisplay∞ 0f(q)iljl(qR)q2dq2P2(cosθR) (C4) =−2 (2π)2/integraldisplay∞ 0f(q)j2(qR)q2dq2P2(cosθR)=f(R)2P2(cosθR), (C5) where jl(qR) are the spherical Bessel functions and the function f(R) is defined by the last equality. [1] A. Steane, Rep. Prog. Phys. 61,117(1998 ). [2] L. You and M. S. 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PhysRevB.84.115302.pdf

PHYSICAL REVIEW B 84, 115302 (2011) Ab initio computation of the energies of circular quantum dots M. Pedersen Lohne,1G. Hagen,2,3M. Hjorth-Jensen,4S. Kvaal,5and F. Pederiva6 1Department of Physics, University of Oslo, N-0316 Oslo, Norway 2Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 3Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37831, USA 4Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway 5Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway 6Dipartimento di Fisica,Universit `a di Trento, and I.N.F .N., Gruppo Collegato di Trento, I-38123 Povo, Trento, Italy (Received 26 September 2010; revised manuscript received 21 December 2010; published 8 September 2011) We perform coupled-cluster and diffusion Monte Carlo calculations of the energies of circular quantum dots up to 20 electrons. The coupled-cluster calculations include triples corrections and a renormalized Coulombinteraction defined for a given number of low-lying oscillator shells. Using such a renormalized Coulombinteraction brings the coupled-cluster calculations with triples correlations in excellent agreement with thediffusion Monte Carlo calculations. This opens up perspectives for doing ab initio calculations for much larger systems of electrons. DOI: 10.1103/PhysRevB.84.115302 PACS number(s): 73 .21.La, 71 .15.−m, 31.15.bw, 02 .70.Ss I. INTRODUCTION Strongly confined electrons offer a wide variety of complex and subtle phenomena, which pose severe challenges toexisting many-body methods. Quantum dots, in particular (thatis, electrons confined in semiconducting heterostructures),exhibit, due to their small size, discrete quantum levels. Theground states of, for example, circular dots show similar shellstructures and magic numbers as seen for atoms and nuclei.These structures are particularly evident in measurements ofthe change in electrochemical potential due to the additionof one extra electron /Delta1 N=μ(N+1)−μ(N). Here, Nis the number of electrons in the quantum dot, and μ(N)= E(N)−E(N−1) is the electrochemical potential of the system. Theoretical predictions of /Delta1Nand the excitation energy spectrum require accurate calculations of ground-stateand excited-state energies. The above-mentioned quantum mechanical levels can, in turn, be tuned by means of, for example, the application of various external fields. The spins of the electrons in quantum dots provide a natural basis for representing so-called qubits. 1 The capability to manipulate and study such states is evidencedby several recent experiments (see, for example, Refs. 2 and 3). Coupled quantum dots are particularly interesting since so-called two-qubit quantum gates can be realized by manipulating the exchange coupling, which originates fromthe repulsive Coulomb interaction and the underlying Pauliprinciple. For such states, the exchange coupling splits singlet and triplet states, and depending on the shape of the confining potential and the applied magnetic field, one can allow forelectrical or magnetic control of the exchange coupling. Inparticular, several recent experiments and theoretical investiga-tions have analyzed the role of effective spin-orbit interactions in quantum dots 4–7and their influence on the exchange coupling. A proper theoretical understanding of the exchange cou- pling, correlation energies, ground-state energies of quantumdots, the role of spin-orbit interactions, and other proper-ties of quantum dots as well requires the development ofappropriate and reliable theoretical few- and many-body methods. Furthermore, for quantum dots with more than twoelectrons and/or specific values of the external fields, thisimplies the development of few- and many-body methodswhere uncertainty quantifications are provided. For mostmethods, this means providing an estimate of the error dueto the truncation made in the single-particle basis and thetruncation made in limiting the number of possible excitations.For systems with more than three or four electrons, ab initio methods that have been employed in studies of quantum dotsinclude variational and diffusion Monte Carlo, 8–11path integral approaches,12large-scale diagonalization (full configuration interaction),13–16and to a very limited extent coupled-cluster theory.17–20Exact diagonalization studies are accurate for a very small number of electrons, but the number of basisfunctions needed to obtain a given accuracy and the com-putational cost grow very rapidly with electron number. Inpractice, they have been used for up to eight electrons, 13,14,16 but the accuracy is very limited for all except N/lessorequalslant3 (see, for example, Refs. 15and21). Monte Carlo methods have been applied up to N=24 electrons.10,11Diffusion Monte Carlo, with statistical and systematic errors, provides, in principle,exact benchmark solutions to various properties of quantumdots. However, the computations start becoming rather timeconsuming for larger systems. Hartree, 22restricted Hartree- Fock, spin- and/or space-unrestricted Hartree-Fock23–25and local spin-density, and current density functional methods26–29 give results that are satisfactory for a qualitative understanding of some systematic properties. However, comparisons withexact results show discrepancies in the energies that aresubstantial on the scale of energy differences. Another many-body method with the potential of providing reliable error estimates and accurate results is coupled-clustertheory, with its various levels of truncations. Coupled-clustertheory is the method of choice in quantum chemistry,atomic and molecular physics, 17,18,30and has recently been applied with great success in nuclear physics as well (see,for example, Refs. 31–34). In nuclear physics, with our spherical basis codes, we expect now to be able to perform 115302-1 1098-0121/2011/84(11)/115302(15) ©2011 American Physical SocietyM. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) ab initio calculations of nuclei up to132Sn with more than 20 major oscillator shells. The latter implies dimensionalities ofmore than 10 100basis Slater determinants, well beyond the reach of the full configuration-interaction approach. Coupled-cluster theory offers a many-body formalism that allowsfor systematic expansions and error estimates in terms oftruncations in the basis of single-particle states. 35The cost of the calculations scale gently with the number of particlesand single-particle states, and we expect to be able to studyquantum dots up to 50 electrons without a spherical symmetry.The main advantage of the coupled-cluster method over, say,full configuration approaches relies on the fact that it offers anattractive truncation scheme at a much lower computationalcost. It preserves, at the same time, important features such assize extensivity. The aim of this work is to apply coupled-cluster theory with the inclusion of triples excitations through the highly ac-curate and efficient /Lambda1-coupled-cluster singles-doubles (triples) [/Lambda1-CCSD(T)] approach 36–39for circular quantum dots up toN=20 electrons, employing different strengths of the applied magnetic field. The results from these calculationsare compared in turn with, in principle, exact diffusion MonteCarlo calculations. Moreover, this paper introduces a techniquewidely applied in the nuclear many-body problem, namely,that of a renormalized two-body Coulomb interaction. Insteadof using the free Coulomb interaction in an oscillator basis,we diagonalize the two-electron problem exactly using atailor-made basis in the center-of-mass frame. 15The obtained eigenvectors and eigenvalues are used, in turn, to obtain, viaa similarity transformation, an effective interaction definedfor the lowest 10–20 oscillator shells. These shells define oureffective Hilbert space where the coupled-cluster calculationsare performed. This technique has been used with great successin the nuclear many-body problem, in particular, since thestrong repulsion at short interparticle distances of the nuclearinteractions requires a renormalization of the short-rangepart. 40,41With this renormalized Coulomb interaction and coupled-cluster calculations with triples excitations includedthrough the /Lambda1-CCSD(T) approach, we obtain results in close agreement with the diffusion Monte Carlo calculations. Thisopens up many interesting avenues for ab initio studies of quantum dots, in particular, for systems beyond the simplecircular quantum dots. This paper is organized as follows. Section IIintroduces (i) the Hamiltonian and interaction for circular quantum dots,(ii) the basic ingredients for obtaining an effective interactionusing a similarity-transformed Coulomb interaction, then(iii) a brief review of coupled-cluster theory and the /Lambda1- CCSD(T) approach, and finally (iv) the corresponding detailsbehind the diffusion Monte Carlo calculations. In Sec. III, we present our results, whereas Sec. IVis devoted to our conclusions and perspectives for future work. II. COUPLED-CLUSTER THEORY AND DIFFUSION MONTE CARLO We present first our Hamiltonian in Sec. II A; thereafter, we discuss how to obtain a renormalized two-body interactionin an effective Hilbert space. In Sec. II C, we present ourcoupled-cluster approach, and finally in Sec. II D, we briefly review our diffusion Monte Carlo approach. A. Physical systems and model Hamiltonian We will assume that our problem can be described entirely by a nonrelativistic many-electron Hamiltonian ˆH, resulting in the Schr ¨odinger equation ˆH|/Psi1/angbracketright=E|/Psi1/angbracketright, (1) with|/Psi1/angbracketrightbeing the eigenstate and Ethe eigenvalue. The many-electron Hamiltonian is normally written in terms ofa noninteracting part ˆH 0and and interacting part ˆV, namely, ˆH=ˆH0+ˆV=N/summationdisplay i=1ˆhi+N/summationdisplay i<jˆvij, where ˆH0is the (one-body) Hamiltonian of the noninteracting system, and ˆVdenotes the (two-body) Coulomb interaction. In general, the Schr ¨odinger equation ( 1) can not be solved exactly. We define the reference Slater determinant |/Phi10/angbracketrightas the ground state of the noninteracting system by filling allthe lowest-lying single-particle orbits. Since we will limitourselves to systems with filled shells, this may be a goodapproximation, in particular, if the single-particle field is thedominating contribution to the total energy. The noninteractingSchr ¨odinger equation reads as ˆH 0|/Phi1/angbracketright=e0|/Phi10/angbracketright, (2) where ˆH0=N/summationdisplay i=1ˆhi=N/summationdisplay i=1[ˆti+ˆvcon(ri)]. The terms ˆtiand ˆvcon(ri) are the kinetic-energy operator and the confining potential (from an external applied potential field) ofelectron i, respectively. The vector r irepresents the position in two dimensions of electron i. Due to the identical and fermionic nature of electrons, the eigenstates of Eq. ( 2)a r e Slater determinants, with the general form |/Phi1/angbracketright=|ijk . . . m /angbracketright=ˆa† ia† jˆa† k...ˆa† m|0/angbracketright, with ˆa†being standard fermion creation operators (and ˆabeing annihilation operators). The single-particle eigenstates |i/angbracketright= ˆa† i|0/angbracketrightand eigenenergies εiare given by the solutions of the one-particle Schr ¨odinger equation governed by the operator ˆhi. Since the total energy of the noninteracting system is given by the sum of single-particle energies /epsilon1i,w eh a v e e0=N/summationdisplay i=1εi, the reference determinant |/Phi10/angbracketrightis obviously the Slater deter- minant constructed from those orbitals with single-particleenergies that yield the lowest total energy. In the particle-holeformalism, orbitals in the occupied space are referred to ashole states, while orbitals in the virtual space are denotedparticle states. In principle, any complete and orthogonalsingle-particle basis can be used. However, since our coupled-cluster approach involves the solution of a set of nonlinear 115302-2Ab INITIO COMPUTATION OF THE ENERGIES OF ... PHYSICAL REVIEW B 84, 115302 (2011) equations, it is preferable to start from a basis that produces a mean-field solution not too far away from the “exact” and fullycorrelated many-body solution. Therefore, our main resultswill be obtained using the Hartree-Fock basis as a starting pointfor our coupled-cluster calculations. The Hartree-Fock basis isobtained from a linear expansion of harmonic-oscillator basisfunctions, such that the expectation value of the Hamiltonianis minimized. For the diffusion Monte Carlo calculations, it is also necessary to start from a model wave function that is usedas importance function in the sampling, as we will discusslater. The Slater determinant part, in this case, is builtstarting from the self-consistent orbitals generated in a localdensity approximation calculation in order to include as muchinformation as possible about both exchange and correlationeffects at the one-body level. Explicit two-body correlationsare then included as an elaborate Jastrow factor; see Sec. II D for further details. Our model Hamiltonian 42for a quantum dot consists of a two-dimensional system of Nelectrons moving in the z= 0 plane, confined by a parabolic lateral confining potentialV con(r). The Hamiltonian is ˆH=N/summationdisplay i=1/parenleftbigg −¯h2 2mem∗∇2 i+Vcon(ri)/parenrightbigg +e2 /epsilon1N/summationdisplay i<j1 |ri−rj|. In the above equation, m∗is a parameter relating the bare electron mass to an effective mass, and /epsilon1is the dielectric constant of the semiconductor. In the following (if notexplicitly specified otherwise), we will use effective atomicunits, defined by ¯ h=e 2//epsilon1=mem∗=1. In this system of units, the length unit is the Bohr radius a0times/epsilon1/m∗, and the energy has units of Hartrees times m∗//epsilon12. As an example, for the GaAs quantum dots, typical values are /epsilon1=12.4 andm∗=0.067. The effective Bohr radius a∗ 0and effective Hartree H∗are/similarequal97.93˚A and /similarequal11.86 meV , respectively. In this work, we will consider circular dots only with N=2, 6, 12, and 20 electrons confined by a parabolic potentialV con(r)=mem∗ω2r2/2. The numbers N=2, 6, 12, and 20 are so-called magic numbers corresponding to systems withclosed harmonic-oscillator shells, and hopefully a single-reference Slater determinant yields a good starting point forour calculations. Although the emphasis here is on closed-shellsystems, we show also results for systems with one particleattached to or removed from a closed-shell system for N=3, 5, 7, 11, and 13. The one-body part of our Hamiltonian becomes ˆH 0=N/summationdisplay i=1/parenleftbigg −1 2∇2 i+ω2 2r2 i/parenrightbigg , whereas the interacting part is ˆV=N/summationdisplay i<j1 |ri−rj|. The unperturbed part of the Hamiltonian yields the single- particle energies /epsilon1i=ω(2n+|m|+1), (3)where n=0,1,2,3,... andm=0,±1,±2,... . The index iruns from 0 ,1,2,... . The shell structure is clearly deduced from this expression. We define Ras the shell index. We will denote the shell with the lowest energy as R=1, the shell with the second lowest energy as R=2, and so forth. Hence, Ri≡/epsilon1i−1 ω(i=1,2,3,...). (4) In the calculations, we limit ourselves to values of ω= 0.28 a.u. (atomic units), ω=0.5 a.u., and ω=1.0a . u .F o r higher values of the oscillator frequency, the contribution tothe energy from the single-particle part dominates over thecorrelation part. The value ω=1.0 is an intermediate case, which also allows for comparison with Taut’s exact solutionforN=2( s e eR e f . 43), while ω=0.5 and 0 .28 represent cases where correlations are stronger, due to the lower averageelectron density in the dot. B. Effective interaction Whenever a single-particle basis is introduced in order to carry out a many-body calculation, it must be truncated.The harmonic-oscillator basis is the de facto standard for quantum dots and nuclear structure calculations. In nuclei, theintrinsic Hamiltonian is most easily treated using this basis,and for quantum dots, the confining potential is to a goodapproximation harmonic. However, the discrete Hilbert space Hobtained from such a truncation grows exponentially with the number of particles.For example, allowing nsingle-particle states and Nparticles, dim(H)=/parenleftbiggn N/parenrightbigg . As an example, if we distribute N=6 electrons in the total number of single-particle states defined by 10 major oscillatorshells, we have n=110, resulting in dim( H)≈2.3×10 13 Slater determinants. This number is already beyond the limit of present full configuration-interaction approaches. In ourcoupled-cluster calculations, we perform studies up to some20 major shells. For 20 shells, the total number of single-particle states is n=420, for which dim( H)≈1.3×10 18, well beyond reach of standard diagonalization methods in theforeseeable future. For 20 electrons in 20 shells, the numberof Slater determinants is much larger, 7 .6×10 33in total. But, even if we could run large configuration-interaction calculations, the convergence of the computed energies asa function of the chosen single-particle basis is slow for aharmonic-oscillator basis, mainly due to the fact that thisbasis does not properly take into account the cusp condition at|r i−rj|=0 of the Coulomb interaction. In fact, the error /Delta1E in the energy for a quantum-dot problem, when increasing thedimensionality to one further shell with a harmonic-oscillator(HO) basis, behaves like /Delta1E∼O/parenleftbig R −k+δ−1 HO/parenrightbig . (5) Here,kis the number of times a given wave function /Psi1may be differentiated weakly, δ∈[0,1) is a constant, and RHOis the last oscillator shell. The derivation of the latter relationis detailed in Ref. 21, together with extensive discussions of the convergence properties of quantum-dot systems. For 115302-3M. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) the ground state of the two-electron quantum dot, we have precisely k=1, while for higher electron numbers, one observes k=O(1). This kind of estimate tells us that an approximation using only a few HO eigenfunctions necessarilywill give an error depending directly on the smoothness k. Although the coupled-cluster method allows for the in- clusion of much larger single-particle spaces, the slow con-vergence of the energy seen in full configuration-interactioncalculations applies to this method as well as it approximatesthe configuration-interaction solution using the same set ofsingle-particle functions. For an overview of coupled-clustererror analysis, see Refs. 35and44. One way to circumvent the dimensionality problem is to introduce a renormalized Coulomb interaction ˆV effdefined for a limited number of low-lying oscillator shells. Such tech-niques have been widely used in nuclear many-body problems(see, for example, Refs. 15,45and46). For quantum dots, this was first applied to a configuration interaction calculation byNavratil et al. , 47albeit for a different quantum-dot model. But, the potential of this method has not been explored fully, exceptfor recent preliminary studies in Refs. 15,21, and 48, which demonstrate a significant improvement of the eigenvalues.Furthermore, we expect that the potential of this methodis of even greater interest when linked up with an efficientmany-body method such as the coupled-cluster approach. The recipe for obtaining such an effective interaction is detailed in several works (see, for example, Refs. 15,46, and 49). Here, we give only a brief overview. The Hilbert space His divided into two parts PHandQH, where Pand is the orthogonal projector onto the smaller, effective model space, and Q=1−P. Note here that PH will be the space in which we do our many-body computations,andHis, in principle, the whole untruncated Hilbert space. The interaction operator ˆVis considered a perturbation, and we introduce a convenient complex parameter zand study ˆH(z)= ˆH 0+zˆV. Setting z=1 recovers the original Hamiltonian. Consider a similarity transformation of ˆH(z) defined by ˜H(z)≡e−X(z)ˆH(z)eX(z), (6) where the operator X(z) is such that the property Q˜H(z)P=0( 7 ) holds. Equation ( 6) must not be confused with equations from coupled-cluster theory. The idea is that X(z) should be determined from perturbation theory, which gives an analyticoperator function with X(0)=0. The most important consequences of these equations are that (i) ˜Hhave identical eigenvalues with ˆH, (ii) that there areD=dim(PH) eigenvalues, the eigenvectors of which areentirely in the model space PH. Thus, the effective Hamiltonian defined by ˆH eff(z)≡P˜H(z)P (8) is a model-space operator with Dexact eigenvalues. At z=0, these are the unperturbed eigenvalues, and these are continuedanalytically as zapproaches z=1. Equations ( 6) and ( 7) are not sufficient to determine X(z) uniquely. The order-by-order expansion of X(z)m u s t be supplied with side conditions. One of the most popularconditions is that X(z) †=−X(z) such that ˜H(z) is Hermitian, and, additionally, that the effective eigenvectors are as close as possible to the exact eigenvectors , i.e., that the quantity /Delta1 defined by /Delta1≡D/summationdisplay k=1/bardbl|/Psi1k/angbracketright−|/Psi1eff,k/angbracketright/bardbl2(9) is minimized, where |/Psi1eff,k/angbracketrightare the eigenvectors of ˆHeff(see Ref. 15). One can obtain a formula for X(z) in this case, namely, X=artanh( ω−ω†), where ω=QωP is the operator such that exp( ω)P|/Psi1k/angbracketright= |/Psi1k/angbracketright. Order-by-order expansion of ˜H(z) reveals that it contains m-body terms for all m/lessorequalslantN, even though ˆVonly contains two-body interactions. However, the many-body terms can beshown to be of lower order 49inz. By truncating ˜H(z)a t terms at the two-body level, we obtain the so-called subcluster approximation to the effective Hamiltonian. This can be computed by exact diagonalization of the two-body problem,a simple task for the quantum-dot problem. 50 The one-body part of ˜His always H0, so it is natural to define the effective interaction by Heff=H0+Veff. (10) The reader should, however, keep in mind that the sub- cluster approximation always produces missing many-bodycorrelations when inserted in a many-body context. The sizeof this source of error can only be quantified a posteriori , either by comparison with experiment and/or exact calculations (see,for example, Ref. 51) for a discussion on missing many-body physics and the nuclear many-body problem. C. Coupled-cluster method The single-reference coupled-cluster theory is based on the exponential ansatz for the ground-state wave function of theN-electron system |/Psi1 0/angbracketright=eT|/Phi10/angbracketright, where Tis the cluster operator (an N-particle– N-hole ex- citation operator) and |/Phi10/angbracketrightis the corresponding reference determinant (defining our chosen closed-shell system orvacuum) obtained by performing some mean-field calculationor by simply filling the Nlowest-energy single-electron states in two dimensions. The operator Tis a simple many-body excitation operator, which in all standard coupled-cluster approximations is trun-cated at a given (usually low) M-particle– M-hole excitation levelM<N , with Nbeing the number of electrons. If all excitations are included up to the N-particle– N-hole set of Slater determinants, one ends up with solving the full problem.The general form of the truncated cluster operator, defininga standard single-reference coupled-cluster approximationcharacterized by the chosen excitation level M,i s T=T 1+T2+T3+···+ TM, (11) 115302-4Ab INITIO COMPUTATION OF THE ENERGIES OF ... PHYSICAL REVIEW B 84, 115302 (2011) where Tk=1 (k!)2/summationdisplay i1,...,i k;a1,...,a kta1...ak i1...ikˆa† a1...ˆa† akˆaik...ˆai1. Here and in the following, the indices i,j,k,... label occu- pied single-particle orbitals while a,b,c,... label unoccupied orbitals. The unknown amplitudes ta i,tab ij, etc., in the last equation are determined from the solution of the coupled-cluster equations discussed below. For a truncated Toperator, we will use the notation T(M), where Mrefers to highest possible particle-hole excitations. As an example, we list here the expressions for one- particle–one-hole, two-particle–two-hole, and three-particle–three-hole operators, labeled T 1,T2, andT3, respectively, T1=/summationdisplay i<εf/summationdisplay a>εfta iˆa† aˆai, (12) and T2=1 4/summationdisplay ij<ε f/summationdisplay ab>ε ftab ijˆa† aˆa† bˆajˆai, (13) and finally T3=1 36/summationdisplay ijk<ε f/summationdisplay abc>ε ftabc ijkˆa† aˆa† bˆa† cˆakˆajˆai. (14) We will in this paper limit ourselves to a single reference Slater determinant /Phi10. The cluster amplitudes ta1...an i1...inare determined by solving a coupled system of nonlinear and energy-independent algebraicequations of the form /angbracketleftbig /Phi1a1...an i1...in/vextendsingle/vextendsingle¯H|/Phi10/angbracketright=0,i 1<···<in,a 1<···<an(15) where n=1,..., M . Here, ¯H=e−T(M)ˆHeT(M)=(ˆHeT(M))C (16) is the similarity-transformed Hamiltonian of the coupled- cluster theory truncated at M-particle– M-hole excitations and the subscript Cdenotes the connected part of the corresponding operator expression, and |/Phi1a1...an i1...in/angbracketright≡aa1...aanain...a i1|/Phi1/angbracketright are the n-particle– n-hole or n-tuply excited determinants relative to reference determinant |/Phi10/angbracketright. The Hamiltonians ¯H and ˆHare normal ordered. If we limit ourselves to include only one-particle–one-hole and two-particle–two-hole excitations, what is known ascoupled cluster of singles and doubles (CCSD), the methodcorresponds to M=2, and the cluster operator T (N)is approximated by T(M)=T(2)=T1+T2, (17) given by the operators of Eqs. ( 12) and ( 13). The standard CCSD equations for the singly and doubly excited cluster amplitudes ti aandtij ab, defining T1andT2, respectively, can be written as /angbracketleftbig /Phi1a i/vextendsingle/vextendsingle¯H(CCSD) |/Phi10/angbracketright=0 (18) and /angbracketleftbig /Phi1ab ij/vextendsingle/vextendsingle¯H(CCSD) |/Phi1/angbracketright=0,i < j ,a < b (19)where ¯H(CCSD) =¯H=e−T(2)ˆHeT(2)=(ˆHeT(2))C (20) is the similarity-transformed Hamiltonian of the CCSD ap- proach and the subscript Cstands for connected diagrams only. The system of coupled-cluster equations is obtained in the following way. We first insert the coupled-cluster wavefunction |/Psi1 0/angbracketrightinto the N-body Schr ¨odinger equation ˆH|/Psi10/angbracketright=/Delta1E 0|/Psi10/angbracketright, (21) where /Delta1E 0=E0−/angbracketleft/Phi10|ˆH|/Phi10/angbracketright is the corresponding energy relative to the reference energy /angbracketleft/Phi10|ˆH|/Phi10/angbracketright, and premultiply both sides on the left by e−T(N)to obtain the connected-cluster form of the Schr ¨odinger equation ¯H|/Phi1/angbracketright=/Delta1E 0|/Phi1/angbracketright, (22) where ¯H=e−T(2)ˆHeT(2)=(HeT(2))C (23) is the similarity-transformed Hamiltonian. Next, we project Eq. ( 22), in which Tis replaced by its approximate form T(M)[ E q .( 11)] onto the excited determinants |/Phi1a1...an i1...in/angbracketright, corresponding to the M-particle– M- hole excitations included in TM. The excited determinants |/Phi1a1...an i1...in/angbracketrightare orthogonal to the reference determinant |/Phi10/angbracketright, so that we end up with nonlinear and energy-independentalgebraic equations of the form of Eq. ( 15). Once the system of equations [Eq. ( 15)] is solved for T M orti1...ina1...an[or Eqs. ( 18) and ( 19) are solved for T1andT2orti a andtij ab], the ground-state coupled-cluster energy is calculated using the equation E0=/angbracketleft/Phi10|ˆH|/Phi10/angbracketright+E0/Delta1=/angbracketleft/Phi10|ˆH|/Phi10/angbracketright+/angbracketleft/Phi10|¯H|/Phi10/angbracketright.(24) It can easily be shown that if Hcontains only up to two-body interactions and 2 /lessorequalslantM/lessorequalslantN, we can write E0=/angbracketleft/Phi10|ˆH|/Phi10/angbracketright+/angbracketleft/Phi10|/bracketleftbigˆH/parenleftbig T1+T2+1 2T2 1/parenrightbig/bracketrightbig C|/Phi10/angbracketright. (25) In other words, we only need T1andT2clusters to calculate the ground-state energy E0of the N-body ( N/greaterorequalslant2) system, even if we solve for other cluster components Tnwithn> 2. As long as the Hamiltonian contains up to two-body interactions,the above energy expression is correct even in the exact case,when the cluster operator Tis not truncated (see, for example, Refs. 17,18, and 30for proof). The nonlinear character of the system of coupled-cluster equations of the form of Eq. ( 15) does not mean that the resulting equations contain very high powers of T M.F o r example, if the Hamiltonian ˆHdoes not contain higher-than- pairwise interactions, the CCSD equations for the T1andT2 clusters, or for the amplitudes ti aandtij abthat represent these clusters, become /angbracketleftbig /Phi1a i/vextendsingle/vextendsingle/bracketleftbigˆH/parenleftbig 1+T1+T2+1 2T2 1+T1T2+1 6T3 1/parenrightbig/bracketrightbig C|/Phi1/angbracketright=0, (26) 115302-5M. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) /angbracketleftbig /Phi1ab ij/vextendsingle/vextendsingle/bracketleftbigˆH/parenleftbig 1+T1+T2+1 2T2 1+T1T2+1 6T3 1 +1 2T2 2+1 2T2 1T2+1 24T4 1/parenrightbig/bracketrightbig C|/Phi1/angbracketright=0. (27) The explicitly connected form of the coupled-cluster equations, such as Eqs. ( 15)o r( 26) and ( 27), guarantees that the process of solving these equations leads to connected termsin cluster components of Tand connected terms in the energy E 0, independent of the truncation scheme Mused to define TM. The absence of disconnected terms in TMandE0is essential to obtain the rigorously size-extensive results.17,18It is easy to extend the above equations for the cluster amplitudes toinclude triples excitations, leading to the so-called CCSDT(Ref. 52) hierachy of equations. Defining f=/summationdisplay pqfpq{a+ paq} withfpqthe Fock matrix elements and W=1 4/summationdisplay pqrs/angbracketleftpq||rs/angbracketright{a+ pa+ qaras}, where /angbracketleftpq||rs/angbracketrightare antisymmetrized two-body matrix el- ements, the extension to triples gives the followingequations for the amplitudes with one-particle–one-holeexcitations: /angbracketleftbig /Phi1 a i/vextendsingle/vextendsingle[fT1+f/parenleftbig T2+1/2T2 1/parenrightbig +WT 1+W/parenleftbig T2+1/2T2 1/parenrightbig +W/parenleftbig T1T2+1/6T3 1+T3)/bracketrightbig C|/Phi1/angbracketright=0, and with two-particle–two-hole excitations /angbracketleftbig /Phi1ab ij/vextendsingle/vextendsingle/bracketleftbig fT1+f(T3+T2T1)+W+WT 1+W/parenleftbig T2+1/2T2 1/parenrightbig +W/parenleftbig T1T2+1/6T3 1+T3/parenrightbig +W/parenleftbig T1T3+1/2T2 2+1/2T2T2 1+1/24T4 1/parenrightbig/bracketrightbig C|/Phi1/angbracketright=0, and, with three-particle–three-hole excitations, we end up with /angbracketleftbig /Phi1abc ijk/vextendsingle/vextendsingle/bracketleftbig fT3+f/parenleftbig T3T1+1/2T2 2/parenrightbig +WT 2+W(T3+T1T2) +W/parenleftbig 1/2T2+T3T11/2T2 1+T1/parenrightbig +W/parenleftbig T2T3+1/2T2 2T1+1/2T3T2 1+1/6T2T3 1/parenrightbig/bracketrightbig C|/Phi1/angbracketright=0. Different approximations to the solution of the triples equa- tions yield different CCSDT approximations. The CCSDmethod scales (in terms of the most computationally expensivecontributon) as n 2 on4u, where n0represents the number of occupied orbitals and nuthe number of unoccupied single- particle states. The full CCSDT scales as n3 on5u. Coupled-cluster theory with inclusion of full triples CCSDT is usually considered to be too computationally expensivein most many-body systems of considerable size. Therefore,triples corrections are usually taken into account perturba-tively using the noniterative CCSD(T) approach describedin Ref. 53. Recently, a more sophisticated way of including the full triples is known as the /Lambda1-CCSD(T) approach. 36–39 In the /Lambda1-CCSD(T) approach, the left-eigenvector solution of the CCSD similarity-transformed Hamiltonian is utilized in thecalculation of a noniterative triples correction to the coupled-cluster ground-state energy. The left-eigenvalue problem is given by /angbracketleft/Phi1 0|/Lambda1¯H=E/angbracketleft/Phi10|/Lambda1, (28) were/Lambda1denotes the de-excitation cluster operator /Lambda1=1+/Lambda11+/Lambda12, (29) /Lambda11=/summationdisplay i,aλi aaaa† i, (30) /Lambda12=1 4/summationdisplay i,j,a,bλij ababaaa† ia† j. (31) The unknowns, λi aandλij ab, result from the ground-state solution of the left-eigenvalue problem ( 28). Using a single- particle basis that diagonalizes the Fock matrix fwitin the hole-hole andparticle -particle blocks simultaneously, and utilizing the λi aandλij abde-excitation amplitudes together with the cluster amplitudes ta iandtab ij, we get the noniterative /Lambda1- CCSD(T) energy correction to the coupled-cluster correlationenergy (see Refs. 36–39for more details) /Delta1E 3=1 (3!)2/summationdisplay ijkabc/angbracketleft/Phi10|/Lambda1(fhp+W)N/vextendsingle/vextendsingle/Phi1abc ijk/angbracketrightbig ×1 γabc ijk/angbracketleftbig /Phi1abc ijk/vextendsingle/vextendsingle(WNT2)C|/Phi10/angbracketright. (32) Here, fhpdenotes the part of the normal-ordered one-body Hamiltonian that annihilates particles and creates holes, while γabc ijk≡fii+fjj+fkk−faa−fbb−fcc (33) is expressed in terms of the diagonal matrix elements of the normal-ordered one-body Hamiltonian f. In the case of Hartree-Fock orbitals, the one-body part of the Hamiltonianis diagonal and f hpvanishes. The state |/Phi1abc ijk/angbracketrightis a three- particle–three-hole excitation of the reference state. For afurther discussion of various approximations to the triplescorrelations, see, for example, Refs. 17and18. In this paper, we focus on the CCSD, the CCSD(T), and the/Lambda1-CCSD(T) approaches, using either a renormalized or an unrenormalized interaction. In order to avoid an iterativesolution of the CCSD(T) and /Lambda1-CCSD(T) equations, we start from a self-consistent Hartree-Fock basis such that the Fockmatrix fis diagonal. Using such a basis, the computational cost of the CCSD(T) and /Lambda1-CCSD(T) energy corrections is n 3 on4unumber of cycles, done only once. It is also important to keep in mind, in particular, that when linking our coupled-cluster theory with Monte Carlo approaches, a wave function-based method such as coupled-cluster theory is defined withina specific subset of the full Hilbert space. In our case, theHilbert space will be defined by all possible many-body wavefunctions, which can be constructed within a certain numberof the lowest-lying single-particle states. D. Diffusion Monte Carlo The diffusion Monte Carlo method seeks the solution of the equation ∂τ|/Psi1(R,τ)/angbracketright=[ˆH−E0]|/Psi1(R,τ)/angbracketright, (34) 115302-6Ab INITIO COMPUTATION OF THE ENERGIES OF ... PHYSICAL REVIEW B 84, 115302 (2011) where Rcollectively indicates the degrees of freedom of the system (the 3 Nelectron coordinates, in this case). By expanding the state |/Psi1(R,τ)/angbracketrighton the basis of eigenstates |φn/angbracketright ofˆH, a formal solution of Eq. ( 34) is given by |/Psi1(R,τ)/angbracketright=e−(ˆH−E0)τ|/Psi1(R,0)/angbracketright =/summationdisplay ne−(ˆH−E0)τ|φn/angbracketright/angbracketleftφn|/Psi1(R,0)/angbracketright =/summationdisplay ne−(ˆEn−E0)τ|φn/angbracketright/angbracketleftφn|/Psi1(R,0)/angbracketright (35) from which it is evident that, for τ→∞ , the only surviving component is the ground state of ˆH. Equation ( 34) can be numerically solved by expanding the state to be evolved ineigenstates |R i/angbracketrightof the position operator (called “walkers”), so that the evolution reads as /summationdisplay i/angbracketleftRi|/Psi1(R,τ)/angbracketright=/summationdisplay i/angbracketleftR|e−(ˆH−E0)τ|R/prime i/angbracketright/angbracketleftR/prime i|/Psi1(R/prime,0)/angbracketright. (36) Formally, in terms of the Green’s function solution of Eq. ( 34), the solution can be written as /Psi1(R,τ)=/integraldisplay G(R/prime,R,τ)/Psi1(R/prime,0)dR/prime. (37) The Green’s function G(R/prime,R,τ)=/angbracketleftR|exp[−(ˆH−E0)]|R/angbracketright is, in general, unknown. However, in the limit /Delta1τ→0, it can be written in the following form: G(R/prime,R,τ)/similarequal/radicalBigg/parenleftbiggmem∗ 2π¯h2/Delta1τ/parenrightbiggd e(R−R/prime)2 2¯h2/mem∗/Delta1τe−[V(R)−E0]/Delta1τ,(38) that is, as the product of the free-particle Green’s function, having the effect of displacing the d-dimensional walkers, and a factor containing the potential, which is interpreted as aweight for the estimators computed at the walker position,and a probability for the walker itself to generate one ormore copies of itself in the next generation. Due to thedivergence of the potential at the origin, it is necessary tomodify the algorithm, introducing the so-called “importancesampling.” In practice, the sampled distribution is modifiedby multiplying by an approximate solution of the Schr ¨odinger equation /Psi1 T(R), which is usually determined by a variational Monte Carlo calculation /Psi1T(R)/Psi1(R,τ)=/integraldisplay G(R/prime,R,τ)/Psi1T(R) /Psi1T(R/prime)/Psi1T(R/prime)/Psi1(R/prime,0)dR/prime. (39) A final important observation is the fact that the procedure described above is well defined only in the case of a totallysymmetric ground state. For a many-fermion system, it wouldbe necessary, in principle, to project on an excited state ofthe Hamiltonian, which leads to a severe instability of thevariance on the energy estimation. This problem is usuallytreated by artificially imposing, as an artificial boundarycondition, that the solution vanishes on the nodes of the trialfunction /Psi1 T(fixed-node approximation). Many other technical details enter the real calculation. A thorough description of thediffusion Monte Carlo (DMC) algorithm, as implemented forthe calculations of this paper, can be found in Ref. 54.The fixed-node DMC calculations depend on the quality of the trial wave function /Psi1 T(R), which is usually built starting from a parametrized ansatz. The values of the parametersare computed by minimizing the expectation value of theHamiltonian on /Psi1 T(R). The trial wave functions we use have the form10 /Psi1(R)L,S=exp[φ(R)]Nconf/summationdisplay i=1αi/Xi1L,S i(R), (40) where the αiare variational parameters. Because in this paper we are considering only closed-shell dots that have L=0 and S=0, the sum in Eq. ( 40) reduces to a single term /Xi1L=0,S=0=D↑D↓, (41) where the Dχare Slater determinants of spin-up and spin-down electrons, using orbitals from a local density approximationcalculation with the same confining potential and the samenumber of electrons. The function exp[ φ]i nE q .( 40)i sa generalized Jastrow factor of the form lφ(R)= N/summationdisplay i=1/bracketleftBigg6/summationdisplay k=1γkJ0/parenleftbiggkπri Rc/parenrightbigg/bracketrightBigg +N/summationdisplay i<j1 2/parenleftbiggaijrij 1+b(ri)rij+aijrij 1+b(rj)rij/parenrightbigg ,(42) where b(r)=bij 0+bij 1tan−1[(r−Rc)2/2Rc/Delta1]. (43) It explicitly includes one- and two-body correlations and effective many-body correlations via the spacial dependenceofb(r). The quantity R crepresents an “effective” radius of the dot, which is optimized in the variational procedure. The b0and b1parameters depend only on the relative spin configuration of the pair ij. The parameters aijare fixed in order to satisfy the cusp conditions, that is, the condition of finiteness ofthe local energy ˆH/Psi1//Psi1 forr ij→0. For a two-dimensional system, aij=1 if the electron pair ijhas antiparallel spin, and aij=1/3 otherwise. The dependence of aijon the relative spin orientation of the electron pair introduces spin contaminationinto the wave function. However, the magnitude of the spincontamination and its effect on the energy has been shown tobe totally negligible in the case of well-optimized atomic wavefunctions, 55and we expect that to be true here as well. Also, the coefficients γkin the one-body term, the coeffi- cients /Delta1,b0, andb1in the two-body term, and the coefficients αimultiplying the configuration state functions are optimized by minimizing the variance of the local energy.56 III. RESULTS We start our discussion with the results for the two-electron system since these can, for certain values of the oscillatorfrequency, be compared with the exact results of Taut. 43 These results are presented in the next section using botha renormalized two-body Coulomb interaction and the bareCoulomb interaction. Thereafter, we present coupled-clusterresults with singles and doubles excitations for systems withN=6 and 12 electrons with the bare Coulomb interaction. 115302-7M. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) The slow convergence as a function of the number of oscillator shells with the bare interaction serves to motivate theintroduction of an effective Coulomb interaction. In the mainresult section, we present CCSD, CCSD(T), and /Lambda1-CCSD(T) results for N=6, 12, and 20 electrons using an effective two-body Coulomb interaction and compare with diffusionMonte Carlo (DMC) calculations for the same systems. A. Results for two electrons In this section, we limit our attention to the two-electron system and compare our DMC results with coupled-cluster cal-culations with CCSD correlations only. The results presentedhere serve to demonstrate the reliability of using an effectiveCoulomb interaction. The CCSD approach gives the exact eigenvalues for the two-particle system. We have employed a standard harmonic-oscillator basis using the frequencies ω=0.5 and 1.0 a.u. Our results are listed in Table I. The variable Rrepresents the num- ber of oscillator shells in which the effective interaction caserepresents the model space for which the effective Coulombinteraction is defined. The calculations labeled CCSD- V represent the results obtained with the unrenormalized or bareCoulomb interaction, while the shorthand CCSD- V effstands for the results obtained with an effective interaction. Sincethe latter, irrespective of size of model space (number oflowest-lying oscillator shells in our case) always gives theexact lowest-lying eigenvalues by construction (a similaritytransformation preserves always the eigenvalues), these results TABLE I. Ground-state energies for two electrons in a circular quantum dot within the CCSD approach with (CCSD- Veff)a n d without (CCSD- V) an effective Coulomb interaction. The diffusion Monte Carlo (DMC) results are also included. For ω=1, Taut’s exact result from Ref. 43is 3 a.u. All energies are in atomic units. There are no triples corrections for the two-body problem. The variable R represents the number of oscillator shells. ωR CCSD- V CCSD- Veff DMC 0.5 2 1.786 914 1.659 772 4 1.673 874 1.659 7726 1.667 259 1.659 772 8 1.664 808 1.659 772 10 1.663 535 1.659 77212 1.662 762 1.659 772 14 1.662 244 1.659 772 16 1.661 875 1.659 77218 1.661 599 1.659 772 20 1.661 378 1.659 772 1.659 75(2) 1.0 2 3.152 329 3.000 000 4 3.025 232 3.000 000 6 3.013 627 3.000 0008 3.009 237 3.000 000 10 3.000 895 3.000 000 12 3.000 654 3.000 00014 3.000 505 3.000 000 16 3.000 406 3.000 000 18 3.000 335 3.000 00020 3.000 282 3.000 000 3.000 00(3)are unchanged as a function of the number of oscillator shells R. For the two-body problem, coupled-cluster theory at the level of singles and doubles excitations yields the same as exactdiagonalization in the same two-particle space. In our case,the number of two-body configurations is given by all allowedconfigurations that can be constructed by placing two particlesin the single-particle orbits defined by the given numberof oscillator shells R.F o rω=1.0 a.u., Taut’s exact result from Ref. 43is reproduced. The noninteracting part of the Hamiltonian gives a contribution of 2 a.u. to the ground-stateenergy, while the two-particle interaction results in 1 a.u. We notice also that the DMC results agree perfectly (within six leading digits) with our CCSD- V effcalculations. The standard error in the DMC calculations is given in parentheses. If, on the other hand, we use the bare Coulomb interaction, we see that the convergence of the CCSD- Vresults as a function of Ris much slower and in line with the analysis of Ref. 21and our discussion in Sec. II B. One needs at least some 16–20 major oscillator shells (between 272 and 420single-particle states) in order to get a result within three to fourleading digits close to the exact answer. The slow convergenceof the bare interaction for the two-electron problem may beeven more prevalent in a many-body system, in particular,for small values of ω, where correlations are expected to be more important. With more particles, we may expect evenworse convergence. In Table II, we present for the case of ω=1.0 a.u. CCSD results for N=6 and 12 electrons. The bare Coulomb interaction in an oscillator basis is used. Thediffusion Monte Carlo results are for N=6, 20.1597(2) a.u., and for N=12, 65 .700(1) a.u. Using the bare interaction thus results in a slow convergence, as will be demonstrated inthe next section. The result of 20 .1737 a.u. obtained with an effective Coulomb at the CCSD level for N=6 and R=20 is much closer to the DMC result, as can be seen fromTable III. These results serve the aim of motivating the introduction of an effective two-particle interaction. In thenext section, we will make further comparisons between ourresults with and without an effective interaction. In particular, TABLE II. Ground-state energies for N=6 and 12 electrons in a circular quantum dot within the CCSD approach using the bare Coulomb interaction. All energies are in atomic units. There are no triples corrections. Results are presented for an oscillator frequencyω=1.0 a.u. The variable Rrepresents the number of oscillator shells. ForN=12, the first three shells are filled and there are no results for two shells only. RN =6 N=12 2 22.219 813 3 21.419 889 73.765 5494 20.421325 70.297531 6 20.260 893 66.452 006 8 20.221 750 65.889 32410 20.216 128 65.887 965 12 20.206 257 65.848 353 14 20.199 986 65.825 018 16 20.195 658 65.809 710 18 20.192 497 65.798 90220 20.189 900 65.789 460 115302-8Ab INITIO COMPUTATION OF THE ENERGIES OF ... PHYSICAL REVIEW B 84, 115302 (2011) TABLE III. Ground-state energies for N=6 electrons in a circular quantum dot within various coupled-cluster approximations utilizing an effective Coulomb interaction and the diffusion Monte Carlo (DMC) approach. The coupled-cluster results have been obtained with an effective two-body interaction using a self-consistent Hartree-Fock basis and the CCSD, the CCSD(T), and the /Lambda1-CCSD(T) approaches discussed in the text. EHFis the Hartree-Fock energy, while Rstands for the number of major oscillator shells. All energies are in atomic units. ωR E HF CCSD CCSD(T) /Lambda1-CCSD(T) DMC 0.28 10 7.9504 7.6241 7.6032 7.6064 12 7.9632 7.6245 7.6023 7.605714 7.9720 7.6247 7.6016 7.6052 16 7.9785 7.6249 7.6012 7.6048 18 7.9834 7.6251 7.6008 7.604620 7.9872 7.6252 7.6006 7.6044 7.6001(1) 0.5 10 12.1927 11.8057 11.7871 11.7892 12 12.2073 11.8055 11.7858 11.788014 12.2173 11.8055 11.7850 11.7873 16 12.2246 11.8055 11.7845 11.7868 18 12.2302 11.8055 11.7841 11.786420 12.2346 11.8055 11.7837 11.7862 11.7888(2) 1.0 10 20.6295 20.1766 20.1623 20.1633 12 20.6461 20.1753 20.1602 20.161214 20.6576 20.1746 20.1589 20.1600 16 20.6659 20.1742 20.1580 20.1592 18 20.6723 20.1739 20.1574 20.158620 20.6773 20.1737 20.1570 20.1582 20.1597(2) we will try to extract convergence criteria for both approaches and link our numerical results with the predictions made byKvaal in Eq. ( 5). B. Results with an effective Coulomb interaction We present here our final and most optimal results for N=6, 12, and 20 electrons using the CCSD, the CCSD(T), and the /Lambda1-CCSD(T) approaches. We list the CCSD(T) triples results as well. This method has for a long time been consideredas the calculational gold standard in quantum chemistry dueto its low computational cost and accuracy. We emphasize,however, that the /Lambda1-CCSD(T) approach is an improvement of the standard CCSD(T) approach, and should therefore beconsidered as our best and most accurate coupled-clustercalculation in this work. In all calculations, we employ aneffective Coulomb interaction and a self-consistent Hartree-Fock basis for different values of the oscillator frequencyωand the model space R. The results are compared with diffusion Monte Carlo calculations. 57In addition to the values ofω=1.0 and 0.5, which serve more as a reference for earlier calculations, we present results for ω=0.28 a.u., which corresponds to 3.32 eV , a frequency which shouldapproximate the experimental situation in Ref. 58.T h er o l e of correlations is also more important for smaller valuesofω, allowing us therefore to test the reliability of our single-reference CCSD and /Lambda1-CCSD(T) calculations. As the system becomes more and more correlated, contributionsfrom clusters beyond the T(3) (beyond three-particle–three- hole correlations) level might become non-negligible. Forvalues of ω> 1, the single-particle part of the Hamil- tonian dominates and correlations play a less prominentrole. Our results for N=6, 12, and 20 electrons are displayed in Tables III,IV, and V, respectively. We present also themean-field energies (that is, the Hartree-Fock ground-state energies). These are labeled as E HFin the tables. For all values of ωwithR=20 major oscillator shells, our best coupled-cluster results, the /Lambda1-CCSD(T) calculations, are very close to the diffusion Monte Carlo calculations. Even for 10major shells, the results are close to the DMC calculations,suggesting thereby that the usage of an effective interactionprovides a better starting point for many-body calculations.The convergence of the coupled-cluster calculation in terms ofthe number of major oscillator shells is also better than the re-sults shown in Table IIwith the bare Coulomb interaction. This discussion will be further elaborated at the end of this section. InR=20 major shells, the /Lambda1-CCSD(T) results are very close to the DMC results. As an example, consider the ω=1 results for N=6 in Table III. The CCSD result is 20.1737 a.u., while the /Lambda1-CCSD(T) number is 20.1582 a.u. The corresponding DMC energy is 20.1597(2) and very close to our/Lambda1-CCSD(T) result. With R=20 shells, our coupled-cluster calculations are almost converged at the level of the fifthor sixth number after the decimal point. At the end of this section, we discuss the convergence properties of the various coupled-cluster approaches as functions of the number ofoscillator shells R. In Tables III,IV, and V, we see that the CCSD(T) results are in most cases overshooting the diffusion Monte Carlo results.From numerous coupled-cluster studies in quantum chemistry,it has been found that CCSD(T) tends to overestimate the roleof triples and thereby often overshoots the exact energy. The/Lambda1-CCSD(T) approach has, on the other hand, been found to give highly accurate correlation energies, and even in somecases performing better than the full CCSDT approach (seeRefs. 36–39). This is also consistent with our findings for the CCSD(T) and /Lambda1-CCSD(T) correlation energies in quantum dots. 115302-9M. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) TABLE IV . Same caption as in Table IIIexcept the results are for N=12 electrons. ωR E HF CCSD CCSD(T) /Lambda1-CCSD(T) DMC 0.28 10 26.3556 25.7069 25.6445 25.6540 12 26.3950 25.7066 25.6388 25.649114 26.4221 25.7074 25.6363 25.6470 16 26.4410 25.7081 25.6346 25.6456 18 26.4551 25.7085 25.6334 25.644620 26.4659 25.7089 25.6324 25.6439 25.6356(1) 0.5 10 39.9948 39.2218 39.1659 39.1721 12 40.0409 39.2203 39.1599 39.166714 40.0709 39.2197 39.1565 39.1635 16 40.0922 39.2195 39.1543 39.1615 18 40.1080 39.2194 39.1527 39.160120 40.1202 39.2194 39.1516 39.1591 39.159(1) 1.0 10 66.6596 65.7552 65.7118 65.7149 12 66.7106 65.7484 65.7017 65.705114 66.7445 65.7449 65.6961 65.6996 16 66.7686 65.7430 65.6926 65.6963 18 66.7867 65.7417 65.6903 65.694120 66.8006 65.7409 65.6886 65.6924 65.700(1) Let us briefly discuss the error in our coupled-cluster calculations. There are two sources of error, the first comingfrom the finite size of the single-particle basis, and theother from truncation of the cluster amplitude Tat the T(3) excitation level (three-particle–three-hole excitations). We arepresently not able to provide a mathematical error estimate ontruncations in terms of the number of particle-hole excitationoperators in the cluster operator T. However, several studies from quantum chemistry (see Ref. 18and references therein) and in nuclear physics 32,34have shown that the CCSD approach gives about 90% of the correlation energy, while CCSDTgives about 99% of the full correlation energy. Assuming thatthe DMC results are to be considered as exact results, wecan calculate the percentage of correlation energy our CCSD and/Lambda1-CCSD(T) calculations give for different numbers of electrons Nand values ωof the confining harmonic-oscillator potential. In Table VI, we list the amount (in percentage) of correlation energy obtained at the CCSD and /Lambda1-CCSD(T) level; the coupled-cluster calculations were done in a modelspace of R=20 major oscillator shells. As we see from Table VI, the CCSD approximation gives 90%, or more, of the full correlation energy, while the/Lambda1-CCSD(T) approximation is at the level of 99%–100% of the full correlation energy for R=20. The CCSD approximation is clearly performing better for larger values ωof the confining potential, but this is expected since the system becomes TABLE V . Same caption as in Table IIIexcept the results are for N=20 electrons. ωR E HF CCSD CCSD(T) /Lambda1-CCSD(T) DMC 0.28 10 63.2588 62.2851 62.1802 62.1946 12 63.2016 62.0772 61.9503 61.969214 63.2557 62.0634 61.9265 61.9466 16 63.3032 62.0646 61.9214 61.9423 18 63.3369 62.0656 61.9181 61.939520 63.3621 62.0664 61.9156 61.9375 61.922(2) 0.5 10 95.2872 94.0870 93.9864 93.9971 12 95.3407 93.9963 93.8818 93.8944 14 95.4164 93.9921 93.8700 93.8833 16 95.4676 93.9904 93.8632 93.8771 18 95.5043 93.9895 93.8588 93.873020 95.5320 93.9891 93.8558 93.8702 93.867(3) 1.0 10 157.4356 156.0128 155.9324 155.9381 12 157.5613 155.9868 155.8978 155.9042 14 157.6437 155.9740 155.8795 155.8863 16 157.7002 155.9669 155.8687 155.8758 18 157.7413 155.9627 155.8618 155.869020 157.7725 155.9601 155.8571 155.8646 155.868(6) 115302-10Ab INITIO COMPUTATION OF THE ENERGIES OF ... PHYSICAL REVIEW B 84, 115302 (2011) TABLE VI. Percentage of correlation energy at the CCSD level (/Delta1E 2) and at the /Lambda1-CCSD(T) level ( /Delta1E 3), for different numbers of electrons Nand values of the confining harmonic potential ω.A l l numbers are for R=20. ω=0.28 ω=0.5 ω=1.0 N/Delta1 E 2 /Delta1E 3 /Delta1E 2 /Delta1E 3 /Delta1E 2 /Delta1E 3 6 94% 99% 96% 100 % 97% 100% 12 91% 99% 94% 100 % 96% 100% 20 90% 99% 93% 100% 95% 100% less and less correlated for larger values of ω. This shows that our coupled-cluster calculations of circular quantumdots are within or even better than the accuracy seen indifferent applications in both quantum chemistry and nuclearphysics. The fact that the (perturbative inclusions of triples)CCSD(T) and the /Lambda1-CCSD(T) methods work that well even for small values of the oscillator energy, is due to the fact thatcorrelations beyond the Hartree-Fock and CCSD levels arestill small, not exceeding 10% of the correlation energy. As previously discussed, DMC results reported in this paper are still affected by the fixed-node approximation.The extent of the error only depends on the nodal surfaceof the wave function. Because we use a single product ofSlater determinants, given the circular symmetry of the dotsconsidered, the nodes depend only on the set of single-particlefunctions used. Previous tests performed changing the set ofsingle-particle orbitals show that differences are of the orderof one millihartrees or less. 10The optimization of the Jastrow factor only influences the variance of the energy, which istypically of the order of 0 .5% of the total energy. Therefore, for circular quantum dots, we can conclude, assuming thatthe DMC calculations are as close as possible to the exactenergies, that with an effective two-body interaction, a finitebasis set of R=20 major oscillator shells, and at most three-particle–three-hole correlations in the cluster amplitude,the remaining many-body effects are almost negligible as weare within 99%–100% of the full correlation energy. In order to study the role of correlations as a function of the oscillator frequency ωand the number of electrons, we define the relative energy /epsilon1=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleE DMC−/angbracketleftˆH0/angbracketright EDMC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (44) where /angbracketleftˆH 0/angbracketrightis the expectation value of the one-body operator, the so-called unperturbed part of the Hamiltonian. For N=6, this corresponds to an expectation value /angbracketleftˆH0/angbracketright=10ωfor the one-body part of the Hamiltonian, while for N=12 and 20, the corresponding numbers are /angbracketleftˆH0/angbracketright=28ωand/angbracketleftˆH0/angbracketright=60ω, respectively. Assuming that the diffusion Monte Carlo resultsare as close as possible to the true eigenvalues, the quantity /epsilon1 measures the role of the two-body interaction and correlationscaused by this part of the Hamiltonian as functions of ωand N, the number of electrons. The results for /epsilon1are shown in Fig.1. Results for N=2 are also included. We see from this figure that the effect of the two-body interaction becomes increasingly important as we increasethe number of particles. Moreover, the interaction is more FIG. 1. Relative correlation energy /epsilon1defined in Eq. ( 44)f o r different values of ¯ hωand number of electrons. The DMC numbers are obtained from Tables I–VusingR=20. important for the smaller values of the oscillator frequency ω. This is expected since the contribution from the one-bodyoperator is reduced due to smaller values of ω. Including more electrons obviously increases the contribution fromthe two-body interaction. Since our optimal coupled-clusterresults are very close to the DMC results, almost identicalresults are obtained if we replace the DMC results with the/Lambda1-CCSD(T) results. We can also study the role of correlations beyond the Hartree-Fock energy E HF. In order to do this, we relate the Hartree-Fock energy EHFin Tables III–Vto the optimal coupled-cluster calculation, namely, the /Lambda1-CCSD(T) results. The relative difference between these quantities conveysthereby information about correlations beyond the mean-fieldapproximation. This relative measure is defined as χ=/vextendsingle/vextendsingle/vextendsingle/vextendsingleE /Lambda1-CCSD(T) −EHF E/Lambda1-CCSD(T)/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (45) The results are shown in Fig. 2forN=6, 12, and 20. We see from this figure that correlations beyond the Hartree-Fock levelare important for few particles and low values of ω. Increasing the number of electrons in the circular dot decreases the roleof correlations beyond the mean-field approximation, a feature FIG. 2. Relative correlation energy χdefined in Eq. ( 45)f o r different values of ¯ hωand number of electrons. The numbers are obtained from Tables III–VusingR=20. 115302-11M. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) which can be understood from the fact that, for larger systems, multiparticle excitations across the Fermi level decrease inimportance. This is due to the fact that the single-particle wavefunctions for many states around the Fermi level have morethan one node, resulting in normally smaller matrix elements.Stated differently, with an increasing number of electrons,the particles close to the Fermi level are more apart fromeach other, in particular, for those particles that occupy statesaround and above the Fermi level. The consequence of thisis that correlations beyond the Hartree-Fock level decrease inimportance when we add more and more particles. This meansin turn that, for larger systems, mean-field methods are rathergood approximations to systems of many interacting electronsin quantum dots. Similar features are seen in nuclei. For lightnuclei, correlations beyond the mean field are very importantfor ground-state properties, whereas for heavy nuclei suchas 208Pb, mean-field approaches provide a very good starting point for studying several observables. The reader should, however, note that here we have limited our attention to ground-state energies only. Whether ourconclusions about the role of correlations pertain to quantitiessuch as, say, spectroscopic factors remains to be studied. We study now in more detail the convergence properties of our coupled-cluster approaches, in particular, we will relateour/Lambda1-CCSD(T) and CCSD results with the diffusion Monte Carlo results and study the dependence on R. This analysis will be performed with and without an effective Coulombinteraction. The reason for doing this is that we wish tostudy whether the convergence criterion of Eq. ( 5), derived for a full configuration-interaction analysis, applies to variouscoupled-cluster truncations as well. Furthermore, we wishto see whether our calculations with an effective interactionconverge faster as a function of Rcompared to a calculation with the bare interaction. We compute the following quantities: log 10/epsilon1CCSD (R)=log10/vextendsingle/vextendsingle/vextendsingle/vextendsingleE CCSD (R)−EDMC EDMC/vextendsingle/vextendsingle/vextendsingle/vextendsingle(46) and log 10/epsilon1/Lambda1-CCSD(T) (R)=log10/vextendsingle/vextendsingle/vextendsingle/vextendsingleE /Lambda1-CCSD(T) (R)−EDMC EDMC/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(47) In Fig. 3, we plot the results for N=20 electrons and ω=0.5. We have chosen these values since they represent one of the cases where the /Lambda1-CCSD(T) results are always above the DMC results and we have no crossing betweenthese two sets of calculations. The CCSD results, on the otherhand, are always, for all cases reported here, above the DMCresults. This means that the trend seen in Fig. 3for the CCSD calculations applies to all cases listed in Tables III–V, while for the /Lambda1-CCSD(T) calculations, these results are similar for all cases except for N=6 andω=0.5 and 1.0; N=12 and ω=1.0; and N=20 and ω=1.0. In these cases, the results atR=20 are slightly below the DMC results. However, the agreement is still excellent. The interesting feature to notein Fig. 3is that the CCSD results change marginally after R=12 for N=20, and there is essentially very little to gain beyond 20 major shells. With the present accuracy ofthe DMC results, we can conclude that the CCSD results FIG. 3. Relative correlation energy /epsilon1defined in Eqs. ( 46)a n d( 47) for different values of R. The values displayed here are for N=20 andω=0.5. The numbers are obtained from Table V. We include also the /Lambda1-CCSD(T) results obtained with the bare interaction. reach, at most, a relative error of approximately 10−3and that it stays almost stable from R=12 shells. The relative error with respect to the Monte Carlo results does not changemuch. This applies to all CCSD results. This tells us clearly thatthere are important correlations beyond two-particle–two-holeexcitations and that these correlations do not stabilize aftersome few shells. Furthermore, the slope of the /Lambda1-CCSD(T) calculations is much more interesting and resembles the slopeof the configuration interaction analysis of Ref. 21with an effective interaction. For the ground states of three to fiveelectrons, Kvaal found in Ref. 21a slope of approximately α=−4t o−5 for a parametrization log 10/epsilon1≈c+αlog10R for the ground-state energies of various N-electron quantum dots. The variable cis a constant. Our slopes vary between α=−4 and−6, resulting in a relative error of approximately 10−5atR=20 for the results in Fig. 3. The slope of the /Lambda1- CCSD(T) result is α=−4.93. The reader should note that the DMC results can not reach a higher precision. The slope of theCCSD calculation with an effective interaction is α=−0.67 afterR=12. In the same figure, we plot also the /Lambda1-CCSD(T) results obtained without an effective Coulomb interaction, that is,with the bare interaction only. These results are labeled aslog 10/epsilon1/Lambda1-CCSD(T) (R)−bare. A Hartree-Fock basis was used in this case as well in order to obtain converged solutions forthe/Lambda1-CCSD(T) equations. We see in this case that the convergence is much slower, resulting in a slope given byα=−2.58, a result not far from the analysis of Ref. 21 for the bare interaction. Figure 4exhibits a similar trend, except that here we present results for N=12 electrons andω=0.5. The slope of the /Lambda1-CCSD(T) results is now α=−6.38 with an effective interaction and α=−1.81 with a bare Coulomb interaction. We notice again that the CCSDresults saturate around R=12 major shells. These results are very interesting as they show that the usage of an effectiveinteraction can really speed up the convergence of the energyas a function of the number of shells. Furthermore, these resultstell us also that correlations beyond the singles and doublesapproach are simply necessary. The convergence behavior of 115302-12Ab INITIO COMPUTATION OF THE ENERGIES OF ... PHYSICAL REVIEW B 84, 115302 (2011) FIG. 4. Relative correlation energy /epsilon1defined in Eqs. ( 46)a n d( 47) for different values of R. The values displayed here are for N=12 andω=0.5. The numbers are obtained from Table IV. We include also the /Lambda1-CCSD(T) results obtained with the bare interaction. the/Lambda1-CCSD(T) results resembles, to a large extent, those of a full configuration-interaction approach with and without aneffective interaction. Although we can extract similar conver-gence behaviors as those predicted in Ref. 21as functions of a truncation in the single-particle basis, the challenge isto provide more rigid mathematical convergence criteria fortruncations in the number of particle-hole excitations. Here,we can only justify a posteriori that triples corrections are necessary. Work along these lines is in progress. Using equation-of-motion coupled-cluster method, as dis- cussed in detail in, for example, Refs. 17,18,33, and 34, we can go beyond quantum dots with closed-shell configurations andcompute properties such as electrochemical potentials, addi-tion spectra, and excited states. For the sake of completeness,we list in Table VIIthe electrochemical potentials μ(N)= E(N)−E(N−1) for N=3, 6, 7, 12, and 13 electrons, calculated with the electron attached and ionization potentialequation-of-motion coupled-cluster method (CC) and dif-fusion Monte Carlo (DMC). We have chosen a frequencyω=0.28, since this frequency is the one that involves the strongest degrees of correlations. It is also the frequency thatexhibits the largest deviations between our coupled-clusterresults and the diffusion Monte Carlo calculations. We seefrom Table VIIthat the agreement between the two different many-body methods is indeed very good, with differencesof the order of 0 .02 in most cases. The spin assignments for the ground states with both methods are also the same.ForN=3, the ground state has orbital momentum projection M=1 and total spin S=1/2; forN=5, the corresponding quantum numbers are M=1 and S=1/2; for N=7, we haveM=2 and S=1/2; for N=11, we obtain M=0 andS=1/2; and finally for N =13, we have M=3 and S=1/2. These results demonstrate that the coupled-cluster method can be extended to open-shell systems. With therecent development of two-particle-attached and two-particle-removed coupled-cluster methods (see Ref. 59), we are now in the position where one can also study quantum-dot systemssuch as N=8 or 18, or for larger electron systems as well. A more in-depth analysis of our one-particle-attached andone-particle-removed methods will be presented in Ref. 60.TABLE VII. The electrochemical potentials μ(N)=E(N)− E(N−1) for N=3, 6, 7, 12, and 13 electrons computed with the electron attached and ionization potential equation-of-motion coupled-cluster (CC) method and the diffusion Monte Carlo (DMC)method. A frequency of ω=0.28 has been used. The absolute value of the energy difference between the two many-body approaches is listed in the final column as |/Delta1E|. All coupled-cluster results have been obtained for R=20. CC DMC /Delta1E E(3)−E(2) 1.2284 1.2123(1) 0.0161(1) E(6)−E(5) 2.0438 2.0663(1) 0.0225(1) E(7)−E(6) 2.4528 2.4341(1) 0.0187(1) E(12)−E(11) 3.5420 3.5618(1) 0.0198(1) E(13)−E(12) 3.8738 3.8582(1) 0.0156(1) IV . CONCLUSIONS AND PERSPECTIVES We have shown in this paper that coupled-cluster calcu- lations that employ an effective Coulomb interaction and aself-consistent Hartree-Fock single-particle basis reproduceexcellently diffusion Monte Carlo calculations, even for verylow oscillator frequencies. This opens up many interesting per-spectives, in particular, since our coupled-cluster calculationsare rather inexpensive from a high-performance computingstandpoint. Properties such as addition spectra and excitedstates can be extracted using equation-of-motion-based tech-niques (see, for example, Refs. 17,18,33, and 34). In Refs. 33 and 34, addition spectra of nuclei in the chain of oxygen isotopes have been calculated using the particle-attached orparticle-removed equation-of-motion method (Refs. 17and 18). We have also performed preliminary calculations of addition spectra for quantum dots using the above closed-shellsystems for N=3, 5, 7, 11, and 13, obtaining a very good agreement with the diffusion Monte Carlo results listed inRefs. 10and 61. Combining the one-particle-attached and -removed method with our recently developed two-particle-attached and -removed coupled-cluster methods, 59we can compute almost all circular quantum dots up N=22 electrons except for dots with N=9 and N=15–17 electrons. These results will be presented elsewhere.60 Furthermore, since our codes run in an uncoupled basis, one can also study other trapping potentials than the standardharmonic-oscillator potential. A time-dependent formulationof coupled-cluster theory may even allow for studies oftemporal properties of quantum dots such as the effect of atime-dependent perturbation. For circular dots, we found that with the inclusion of triples correlations, there are, for all systems studied, indicationsthat many-body correlations beyond three-particle–three-holeexcitations in the coupled-cluster amplitude Tare negligi- ble. We observe also that for systems with more particles,correlations beyond the Hartree-Fock level tend to decrease.Thus, although we are able to extend ab initio coupled-cluster calculations of quantum dots to systems up to 50 electrons,a mean-field description will probably convey most of theinteresting physics. 115302-13M. PEDERSEN LOHNE et al. PHYSICAL REVIEW B 84, 115302 (2011) With two popular and reliable many-body techniques such as coupled-cluster theory and diffusion Monte Carlo resulting in practically the same energies, one is in the positionwhere one can extract almost exact density functionals forquantum-dot systems. This allows for important comparisonswith available density functionals for quantum dots. Finally,we have also noted that triples correlations are necessary inorder to obtain correct results. 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PhysRevB.82.235301.pdf

Cascading enables ultrafast gain recovery dynamics of quantum dot semiconductor optical amplifiers Niels Majer, Kathy Lüdge, and Eckehard Schöll Institut für Theoretische Physik, Technische Universität Berlin, D-10623 Berlin, Germany /H20849Received 27 August 2010; revised manuscript received 29 October 2010; published 1 December 2010 /H20850 In this work the ultrafast gain recovery dynamics of a quantum dot semiconductor optical amplifier is investigated on the basis of semiconductor Bloch equations including microscopically calculated carrier-carrierscattering rates between the two-dimensional carrier reservoir and the confined quantum dot ground and firstexcited state. By analyzing the different scattering contributions we show that the cascading process makes amajor contribution to the ultrafast recovery dynamics. DOI: 10.1103/PhysRevB.82.235301 PACS number /H20849s/H20850: 78.67.Hc, 42.55.Px, 42.60.Rn I. INTRODUCTION Optoelectronic devices based on semiconductor quantum dots /H20849QDs /H20850are promising candidates for future high-speed telecom applications with low operation currents, high tem-perature stability, low chirp, and ultrafast gain recovery dy-namics and hence pattern effect free amplification at high bitrates. Efficient carrier-carrier and carrier-phonon scatteringprocesses into and between the discrete confined QD levelsplay a crucial role in the operation characteristics of QD-based devices and numerous experimental and theoretical in-vestigations have addressed these topics. The simplest theo-retical approach to carrier scattering is based on time-dependent perturbation theory containing energy conserving /H9254functions in terms of single-particle energies of the initial and final states.1,2In more elaborate quantum kinetic descriptions3,4the requirement of strict energy conservation is absent, which helps to resolve such issues as the phononbottleneck 5,6present in the simpler perturbative approaches. In this paper we investigate the performance of QD semi- conductor optical amplifiers /H20849SOAs /H20850by analyzing the gain recovery dynamics in response to ultrashort /H20849150 fs /H20850input pulses. Under high-injection currents the dominant contribu-tion to carrier scattering into the QDs is given by carrier-carrier scattering /H20849Auger /H20850processes which we calculate using a time-dependent perturbative approach. Carrier-phononscattering is implicitly included through the assumption offast thermalization of carriers in the carrier reservoir. By in-cluding carrier-carrier scattering processes between the con-fined QD ground state /H20849GS/H20850and first excited state /H20849ES/H20850and the carrier reservoir, we extend our previous model whichwas restricted to the QD ground state. 7–10The carrier reser- voir is modeled by a two-dimensional /H208492D/H20850quantum well /H20849QW /H20850into which carriers are injected and then scatter into the discrete QD levels. There is an ongoing discussion in theliterature 11–14whether direct capture or relaxation processes dominate the recovery dynamics of QD SOAs. Inexperiments 11,15,16the gain recovery was found to be very fast /H20849/H11011ps/H20850. Including only the QD ground state in the carrier kinetics cannot account quantitatively for this behavior.Therefore in typical rate equation approaches a multitude ofQD levels are considered 12using numerical fits with constant transition rates between QW states and QD levels. In ourapproach we incorporate microscopically calculated Augerscattering contributions for transitions between GS, ES, and QW. This allows us to quantify the strength of the differentscattering processes and we find that the cascading processvia the excited state to the ground state of the quantum dotmakes the major contribution to the ultrafast gain recoverydynamics observed in QD SOAs. The paper is organized as follows. After introducing the theoretical model in Sec. IIwe present the microscopic ap- proach to Auger scattering in Sec. III. In Sec. IVwe discuss the impact of the various scattering channels to the gain re-covery dynamics of the QD SOA and conclude in Sec. V. II. MODEL The theoretical model used to describe the QD SOA is based on a Bloch equation approach17for the coupled dy- namics of the interband polarization and the electron /H20849e/H20850and hole /H20849h/H20850occupation probabilities of the QDs. We consider optical transitions between the electron and hole GS as wellas the first ES of the QDs. Inhomogeneous broadening,which occurs in real devices due to fluctuations in QD sizeand material composition and directly affects the energy lev-els, is accounted for by assuming a Gaussian size distributionaround a central ground-state transition frequency /H92750with standard deviation /H9254/H9275. The spectral QD density is then given byN/H20849/H9275/H20850=NQD /H208812/H9266/H9254/H9275exp/H20851−/H20849/H9275−/H92750/H208502 2/H9254/H92752/H20852and the total QD density NQDis approximated by a sum over a finite number of subensembles NQD=/H20858jNj=/H20858jN/H20849/H9275j/H20850/H9004/H9275, where /H9004/H9275denotes the spectral width of the QD subgroups. Furthermore, the separate dy-namics of electrons and holes in the 2D carrier reservoirsurrounding the QDs is taken into account in our modelingapproach. Assuming an input light field of the form E/H20849t/H20850 = 1 2E0/H20849t/H20850/H20851exp/H20849i/H9275Lt/H20850+exp /H20849−i/H9275Lt/H20850/H20852, where E0/H20849t/H20850is the slowly time varying envelope of the electric field and /H9275Lis the carrier-wave frequency, the set of equations in the usualslowly varying envelope and rotating wave approximationtake the following form: /H11509pmj /H11509t=−i/H9254/H9275mjpmj−i/H9024 2/H20849fe,mj+fh,mj−1/H20850−1 T2pmj, /H208491/H20850 /H11509fe,mj /H11509t=−I m /H20851/H9024pmj/H11569/H20852−Rsp+/H20879/H11509fe,mj /H11509t/H20879 col, /H208492/H20850PHYSICAL REVIEW B 82, 235301 /H208492010 /H20850 1098-0121/2010/82 /H2084923/H20850/235301 /H208496/H20850 ©2010 The American Physical Society 235301-1/H11509fh,mj /H11509t=−I m /H20851/H9024pmj/H11569/H20852−Rsp+/H20879/H11509fh,mj /H11509t/H20879 col, /H208493/H20850 /H11509we /H11509t=j/H20849t/H20850 e0−R˜sp−2/H20858 m,jNj/H20879/H11509fe,mj /H11509t/H20879 col, /H208494/H20850 /H11509wh /H11509t=j/H20849t/H20850 e0−R˜sp−2/H20858 m,jNj/H20879/H11509fh,mj /H11509t/H20879 col. /H208495/H20850 Equations /H208491/H20850–/H208493/H20850constitute the semiconductor Bloch equa- tions and Eqs. /H208494/H20850and /H208495/H20850are the dynamic equations for the QW carriers. The superscript jdenotes the jth subgroup of QDs of the inhomogeneously broadened ensemble. The QDground state and excited state are labeled by m. The micro- scopic polarization p mjis a dimensionless quantity describing the probability of an optical transition between the respectiveelectron and hole levels. T 2is the dephasing time of the optical polarization, which accounts for coherence loss through scattering processes. fe,mjand fh,mjare the electron and hole occupation probabilities of the jth subensemble of themth QD level, and weandwhare the electron and hole densities in the carrier reservoir, respectively. The total QDcarrier density is obtained by summing over all subgroups, n b,m=2/H20858jfb,mjNjforb=e,h. The factor of 2 arises due to spin degeneracy of the QD levels. The total macroscopic polar-ization density Pis obtained by summing over all states and subensembles, P=/H20858 j,mNj d/H9262pmj, where dis the thickness of the active region. The detuning of the input light field frequency /H9275Lto the frequency /H9275mjof the optical transition of the respec- tive QD level is given by /H9254/H9275mj=/H20849/H9275mj−/H9275L/H20850. The Rabi fre- quency of the QD transitions with associated dipole moment /H9262/H20849assumed to be equal for all QDs /H20850is/H9024/H20849t/H20850=/H9262 /H6036E/H20849t/H20850. Rsp/H20849fe,mj,fh,mj/H20850=Wmfe,mjfh,mjis the spontaneous recombination rate with the Einstein coefficient Wm=/H92622/H20881/H9280bg 3/H9266/H92800/H6036/H20849/H9275m c/H208503/H20849back- ground dielectric constant /H9280bg, elementary charge e0, and vacuum velocity of light c/H20850. An important contribution to the dynamics of QD SOAs is the nonradiative carrier-carrierscattering between confined QD states and continuous 2D QW states denoted by /H20849 /H11509//H11509t/H20850fb,mj/H20841col. The electric current den- sityj/H20849t/H20850is injected into the QW. The spontaneous QW band- band recombination rate is given by R˜sp=BSwewhwith rate constant BS. The InGaAs dot-in-a-well structure is modeled by a parabolic QW band structure with effective masses me =0.043 m0and mh=0.45 m0. The in-plane part of the QD wave functions is approximated by 2D harmonic-oscillatoreigenfunctions, whereas the QW wave functions are approxi-mated by orthogonalized plane waves. The confinement po-tential in zdirection is treated in an effective-well-width ap- proximation using an infinite potential barrier with width L =8 nm. A schematic energy diagram of the QD-QW systemis shown in Fig. 1/H20849a/H20850. The input pulse amplitude E 0/H20849t/H20850is modeled as a Gaussian with a full width at half maximum/H20849FWHM /H20850of 150 fs. The parameters used in the simulation are listed in Table I.III. AUGER SCATTERING PROCESSES In a device operated under high-injection currents carrier- carrier scattering makes, next to carrier-phonon scattering, animportant contribution to the interlevel carrier dynamics. Inthis paper we focus on an operating regime well above trans-parency, where we assume Coulomb scattering to make thedominant contribution to carrier scattering and therefore re-strict the QD-QW scattering contributions to those resultingfrom carrier-carrier scattering. Carrier-phonon effects remainimplicitly included in the fast thermalization of QW carriers.For the evaluation of the carrier-carrier scattering contribu-tions we remain within the Markov limit since the total statespace is in this case large enough to provide efficient scat-tering already with strict energy conservation /H20849as opposed to carrier-phonon scattering, where a mismatch of the transitionenergy to the LO-phonon energy prohibits efficient scatteringwithin the Markov limit and quantum kinetic theories predict relaxation direct capture(b) (a) Energy (c)/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright zIII II II V I II III IVES GS hν∆EQD e,0 ∆EQD h,0Sin,rel e Sin,rel h Sin,rel/prime e Sin,rel/prime/prime eQW QW∆e ∆hSin,cap e,GS Sin,cap e,ES Sin,cap h,GS Sin,cap h,ESSin,cap e,GS Sin,cap e,ES FIG. 1. /H20849Color online /H20850/H20849a/H20850Energy diagram of the QD-QW sys- tem. /H20849b/H20850Direct electron-capture processes from the QW to the QD ground state /H20849I,II/H20850and first-excited state /H20849III,IV /H20850. Panels I, III and panels II, IV show pure e-eand mixed e-hscattering processes, respectively. /H20849c/H20850Electron relaxation processes to the QD ground state. /H20849blue/dark gray arrows denote electron transitions also in the valence band /H20850. TABLE I. Numerical parameters used in the simulation unless stated otherwise. Symbol Value Symbol Value /H9004Ee,m=0QD210 meV /H9004Ee,m=/H110061QD146 meV /H9004Eh,m=0QD50 meV /H9004Eh,m=/H110061QD44 meV /H9004e 64 meV /H9004h 6m e V d 4n m /H9262 0.6e0nm /H6036/H9254/H9275 4.25 meV /H6036/H9004/H9275 0.36 meV /H6036/H92750=/H6036/H9275L 0.96 eV BS540 ns−1nm2 /H9280bg 14.2 NQD1010cm−2 me 0.043 m0 mh 0.45m0 Wm=0,Wm=/H110061 0.7, 0.84 ns−1FWHM 150 fs T2 25 fs /H9008 /H9266 j0 1A c m−2j 1,...,70MAJER, LÜDGE, AND SCHÖLL PHYSICAL REVIEW B 82, 235301 /H208492010 /H20850 235301-2larger scattering rates /H20850. The collision term in Eqs. /H208492/H20850–/H208495/H20850is given in the Markov limit up to second order in the screenedCoulomb potential Vby 2,18 /H20879/H11509fb,mj /H11509t/H20879 col=2/H9266 /H6036/H20858 /H92631/H92632/H92633V/H92631/H92633/H92632m/H20851V/H92631/H92633/H92632m/H11569−V/H92633/H92631/H92632m/H11569/H20852 /H11003/H9254/H20849/H9280m−/H9280/H92631+/H9280/H92632−/H9280/H92633/H20850 /H11003/H20851f/H92631/H208491−f/H92632/H20850f/H92633/H208491−fb,mj/H20850 −/H208491−f/H92631/H20850f/H92632/H208491−f/H92633/H20850fb,mj/H20852/H20849 6/H20850 with b/H33528/H20853e,h/H20854. In these Auger-type scattering events two car- riers scatter from initial states /H92631/H20849with energy /H9280/H92631/H20850and/H92633to final states mand/H92632, respectively, and vice versa. The calculation involves screened Coulomb matrix elementsfor the direct and exchange interaction /H20841V /H92631/H92633/H92632m/H208412and V/H92631/H92633/H92632mV/H92633/H92631/H92632m/H11569, respectively. Details of the explicit form of the Coulomb matrix elements and the involved wave func- tions can be found in the Appendix. The index mrepresents the quantum number of the 2D angular momentum of theconfined QD states, m=0 and m/H33528/H208531,−1 /H20854for the GS and ES, respectively. The /H9254function in Eq. /H208496/H20850ensures energy con- servation. The occupation probability of state /H9263iis denoted byf/H9263i. Figures 1/H20849b/H20850and1/H20849c/H20850give a systematic overview of all processes leading to in-scattering into the QD electron levels. The blue /H20849dark gray /H20850arrows denote electron transi- tions of the scattering partners. Panels I and III show puree-eprocesses while panels II and IV display mixed e-hpro- cesses. The corresponding processes for in-scattering into theQD hole levels are obtained by exchanging all electron andhole states. The out-scattering processes are obtained by in-verting all arrows of the electron transitions. The exchangeprocesses of pure e-ecapture processes contributing to the scattering rates are not shown since there is no qualitativedifference to the direct processes. In case of mixed e-hpro- cesses /H20849II, IV /H20850the exchange processes lead to transitions across the band gap that are neglected since they are unlikelyto occur. Note that the process shown in panel III of Fig. 1/H20849c/H20850 is the exchange process of the one in panel I. In the follow-ing we decompose the scattering rate into contributionsoriginating from direct carrier capture from the QW into theGS and ES /H20849R cap/H20850and relaxation processes from ES to GS /H20849Rrel/H11032,Rrel/H11033/H20850as shown in Figs. 1/H20849b/H20850and 1/H20849c/H20850, respectively. The relaxation processes are split into scattering events with one intra-QD transition and a QW transition /H20849Rrel/H11032/H20850and pro- cesses where both involved carriers perform transitions from QW states to QD states /H20849Rrel/H11033/H20850, i.e., /H20849/H11509//H11509t/H20850fb,mj/H20841col=Rcap +Rrel/H11032+Rrel/H11033. Processes involving three QD states are ne- glected since energy conservation allows at most one kstate in the QW to be involved thus inhibiting efficient scattering. A. Direct capture The contribution to Eq. /H208496/H20850from direct capture processes /H20851Fig. 1/H20849b/H20850/H20852can be expressed asRcap=Sb,min,cap/H208491−fb,mj/H20850−Sb,mout,capfb,mj/H208497/H20850 with the in-scattering rate /H20849k1→m,k3→k2/H20850, where states in the QW are labeled by the in-plane carrier momentum ki, Sb,min,cap=2/H9266 /H6036/H20858 k1k2k3b/H11032Vk1k3k2m/H208512Vk1k3k2m/H11569−/H9254b,b/H11032Vk3k1k2m/H11569/H20852 /H11003/H9254/H20849/H9280mb−/H9280k1b+/H9280k2b/H11032−/H9280k3b/H11032/H20850fk1/H208491−fk2/H20850fk3. /H208498/H20850 The respective out-scattering rate is obtained by the substi- tution fki→/H208491−fki/H20850in Eq. /H208498/H20850. Assuming quasi-equilibrium within the QW system, the in- and out-scattering rates are related to each other via detailed balance,7 Sb,min,cap=Sb,mout,capexp/H20873/H11007/H9004Eb,mQD/H11006FbQW kT/H20874, /H208499/H20850 where FbQWare the quasi-Fermi levels in the QW and the upper and lower signs refer to b=eand b=h, respectively. The calculated direct capture rates as a function of the QWcarrier densities are shown in Figs. 2/H20849a/H20850and2/H20849b/H20850. For low densities the in-scattering rates show a quadratic increasewith growing QW carrier densities as expected from massaction kinetics. With increasing w bdeviations from this be- havior become apparent eventually leading to a decrease inthe scattering rates due to Pauli blocking /H20851see red dashed- dotted line in Fig. 2/H20849a/H20850/H20852. The out-scattering rates are charac- terized by a sharp increase followed by a decrease due toPauli blocking at higher densities. B. Relaxation processes The relaxation processes shown in Fig. 1/H20849c/H20850describe a redistribution of carriers within the intra-QD levels. The in-scattering rate /H20849m 1→m,k3→k2/H20850for processes I and II with00.51(a)Sin,cap e,GS Sin,cap h,GS Sin,cap e,ES Sin,cap h,ES we(1011cm−2)00.10.2 x10000x2000(b)Sout,cap e,GS Sout,cap h,GS Sout,cap e,ES Sout,cap h,ES 0 10 20 30 40 we(1011cm−2)0123 x5(c) Sin,rel/prime e Sin,rel/prime h Sin,rel/prime/prime e Sin,rel/prime/prime h 0 10 20 30 40 we(1011cm−2)0123 x10 x100x2(d) Sout,rel/prime e Sout,rel/prime h Sout,rel/prime/prime e Sout,rel/prime/prime hscattering rates ( ps−1) FIG. 2. /H20849Color online /H20850Auger scattering rates of the QD-QW system vs QW electron density we/H20849wh/we=1.5 /H20850./H20851/H20849a/H20850and /H20849b/H20850/H20852Direct capture rates. /H20851/H20849c/H20850and /H20849d/H20850/H20852Intra-QD relaxation rates. Parameters see Table I.CASCADING ENABLES ULTRAFAST GAIN RECOVERY … PHYSICAL REVIEW B 82, 235301 /H208492010 /H20850 235301-3m1/H33528/H20853−1,1 /H20854,m=0, contributing to Eq. /H208496/H20850, is given by Sb,min,rel/H11032=2/H9266 /H6036/H20858 k2k3b/H11032Vm1k3k2m/H208512Vm1k3k2m/H11569−/H9254b,b/H11032Vk3m1k2m/H11569/H20852 /H11003/H9254/H20849/H9280mb−/H9280m1b+/H9280k2b/H11032−/H9280k3b/H11032/H20850/H208491−fk2/H20850fk3, Rrel/H11032=Sb,min,rel/H11032fb,m1j/H208491−fb,mj/H20850−Sb,mout,rel/H11032/H208491−fb,m1j/H20850fb,mj. /H2084910/H20850 The dynamical equations for the processes III and IV /H20849Rrel/H11033/H20850 in Fig. 1/H20849c/H20850can be obtained in a similar fashion as Eq. /H2084910/H20850 with the difference that pure /H20849e-e,h-h/H20850and mixed /H20849e-h/H20850pro- cesses have to be separated, as the QD occupation factorsinvolve the carrier types of both involved scattering partners.The rate for the mixed case is vanishingly small and theshown rate is the one for the pure case. For electrons it reads: S e,min,rel/H11033=2/H9266 /H6036/H20858 k1k2Vk1m1k2m/H208512Vk1m1k2m/H11569−Vm1k1k2m/H11569/H20852 /H11003/H9254/H20849/H9280me−/H9280m1e+/H9280k2e−/H9280k1e/H20850/H208491−fk2/H20850fk1, Rrel/H11033=Se,min,rel/H11033fe,m1j/H208491−fe,mj/H20850−Se,mout,rel/H11033/H208491−fe,m1j/H20850fe,mj. /H2084911/H20850 The calculated rates are shown in Figs. 2/H20849c/H20850and2/H20849d/H20850. The relaxation rates of type Srel/H11032are characterized by a sharp increase and a decrease at higher densities due to the effectof Pauli blocking. Note that in- and out-scattering rates only differ by a constant factor S e/hout,rel=Se/hin,relexp/H20851−/H9004e/h//H20849kBT/H20850/H20852re- sulting from detailed balance. IV . GAIN RECOVERY DYNAMICS A key parameter determining the performance of an SOA is the gain recovery time after an input light pulse with pulsearea /H9258=/H20848/H9024dthas depleted the carriers. This can be deter- mined by a pump-probe experiment, where a probe pulsemeasures the gain after a delay time /H9270with respect to the pump pulse. The gain of the probe pulse can be approxi- mated by g/H20849/H9275,/H9270/H20850=−/H9275Im/H20851Pprobe /H20849/H9275,/H9270/H20850 Eprobe /H20849/H9275/H20850/H20852. Here, Pprobe andEprobe are the Fourier amplitudes of the macroscopic polarization and the probe pulse electric field, respectively. The inputlight field is chosen as resonant to the central ground-statetransition of the QDs. Our microscopic approach now allowsfor an analysis of the influence of different scattering chan-nels on the gain recovery dynamics of a QD SOA. In Figs. 3 and4we show the calculated gain recovery and the associ- ated carrier dynamics for three different scenarios sketchedin Fig. 3/H20849b/H20850. Scenario 1 allows only direct capture to the QD ground and excited state while scenario 2 allows the carriersto relax to the QD ground state via a cascading process.Scenario 3 includes both cascading and direct capture. Figure3/H20849a/H20850shows the normalized gain in dependence of the pump- probe delay time /H9270for the three scenarios depicted in Fig. 3/H20849b/H20850. All curves show maximum gain depletion for a pump-probe delay time /H9270=0. Scenario 2 deviates only slightly from scenario 3 in the intermediate recovery stage, whereas thecoherent recovery phase /H20849 /H9270/H11349100 fs /H20850perfectly coincides. If the relaxation path from the excited state to the ground stateis blocked /H20849scenario 1 /H20850the recovery dynamics drastically de- teriorates compared to the two other scenarios. The initialgain depletion around /H9270=0 is stronger and the normalized gain recovers much more slowly than in the other scenariosfor /H9270/H110220. The population dynamics of the QD electrons and holes for scenario 1 with pump-probe delay /H9270=0 is shown in Figs. 4/H20849a/H20850and4/H20849b/H20850, respectively, while Figs. 4/H20849c/H20850and4/H20849d/H20850 depict the electron and hole dynamics for scenario 3. In Figs.4/H20849a/H20850and4/H20849b/H20850, the resonant interaction of the QD ground state with the incoming light field causes a strong carrier depletionboth of electrons and holes, whereas the off-resonant ES ex-periences only little light-matter interaction and the relax-ation path to the GS is blocked. In Figs. 4/H20849c/H20850and4/H20849d/H20850relax- ation processes between the ES and GS are allowed resultingin a strong decrease in the ES carrier populations and a fasterrefilling of the GS compared to scenario 1. The correspond-ing time evolution of the QW carrier densities for scenarios 1and 3 is plotted in Fig. 5. The recovery dynamics of the carrier reservoir is on a nanosecond rather than a picosecond 2 0 2 4 6 8 d e l a y t i m e
( p s )0 . 00 . 20 . 40 . 60 . 81 . 0n o r m a l i z e d g a i n 3 2 1(a) (b) 2 31 QW QWQW FIG. 3. /H20849Color online /H20850Gain recovery dynamics for scattering scenarios 1 /H20849direct capture /H20850,2/H20849cascading /H20850, and 3 /H20849all/H20850as sketched in panel /H20849b/H20850/H20849Auger electrons omitted /H20850. Parameters: j/H1101510j0, see also Table I. 0.850.900.951.00fe(a)scenario 1 e,GS e,ES 0 2 4 6 8 t(ps)0.10.15fh(b)h,GS h,ES(c)scenario 3 0 2 4 6 8 t(ps)(d) FIG. 4. /H20849Color online /H20850Average /H20851/H20849a/H20850and /H20849b/H20850/H20852electron and /H20851/H20849b/H20850 and /H20849d/H20850/H20852hole population vs time for pump-probe delay /H9270=0 for direct capture processes only /H20851scenario 1 in Fig. 3/H20849b/H20850/H20852and with all Auger processes /H20851scenario 3 in Fig. 3/H20849b/H20850/H20852, respectively. Red/gray /H20849black /H20850solid curves are ground-state electron /H20849hole /H20850populations, and green/light gray /H20849blue/dark gray /H20850dashed curves are excited- state populations. Parameters as in Fig. 3.MAJER, LÜDGE, AND SCHÖLL PHYSICAL REVIEW B 82, 235301 /H208492010 /H20850 235301-4time scale /H20849as for the QD carriers /H20850. Again, comparing sce- narios 1 and 3 the carrier dynamics differs qualitatively. Withthe additional relaxation channel /H20849scenario 3 /H20850, the carrier drain from the QW is stronger, the maximum depletion ishigher, and it occurs earlier. This is due to stronger accumu-lated scattering into the QDs. The nonlinearity of carrier-carrier scattering, especially due to Pauli blocking effects,could potentially shift the relative importance of differentscattering channels for changing QW carrier densities. InFig. 6the gain recovery dynamics is plotted for different injection current densities of j=5j 0and j=20j0. While the gain recovers faster in the case of higher injection current/H20851panel /H20849b/H20850/H20852, the cascading process remains the key channel in the gain recovery. The direct capture process in panel /H20849a/H20850is omitted because for the given injection current density theQD ground state is not in an inverted state. V . CONCLUSION In conclusion, our systematic microscopic analysis of dif- ferent Auger scattering channels, including confined QDground and excited states and extended QW states, combinedwith a full nonlinear simulation of the coupled polarizationand population dynamics of carriers has established that cas-cading Coulomb scattering processes from the carrier reser-voir via the QD excited state into the QD ground state con-stitute the major contribution to the ultrafast recoverydynamics of QD SOAs. ACKNOWLEDGMENTS This work was supported by DFG in the framework of Sfb 787. We thank A. Knorr and U. Woggon for useful dis-cussions. In addition we thank A. Wilms for helpful com-ments regarding the calculation of the Coulomb scatteringrates. APPENDIX: MATRIX ELEMENTS AND ORTHOGONALIZATION PROCEDURE The matrix elements of the Coulomb potential from Eq. /H208496/H20850, V/H92631/H92633/H92632/H9263=/H20885d3rd3r/H11032/H9023/H9263/H11569/H20849r/H20850/H9023/H92632/H11569/H20849r/H11032/H20850e02 4/H9266/H92800r/H9023/H92631/H20849r/H20850/H9023/H92633/H20849r/H11032/H20850 /H20849A1/H20850 involve the single-particle wave functions /H9023v/H20849r/H20850of electrons and holes in the confinement potential of the QD-QW sys-tem. Under the assumption that the wave functions can be separated into an in-plane component and a perpendicular /H20849z/H20850 component /H9023/H9263/H20849r/H20850=/H9278lb/H20849/H9267/H20850/H9264/H9268b/H20849z/H20850ub/H20849r/H20850, where ub/H20849r/H20850are the lat- tice Bloch functions and land/H9268the quantum numbers of the in-plane and z-component wave functions, respectively, Eq. /H20849A1/H20850can be expressed as2 V/H92631/H92633/H92632/H9263=1 A/H20858 qV/H92681/H92683/H92682/H9268b,b2/H9254b,b1/H9254b2,b3 /H11003/H20855/H9278lb/H20841e−iq·/H9267/H20841/H9278l1b1/H20856/H20855/H9278l2b2/H20841eiq·/H9267/H20841/H9278l3b3/H20856, /H20849A2/H20850 where qis the two-dimensional in-plane carrier momentum and V/H92681/H92683/H92682/H9268b,b2 =e02 2/H92800q/H20885dzdz /H11032 /H11003/H9264/H9268b/H20849z/H20850/H11569/H9264/H92682b/H11032/H20849z/H11032/H20850/H11569e−q/H20841z−z/H11032/H20841/H9264/H92683b/H11032/H20849z/H20850/H9264/H92681b/H20849z/H11032/H20850./H20849A3/H20850 In a simple approach to the QD confinement we approxi- mate the z-confinement potential by an infinite-height poten- tial barrier of width L=2d, where dis the thickness of the QW layer. For the in-plane confinement potential of the QDswe adopt the model of a two-dimensional harmonic oscilla-tor allowing for an analytic solution of the wave functionoverlap integrals in Eq. /H20849A2/H20850. The calculation of realistic single-particle wave functions is a task of its own and re-quires detailed knowledge of the QD geometry and the ma-terial composition and is beyond the scope of this paper. In aquantum well containing no QDs the in-plane component of the wave functions would be plane waves /H9278k0/H20849/H9268/H20850 =/H208491//H20881A/H20850eik·/H9268. In the presence of QDs the QW wave func- tions are altered in the vicinity of the QDs due to localchanges in the QD-QW potential. To describe the combinedsystem we therefore use plane waves orthogonalized withrespect to the QD states /H20849OPW /H20850 /H20841 /H9278k/H20856=1 Nk/H20873/H20841/H9278k0/H20856−/H20858 /H9251/H20841/H9278/H9251/H20856/H20855/H9278/H9251/H20841/H9278k0/H20856/H20874, /H20849A4/H20850 where Nkis the normalization constant. The summation runs over all localized QD states. Assuming an ensemble of iden-tical QDs located at positions /H20853 /H9268i/H20854with nonoverlapping−2 0 2 4 6 8 dela ytimeτ(ps)0.00.51.0norma lizedgain (a) 3 2 0 2 4 6 8 delay time τ(ps)(b) 3 2 1 FIG. 6. /H20849Color online /H20850Gain recovery dynamics for scattering scenarios 1 /H20849direct capture /H20850,2/H20849cascading /H20850, and 3 /H20849all/H20850as sketched in panel /H20849b/H20850of Fig. 3for /H20849a/H20850: injection current density j=5j0and /H20849b/H20850: injection current density j=20j0. Parameters: see Table I.0 500 1000 t[ps ]scenar io3 1.291.3 wh(1011cm−2) (b) −200 0 500 1000 t(ps )0.830.84we(1011cm−2)scenar io1 (a) FIG. 5. /H20849Color online /H20850/H20849a/H20850Time traces of we/H20849red solid, left axis /H20850 and wh/H20849black dashed, right axis /H20850for scenario 1 of Fig. 3/H20849b/H20850./H20849b/H20850 Same for scenario 3. Parameters as in Fig. 3.CASCADING ENABLES ULTRAFAST GAIN RECOVERY … PHYSICAL REVIEW B 82, 235301 /H208492010 /H20850 235301-5wave functions /H9278mi/H20849/H9268/H20850the summation runs over /H9251=/H20849m,i/H20850.B y construction, the OPW states are orthogonal to all QD states.The orthogonality of different OPW states can be obtainedfor an ensemble of randomly distributed QDs in the largearea limit, meaning that both the in-plane area A→/H11009and the number of QDs N→/H11009such that the QD density remains constant /H20849N/A=const /H20850. In that case only the QW densities appear in the evaluation of Eq. /H20849A2/H20850. 1H. C. Schneider, W. W. Chow, and S. W. Koch, Phys. Rev. B 64, 115315 /H208492001 /H20850. 2T. R. Nielsen, P. Gartner, and F. Jahnke, Phys. Rev. B 69, 235314 /H208492004 /H20850. 3M. Lorke, T. R. Nielsen, J. Seebeck, P. Gartner, and F. Jahnke, Phys. Rev. B 73, 085324 /H208492006 /H20850. 4M. Lorke, F. Jahnke, and W. W. Chow, Appl. Phys. Lett. 90, 051112 /H208492007 /H20850. 5J. Seebeck, T. R. Nielsen, P. Gartner, and F. Jahnke, Phys. Rev. B 71, 125327 /H208492005 /H20850. 6H. Kurtze, J. Seebeck, P. Gartner, D. R. Yakovlev, D. Reuter, A. D. Wieck, M. Bayer, and F. Jahnke, Phys. Rev. B 80, 235319 /H208492009 /H20850. 7K. Lüdge and E. Schöll, IEEE J. Quantum Electron. 45, 1396 /H208492009 /H20850. 8K. Lüdge and E. Schöll, Eur. Phys. J. D 58, 167 /H208492010 /H20850. 9C. Otto, K. Lüdge, and E. Schöll, Phys. Status Solidi B 247, 829 /H208492010 /H20850. 10M. Wegert, N. Majer, K. Lüdge, S. Dommers-Völkel, J. Gomis- Bresco, A. Knorr, U. Woggon, and E. Schöll, Semicond. Sci. Technol. 26, 014008 /H208492011 /H20850.11J. Gomis-Bresco, S. Dommers, V . V . Temnov, U. Woggon, J. Martinez-Pastor, M. Laemmlin, and D. Bimberg, IEEE J. Quan- tum Electron. 45, 1121 /H208492009 /H20850. 12J. Kim, C. Meuer, D. Bimberg, and G. Eisenstein, Appl. Phys. Lett. 94, 041112 /H208492009 /H20850. 13A. V . Uskov, T. W. Berg, and J. Mørk, IEEE J. Quantum Elec- tron. 40, 306 /H208492004 /H20850. 14G. Bertrand, C. Delage, M. Bafleur, N. Nolhier, J.-M. Dorkel, Q. Nguyen, N. Mauran, D. Trémouilles, and P. Perdu, IEEE J. Solid-State Circuits 36, 1373 /H208492001 /H20850. 15J. Gomis-Bresco, S. Dommers, V . V . Temnov, U. Woggon, M. Laemmlin, D. Bimberg, E. Malic, M. Richter, E. Schöll, and A.Knorr, Phys. Rev. Lett. 101, 256803 /H208492008 /H20850. 16P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and D. Bimberg, IEEE Photon. Technol. Lett. 12, 594 /H208492000 /H20850. 17W. W. Chow and S. W. 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PhysRevB.73.094123.pdf

Empirical tight-binding model for titanium phase transformations D. R. Trinkle,1,2M. D. Jones,3,2R. G. Hennig,4S. P. Rudin,2R. C. Albers,2and J. W. Wilkins4 1Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright Patterson Air Force Base, Dayton, Ohio 45433-7817, USA 2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 3State University of New York, Buffalo, New York 14260, USA 4Ohio State University, Columbus, Ohio 43210, USA /H20849Received 26 February 2005; revised manuscript received 13 December 2005; published 28 March 2006 /H20850 For a previously published study of the titanium hexagonal close packed /H20849/H9251/H20850to omega /H20849/H9275/H20850transformation, a tight-binding model was developed for titanium that accurately reproduces the structural energies and elec-tron eigenvalues from all-electron density-functional calculations. We use a fitting method that matches thecorrectly symmetrized wave functions of the tight-binding model to those of the density-functional calculationsat high symmetry points. The structural energies, elastic constants, phonon spectra, and point-defect energiespredicted by our tight-binding model agree with density-functional calculations and experiment. In addition, amodification to the functional form is implemented to overcome the “collapse problem” of tight binding,necessary for phase transformation studies and molecular dynamics simulations. The accuracy, transferability,and efficiency of the model makes it particularly well suited to understanding structural transformations intitanium. DOI: 10.1103/PhysRevB.73.094123 PACS number /H20849s/H20850: 71.15.Nc, 61.72.Ji, 63.20. /H11002e, 62.20.Dc I. INTRODUCTION Titanium is a useful starting material for many structural alloys;1however, the formation of the high-pressure omega phase is known to lower toughness and ductility.2The atom- istic mechanism of the transformation from the room tem-perature /H9251phase /H20851hexagonal close packed /H20849hcp /H20850/H20852to the high- pressure /H9275was recently elucidated by Ref. 3. The explication of the /H9251→/H9275atomistic transformation relied on the compari- son of approximate energy barriers for nearly 1000 different6- and 12-atom pathways. That study required the use of anaccurate and efficient interatomic potential model: In thiscase, a tight-binding model reparametrized using all-electrondensity-functional calculations. After reparametrizing, we modify the functional form of tight binding for small interatomic distances to overcome thecollapse problem. This ensures that the potential is suitablefor phase transition studies and molecular dynamicssimulations. The collapse problem for tight-bindingmodels is caused by unphysically large overlap at smalldistances creating a low energy binding state; by modifyingthe functional form using short-range splining, the collapseproblem can be avoided. This paper provides the details ofthe model used in the previous phase transformation study ofRef. 3 and describes a general solution to the collapse prob-lem. Tight binding /H20849TB /H20850is a parametrized electronic structure method for calculation of total energies and atomic forcesfor arbitrary structures. It is an empirical model that canreproduce density-functional results for a range of structuresyet requires orders of magnitude less computational effort.The parameters of the model are determined by fittingto a database and the range of applicability is determinedby comparison to structures not in the database. Theend result is a model that balances three competingproperties—efficiency, accuracy, and transferability—which make it applicable to a variety of important structures. We fit our model to total energies and electron eigenval- ues for several crystal structures over a range of volumes toproduce a transferable model for the study of the /H9251→/H9275 transformation in Ti.3The potential has been successfully used to compute and sort possible transformation pathways,4 comparing favorably with generalized gradient approxima-tion /H20849GGA /H20850calculations in accuracy. Our fitting database is chosen to sample a large portion of the available phase spaceof parameters while constraining those parameters as muchas possible. The resulting model reproduces total energies,elastic constants, phonons, and point defects; all of which arenecessary for transformation modeling. The computationalefficiency allows simulations of length and time scales thatare inaccessible with GGA. In addition, the functional formsare modified for small distances to overcome the unphysicalcollapse problem; this is necessary for phase transitions andmolecular dynamics which sample small interatomic dis-tances. Moreover, the modification presented is applicable toother nonorthogonal tight-binding models without modifyingexisting parameters, hence extending their range of applica-bility. Section II describes tight binding as a parametrized elec- tronic structure method, the functional forms for titanium,the modifications for short distances, our fitting database, and our method of optimization. Section III gives the opti-mized parameters, and tests our model against total energies,elastic constants, phonons, and point defect formation ener-gies for /H9251,/H9275, and body-centered cubic /H20849bcc /H20850Ti. The point defect formation energies are used to compare our param-eters to those of Mehl and Papaconstantopoulos 5and Rudin et al. ,6and to demonstrate the efficacy of our modification of the short-range Hamiltonian and overlap functions.PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 1098-0121/2006/73 /H208499/H20850/094123 /H208499/H20850/$23.00 ©2006 The American Physical Society 094123-1II. METHODOLOGY A. Tight-binding formulation Electronic structure methods separate the total energy of a crystal into an ionic contribution and an electronic contribu-tion derived as the solution to a Hamiltonian problem. Treat-ing electrons as noninteracting fermionic quasiparticles per-mits an appropriate one-particle solution. 7To numerically solve the electronic problem requires a set of basis functions /H9278i, in terms of which the matrix Hijof the Hamiltonian op- erator and overlap matrix Sijare Hij=/H20855/H9278i/H20841Hˆ/H20841/H9278j/H20856,Sij=/H20855/H9278i/H20841/H9278j/H20856. These matrices give the eigenvalue equation H/H9274n=/H9280nS/H9274n, /H208491/H20850 where the electronic contribution to the total energy includes the term 2/H20858 /H9280n/H11021EF/H9280n, with Fermi energy EF. The Hamiltonian contains information about the wave function solutions themselves /H20849e.g., density- functional theory /H20850. Typically, the wave functions must be found self-consistently, which increases the computationalrequirements. In the tight-binding method, approximate Hamiltonian and overlap matrices are constructed by assuming atom-centered orbitals in a two-center approximation. This tech-nique is related to the linear combination of atomic orbitals/H20849LCAO /H20850method, which uses a basis /H9278iof solutions to the isolated atomic Schrödinger equation up to some energy and angular momentum quantum numbers /H20849nl/H20850:/H9278nlm/H20849r/H6023/H20850 =fnl/H20849/H20841r/H20841/H20850Ylm/H20849r/H6023//H20841r/H20841/H20850. Tight-binding Hamiltonian and overlap functions are calculated independently of the local environ- ment which increases efficiency but at the expense of trans-ferability. Empirical tight-binding eliminates explicit basis functions from the problem and parametrizes the Hamiltonian andoverlap matrices in terms of simple two-center integrals. 8 The basis is chosen to be angular momentum solutions lmup to some maximum lvalue: For a maximum l=1 we use s,px, py, and pzas the basis functions; for a maximum of l=2, we add in the five dorbitals dxy,dyz,dzx,dx2−y2, and d3z2−r2. The Hamiltonian and overlap matrices are written as sums of pa- rametrized functions h¯lm,l/H11032m/H11032/H20849r/H6023/H20850and s¯lm,l/H11032m/H11032/H20849r/H6023/H20850where r/H6023=R/H6023i −R/H6023jis the separation between two atoms iandj. The two- center approximation allows these functions to be simplifiedfurther according to the angular momentum components of the basis. 8For example, h¯pz,pz/H20849r/H6023/H20850separates into two symme- trized integrals h¯pz,pz/H20849r/H6023/H20850=hpp/H9268/H20849r/H20850cos2/H9258z+hpp/H9266/H20849r/H20850sin2/H9258z, where /H9258zis the angle between r/H6023and the zaxis. The higher rotational angular momentum integral hpp/H9254/H20849r/H20850is zero because aporbital has a maximal azimuthal quantum number of 1 along the zaxis. The integrals hpp/H9268/H20849r/H20850andhpp/H9266/H20849r/H20850are func-tions of only the distance of separation r=/H20841R/H6023j−R/H6023j/H11032/H20841. We write each Hamiltonian and overlap integral in these symmetrized functions; for a model with an spdbasis, there are ten inte- grals /H20849forhands/H20850to be determined: /H20849ss/H9268/H20850,/H20849sp/H9268/H20850,/H20849pp/H9268/H20850, /H20849pp/H9266/H20850,/H20849sd/H9268/H20850,/H20849pd/H9268/H20850,/H20849pd/H9266/H20850,/H20849dd/H9268/H20850,/H20849dd/H9266/H20850, and /H20849dd/H9254/H20850. The Hamiltonian and overlap matrices are then computed for an arbitrary atomic arrangement. In empirical tight binding, thetotal energy of the system is given by the eigenvalues /H9280nof Eq. /H208491/H20850and an ionic contribution EtotalTB=2/H20858 /H9280n/H11021EF/H9280n+V/H20849Rnuclei /H20850, where Vdoes not depend on the electronic states of the sys- tem. We use functional forms developed at the U.S. Naval Re- search Laboratory /H20849NRL /H20850, Washington, D.C., that do not use an explicit external pair potential but instead hasenvironment-dependent on-site energies. 9–11Without the pair potential V/H20849Rnuclei /H20850, the total energy is the sum of the occu- pied electron eigenvalues. Accommodating the lack of a pair potential requires a constant shift in the electron eigenvaluesin the fit database. The on-site Hamiltonian elements /H9280s,/H9280p, and/H9280dare not constants, but rather, depend on the distances of neighboring atoms to approximate three-body terms.12 The onsite energies /H9280l,iare functions of the “local density” /H9267i with four parameters /H9280l,i=al+bl/H9267i2/3+cl/H9267i4/3+dl/H9267i2, /H208492/H20850 where /H9267i=/H20858 j/HS11005iexp /H20849−/H92612rij/H20850fc/H20849rij/H20850. /H208493/H20850 The smooth cut-off function fc/H20849r/H20850is fc/H20849r/H20850=/H208751 + exp/H20873r−R0 l0/H20874/H20876−1 . /H208494/H20850 The intersite functions hll/H11032m/H20849r/H20850andsll/H11032m/H20849r/H20850are given by three parameters each hll/H11032m/H20849r/H20850=/H20849ell/H11032m+fll/H11032mr/H20850exp /H20849−gll/H11032m2r/H20850fc/H20849r/H20850, /H208495/H20850 sll/H11032m/H20849r/H20850=/H20849e¯ll/H11032m+f¯ll/H11032mr/H20850exp /H20849−g¯ll/H11032m2r/H20850fc/H20849r/H20850. The squared parameters gll/H11032mandg¯ll/H11032mguarantee the expo- nential terms to decay with increased distance. The overlap and Hamiltonian functions have an unfortu- nate behavior for small distances rwhich can lead to cata- strophic failure in the Hamiltonian problem. The functionalform in Eq. /H208495/H20850is exponentially damped as rgrows; in re- verse, this means that our intersite functions grow exponen- tially as rbecomes small. As horsbetween two atoms grow in magnitude they increase the bonding between the two re-spective atoms; as /H20841s/H20841→1 the energy of the bond grows as 1//H208491− /H20841s/H20841/H20850. When the bond energy grows, the bonding state is populated while the antibonding state is not; this results in a net attractive force between the two atoms. As the inter-atomic distance shrinks, the entire overlap matrix Sceases to be positive definite, and the Hamiltonian problem of Eq. /H208491/H20850 is no longer solvable. This causes the “collapse problem” inTRINKLE et al. PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-2molecular dynamics: Two atoms come close to each other and see a large attractive force that pulls them towards eachother until Sis not positive definite. In actuality, the Hamil- tonian problem is not meaningful even before S is not posi- tive definite, because the model predicts a bond with an un-physically low energy. In a real material, the growth inbonding is counteracted by Coulumb repulsion: a two-electron term that is not included in the tight-binding formal-ism. Short-range splining. To resolve this, we modify the in- tersite functions to keep the overlap matrix Spositive defi- nite. Because our fitting database includes only interatomicdistances larger than some minimum distance R min, the func- tional form is guaranteed to be correct only for r/H11022Rmin. Below Rmin, we smoothly interpolate both hll/H11032m/H20849r/H20850and sll/H11032m/H20849r/H20850to a constant value. This choice guarantees that the results for the fitting database are independent of the inter- polation function. The interpolation is performed with aquartic spline, from r=R mindown to r=Rmin−/H9268; below Rmin−/H9268, the function takes on a constant value. We choose spline values to enforce continuity of value and the first andsecond derivatives; the final functions for both h ll/H11032m/H20849r/H20850and sll/H11032m/H20849r/H20850are hinter. /H20849r/H20850=/H20902h/H20849r/H20850 :r/H11022Rmin, hspline /H208490/H20850 :r/H11021Rmin−/H9268, hspline /H20849r−Rmin+/H9268/H20850:otherwise,/H208496/H20850 where hspline /H20849u/H20850=h0−1 2/H9268h0/H11032+1 12/H92682h0/H11033+/H20873/H9268h0/H11032−1 3/H92682h0/H11033/H20874u3 /H92683 +/H20873−1 2/H9268h0/H11032+1 4/H92682h0/H11033/H20874u4 /H92684, /H208497/H20850 foruin/H208510,/H9268/H20852, and h0,h0/H11032, and h0/H11033are the value, first, and second derivative of h/H20849r/H20850atRmin. Figure 1 shows this inter- polation schematically. While we smoothly interpolate hll/H11032mandsll/H11032m,w e retain the environment-dependent on-site terms; this has the effect of reducing the strength of bonding while the on-site energy continues to grow—effectively producinga pair repulsion between atoms at small distances. Figure 2 illustrates the collapse problem for the Ti dimer and how short-range splining stabilizes the model for smalldistances. As the distance between the two atoms decreases,a bonding state with an artificially low energy decreases thedimer energy. The precipitous drop in the energy of thisbonding state is due to an increase in the overlap; at 1.92 Å,the overlap matrix becomes nonpositive definite, and theeigenproblem is no longer solvable. A /H9268value of 0.5 or 0.25 Å makes the dimer stable; this is necessary but not suf-ficient to solve the collapse problem for all cases. Our parametrization has 74 parameters to be optimized, plus 3 fixed parameters. The cut-off function f c/H20849r/H20850has two fixed parameters R0andl0, while the minimum distance Rmin is set by the database. There are ten Hamiltonian and ten overlap functions, each with 3 parameters for a total of 60parameters. The three on-site energy functions have four pa-rameters each, and a single parameter /H9261for the density gives 13 parameters. Finally, the short-range spline range param-eter /H9268is determined using the dimer, and testing with mo- lecular dynamic calculations and defect relaxations.13 B. Fitting database We compile a database of electronic structure calculations of several crystal structures using full-potential linearizedaugmented plane wave /H20849FLAPW /H20850calculations 14with the WIEN97 program suite.15We use the generalized gradient ap- proximation /H20849GGA /H20850for the exchange-correlation energy.16 The sphere radius is RMT=2.0 bohr=1.06 Å; there is a neg- ligible charge leakage of 10−8electrons. The plane-wave cut- offKmaxis given by RMTKmax=9; this corresponds to an en- ergy cut off of 275 eV. The energy cut off is not as large asrequired in a typical pseudopotential calculation because the FIG. 1. Interpolated intersite function with short-range spline. The parametrized function h/H20849r/H20850grows exponentially as rap- proaches zero, though the function is only sampled in the fittingdatabase down to R min.A t r=Rmin, we replace the function with a quartic spline that matches the value, first, and second derivatives atR min; the dashed curve shows the growth of the original function. The spline smoothly goes to a constant value in a width of /H9268. Only one adjustable parameter /H9268is added to the entire fitting database, as /H9268is the same for all functions. FIG. 2. Energy of Ti dimer calculated with tight binding using short-range splining. Without any short-range splining, the overlapmatrix becomes artificially large, creating a bonding state with verylow energy at small distances; at 1.92 Å, the tight-binding dimerHamiltonian problem becomes unsolvable. As described in the text,by short-range splining of the Hamiltonian and overlap functions,the model is stable and becomes repulsive at small distances.EMPIRICAL TIGHT-BINDING MODEL FOR TITANIUM ¼ PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-3plane waves are only used in the interstitial regions away from atom centers. The charge density is expanded in a Fou-rier series; the largest magnitude vector in the expansionG maxis 18 bohr−1/H2084934 Å−1/H20850. Local orbitals are used for the s, p, and dsolutions inside the spheres.14Our core configura- tion is Mg with semicore 3 pstates represented by the local p orbitals; our 4 s,3d, and 4 pstates are the valence orbitals. A Fermi-Dirac smearing of 20 mRyd /H20849272 meV /H20850is used to cal- culate the total energy.17 Table I shows a summary of the fitting database; it con- sists of the total energies and eigenvalues on a k-point grid for several crystal structures. Five structures are used: simplecubic /H20849sc/H20850, body-centered cubic /H20849bcc /H20850, face-centered cubic /H20849fcc /H20850, hexagonal closed-packed /H20849 /H9251/H20850, and omega /H20849/H9275/H20850. The three cubic structures are calculated over a range of volumes, while the hexagonal structures are calculated only atFLAPW equilibrium volumes and c/aratios. For each struc- ture, we fit to the total FLAPW energy of the structure andelectron eigenvalues shifted by a constant on a k-point grid. The FLAPW electron eigenvalues for each structure areshifted by a constant amount so that the sum of occupiedlevels equals the total energy; fitting the shifted eigenvalueswill allow our model to reproduce the correct relative elec- tron energies and total energies without a pair potential. Wecalculate the nine lowest bands per atom above the semicore3pstates; these represent the 4 s,3d, and 4 pstates both be- low and above the Fermi level. We use the lowest six bandsat each kpoint for fitting the cubic structures, nine bands for /H9251, and 12 bands for /H9275. In addition to eigenvalues on a regular grid, we include eigenvalues at high symmetry points and directions in theBrillouin zone to aid in fitting. 20,21For the three cubic struc- tures, we calculate the eigenvalues at several high-symmetrypoints and directions /H2084910 for bcc and sc, and 12 for fcc /H20850and then decompose the electronic wave functions in terms of thesymmetry character of the eigenvalues. 22Again, we use the lowest six states for the high-symmetry points. We are care-ful not to fit too many eigenvalues at high-symmetry points,since the lowest nine bands in the GGA band structure maynot correspond to those predicted in our spdbasis. 23 Because our fit includes the electron eigenvalues, we ex- pect our model to reproduce both total energies and energyderivatives. Phonons and elastic constants can be written interms of the forces on atoms due to small displacements; theHellman-Feynman theorem relates the force on an atom R ito the eigenvalues as Fi=−2 /H20858 /H9255n/H11021EF/H20883/H9274n/H20879/H11509Hˆ /H11509Ri/H20879/H9274n/H20884. Thus, the electron eigenvalues of the bulk crystal contain information about phonons and elastic constants. C. Optimization of parameters The parameters are optimized to minimize the mean squared error. We use the nonlinear least-squares minimiza-tion method of Levenberg-Marquardt with a numericalJacobian. 24We weight each kpoint by unity, and the result- ing total energy by 200; accordingly the total energies areweighted approximately the same as the k-point data. We initialize our parameters using the Hamiltonian and overlapvalues for Ti from Ref. 20 adapted to our functional form.We then fit only the environment-dependent on-site terms tothe band structure of the cubic elements. After an initial fit isfound, we include the hopping terms in the optimization. Weproceed using only the cubic band structure, then the cubicband structure and total energies, and finally all structuresand energies. After a new minimum is found, we check eachfunction to see if the minimization has made the exponentialterm g /H20849ll/H11032m/H20850too large; this corresponds to making the entire function approximately zero over the sampled range of r values. We remedy this by resetting the e,f, and gparam- eters to 0, 0, and 0.5. Several fitting runs are performed untilthe entire fit set is accurately reproduced. III. RESULTS A. Parameters and fitting residuals Table II lists the parameters of the optimized tight-binding model. Figure 3 shows the hopping integrals h/H20849ll/H11032m/H20850/H20849r/H20850andTABLE I. Crystal structures used in tight-binding fitting data- base. Five different crystal structures are used, with five volumesfor each of the cubic crystal structures. The lattice constant a 0, volume per atom, and nearest neighbor distance and multiplicity foreach structure is listed. The equilibrium lattice constant for eachstructure is denoted by a. The same k-point mesh is used for all volumes of a given structure, and is constructed using the prescrip-tion of Refs. 18 and 19. The smallest distance to appear in thisfitting database is R min=2.350 Å. Structure a0/H20849Å/H20850 V/H20849Å3/H20850 nn /H20849Å/H20850 k-point mesh bcc 2.887 12.03 2.500 /H110038 Shifted 5 /H110035/H110035 3.060 14.32 2.650 /H110038 /H2084944 points /H20850 3.281a17.66 2.841 /H110038 3.406 19.76 2.950 /H110038 3.579 22.93 3.100 /H110038 fcc 3.747 13.16 2.650 /H1100312 Unshifted 5 /H110035/H110035 3.960 15.52 2.800 /H1100312 /H2084947 points /H20850 4.127a17.57 2.919 /H1100312 4.384 21.06 3.100 /H1100312 4.596 24.27 3.250 /H1100312 sc 2.350 12.98 2.350 /H110036 Shifted 5 /H110035/H110035 2.500 15.62 2.500 /H110036 /H2084935 points /H20850 2.645a18.50 2.645 /H110036 2.800 21.95 2.800 /H110036 2.950 25.67 2.950 /H110036 /H9251 2.952a17.69 2.952 /H1100312 Unshifted 5 /H110035/H110032 /H20849c/a=1.588 /H20850/H20849 42 points /H20850 /H9275 4.600a17.23 2.656 /H110033 Unshifted 3 /H110033/H110034 /H20849c/a=0.613 /H20850/H20849 35 points /H20850 aFLAPW equilibrium lattice constant.TRINKLE et al. PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-4s/H20849ll/H11032m/H20850/H20849r/H20850for a range of volumes; the Rminin the database is 2.35 Å, and we interpolate each function to a constant value below Rmin. Finally, Fig. 4 shows the environment-dependent onsite energies as a function of volume for an hcp crystalwith c/a=1.588. To use the potential for phase-transformation studies, /H9268 was determined by testing the stability of /H208491/H20850the dimer, /H208492/H20850 molecular dynamics runs, and /H208493/H20850defect relaxations. While the lowest energy pathways studied by Ref. 3 have distancesof the closest approach of 2.6 Å, there were possiblepathways where atoms approached within 2.3 Å of eachother. Without short-range splining, calculations of energies of structures with distances below our R minvalue can become problematic. Initially, a /H9268value of 0.529 Å was chosen based on the dimer; however, defect relaxationTABLE II. Tight-binding parametrization for titanium. The on- site parameters are given for the s,p, and dorbitals. Each term is density dependent; the parameter in the density dependence is /H9261. The cutoff function has fixed parameters R0andl0. Next, the inter- site Hamiltonian and overlap elements are given for each of the tensymmetrized /H20849ll /H11032m/H20850combinations. Below Rmin, each intersite func- tion is smoothly interpolated to a constant value over the range /H9268. al/H20849eV/H20850 bl/H20849eV/H20850 cl/H20849eV/H20850 dl/H20849eV/H20850 s: −3.272 /H110031003.714 /H110031028.029 /H110031037.879 /H11003104 p: 4.974 /H110031003.747 /H11003101−1.874 /H110031032.721 /H11003104 d: 3.632 /H1100310−13.238 /H110031018.877 /H110031019.355 /H11003102 /H9280l,i=al+bl/H9267i2/3+cl/H9267i4/3+dl/H9267i2/H208492/H20850 /H9267i=/H20858 j/HS11005iexp /H20849−/H92612rij/H20850fc/H20849rij/H20850:/H92612=/H208490.3620 Å /H20850−1/H208493/H20850 fc/H20849r/H20850=/H208751 + exp/H20873r−R0 l0/H20874/H20876−1 :R0= 6.615 Å l0= 0.2646 Å/H208494/H20850 e/H20849ll/H11032m/H20850 f/H20849ll/H11032m/H20850 1/g/H20849ll/H11032m/H208502/H20849Å/H20850 ss/H9268: h=−1.086 /H11003102eV −3.900 /H11003103eV/Å 0.3277 s=9.277 −2.624 Å−10.8357 sp/H9268: h=−1.793 /H11003103eV 8.066 /H11003102eV/Å 0.4926 s=−11.81 0.02523 Å−10.5993 pp/H9268: h=−4.865 /H11003102eV 1.816 /H11003102eV/Å 0.6929 s=0.08093 −1.351 Å−11.036 pp/H9266: h=1.202 /H11003101eV −8.252 /H11003100eV/Å 0.8925 s=4.478 −0.2899 Å−10.8026 sd/H9268: h=−5.537 /H11003102eV 3.096 /H11003102eV/Å 0.4772 s=−4.331 −5.085 Å−10.4498 pd/H9268: h=−2.338 /H11003102eV 9.994 /H11003101eV/Å 0.6321 s=0.02557 −3.383 Å−10.5728 pd/H9266:h=−4.979 /H11003100eV 7.855 /H1100310−1eV/Å 1.617 s=0.1943 2.308 Å−10.5882 dd/H9268: h=1.706 /H11003102eV −1.150 /H11003102eV/Å 0.5266 s=−0.9905 0.7605 Å−10.7990 dd/H9266: h=9.920 /H11003100eV 3.538 /H11003101eV/Å 0.5366 s=−1.490 −1.498 Å−10.5213 dd/H9254: h=1.109 /H11003103eV −6.205 /H11003102eV/Å 0.3340 s=15.58 −5.276 Å−10.4412 /H20853h,s/H20854/H20849ll/H11032m/H20850/H20849r/H20850=/H20849e/H20849ll/H11032m/H20850+f/H20849ll/H11032m/H20850r/H20850exp /H20849−g/H20849ll/H11032m/H208502r/H20850fc/H20849r/H20850/H208495/H20850 Rmin=2.350 Å, /H9268=0.265 Å /H208496/H20850 FIG. 3. Tight-binding intersite Hamiltonian and overlap func- tions. The parametrized hopping integrals are shown for distancesfrom 2.4 to 3.5 Å. The R minin the fit is 2.35 Å; below this, these functions are smoothly interpolated to a constant value. Circles rep-resent /H9268integrals, squares /H9266integrals, and triangles /H9254integrals; black is for s, dark gray for p, and light gray for d. FIG. 4. Tight-binding onsite energy terms for hcp structure. The onsite energies are environment dependent in our model; we showthe variation with respect to the volume of an hcp crystal withc/a=1.588. The low volume of 10 Å 3has a lattice constant of 2.44 Å, and the high volume of 25 Å3has a lattice constant of 3.31 Å. The equilibrium hcp volume is 17.56 Å3.EMPIRICAL TIGHT-BINDING MODEL FOR TITANIUM ¼ PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-5calculations showed that a value of 0.265 Å was necessary to ensure stability for some of the point defects. Table III lists the errors in our tight-binding model with respect to the fitting database. Our average total energy er-rors are approximately 1 meV; root-mean square errors inthek-point energies are approximately 100 meV. The tight- binding parametrization adequately reproduces the databaseenergetics. To test transferability, we compare to propertiesoutside of this database; nearly all of the following resultsare for structures not included in the fit database. B. Total energies Figure 5 shows the tight-binding total energy as a function of volume for /H9251and/H9275. These curves were not included in the fitting database; only the two points indicated. We repro-duce both the slightly lower energy of /H9275over/H9251predicted bypseudopotential methods3and FLAPW calculations, as well as the slightly lower equilibrium volume of /H9275. The three cubic structures were included in the fit and have errors onthe order of 3 meV/atom /H20849c.f., Table III /H20850. This shows a wide range of applicability for our model under pressure. C. Elastic constants and phonons Table IV shows the equilibrium lattice constants and elas- tic constants for /H9251,/H9275, and bcc for our tight-binding model.TABLE III. Fitting errors in total energy and kpoints for tight- binding model. For each structure, we report the absolute error inthe total energy /H20849first line /H20850and the rms error in all k-point energies in the fit set /H20849second line /H20850. The total energy errors are on the order of 1 meV, while the rms band-structure errors are on the order of100 meV. Low volume /H20849meV /H20850Equilibrium /H20849meV /H20850High volume /H20849meV /H20850 bcc 1.64 0.957 4.31 200 104 110 fcc −1.79 1.25 −0.821 136 87.1 114 sc −0.0190 −0.115 −1.60 435 195 140 /H9251 −1.66 69.1 /H9275 −0.00993 67.9TABLE IV. Lattice parameters and elastic constants in units of GPa for /H9251,/H9275, and bcc Ti from tight binding, GGA, and experiment. GGA corresponds to the elastic constants found using VASP /H20849Refs. 25 and 26 /H20850. The experimental /H9251elastic constants are measured at 4K /H20849Ref. 31 /H20850, and the bcc elastic constants at 1238 K /H20849Ref. 32 /H20850. Our tight-binding model reproduces the GGA elastic constant com-binations that preserve the symmetry of the structure /H20849e.g., C 11 +C12/H20850, but has larger error with those that break it /H20849e.g., C44/H20850. The deviation between the bcc experimental elastic constants and ourcalculations is due to the high temperature needed to stabilize thebcc structure in Ti. a/H20849Å/H20850 c/H20849Å/H20850 C 11 C12 C13 C33 C44 Tight binding /H9251 2.94 4.71 155 91 79 173 65 /H9275 4.58 2.84 184 90 52 261 100 bcc 3.27 — 87 112 — — 31 GGA /H9251 2.95 4.68 172 82 75 190 45 /H9275 4.59 2.84 194 81 54 245 54 bcc 3.26 — 95 110 — — 42 Experiment /H9251 2.95 4.68 176 87 68 191 51 bcc 3.31 — 134 110 — — 36 FIG. 5. /H20849Color online /H20850Comparison of tight-binding energy as a function of volume for /H9251and/H9275with first principles data. The two filled points were included in the fit; the lines are FLAPW totalenergies. Our tight-binding model reproduces the fit data—slightlylower ground-state energy and equilibrium volume for /H9275—and the equation of state of the full-potential calculations. FIG. 6. Comparison of tight-binding phonons for the /H9251phase with experimental phonon data. The crosses are the experimentalphonon frequencies at 295 K /H20849Ref. 34 /H20850. The deviation from the experimental values at small qcorresponds to the mismatch in the /H9251 elastic constants. Our tight-binding model does well for the high- energy optical and acoustic branches which are important for mod-eling the /H9251→/H9275transformation.TRINKLE et al. PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-6The GGA numbers correspond to the elastic constants found using VASP.25,26Elastic constant combinations which do not break symmetry such as C11+C12,C13,C33in the hexagonal crystals, and C11+2C12in bcc are reproduced within ap- proximately 10%. However, the symmetry breaking elasticconstant combinations such as C 11−C12andC44have larger errors. It is worth noting that none of this data, except for thebulk modulus of bcc, appears in any form in the fitting da-tabase; the agreement is a consequence of reproducing theelectron eigenvalues. We calculate phonons using the direct-force method. 27–30 We calculate the forces on all atoms in a supercell where oneatom at the origin is displaced by a small amount. The nu-merical derivative of the forces with respect to the displace-ment distance approximates the force constants folded withthe translational symmetry of the supercells. The Fourier transform of the force constants gives the dynamical matrix,and its eigenvalues give the phonon frequencies. 33Forqvec- tors commensurate with the supercell, the phonon frequen-cies are exact; for incommensurate qvectors, the calculated phonons are a Fourier interpolation between exact values.Our supercells are 4 /H110034/H110033 for /H9251,3/H110033/H110034 for /H9275, and 4 /H110034/H110034 simple cubic cell for bcc; in all cases, a 2 /H110032/H110032 k-point mesh is used in the supercell. Again, none of the following data is included in the fit database. Figures 6–8 are the predicted phonon dispersions for our tight-binding model /H20849TB /H20850, calculated at the equilibrium vol- umes for each structure. The /H9251phonons match the experi- mental values well for the high energy optical and acousticbranches; these are important for modeling the shuffle duringmartensitic transformation. The deviation from experimentfor small qcorresponds to our mismatch in elastic constants. The largest deviation occurs with a single low branch at theK point. The /H9275phonons are expectedly stiffer along the c axis than in the basal plane due to the low c/aratio. The bcc phonons show phonon instabilities corresponding to thebcc→ /H9275transformation /H20849L-2/H114083/H20851111 /H20852phonon /H20850and the bcc →/H9251transformation /H20849T-/H20851011 /H20852branch /H20850.35The stability of the /H208510/H9264/H9264/H20852direction near /H9003is an artifact of the direct-force method in both /H20849TB /H20850and GGA; the true spectra has a single imaginary branch for small /H9264values. Using an 8 /H110038/H110038 simple-cubic bcc supercell in TB to derive the force con-stants removes the artificial stability; such a cell is prohibi-tively large to compute with GGA, but is expected to pro-duce the correct behavior near /H9003as well. Throughout theTABLE V. Point defect energies in electron volts for /H9251and/H9275Ti from tight binding for different parametrizations. GGA refers to thedefect formation energies calculated with VASP; TB to the param- etrization in this work; NRL to the parametrization by Mehl andPapaconstantopoulos; 5and LANL to the parametrization by Rudin et al.6NN refers to the distance of closest approach for two atoms in each defect in TB. The formation energies are calculated afterrelaxation. The defects marked coll. fell victim to the “collapse problem” during relaxation. The /H9251-tetrahedral site is unstable in both TB and GGA, relaxing to form a dumbbell along the /H208510001 /H20852 direction. The /H9275-hexahedral site is very close to the /H9275-tetrahedral site /H20849Ref. 37 /H20850. Many of the interstitial defects sample small dis- tances, requiring the use of short-range splining to stabilize thedefects. Defect GGA TB NRL LANL NN /H20851Å/H20852 /H9251defects Octahedral 2.58 2.89 1.31 2.55 2.50 ÅTetrahedral Unstable to dumbbellDumbbell- /H208510001 /H20852 2.87 2.81 1.81 coll. 2.18 Å Vacancy 2.03 1.88 1.51 1.92 2.83 ÅDivacancy-AB 3.92 3.83 3.73 3.68 2.81 Å /H9275defects Octahedral 3.76 4.11 3.20 3.67 2.30 ÅTetrahedral 3.50 3.58 2.86 coll. 2.21 Å Hexahedral 3.49 3.86 2.88 4.37 2.28 ÅVacancy-A 2.92 2.85 2.99 3.25 2.60 ÅVacancy-B 1.57 1.34 1.01 1.90 2.62 Å FIG. 7. Predicted /H9275phonons from tight binding. As expected from the c/aratio of 0.620, the phonon modes are stiffer along the /H2085100/H9264/H20852direction than the basal plane directions /H20851/H926400/H20852and /H208510/H92640/H20852. FIG. 8. Comparison of tight-binding bcc phonons with calcu- lated spectra using GGA. At T=0, the bcc phase in Ti is unstable, as shown by the imaginary phonon frequencies. The agreement be-tween the phonons calculated using the current model and thosewith GGA is good, for both stable and unstable phonons. The de-viation at the Ppoint indicates too much stiffness in the TB model for motion of /H20851111 /H20852chains in bcc compared to GGA. The dip in the /H20851 /H9264/H9264/H9264 /H20852branch is near the L-2 /H114083/H20851111 /H20852phonon, which corresponds to the bcc→/H9275transformation pathway. The imaginary phonon for T- /H20851011 /H20852corresponds to the bcc →/H9251transformation mechanism /H20849Ref. 35/H20850.EMPIRICAL TIGHT-BINDING MODEL FOR TITANIUM ¼ PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-7Brillouin zone, the TB phonons agree well with the GGA phonons. The stability of the P phonon in TB indicates toomuch stiffness for the motion of /H20851111 /H20852chains in bcc; how- ever, the phonons in bcc are stabilized at high temperatures /H20849/H110111200 K /H20850by strong anharmonicity. 36The importance of this deviation will need to be investigated for high tempera- ture martensitic transformations from bcc. D. Point defects Table V shows the formation energies of point defects for /H9251and/H9275at the equilibrium volumes for our tight-binding model. All /H9251calculations are performed with a 4 /H110034/H110033/H2084996 atom /H20850supercell and all /H9275with a 3 /H110033/H110034/H20849108 atom /H20850super- cell, using the original lattice constants for both TB andGGA. The atoms were relaxed at fixed volume to forces ofless than 5 meV/Å, and the reference /H9251and/H9275energies were computed using the same supercell and k-point meshes /H208492 /H110032/H110032 in the supercell, 20 meV smearing /H20850. No point defect information is included in the initial fit; we reproduce theGGA formation energies for all of the point defects consid-ered. This indicates that our tight-binding model is appli-cable to the study of the /H9251→/H9275transformation path, where atoms move out of their equilibrium configurations and oftenclose to one another. The formation energies of point defects shows some im- provement of our model over two existing models. 5,6The potential by Rudin et al. uses the same functional forms as our potential without short-range splining for hopping andoverlap functions; Mehl and Papaconstantopoulos use thesame onsite function form, but adds additional quadratic pa-rameters to the hopping and overlap functions in Eq. /H208495/H20850. All three potentials use the same on-site functional forms. For allthree potentials, the binding energies versus volume, elasticconstants, and phonons are similar, though Rudin’s more ac-curately captures the low frequency /H9251phonons. However, point defect formation energies compare better with GGAusing our tight-binding parametrization, showing an agree-ment of 13% in formation energy for a variety of defects. The short distances sampled by the point defects empha- size the need for short-range splining of both the overlap andHamiltonian functions. The collapse of two defects in the Rudin et al. model is due to the growth of the overlap ma- trices; the lower energies predicted by Mehl and Papacon-stantopoulus could be due to overly large overlap elements atshort distances as well. Interstitial defects, like phase trans-formation pathways, can sample interatomic distancessmaller than the smallest distance included in the fitting da-tabase; without short-range splining, this can lead to artifi-cially lower energies, or even collapse. Without short-rangesplining, all three tight-binding parametrizations fail for theTi dimer at small distances: 1.92 Å for this work, 1.76 Å forMehl and Papaconstantopoulos, and 1.28 Å for Rudin et al. The use of short-range splines provides a solution to thecollapse problem for non-orthogonal tight-binding models. IV . CONCLUSION We present an accurate and transferable tight-binding model with parameters determined by density-functional cal-culations. It reproduces density-functional structural energieswith pressure, elastic constants, phonons, and point defectenergies. The efficiency compared to GGA allows access tolarger length- and time-scales with a small sacrifice in accu-racy. By fixing the short-range behavior of the potential,point defects can be accurately computed, which allows thecalculation of energy barriers for phase transformation path-ways. The wide range of applicability makes it particularlywell suited to the study of martensitic phase transformations,such as /H9251→/H9275.3Short-range splines represent a solution to the potential collapse problem of non-orthogonal tight-binding models allowing an increase in the range of applica-bility, without reoptimizing existing parameters. ACKNOWLEDGMENTS D.R.T. thanks Los Alamos National Laboratory for its hospitality and acknowledges support from the Ohio StateUniversity. This research is supported by DOE Grant Nos.DE-FG02-99ER45795 /H20849OSU /H20850and W-7405-ENG-36 /H20849LANL /H20850. Computational resources were provided by the Ohio Supercomputing Center and NERSC. 1Titanium and Titanium Alloys: Fundamentals and Applications , edited by C. Leyens and M. Peters, /H20849Wiley-VCH, Weinheim, 2003 /H20850. 2S. K. Sikka, Y. K. Vohra, and R. Chidambaram, Prog. Mater. Sci. 27, 245 /H208491982 /H20850. 3D. R. Trinkle, R. G. Hennig, S. G. Srinivasan, D. M. Hatch, M. D. Jones, H. T. Stokes, R. C. Albers, and J. W. Wilkins, Phys. Rev.Lett. 91, 025701 /H208492003 /H20850. 4D. R. Trinkle, D. M. Hatch, H. T. Stokes, R. G. Hennig, and R. C. Albers, Phys. Rev. B 72, 014105 /H208492005 /H20850. 5M. J. Mehl and D. A. Papaconstantopoulos, Europhys. Lett. 60, 248 /H208492002 /H20850. 6S. P. Rudin, M. D. Jones, and R. C. Albers, Phys. Rev. B 69, 094117 /H208492004 /H20850.7W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 /H208491965 /H20850. 8J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 /H208491954 /H20850. 9M. J. Mehl and D. A. Papaconstantopoulos, Phys. Rev. B 54, 4519 /H208491996 /H20850. 10R. E. Cohen, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 50, 14694 /H208491994 /H20850. 11S. H. Yang, M. J. Mehl, and D. A. Papaconstantopoulos, Phys. Rev. B 57, R2013 /H208491998 /H20850. 12The energy of an orbital lat atom iin the two-body approxima- tion is /H20855/H9278l,i/H20841Hˆ/H20841/H9278l,i/H20856; the first three-body correction is to include hopping from atom itojand back to i. 13Different values of /H9268could be used for each Hamiltonian and overlap element; we use a single parameter for simplicity. 14D. J. Singh, Planewaves, Pseudopotentials and the LAPW MethodTRINKLE et al. PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-8/H20849Kluwer Academic, Boston, 1994 /H20850. 15P. Blaha, K. Schwart, and J. Luitz, WIEN97: A Full Potential Linearized Augmented Plane Wave Package for CalculatingCrystal Properties /H20849Technical Universität Wien, Austria, 1999 /H20850. 16J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 /H208491996 /H20850. 17The large smearing was used to reduce error introduced with a small number of kpoints; Ref. 21 used smaller smearings with- out adverse effects. 18D. J. Chadi and M. L. Cohen, Phys. Rev. B 8, 5747 /H208491973 /H20850. 19H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 /H208491976 /H20850. 20D. A. Papaconstantopoulos, Handbook of the Band Structure of Elemental Solids /H20849Plenum, New York, 1986 /H20850. 21M. D. Jones and R. C. Albers, Phys. Rev. B 66, 134105 /H208492002 /H20850. 22J. F. Cornwell, Group Theory and Electronic Energy Bands in Solids /H20849North-Holland, Amsterdam, 1969 /H20850. 23For example, in a bcc lattice the /H900325states are lower in energy than the /H900315states; however, the /H900325state corresponds to the f orbitals x/H20849y2−z2/H20850etc., while the /H900315states correspond to por- bitals x,y,z. Hence, we only fit the lowest 6 states at /H9003to exclude the /H900325states from the fit. 24D. W. Marquardt, J. Soc. Ind. Appl. Math. 11, 431 /H208491963 /H20850.25G. Kresse and J. Hafner, Phys. Rev. B 47, R558 /H208491993 /H20850. 26G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 /H208491996 /H20850. 27K. Kunc and R. M. Martin, Phys. Rev. Lett. 48, 406 /H208491982 /H20850. 28S. Wei and M. Y. Chou, Phys. Rev. Lett. 69, 2799 /H208491992 /H20850. 29W. Frank, C. Elsässer, and M. Fähnle, Phys. Rev. Lett. 74, 1791 /H208491995 /H20850. 30K. Parlinski, Z.-Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 /H208491997 /H20850. 31G. Simmons and H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties /H20849MIT Press, Cambridge, MA, 1971 /H20850. 32W. Petry, A. Heiming, J. Trampenau, M. Alba, C. Herzig, H. R. Schober, and G. Vogl, Phys. Rev. B 43, 10933 /H208491991 /H20850. 33N. W. Ashcroft and N. D. Mermin, Solid State Physics /H20849Saunders College, Philadelphia, 1976 /H20850. 34C. Stassis, D. Arch, B. N. Harmon, and N. Wakabayashi, Phys. Rev. B 19, 181 /H208491979 /H20850. 35W. G. Burgers, Physica /H20849Utrecht /H208501, 561 /H208491934 /H20850. 36K. M. Ho, C. L. Fu, and B. N. Harmon, Phys. Rev. B 29, 1575 /H208491984 /H20850. 37R. G. Hennig, D. R. Trinkle, J. Bouchet, S. G. Srinivasan, R. C. Albers, and J. W. Wilkins, Nat. Mater. 4, 129 /H208492005 /H20850.EMPIRICAL TIGHT-BINDING MODEL FOR TITANIUM ¼ PHYSICAL REVIEW B 73, 094123 /H208492006 /H20850 094123-9

PhysRevB.82.224416.pdf

Interplay between the Kondo effect and randomness: Griffiths phase in MxTiSe 2 (M=Co, Ni, and Fe) single crystals M. Sasaki,1A. Ohnishi,1T. Kikuchi,1M. Kitaura,1Ki-Seok Kim,2and Heon-Jung Kim3,* 1Department of Physics, Faculty of Science, Yamagata University, Kojirakawa, Yamagata 990-8560, Japan 2Asia Pacific Centre for Theoretical Physics, POSTECH, Pohang, Gyeongbuk 790-784, Republic of Korea 3Department of Physics, College of Natural Science, Daegu University, Gyeongbuk 712-714, Republic of Korea /H20849Received 18 November 2010; published 15 December 2010 /H20850 We investigate the interplay between the Kondo effect and randomness in MxTiSe 2/H20849M=Co, Ni, and Fe /H20850 single crystals. Although the typical low- Tupturn of resistivity implies the Kondo effect around the single-ion Kondo temperature TK, positive magnetoresistance linearly proportional to the magnetic field and the power- law scaling of magnetization suggest the forbidden coexistence between Kondo effect and time-reversal sym-metry breaking. This puzzling result is resolved by the Griffiths scenario—disorder-induced distribution of the Kondo temperature produces an effective Kondo temperature /H20849T¯K/H20850much lower than TK, allowing unscreened local moments above T¯Kand resulting in non-Fermi-liquid properties in MxTiSe 2below the percolation thresh- old/H20849x/H11021xc/H20850. DOI: 10.1103/PhysRevB.82.224416 PACS number /H20849s/H20850: 75.20.Hr, 71.27. /H11001a, 72.15.Qm Interplay between the Kondo effect and randomness is one of the central interests in modern condensed-matterphysics, such as for heavy-fermion systems and dilute mag-netic semiconductors, where non-Fermi-liquid physics ap-pears in both the magnetic properties and electrical transportbeyond the Fermi-liquid theory. 1In particular, disorder- induced distribution of the Kondo temperature down to T =0 K allows unscreened local moments, resulting in non-Fermi-liquid physics—a typical example of the quantumGriffiths effect. 2 In this study we intercalate TiSe 2with magnetic 3 dtran- sition metals /H20849Fe, Co, and Ni /H20850and investigate the possible quantum Griffiths phase from both magnetic and transportproperties. Layered compounds Ti X 2/H20849X=S, Se, and Te: chalcogen /H20850are well known for their low-dimensional elec- tronic structures that are easily intercalated with guestatoms, 3,4which produce two-dimensional semiconductors /H20849TiS 2/H20850, semimetals /H20849TiSe 2/H20850, and metals /H20849TiTe 2/H20850, depending on the degree of electron transfer from the chalcogens toTi 3dorbitals and of the p-dhybridization. In particular, TiSe 2displays an unexpected charge-density wave /H20849CDW /H20850 beyond the conventional nesting picture, where the Fermisurface consists of six ellipsoidal pockets. It is fascinatingthat Cu intercalated TiSe 2/H20849CuxTiSe 2/H20850exhibits superconduc- tivity around 4.2 K with doping dependence similar to that ofT cwith high- Tccuprate superconductors.5 When magnetic 3 dtransition metals /H20849M=Co, Ni, and Fe /H20850 are intercalated into TiSe 2/H20849MxTiSe 2/H20850, they act as spin-flip scattering centers to produce the Kondo effect in the dilutelimit. Increasing the concentration of intercalated magneticimpurities, their random distribution is expected to causequantum Griffiths effects in the disordered Kondo system. 1 When the concentration is above a critical value /H20849xc/H20850, inter- preted as the percolation threshold, Ruderman-Kittel-Kasuya-Yosida /H20849RKKY /H20850interactions between the interca- lated local quantum spins become important in competingwith the Kondo effect, providing a framework to describeheavy-fermion physics. 2Although this is the desired overall picture, the expected non-Fermi-liquid physics is rare in thedisordered Kondo regime /H20849x/H11021xc/H20850.MxTiSe 2with magnetic M provides an opportunity to examine the disordered Kondoeffect systematically. In this paper we investigate the interplay between the Kondo effect and randomness in M xTiSe 2/H20849M=Co, Ni, and Fe/H20850single crystals. Although the typical low- Tupturn of re- sistivity implies the Kondo effect around the single-ionKondo temperature T K, positive magnetoresistance /H20849MR/H20850lin- early proportional to the magnetic field and power-law scal-ing of magnetization deviates from the local Fermi-liquidpicture, suggesting the forbidden coexistence of the Kondoeffect and time-reversal symmetry breaking. We resolve thispuzzling result, based on the quantum Griffiths scenario thatdisorder-induced distribution of the Kondo temperature causes an effective Kondo temperature /H20849T ¯K/H20850much lower than TK, allowing unscreened local moments above T¯Kand result- ing in non-Fermi-liquid properties in MxTiSe 2below the per- colation threshold /H20849x/H11021xc/H110150.07/H20850. On the other hand, MxTiSe 2evolves into the usual metallic phase above xc. In the previous work on Fe xTiSe 2,6the peak in /H9267/H20849T/H20850 curves, characterizing the CDW decreased appreciably whenincreasing the intercalated Fe concentration xup to the criti- cal concentration x c/H208490.065/H11021xcexp/H110210.075 /H20850, leaving anoma- lously high resistivity at low temperature—above xc, the transport properties change dramatically and a metallic stateis recovered. The critical concentration x cis reported to be a point, where a percolation path of Fe clusters forms. Accord-ing to an angle-resolved photoemission spectroscopy study, 7 when the concentration of xis high, Fe intercalation pro- duces a nondispersive impurity-induced band near the Fermilevel. This strongly suggests that the intercalated Fe acts as adopant and also as an impurity. Therefore, well below x c, the high resistivity at low temperature is believed to result fromsingle impurity scattering. In fact, qualitatively similar be-havior is observed in M xTiSe 2/H20849M=Ni and Co /H20850but not in CuxTiSe 2, as will be discussed below. The single crystals of the three-dimensional transition- metal Mintercalation compound MxTiSe 2/H20851M=Ni; x /H113490.089 /H20849/H11022xc/H20850,M=Co; x/H113490.13 /H20849/H11022xc/H20850, and M=Cu; xPHYSICAL REVIEW B 82, 224416 /H208492010 /H20850 1098-0121/2010/82 /H2084922/H20850/224416 /H208495/H20850 ©2010 The American Physical Society 224416-1/H113490.06 /H20849/H11021xc/H20850/H20852were grown by a chemical-vapor transport technique in the presence of iodine as a transport agent.6,8To avoid cointercalation of guest Matoms and constituent Ti atoms into the host TiSe 2, all compounds were grown at relatively low temperature /H20849500 °C /H20850. Above that tempera- ture, Ti atoms are known to self-intercalate into the host.9 Values of the intercalated guest concentration xwere deter- mined by electron-probe microanalysis. X-ray powder-diffraction measurements for M xTiSe 2indicated that these compounds have a 1 T-CdI 2structure. Resistivity measure- ments were performed in the temperature range from 4.2 to300 K using a dc four-probe method. MR measurementswere carried out at 4.2 K and at magnetic fields up to 4.0 T.Zero-field cooled magnetizations were investigated with asuperconducting quantum interference device magnetometerat magnetic fields of 0.02 and 0.2 T. In MR and magnetiza-tion measurements, magnetic fields were applied perpendicu-lar to the layer plane. Figure 1/H20849a/H20850shows the temperature dependence of a resis- tivity component /H9267Aof Cu xTiSe 2, whose phonon and residual contributions are subtracted. The detailed method for analy-sis and raw data are presented in the supplementaryinformation. 10The phonon and residual contributions are es- timated as /H9267bg=/H9251/H9267r, where /H9267ris the resistivity curve above xc and/H9251is the scaling parameter with nearly unity value. In the samples at x/H11022xc, the main contribution for resistivity are the electron-phonon and electron-electron scatterings in high-and low- Tregions, respectively. Although the extracted re- sistivity slightly depends on the parameter /H9251, the main fea- tures are retained. As can be seen in Fig. 1/H20849a/H20850, this method isolates resistivity peaks for Cu xTiSe 2. As the concentration ofxincreases, the height of the peak is reduced and the peak moves to lower temperatures. However, the shape of thepeak, especially below the maximum value, remains un-changed. The scaled curves are shown in Fig. S1 in thesupplementary information. 10Figure 1/H20849a/H20850clearly shows that the CDW is suppressed with increasing x, which is consistentwith previous reports. Quantitatively, this decrease is in good agreement with Ref. 5but not Ref. 11, where the CDW is more rapidly suppressed. Similar CDW suppression was re-ported in Fe xTiSe 2. Below the peak region, /H9267Ais quite small in Cu xTiSe 2. We applied the same analysis to MxTiSe 2/H20849M=Fe, Ni, and Co/H20850. As an example, we present the results for Co xTiSe 2in Fig.1/H20849b/H20850—the results are similar for the other compounds. This figure clearly shows the difference for Cu xTiSe 2. The low- Tupturn of resistivity exists in MxTiSe 2/H20849M=Fe, Ni, and Co/H20850. Since this low-temperature part is observed only in MxTiSe 2/H20849M=Fe, Ni, and Co /H20850, it is not related to the CDW. At reasonably high xatx/H11021xc, the CDW peak is compara- tively small and it is quite clear that the low-temperature partbecomes dominant with high x. Since the peak shape is in- dependent of xfor Cu xTiSe 2, we subtracted this peak numeri- cally, assuming that the shape of CDW peaks is same in all MxTiSe 2compounds, especially below the maximum. We could get a low- Tupturn of resistivity with a clear shape. The low- Tupturns at different values of xshare common features, as shown in Fig. 1/H20849c/H20850. This low-temperature com- ponent increases with decreasing temperature, approximatelyfollowing a logarithmic dependence in Tand, saturated with T ndependence at low temperature as shown in Fig. 1/H20849d/H20850. This overall trend is reminiscent of spin-flip scattering andspin singlet formation in the dilute Kondo system. For the M xTiSe 2compound with nearly independent spins at x/H11021xc, it is quite reasonable to expect the Kondo effect to appear. If the Kondo effect is the origin of the anomalous increase in resistivity at low temperature, its effect should be mani-fested in other measurable quantities, such as MR and mag-netization. In the Kondo system, when spin-flip scattering isdominant, negative MR should appear because of the sup-pression of spin-flip scattering by magnetic fields B. On the other hand, at lower temperatures, when spin singlet isformed, the usual positive B-quadratic MR is anticipated. 12,13 However, what we observe is quite different from these pre- dictions. For Cu xTiSe 2, a positive B-quadratic MR is ob- served, as shown in Fig. 2/H20849a/H20850, implying that MR in this case results from the reduction in the mean-free path by cyclotronmotion. However, for M xTiSe 2/H20849M=Ni and Co /H20850,M Ri s dominantly positive and linear with Batx/H11021xc, whereas it is purely quadratic at x/H11022xc. In fact, MR is quadratic only in low fields and becomes linear in the region of x/H11021xc. This crossover occurs at around 1 T or less and is characterized bythe crossover field B cross. The quadratic MR results from impurity scattering and cyclotron motion of charge carriers. According to recent theoretical investigations,14positive linear MR can be observed in the low-field region when theKondo temperature T Kdistributes down to 0 K; thus, spins are unquenched even at the zero-temperature limit. The MRchanges to negative in high fields in which the Zeeman en-ergy exceeds the energy scale set by the peak value of T K. Note that our maximum field of 4 T corresponds to a tem-perature of 2.4 K while the temperature where the resistancestarts to increase is on the order of 100 K. This verifies thatour MR measurements are performed in the low-field region. The natural consequence of the above scenario is the ex- istence of unquenched spins, even at low temperature. In thiscase, magnetization should follow the power-law behavior of0 50 100 150 200 250 3000.000.050.100.150.200.25 CuxTiSe2 /CID85A/CID14975m/CID58cm /CID14976(a) : 0.015 : 0.039 : 0.049: 0.059 0 50 100 150 200 250 30 00.00.20.40.6Co0.006TiSe2 /CID85 (m/CID58cm ) T(K)(c) /CID545mL/CID545AP/CID545A 0 50 100 150 200 250 3000.00.20.40.60.81.0 T(K)CoxTiSe2 /CID85A/CID14975m/CID58cm /CID14976(b) : 0.006 : 0.019: 0.033: 0.059 10 1000.00.10.20.30.40.5 40 4 300T(K)CoxTiSe2 /CID85mL/CID14975m/CID58cm /CID14976(d) : 0.006 : 0.019: 0.033: 0.059 FIG. 1. /H20849Color online /H20850Anomalous increase in resistivity /H9267Aof /H20849a/H20850CuxTiSe 2and/H20849b/H20850and Co xTiSe 2. The background is subtracted as explained in the text. The /H9267Aof Cu xTiSe 2consisting solely of a CDW peak and the low- Tpart of /H9267Ais almost zero. In contrast, the low- Tpart of /H9267Ais considerably high in Co xTiSe 2. The /H20849red/H20850arrow indicates the CDW transition temperature. /H20849c/H20850The/H9267Ais decom- posed into a CDW peak /H20849/H9267AP, red curve /H20850and a low- Tupturn /H20849/H9267mL, green curve /H20850./H20849d/H20850The temperature dependence of /H9267mLof Co xTiSe 2. MxTiSe 2/H20849M=Fe and Ni /H20850all exhibit similar behavior.SASAKI et al. PHYSICAL REVIEW B 82, 224416 /H208492010 /H20850 224416-2M/H11011/H20885 0/H11009 P/H20849TK/H20850/H20849T−aTK/H20850−1dTK/H11011/H20849T−/H9258/H20850/H9251−1, where P/H20849TK/H20850is the distribution function of TKand/H9251is the exponent, which depends on P/H20849TK/H20850, and /H9258is a small quantity.15The/H9251value becomes smaller as the disorder strength W/tincreases, where Wis the band width broadened by disorder and tis the hopping integral.15This power-law behavior is quite different from what is expected in the usualdilute Kondo system. In the dilute Kondo system, magneti-zation is suppressed below T Kbecause of the formation of a spin singlet of an impurity spin and conduction electrons. In MxTiSe 2/H20849M=Fe, Ni, and Co /H20850, the former behavior is ob- served, as shown in Fig. 3, suggesting that spins remain un- quenched up to a certain temperature—much lower than theKondo temperature of the resistivity data. This implies ahuge distribution of the Kondo temperature. On the otherhand, Cu xTiSe 2follows a simple Curie behavior. The value of/H9251, which is obtained from fitting data in the region of 4 /H11021T/H1102180 K, is around 2. According to the calculations of Miranda and Dobrosavljevic,15this value corresponds ap- proximately to a disorder strength W/tof 0.8. Our /H9251values are not far from the critical value /H20849/H9251=1/H20850, where disorder- driven non-Fermi-liquid behavior is expected. The determi-nation of a more precise /H9251value and of the degree of the criticality needs more precise measurements at T/H110214.2 K. To understand how electrical transport properties change across xc, we present the scattering cross section Aof FexTiSe 2in/H9267/H20849T/H20850/H11011AT2atx/H11022xc, the crossover field Bcrossin MR and /H9004/H9267//H9267at 4 T in Figs. 4/H20849a/H20850–4/H20849c/H20850, respectively. As in FexTiSe 2, above xc, the resistivity follows a metallic behavior inMxTiSe 2/H20849M=Ni and Co /H20850. It is very interesting that this metallic behavior exists only when a percolation path isformed and this implies that a coherent effect become impor-tant in stabilizing this metallic state. Moreover, as seen inFig. 4/H20849a/H20850, A becomes larger, approaching x c. This suggests that the electron-electron scattering becomes stronger nearx=x c. This Fermi-liquid state is abruptly destroyed below xc.I n fact, this change accompanies an evolution of MR, as shown- 4 - 3 - 2 - 10123 40.00.20.40.60.8 CoxTiSe2 /g39/g85//g850(%)(b) - 4 - 3 - 2 - 1012340.00.10.20.30.40.5 /g39/g85//g850(%) B(T)NixTiSe2 (c) : 0.024 : 0.044: 0.065 [x100]: 0.019: 0.059 [x2] : 0.094 [x100] -4 -3 -2 -1 0 1 2 3 40123CuxTiSe2 /g39/g85//g850(%):x=0 : 0.004 [x2]: 0.059 [x50](a) FIG. 2. /H20849Color online /H20850The MR of /H20849a/H20850CuxTiSe 2,/H20849b/H20850CoxTiSe 2, and/H20849c/H20850NixTiSe 2measured at T=4.2 K. While the MR of Cu xTiSe 2 is quadratic with B, the MR of other compounds shows a domi- nantly linear dependence over a wide Brange at x/H11021xc. In the low- field region, this linear MR changes into quadratic—only quadraticMR appear at x/H11022x c.0 2 04 06 08 000..0000..5511..0011..5522..0022..55 0 2 04 06 08 00.00.10.20.30.40.5(b)1//g70(103gT/emu ) T(K)Ni0.017TiSe2 /g68=1 . 8 6/g68=1 . 9 2Fe0.05TiSe20 2 04 06 08 00123 01 0 2 0 3 0 4 00.10.20.30.40.50.6Cu0.015TiSe21//g70(104gT/emu ) T(K)Co0.033TiSe2 Co0.006TiSe2/g68=2 . 41//g70 /CID14975104g/emu /CID14976 T(K)/g68=2 . 3(a) FIG. 3. /H20849Color online /H20850Inverse magnetic susceptibility /H9273−1 =H/MofMxTiSe 2/H20849M=Fe, Ni, Co, and Cu /H20850while Cu xTiSe 2fol- lows a Curie behavior, MxTiSe 2/H20849M=Fe, Ni, and Co /H20850shows a power-law dependence. The solid lines are theoretical fit with theformula M/H11011/H20849T− /H9258/H20850/H9251−1. The values of /H9251are 1.92, 1.86, and 2.3–2.4 for Fe xTiSe 2,N i xTiSe 2, and Co xTiSe 2, respectively. Those of /H9258are −2, 0, and −13 K. 0.00 0.04 0.08 0.12 0.1610-410-21000.71.42.1 01234 x(/g39/g85//g850)4T(%)(c)A(10-5m/g58cm/K2) (a) Q-MR(b)Bcross(T) :C uxTiSe2 :C oxTiSe2 :N ixTiSe2L-MRL-MR xc c FIG. 4. /H20849Color online /H20850/H20849a/H20850The concentration of xdependence of Avalues in Fe xTiSe 2;/H20849b/H20850the crossover field BcrossofMxTiSe 2/H20849M =Fe, Ni, Co, and Cu /H20850from quadratic to linear dependence; and /H20849c/H20850 /H9004/H9267//H9267at 4 T /H20849c/H20850. These quantities change character at x/H11011xc/H20849xc /H110110.07/H20850and this critical concentration is indicated by the dotted line. The gray part is the experimentally determined critical region/H208490.065/H11021x cexp/H110210.075 /H20850, where a percolation path is known to be formed /H20849Ref. 6/H20850.INTERPLAY BETWEEN THE KONDO EFFECT AND … PHYSICAL REVIEW B 82, 224416 /H208492010 /H20850 224416-3in Fig. 4/H20849b/H20850. The region of B-linear MR is quite robust at x /H11021xcbut such a region does not exist at x/H11022xc. In contrast, CuxTiSe 2shows only a positive B-quadratic MR. This is quite reasonable because the B-linear term in MR originates from the Kondo disorder mechanism while the quadraticterm results from impurity scattering and the cyclotron mo-tion of charge carriers. Therefore, the large region of linearMR in M xTiSe 2/H20849M=Ni and Co /H20850implies that the Kondo effect with huge TKdistribution is important at x/H11021xc. This effect disappears concomitant with the formation of a perco-lation path at x=x c. The positive B-linear MR with the power-law scaling of magnetization implies time-reversal symmetry breaking, atleast locally in time, associated with unscreened local mo-ments in the disordered Kondo regime; it seems to recoverover longer time scales to cause the typical local Fermi-liquid picture, associated with the Kondo effect. This leads us to introduce an effective Kondo temperature T ¯Kthat is much lower than the single-ion Kondo temperature TK, origi- nating from the disorder-induced distribution P/H20849TK/H20850of the Kondo temperature. The Kondo effect in resistivity appearsaround T K; however, there are some clusters with unscreened local moments due to randomness /H20849called rare regions in the Griffiths scenario /H20850giving rise to unexpected non-Fermi- liquid behavior above the disorder-average Kondo tempera-ture. This interpretation is completely consistent with the crossover magnetic field B cross, identified with Bcross/H11015T¯K, where the B-linear MR results when B/H11022Bcross, while the typical quadratic Bdependence appears below this energy scale, recovering the local Fermi liquid. This situation changes drastically above xc. When the per- colation path is formed near xc, the disorder-average Kondo temperature is not much different from the single-ion Kondotemperature. As a result, the regime of time-reversal symme-try breaking narrows and the positive B-linear MR disap- pears with the Kondo effect. RKKY correlations can com-pete with the Kondo effect above the percolation threshold,but our experiment tells that the RKKY energy scale issmaller than the Kondo temperature, and not relevant in low-energy physics. It is quite interesting that the magnitude of MR also shows discontinuous change near x c, as shown in Fig. 4/H20849c/H20850. However, we do not understand how it changes, as the num-ber and size of impurity clusters increase at this moment.Since dc magnetization probes the time average of Kondofluctuations, a power-law behavior in magnetization cannotbe so pronounced. Next, the strength of the Kondo disorder effect and differ- ent /H9251values should be discussed. Experimentally, this effect is most pronounced in Fe xTiSe 2among the MxTiSe 2com- pounds /H20849M=Fe, Ni, and Co /H20850and yields the largest low- T upturn of resistivity. This compound even shows a variable-range power-law increase in resistivity in the low- Tregion around x=0.05. These observations suggest that Fe xTiSe 2 near xcis very close to a non-Fermi-liquid state. Two sce- narios might describe the origin of this situation: first, aspointed out by Miranda and Dobrosavljevic, 1the distribution of Kondo impurities can be different in a different MxTiSe 2; therefore, the strongest effect in Fe xTiSe 2is due to the criti- cal distribution of TK, whereas the others are relatively far from the critical distribution. Different distributions ofKondo impurities are possible only when the chemistry oftheMcluster formation is quite different. The second possi- bility is the enhanced Kondo fluctuations due to RKKY in-teraction. This effect is believed to be strongest in Fe xTiSe 2 because Fe has the largest spin and thus, the strongest corre-lation between Fe spins, making Kondo fluctuations morecritical in Fe xTiSe 2. Because the chemistry is not thought to be so different in the low concentration region, the formerscenario is more plausible. In contrast, the distribution ofKondo impurities could be important for the series with fixed MinM xTiSe 2and this may be the origin of the observation that the low- Tupturn is largest at x=0.019 in Co xTiSe 2. Finally, it is interesting to compare linear MR of the Ag1+/H9254Se and Ag 1+/H9254Se/H20849Ref. 16/H20850with the present case. Even though quantum magnetoresistance17and strong inhomogeneity18have been proposed as possible origins for the large linear MR, these mechanisms cannot be applied tothe present case in respect that they do not consistently ex-plain the low- Tupturn in resistivity and the power-law be- havior of magnetization. The interplay between the Kondo effect and randomness seems to allow the forbidden coexistence of Kondo effectand time-reversal symmetry breaking in M xTiSe 2/H20849M=Fe, Ni, and Co /H20850single crystals, as indicated by a positive mag- netoresistance that is linearly proportional to the magneticfield. This unexpected result is naturally resolved introducing an effective Kondo temperature T¯Kmuch lower than the single-ion Kondo temperature TKdue to the disorder aver- age. This quantum Griffiths scenario allows linear magne- toresistance when TK/H11022T/H11022T¯K, while the local Fermi-liquid results at low temperatures T¯K/H11022T, are completely consistent with the crossover magnetic field Bcross/H11015T¯K, where the typi- calB2dependence arises when Bcross/H11022B. More detailed in- vestigations are needed near the percolation threshold andare expected to show interplay between the quantum critical-ity and randomness. ACKNOWLEDGMENTS This research was supported by National Nuclear R&D Program through the National Research Foundation of Korea/H20849NRF /H20850funded by the Ministry of Education, Science and Technology /H20849Grant No. 20100018383 /H20850.SASAKI et al. PHYSICAL REVIEW B 82, 224416 /H208492010 /H20850 224416-4*Corresponding author; hjkim76@daegu.ac.kr 1E. Miranda and V. Dobrosavljevic, Rep. Prog. Phys. 68, 2337 /H208492005 /H20850. 2H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev. Mod. Phys. 79, 1015 /H208492007 /H20850. 3F. Lévy, Intercalated Layered Materials /H20849D. Reidel, Dortrecht, 1979 /H20850. 4M. S. Whittingham, Prog. Solid State Chem. 12,4 1 /H208491978 /H20850. 5E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong, and R. J. Cava,Nat. Phys. 2, 544 /H208492006 /H20850. 6M. Sasaki, A. Ohnishi, T. Kikuchi, M. Kitaura, K. Shimada, and H.-J. Kim, J. Low Temp. Phys. 161, 375 /H208492010 /H20850. 7X. Y. Cui, H. Negishi, S. G. Titova, K. Shimada, A. Ohnishi, M. Higashiguchi, Y. Miura, S. Hino, A. M. Jahir, A. Titov, H.Bidadi, S. Negishi, H. Namatame, M. Taniguchi, and M. Sasaki,Phys. Rev. B 73, 085111 /H208492006 /H20850. 8M. Inoue and H. Negishi, J. Phys. Soc. Jpn. 53, 943 /H208491984 /H20850. 9F. J. Di Salvo, D. E. Moncton, and J. V. Waszczak, Phys. Rev. B14, 4321 /H208491976 /H20850. 10See supplementary material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.82.224416 for the raw data and detailed data analysis. 11D. Qian, D. Hsieh, L. Wray, E. Morosan, N. L. Wang, Y. Xia, R. J. Cava, and M. Z. Hasan, Phys. Rev. Lett. 98, 117007 /H208492007 /H20850. 12J. Kondo, Prog. Theor. Phys. 32,3 7 /H208491964 /H20850. 13J. Ruvalds and Q. G. Sheng, Phys. Rev. B 37, 1959 /H208491988 /H20850. 14F. J. Ohkawa, Phys. Rev. Lett. 64, 2300 /H208491990 /H20850. 15E. Miranda, V. Dobrosavljevic, and G. Kotliar, Phys. Rev. Lett. 78, 290 /H208491997 /H20850; E. Miranda and V. Dobrosavljevic, ibid. 86, 264 /H208492001 /H20850. 16R. Xu, A. Husmann, T. F. Rosenbaum, M.-L. Saboungi, J. E. Enderby, and P. B. Littlewood, Nature /H20849London /H20850390,5 7 /H208491997 /H20850. 17A. A. Abrikosov, Phys. Rev. B 58, 2788 /H208491998 /H20850. 18M. M. Parish and P. B. Littlewood, Nature /H20849London /H20850426, 162 /H208492003 /H20850.INTERPLAY BETWEEN THE KONDO EFFECT AND … PHYSICAL REVIEW B 82, 224416 /H208492010 /H20850 224416-5

PhysRevB.102.075149.pdf

PHYSICAL REVIEW B 102, 075149 (2020) Restoring the continuum limit in the time-dependent numerical renormalization group approach Jan Böker and Frithjof B. Anders Lehrstuhl für Theoretische Physik II, Technische Universität Dortmund Otto-Hahn-Str. 4, 44227 Dortmund, Germany (Received 14 June 2020; revised 17 August 2020; accepted 18 August 2020; published 31 August 2020) The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite-size Wilson chain and a part represented by a set of additional reservoirs, each coupled to oneWilson chain site. These additional reservoirs represent high-energy modes of the environment neglected bythe numerical renormalization group and are required to restore the continuum limit of the original problem.We present a hybrid time-dependent numerical renormalization group approach which combines an accuratenumerical renormalization group treatment of the nonequilibrium dynamics on the finite-size Wilson chain witha Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes theintrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bathdiscretization with a Wilson parameter /Lambda1> 1. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfieldmaster equation. For the numerical solution of this master equation, a Lanczos method is employed whichcouples all energy shells of the numerical renormalization group. The presented hybrid approach is appliedto the real-time dynamics in correlated fermionic QISs. An analytical solution of the resonant-level model servesas a benchmark for the accuracy of the method which is then applied to nontrivial models, such as the interactingresonant-level model and the single-impurity Anderson model. DOI: 10.1103/PhysRevB.102.075149 I. INTRODUCTION Quantum impurity systems (QISs) have been of increas- ing interest in the last two decades due to the advent ofsingle-electron transistors [ 1] and the observation of the Kondo effect in nanodevices [ 2–4] as well as in adatoms [5,6] and molecules [ 7] on surfaces. Charge and spin dy- namics of molecules on surfaces [ 8,9], including inelastic processes [ 10–12] as well as local moment formations and quantum phase transitions in the vicinity of graphene vacan- cies [ 13–16], are only a few examples of many such different realizations. QISs are also of fundamental importance as apart of the dynamical mean field theory [ 17,18] where a correlated lattice problem is mapped onto an effective QIS[19] augmented by a self-consistence condition. On the route to functional nanodevices, the real-time dy- namics of local charge [ 20] or spin degrees of freedom (DOF) [21] sparked theoretical interest in nonequilibrium dynamics of observables in such systems [ 22,23]. Charge-transfer and energy-transfer dynamics in molecular systems have also beeninvestigated for more than two decades [ 24]. The theoretical approaches addressing nonequilibrium dy- namics can be divided into three categories. The first class ofapproaches relies on partitioning the full continuum Hamil-tonian into an exactly solvable part and a residue treated asa perturbation. Amongst those are the Keldysh diagrammaticapproaches [ 25–27] to quantum impurity problems [ 28,29] as well as more advanced functional renormalization group[30,31], real-time renormalization group [ 32], and flow equa- tion methods [ 33,34]. The extension of diagrammatic quan-tum Monte Carlo methods [ 35] to the real-time dynamics suffers from a sign problem [ 36–38] which has been tamed by the worm inch algorithm [ 39] only recently. The second class of approaches replaces the closed continuum problem by afinite-size representation of relevant impurity DOF subject toa Lindblad or Bloch-Redfield master equation [ 24,40]. Such approaches have been proposed for systems that are coupledonly weakly to their environment but also have been extendedto more complex QIS [ 41,42] targeting quantum transport problems out of equilibrium. The latter extension uses theLindblad decay rates as fitting parameters to reproduce thecontinuum limit of the noninteracting part of the originalproblem as accurately as possible. The third class of methodsperforms a mapping of the original continuum problem ontoa discretized representation which is then treated by exactdiagonalization [ 43, 44], pure state propagation [ 45–49]b yt h e time-dependent numerical renormalization group (TD-NRG)[50–54], or the time-dependent density matrix renormaliza- tion group (TD-DMRG) approach [ 55–57]. In this paper, we propose a hybrid TD-NRG approach that combines the virtue of the NRG [ 50,51,58] encoding an accurate representation of equilibrium fixed points witha Bloch-Redfield master equation approach [ 24]t or e s t o r e the original continuum problem. In the previous hybrid TD-NRG algorithms, different numerical methods (TD-NRG andChebyshev polynomials [ 59] or TD-NRG and TD-DMRG [60]) were combined but still operated on a finite one- dimensional chain representation of the Hamiltonian and didnot solve the fundamental limitation of all finite-size rep-resentations: true relaxation and thermalization. In a chain 2469-9950/2020/102(7)/075149(25) 075149-1 ©2020 American Physical SocietyJAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) representation of the problem, the continuity equations de- rived from charge conservation lead to back reflections withinthe Wilson chain [ 59] or at the end of a tight-binding chain [ 61]. We make use of the exact decomposition of the bath continuum into the Wilson chain and augmented reservoirsattached to each chain site. We adopt the proposal [ 62] made in the context of the spin-boson model [ 22] to fermionic baths. In the previous work [ 62], only corrections to the Wilson chain parameters obtained from the real part ofthe bosonic reservoir coupling function were included in thecalculations for the spin-boson model [ 63]. Here, we link the Bloch-Redfield tensor [ 24] to the previously neglected imaginary parts of the fermionic reservoir correlation func-tions: These tensor elements govern the real-time dynamicsof the reduced density matrix by connecting NRG eigenstateson different Wilson shells [ 50,51,58] or NRG iterations. In our algorithm, the static reduced density matrix in the TD-NRG [ 50,51] is replaced by a time-dependent version and its dynamics is generated by the previously neglected reservoirs.Our approach conserves the trace of the density matrix at anytime and approaches thermal equilibrium as the steady-statesolution for any Bloch-Redfield tensor that fulfills the genericdetailed balance condition. Therefore, our approach correctsthe drawback of all finite-size real-time methods, namely,that a true stationary steady state can only be reached in thelimit of an infinite system size that is not accessible for suchmethods. The paper is organized as follows. In Sec. II, we introduce the generic quantum impurity model and derive the exacthybrid Wilson-chain continuum representation of the originalcoupling function in Sec. II C. In Sec. II D, we show that the resulting reservoir coupling functions approach two alternat-ing fixed points: one for the even chain sites and one for theodd chain sites that is typical for fermionic baths [ 64,65]. The proposed hybrid approach is presented in Sec. III.A f t e ra short review of the TD-NRG to introduce the notation, wederive the effect of the additional reservoirs up to secondorder in the fermionic coupling functions in Sec. III B which are used in Sec. III C to obtain the nonequilibrium dynamics of the reduced density matrix providing the essential of thehybrid approach. Some technical details about the imple-mentation are provided in Sec. III D . In Sec. IV, we present the benchmark for our approach by demonstrating excellentagreement between the predictions of the continuum hybridTD-NRG approach and the exact analytic solution of thecharge dynamics in the resonant-level model (RLM) [ 51]. The nonequilibrium dynamics of two correlated models, theinteracting RLM [ 66,67], and the single-impurity Anderson model (SIAM) [ 65] are discussed, and the paper ends with a short summary. II. DISCRETIZATION AND RESTORING OF THE CONTINUUM LIMIT A. Introduction to quantum impurity models Quantum impurity models (QISs) describe the coupling of a strongly interacting quantum impurity Himpwith noninter- acting baths Hbathcomprising either conduction bands [ 64,65]or a bosonic environment [ 22]: H=Himp+Hbath+HI. (1) The term HIdescribes the interaction between the two subsys- tems. Hbathmodels Mdifferent noninteracting and continuous fermionic baths, Hbath=M/summationdisplay ν=1/summationdisplay k/epsilon1kνc† kνckν, (2) with the flavors ν.c† kνcreates a bath electron of flavor νwith the energy /epsilon1kν.νmight label the spin σor the channel αin multiband models. We focus on a coupling HIbetween the two subsystems described by a single particle hybridization, HI=M/summationdisplay ν=1Vν(c† 0νAν+A† νc0ν), (3) where c0νannihilates a local bath state of flavor νdefined as a linear combination of annihilators ckνof bath modes with the eigenenergy /epsilon1kν, c0ν=/summationdisplay kλkνckν, (4) such that c0νfulfils canonical commutation relations. A† ν(Aν) accounts for the linear combination of local orbital creation(annihilation) operators inducing transitions in the impuritythat change the particle number by one. The coupling param-etersλ kνcontain the possible energy-dependent hybridization. By integrating out the bath DOF in a path integral formula- tion of the partition function, it was noted early on [ 22,64,68] that the influence of the bath onto the local impurity dynamicsis fully determined by the coupling function /Delta1 ν(z) defined as /Delta1ν(z)=V2 ν/summationdisplay kλ2 kν z−/epsilon1kν. (5) We will utilize the fact that different types of reservoirs [ 68] yield the same local dynamics as long as they provide theidentical coupling functions /Delta1 ν(z). The spectral function /Gamma1ν(ω)=lim δ→0+Im/Delta1ν(ω−iδ)( 6 ) determines the influence of the νth bath onto the local dy- namics. For nonsymmetric baths [ 22], the real part Re /Delta1ν(ω) causes an additional energy renormalization of impurityeigenenergies. This energy renormalization strongly influ-ences the dynamics close to a local quantum critical point[62,69,70] in the case of bosonic baths but plays a less pronounced role in fermionic baths. B. Discretization of the continuum model The NRG [ 58,64] is one of the powerful methods devel- oped to accurately solve QIS. Within this approach, the bathcontinuum is discretized on a logarithmic mesh controlled bythe parameter /Lambda1> 1. The Hamiltonian is then mapped onto a semi-infinite chain, H NRG=lim N→∞HNRG N, (7) HNRG N=Himp+HI−C+Hchain(N), (8) 075149-2RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) mHimpurity m−1 tm+1 V t0 t FIG. 1. The semi-infinite Wilson chain depicted up to the chain linkm. Hchain(N)=N/summationdisplay m=0M/summationdisplay ν=1/epsilon1mνf† mνfmν +N/summationdisplay m=1M/summationdisplay ν=1tm−1ν(f† mνfm−1ν+f† m−1νfmν), HI−C=M/summationdisplay ν=1Vν(f† 0νAν+A† νf0ν), (9) whose chain topology is depicted in Fig. 1.T h e mth chain site represents an exponentially decreasing energy scale ωm= D/Lambda1−(m−1)/2(1+/Lambda1−1)/2, and the original Hamiltonian is only restored [ 64] in the limit /Lambda1→1+. The tight-binding pa- rameters tmalso decrease exponentially, tm∝/Lambda1−m/2, which establishes the hierarchy of scales in the sequence of finite-size Hamiltonians H NRG m. The bath asymmetry [ 58] mentioned above enters the single particle energies /epsilon1mνof each chain site. This sequence of HNRG m is iteratively diagonalized, dis- carding the high-energy states at each step to maintain amanageable number of states. Thereby, the set of eigenstatesofH NRG m,{|r,e;m/angbracketright}, with the corresponding eigenenergies Em r is partitioned into a set of kept (k) states Sk={ |k,e;m/angbracketright}and a set of states Sd={ |l,e;m/angbracketright}which will be discarded (d) in the next NRG iteration. Since the iteration is stopped at a finitebut arbitrary value m=N, we have augmented the eigenstate |k/angbracketrightat iteration mwith the configuration eof the decoupled rest chain m+1→Nto obtain a complete basis set—for details see Refs. [ 50,51]. The reduced basis set of H NRG m,Sk, thus obtained is expected to faithfully describe the spectrumof the full Hamiltonian on the scale of D m, corresponding [64] to a temperature Tm∼Dmfrom which all thermodynamic expectation values are calculated. The NRG algorithm isstopped at chain length Nwhen the lowest temperature of interest is reached. In the present paper, we will not discuss the explicit construction of such chains as a faithful representation of theoriginal continuous baths and refer the reader to the reviews[56,58,64] on this subject. Here we assume that the NRG framework has already provided us with all chain parameterssuch as nearest-neighbor hopping t mνand orbital energy /epsilon1mν of each chain link to fully characterize any chain depicted in Fig.1. Independently of whether the NRG approach, exact di- agonalization, or the density matrix renormalization group(DMRG) [ 56,71] is used to solve such a finite-size representa- tion of an interacting QIS, these numerical approaches sufferfrom the same fundamental problem: The finite-size chainHamiltonian does not contain any information on the lifetimeof excitations and lacks the mechanism for a locally excited system to relax into the true thermodynamic ground state. This leads to two severe limitations when calculating the spectral functions within the NRG: (i) details at high energiesare lost by overbroadening ( b m∝Dm), even if the peak posi- tion and its spectral weight are calculated correctly within themethod, and (ii) spectral information for frequencies belowthe smallest energy scale, i.e., |ω|<D N, is absent, which limits the accuracy of the NRG for calculating transportproperties [ 58,72,73]. C. Restoring the continuum limit We adapt the approach [ 62] introduced in the context of the spin-boson model [ 22] to fermionic baths to reconstruct the correct hybridization function /Delta1ν(z) for a given Wilson chain. We will drop the flavor index νand restrict ourselves to a single flavor for simplicity. We will restore the flavor indexof the bath modes at the end of this section. Since the influence of the continuous bath onto the local dynamics of the quantum impurity is fully determined by thefunction /Delta1(z), the bath Hamiltonian ˜H bath(1) defined as ˜Hbath(1)=/epsilon10f† 0f0+/summationdisplay k/epsilon1k0c† k0ck0+V0(f† 0c00+c† 00f0) (10) yields the same local dynamics as the original Hbathif the Green’s function (GF) of the original bath Gc0;c† 0(z) is identical to the GF Gf0;f† 0(z), /Delta1(z)=V2Gc0;c† 0(z)=V2Gf0;f† 0(z), (11) and the hybridization in Eq. ( 3) is replaced by HI=V(f† 0A+A†f0). (12) The index 1 in ˜Hbath(1) indicates that Hbathhas been replaced by a new bath coupled to a single auxiliary orbital.This new degree of freedom, f 0, will become the first site of the chain representation of the bath continuum which we willconstruct in the following. Analog to Eq. ( 4), we have defined the new operator c 00of the new reservoir 0, c00=/summationdisplay kλk0ck0,/summationdisplay kλ2 k0=1 (13) as a linear combination of its reservoir modes. The bath Hamiltonian Eq. ( 10) describes a RLM whose GF Gf0;f† 0(z)i sg i v e nb y Gf0;f† 0(z)=1 z−/epsilon10−V2 0Gc00;c† 00(z). (14) The unknown reservoir coupling function /Delta10(z), defined as /Delta10(z)≡V2 0Gc00;c† 00(z), (15) is simply related to /Delta1(z)v i aE q .( 11): /Delta10(z)=z−/epsilon10−1 Gf0;f† 0(z)=z−/epsilon10−V2 /Delta1(z). (16) Since the spectrum of Gc00;c† 00(z) must be normalized to unity, the coupling constant V2 0cannot be chosen freely in the model 075149-3JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) reservoir N(c)(b)(a) reservoir mreservoir m−1 reservoir m−1 bath FIG. 2. The reservoir continuum m−1 (a) is recursively re- placed by a single chain site fmcoupled to a new reservoir degree of freedom c0mby a chain link matrix element Vm, shown in (b), to obtain a continuous fraction representation of the original bath by a (c) finite-size tight-binding chain with the continuous reservoircoupled to the end of the chain as used in DMRG calculations [ 74]. but is determined by the integral πV2 0=/integraldisplay∞ −∞dωIm/Delta10(ω−iδ), (17) where /epsilon10is given by the first momentum of the spectrum of /Delta1(z): /epsilon10=1 πV2/integraldisplay∞ −∞dωωIm/Delta1(ω−iδ). (18) Now we can apply the same arguments as above to the new reservoir /Delta10(z) and substitute it by another RLM comprising the second chain site of a chain coupled to the new reservoir1. Recursively, we replace the previous reservoir m−1a t iteration m, shown in Fig. 2(a), by an effective RLM involving a new reservoir mas depicted in Fig. 2(b).A f t e r N+1 such steps, we obtain a chain of length N+1 which is coupled to a single reservoir Nat the end, as plotted in Fig. 2(c). The resulting chain parameters {V m}and{/epsilon1m}represent a continuous fraction expansion with a finite length whichhas been successfully used in DMRG calculations [ 74]. The proper continuum limit is restored by adding a single addi-tional reservoir coupled to the last chain site whose propertiesare uniquely determined by the original coupling function/Delta1(z). The tight-binding parameters V m, however, always re-main of the order of the original bandwidth Dfor all min this procedure and Wilson chains with their refined built-in energyhierarchy cannot be generated this way. To generate more general chains whose sites are coupled by arbitrary linking matrix elements t m(tm<Vm), we need to supplement the algorithm with another step at each iteration.We assume that at some iteration mthe reservoir property is determined by a coupling function ˜/Delta1 m−1(z) such that the corresponding GF is properly normalized by the coupling t2 m−1: Gc0m−1;c† 0m−1(z)=1 t2 m−1˜/Delta1m−1(z). (19) We will explicitly specify ˜/Delta1m−1(z) below by showing how it is determined by the modified recursion. As before, we replacethe reservoir m−1 by an additional chain site mcoupled to a new reservoir mas depicted in Fig. 2(b). The new reservoir coupling function is obtained by the same recursion, /Delta1 m(z)=z−˜/epsilon1m−t2 m−1 ˜/Delta1m−1(z),(20) where the total coupling matrix element is determined by the integral V2 m=1 π/integraldisplay∞ −∞dωIm/Delta1m(ω−iδ). (21) Since the new coupling function /Delta1m(z) must be proportional to a GF, its real part must vanish for |ω|→∞ as 1/ω. Therefore, the energy /epsilon1mhas to be calculated from the first momentum of ˜/Delta1m−1(z), ˜/epsilon1m=1 πt2 m−1/integraldisplay∞ −∞dωωIm˜/Delta1m−1(ω−iδ), (22) to correctly incorporate the center of mass of the previous reservoir. Although ˜ /epsilon1mis of the same order as the original NRG Wilson chain parameter /epsilon1mobtained by the standard NRG approach to a nonconstant density of states [ 58], we will show below that these values are not identical. To beconsistent, we need to replace /epsilon1 m→˜/epsilon1mas given by the first momentum Eq. ( 22). Therefore, we will only use the sets of {tm}from the NRG approach and replace the Wilson chain energies accordingly: /epsilon1m→˜/epsilon1m. Let us introduce a positive semidefinite but otherwise unspecified cutoff function Fdm(ω) which is continuous, 0 /lessorequalslant Fdm(ω)/lessorequalslant1, and its smooth transition between 0 and 1 oc- curs on the energy scale dm. For spectral functions /Gamma1m(ω)= Im/Delta1m(w−i0+) with nonzero contributions for positive and negative frequencies, which is the typical situation in the caseof fermionic baths, 1we demand Fdm(ω)→/braceleftbigg1f o r |ω|/lessmuchdm 0f o r |ω|/greatermuchdm.(23) We use the cutoff function Fdm(ω) to separate a high-energy part from a low-energy part of the coupling function /Gamma1m(ω), /Gamma1L m(ω)=Fdm(ω)/Gamma1m(ω), (24) /Gamma1H m(ω)=(1−Fdm(ω))/Gamma1m(ω), 1For coupling functions which are nonzero only for ω> 0, as is the case for bosonic baths [ 22],Fdm(ω) must vanish for all ω< 0[62]. 075149-4RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) tt’m0Hmc 0Lmc fmtm m−1reservoir mreservoir m reservoir m−1high energy H low energy L FIG. 3. In the modified recursion for the Wilson chain, we di- vided the reservoir manalog to Fig. 2(b) into its high-energy and low-energy ( HandL, respectively) contributions, which are tailored such that the low-energy part is coupled to fmwith the matrix element tm<Vm. so/Gamma1m(ω)=/Gamma1L m(ω)+/Gamma1H m(ω). This step is schematically shown in Fig. 3. The cutoff energy scale dm∝λ−m/2must be self-consistently determined by the equation t2 m=1 π/integraldisplay∞ −∞dω/Gamma1L m(ω−iδ). (25) T h ep r e c i s ev a l u eo f dmwill depend on the analytical form of the specific cutoff function Fd(ω). The separate Hilbert trans- formation of /Gamma1L m(ω) and/Gamma1H m(ω) yields the corresponding real parts to /Gamma1L m(ω)=Im/Delta1L m(ω−i0+) and/Gamma1H m(ω)=Im/Delta1H m(ω− i0+). Partitioning the new reservoir minto a high- and low- energy part, Hres(m)=HL res(m)+HH res(m), (26) the hybridization to the new chain site malso splits into two parts HI(m)=HL I(m)+HH I(m), (27) each involving only low- and, respectively, high-energy modes: HL I(m)=tm(f† mc0Lm+c† 0Lmfm), (28) HH I(m)=t/prime m(f† mc0Hm+c† 0Hmfm). (29) The high-energy coupling constant t/prime maccounts for the differ- ence between V2 mandt2 m:t/prime m=/radicalbig V2m−t2m. The bath operators c0Lmandc0Hmare a linear combination of these new bath modes, c0Lm=/summationdisplay kλkLmckLm,c0Hm=/summationdisplay kλkHmckHm,(30)and also fulfill fermionic commutation relations. Their corre- sponding GFs are related to the coupling functions: Gc0Lm;c† 0Lm(z)=1 t2m/Delta1L m(z),Gc0Hm;c† 0Hm(z)=1 (t/primem)2/Delta1H m(z). (31) After splitting the coupling function /Delta1m(z) into a low- and high-energy part, we use ˜/Delta1m(z)=/Delta1L m(z) in the next iteration step m→m+1v i aE q .( 20). Therefore, we have identified the coupling function ˜/Delta1m−1(z) introduced in Eq. ( 19)a s the low-energy coupling function of the previous iteration, ˜/Delta1m−1(z)=/Delta1L m−1(z). It should be noted here that Vmis always larger than the desired Wilson chain coupling tmfor any /Lambda1> 1 which ensures that the required reservoirs can be generated for anyWilson chain regardless of the choice of /Lambda1.V 0>t0∀/Lambda1> 1 can be shown analytically (see Appendix A). If our algorithm generated a V2 m<t2 m, we would replace tm→Vmimplying that the chain site mdoes not couple to an auxiliary high energy reservoir, i.e., t/prime m=0. By splitting each coupling function into high- and low- energy modes, the continuous fraction expansion has beenmodified such that by coupling a set of additional high-energyreservoirs H H res(m) to the chain site mof a Wilson chain, the original continuous coupling function is restored. The hybridbath Hamiltonian ˜H bath(N)=Hchain(N)+Hres(N)+HI(N) (32) with the additional reservoirs augmenting the Wilson chain Hchain(N), Hres(N)=N−1/summationdisplay m=0M/summationdisplay ν=1HH res,ν(m)+M/summationdisplay ν=1Hres,ν(N), (33) replaces the original Hbathwithout changing the impurity dynamics. This also defines the coupling HI(N) between the finite-size Wilson chain of length Nand the reservoirs HI(N)=N−1/summationdisplay m=0M/summationdisplay ν=1HH I,ν(m)+M/summationdisplay ν=1HI,ν(N). (34) Note that we have finally restored the flavor index ν, and the last chain site is coupled to the full unsplit reservoirs.The topology of this resulting hybrid Hamiltonian is depictedin Fig. 4. In the limit /Lambda1→1 +,t2 mνapproaches V2 mν.A sa consequence t/prime mν→0, and the high energy reservoirs HH I,ν(m) decouple from the system. In this case, the hybrid HamiltonianEq. ( 32) approaches the DMRG tight-binding chain [ 74] augmented by a single reservoir at the end of the finite-sizechain. The hybrid bath Hamiltonian ˜H bath(N) consists of the following terms: the Wilson chain Hamiltonian Hchain(N) generated by the NRG [ 58], the individual high-energy reser- voirs HH res,ν(m) at the energy scale dmandm<N,t h ef u l l remaining reservoir Hres,ν(N) for each flavor νand, most importantly, the coupling between each Wilson chain site m and the corresponding reservoirs HI(N). 075149-5JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) t1 t2 f0tN−1 tN tV 0bathreservoir 2 reservoir 1 reservoir 0reservoir N−1reservoir N FIG. 4. The original bath is replaced by a Wilson chain where each chain site m<Nis coupled to a high-energy reservoir HH res(m) and the last site is connected to the remaining reservoir Hres(N). D. Reservoir coupling functions /Gamma1ν,m(ω) In principle, the recursion outlined in the previous section can be applied to any coupling function /Gamma1ν(ω). In this paper, however, we restrict ourselves to the simplest case as a startingpoint of the recursion. Considering a constant density of stateswithin the band ω∈[−D,D], the hybridization function takes the form /Gamma1(ω)=/Gamma1 0/Theta1(D−|ω|), (35) with the charge fluctuation scale /Gamma10=πV2 2D. The real part of /Delta1(z) is obtained via a Kramers-Kronig relation. Note that we dropped the bath flavor index νsince we focus on spin degenerate coupling functions in this paper. If at each iteration the reservoir /Delta1(ω) is split into a high- energy part /Delta1H(ω) and a low-energy part /Delta1L(ω) in such a way that the adequate Wilson chain coupling parameters tm are generated, then the reservoir coupling functions become invariant at later iterations mif the frequency as well as the magnitude are rescaled by a factor of√ /Lambda1. The results for these rescaled coupling functions are depicted in Fig. 5. The two panels on the left hand side show the hybridizationfunctions for the even iterations and the two panels on theright hand side for the odd iterations, respectively. Clearly, therecursion rapidly approaches convergence. In deriving the leading order correction to the nonequi- librium dynamics in the presence of these additional reser-voirs, the relaxation matrix acquires contributions of the type/Gamma1 m(Em1 l1−Em2 l2), where the coupling function of the reservoir mmust be evaluated at the energy difference between two NRG eigenenergies of two different energy shells m1and m2. Taking into account the NRG energy hierarchy, we can conclude from Fig. 5that/Gamma1m(Em1 l1−Em2 l2)≈0, if either m1< morm2<m. III. NONEQUILIBRIUM DYNAMICS The main focus of this paper is to derive a hybrid approach to the nonequilibirum dynamics of QISs. It combines the time-dependent renormalization group (TD-NRG) [ 50,51] with a Bloch-Redfield approach [ 24] which incorporates the effect of the couplings to the additional reservoirs neglected in theNRG onto the real-time dynamics.FIG. 5. Spectral functions ¯/Gamma1H/L m(ω)=Im/Delta1H m(ω−i0+)/ωmof the high (at the top) and low (at the bottom) energy reservoirs vs ¯ωm=ω/ω m. A bandwidth of D=100/Gamma10and a discretization parameter /Lambda1=2 have been chosen. The sites of the chain that the particular reservoir is coupled to are counted by m. A. Nonequilibrium dynamics in the discretized model: The TD-NRG To set the stage, we review the TD-NRG which is the starting point of the hybrid approach to nonequilibrium. TheTD-NRG was derived [ 50,51] as an extension of the NRG to access the nonequililibrium dynamics of QISs. The TD-NRGis designed to calculate the full nonequilibrium dynamics ofa QIS after a sudden quench: H(t)=H 0/Theta1(−t)+Hf/Theta1(t), but it is restricted to the discretized representation of theQIS. Recently, it was extended to a series of quenches [ 53] mimicking the discretization of time for a time-dependentHamiltonian H(t). The initial state of the system is assumed to be in thermal equilibrium: ρ 0=e−βH0 Tr[e−βH0]. (36) At time t=0, the Hamiltonian suddenly switches and the time evolution is governed by the Hamiltonian Hf. We assume that the switching time is short compared to all relevanttimescales in the QIS such that it can be viewed as instan-taneous. Then, the time evolution of the density operator isgiven by ρ(t>0)=e −itH fρ0eitH f(37) for a time-independent Hf. Using the complete basis set of the final Hamiltonian the time evolution of any local operator O is given by [ 50,51] /angbracketleftO(t)/angbracketright=N/summationdisplay m=mmintrun/summationdisplay r,seit(Em r−Em s)Om r,sρred s,r(m), (38) where Em randEm sare the NRG eigenenergies of the Hamilto- nian Hfat iteration m/lessorequalslantN.Om r,sis the matrix representation of the operator Oat that iteration m[58].mminis the first iteration at which the many-body Hilbert space is truncatedby the NRG approach. ρ red s,r(m) denotes the reduced density 075149-6RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) matrix, ρred s,r(m)=/summationdisplay e/angbracketlefts,e;m|ρ0|r,e;m/angbracketright, (39) in the basis of the final Hamiltonian where the chain DOF eof the chain sites m/prime>m(which are called the environment here) are traced out. In Eq. ( 38), the restricted sums over rands require that at least one of these states is discarded at iterationm: only the discarded states contribute to the dynamics at iteration m. The kept states |k,e;m/angbracketrightare refined by adding the chain link couplings to larger chain sites: The discardedstates at a later iteration are formed from a linear combinationof this tensor product basis. The temperature T N∝/Lambda1−N/2of the TD-NRG calculation is defined by the length of the NRGWilson chain Nand enters Eq. ( 36). The TD-NRG comprises two simultaneous NRG runs: one for the initial Hamiltonian H 0to compute the initial density operator ρ0of the system in Eq. ( 36) and one for Hfto obtain the approximate eigenbasis governing the time evolution inEq. ( 38). This approach has also been extended to multiple quenches [53], time evolution of spectral functions [ 54], and steady state currents at finite bias [ 52,75,76]. The only error of this method originates from the representation of the bath continuumby a finite-size Wilson chain [ 64] and are essentially well understood [ 59,60]. Up to this point, the type of quench is not specified: The switching of H(t) from H 0toHfcould be a local quench or of global nature. Global quenches in infinitely large systems areconceptually complicated since they would imply an instanta-neous change of a system with an infinite amount of energy. Inexperiments, such changes can only occur on a finite timescaleand they spread at a speed whose upper bound is the speed oflight. Throughout this paper, we focus on local quenches where all fermionic baths share the same chemical potential. Note,however, that a local quench does not necessarily imply arelaxation to a thermal equilibrium in the long-time limit.The most prominent example is a quantum dot coupled totwo leads at different chemical potentials. Such a systemrelaxes into a nonequilibrium current carrying steady stateafter switching on the tunneling matrix elements betweenthe leads and the quantum dot [ 77,78]. Such situations are accessible to our approach presented in the next section butare not investigated in this paper. B. Bloch-Redfield extension of the TD-NRG 1. Introduction Finite-size oscillations remain present in the TD-NRG ex- pectation value /angbracketleftO/angbracketright(t) calculated via Eq. ( 38)e v e nf o r t→∞ depending on the NRG discretization parameters [ 59,60,79]. This is a generic feature of accurately calculating the quantumdynamics in finite-size systems, here on a Wilson chain oflength N+1. We define the averaged steady-state value, /angbracketleftO/angbracketright ∞=lim T→∞1 T/integraldisplayT 0dt/angbracketleftO/angbracketright(t) =N/summationdisplay m=mmintrun/summationdisplay r,sOm r,sρred s,r(m)δEs,Er, (40)predicted by the TD-NRG, implying that randshave to be discarded states. This eliminates the finite-size oscillationspresent in /angbracketleftO/angbracketright(t) in the long-time limit. Only the energy diagonal matrix elements contribute to the steady state, whichhas been extensively discussed in the context of the eigenstatethermalization hypothesis [ 80–83]. Since the contribution of the discarded states of the iter- ations m<Nto the thermodynamic density operator in the NRG is negligibly small, a thermalized averaged steady stateimplies vanishing contributions from all ρ red s,r(m) with m<N and an approach of ρred s,r(N)→δENs,ENrexp(−βEN s)/Z. Within the TD-NRG, the values of the matrix elements ρred s,r(m), however, remain fixed and depend on the initial condition[80–83]. The difference /Delta1O=/angbracketleftO/angbracketright ∞−/angbracketleftO/angbracketrightthquantifies the deviation of the TD-NRG steady-state prediction from thethermodynamic limit /angbracketleftO/angbracketright th. In Sec. II C, we have proven that the Hamiltonian ˜Hbath(N), comprising the Wilson chain with Nchain links and a se- quence of reservoirs, generates the same coupling function/Gamma1 ν(ω) as the original continuum problem. Hence, the Hamil- tonian H/prime(N), H/prime(N)=HNRG N+Hres(N)+HI(N), (41) is equivalent to the original Hamiltonian H, H=Himp+HI+Hbath, (42) prior to the Wilson discretization with respect to its impurity dynamics. H/prime(N) augments the standard NRG Hamiltonian of a chain of length N,HNRG N, with the sum of all additional reservoirs Hres(N) and their couplings to the chain links HI(N) as stated in detail in Eq. ( 34). The TD-NRG [ 50,51] utilizes the standard NRG approxi- mation by replacing the original Hamiltonian with the approx-imation H→H NRG N. The aim of this Sec. III B is to derive a set of coupled differential equations for the dynamics ofthe reduced density matrix ρ red s,r(m)i nE q .( 38):ρred s,r(m)→ ρred s,r(m,t). The physical origin of the time dependency of the reduced density matrix is the coupling of the Wilson chainto a set of reservoirs neglected in the NRG approximation.While the exact solution of ρ red s,r(m,t) in the presence of the additional reservoirs is complicated and impractical toimplement, we gear toward an approximate solution in thespirit of weak coupling theories such as a Bloch-Redfield orLindblad type of master equations [ 24,40]. One can explicitly show [ 24,40] that the dynamics of the diagonal elements of the density matrix defined on a finiteHilbert space of dimension Ddecouples from the off-diagonal dynamics within the Bloch-Redfield or Lindblad approaches.The Liovillian operator has D 2eigenvalues: Dof them deter- mine the decay into the steady state while the other D2−D eigenvalues are complex and always come in pairs λi,λ∗ i, since the density matrix must be Hermitian. Below we derive these two types of differential equations for the diagonal and the off-diagonal matrix elements ofρ red s,r(m,t). We show that for a generic decay tensor, the diagonal matrix elements approach the thermal equilibriumdefined by the full density matrix formulation [ 84]o ft h e NRG while the off-diagonal matrix elements vanish in thelong-time limit. To ensure the conservation of the trace of 075149-7JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) the density operator, the differential equation for the diagonal matrix elements requires a coupling of all energy shells, i.e.,all iterations m. This sets a practical limit to our approach and additional approximations are required since the implementa-tion of the couplings between all energy shells is practicallyimpossible. 2. Derivation of the second-order corrections to the TD-NRG dynamics We initially start from the total density operator in the interaction representation, ρI(t)=eiH0tρ(t)e−iH0t, (43) where H0=HNRG N+Hres(N). The total density operator en- codes the dynamics of the original problem and operates onthe Wilson chain DOF as well as the DOF of the reservoirs.Neglecting the system-reservoir coupling H I(N) and assum- ing a factorized density operator in the contributions of eachsubsystem yields a time-independent density operator whosereduced matrix elements relevant for the local expectationvalues are given by the TD-NRG values ρ red s,r(m). By incor- porating the additional system-reservoir coupling, the densityoperator ρI(t) acquires the time dependency that we cast into a master equation for ρred s,r(m,t). The dynamics of the density operator ρI(t) is governed by the differential equation ∂tρI(t)=i[ρI(t),VI(t)] (44) in the interaction picture, where the system-reservoir coupling takes the form VI(t)=eiH0tHI(N)e−iH0t. (45) For expectation values of local operators, it is sufficient to know ρS(t)=TrR[ρI(t)], where we have traced out all the reservoir degrees of freedom. This operator is acting only onthe Wilson chain or system S. Now we can adapt Eq. ( 44) to derive a Bloch-Redfield equation for the reduced density matrix ρ S(t). The individual steps are carried out in Appendix Band can also be found in textbooks—for example, Ref. [ 24]. The derivation requires a complete eigenbasis [ 24]o ft h e discrete system Hamiltonian HSwhich is equal to HNRG(N). For a given NRG eigenbasis |r,e;m/angbracketrightof the discrete Hamilto- nian HS=HNRG N, the Bloch-Redfield master equation reads ˙ρ1,2(t)=−/summationdisplay 3,4R1,2;3,4(t)ρ3,4(t), (46a) R1,2;3,4(t)=δ2,4/summationdisplay 5/Xi1+ 1,5,5,3(t)+δ1,3/summationdisplay 5/Xi1− 4,5,5,2(t)−/Xi1+ 4,2,1,3(t)−/Xi1− 4,2,1,3(t), (46b) /Xi1+ 1,2,3,4(t)=ei(ω1,2+ω3,4)tN/summationdisplay ˜m=0/summationdisplay ν[Cν,˜m(ω3,4)(f† ν,˜m)1,2(fν,˜m)3,4+¯Cν,˜m(ω3,4)(fν,˜m)1,2(f† ν,˜m)3,4], (46c) /Xi1− 1,2,3,4(t)=ei(ω1,2+ω3,4)tN/summationdisplay ˜m=0/summationdisplay ν[C∗ ν,˜m(ω2,1)(f† ν,˜m)1,2(fν,˜m)3,4+¯C∗ ν,˜m(ω2,1)(fν,˜m)1,2(f† ν,˜m)3,4], (46d) with the energy differences ωi,j=Ei−Ej. The index i∈ {1,2,3,4,5}is a general shortcut notation for the tuple i=(ri,ei,mi), where riis a state label of the NRG state at iteration mi, and eiis an environment degree of free- dom of the remaining N−michain sites. eiωi,jt(f(†) ν,˜m)i,j= /angbracketleftri,ei;mi|f(†) ν,˜m(t)|ri,ei;mi/angbracketrightdenotes the factorisation of the time-dependent matrix element of the ˜ mth chain site into a time-independent part and a time-dependent phase factor.The index ˜ mlabels the reservoir index of the sum over all additional reservoirs in H I(N). The bath coupling functions /Gamma1νm(/epsilon1) derived in Sec. II C enter the expression as the greater and the lesser GF for eachreservoir G > ν,m(τ)/G< ν,m(τ) and fully determine the effects of the reservoirs onto the dynamics of the Wilson chain. Thecorrelation functions C ν,m(ω) and ¯Cν,m(ω) are obtained by a half-sided Fourier transformation, Cν,m(ω)=i/integraldisplay∞ 0dτG> ν,m(τ)e−iωτ, (47a) ¯Cν,m(ω)=−i/integraldisplay∞ 0dτG< ν,m(τ)e−iωτ, (47b)that results from integrating Eq. ( 44) and then substituting the resulting expression for ρI(t) back into Eq. ( 44). Using the definitions of the lesser and the greater GFs introduced inEqs. ( B7), we find C ν,m(ω)+C∗ ν,m(ω)=iG> ν,˜m(−ω), (48a) ¯Cν,m(ω)+¯C∗ ν,m(ω)=−iG> ν,˜m(ω), (48b) which relates these combinations to the Fourier transforma- tion of the equilibrium greater and lesser reservoir couplingfunctions. 3. Secular approximation The objective is to derive a differential equation for the reduced density matrix ρred s,r(m;t) using Eqs. ( 46) and to replace ρred s,r(m) by its solution ρred s,r(m;t). The Bloch-Redfield equations introduced in the previous section serve as a startingpoint for a master equation describing the dynamics of thereduced density matrix ρ red s,r(m;t), which is defined as ρred s,r(m;t)=/summationdisplay e/angbracketlefts,e;m|ρS(t)|r,e;m/angbracketright, (49) 075149-8RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) withρred s,r(m)=ρred s,r(m;t=0) as the initial condition. The in- dex pair ( s,r) can either label both discarded states or contain only one discarded state, so we have to allow for the secondstate to be retained for the next NRG iteration. Both, however,are approximate eigenstates of H S,HS|r,e;m/angbracketright≈Em r|r,e;m/angbracketright andHS|s,e;m/angbracketright≈Em s|s,e;m/angbracketright. In the next step, we apply the secular approximation [24,40]. The remaining explicit time dependency on the r.h.s of Eq. ( 46a) in terms of fast oscillating phases, which only occurs in Eqs. ( 46c) and ( 46d), must vanish, providing the additional energy constraint ei(ω1,2+ω3,4)t→δω1,2,−ω3,4, (50) which is consistent with a slowly varying reduced density ma- trix. As a consequence, the time-dependent tensor R1,2,3,4(t) becomes time independent. For the dynamics of ρred s,r(m;t), only the case m1=m2 is relevant. The resulting condition Em1r1−Em1r2=Em3r3−Em4r4requires the discussion of two cases (given that degeneracies inr1,r2are excluded): For the diagonal elements, r1=r2, immediately m3=m4andr3=r4follow (since it is highly unlikely to find two different eigenstates at different iterationsm 3/negationslash=m4that are energetically degenerate.) Ifr1/negationslash=r2, and thus Em1r1−Em1r2/negationslash=0, the equation can only be fulfilled for m1=m2=m3=m4, since it is very unlikely to find the same energy difference on two different NRGiterations. From this discussion, we draw two important conclu- sions: (i) For the occupation dynamics given by the diagonalelements of the density matrix ρ red I(t) (DDM), we obtain Bloch-Redfield tensor matrix elements R1,2,3,4that couple two different iterations m=m1=m2andm/prime=m3=m4. (ii) The dynamics of the off-diagonal elements of the density matrix(ODDM) is determined by the coupling to the reduced densityoperator within the same energy shell m. C. Dynamics of the reduced density matrix ρred s,r(m) Within the Bloch-Redfield approach [ 24], the DDM de- couple from the ODDM. The DDM describe the occupationdynamics and are coupled by relaxation parameters within thesame iteration index mas well as by terms connecting dif- ferent iterations. These later terms are important for derivinga master equation for the occupation dynamics that satisfiesthe conservation of the trace of the density matrix at alltimes. Guided by the energy separation between the discarded states and the kept states which provide the span of theFockspace for all discarded states at later iterations, we usethe approximation /angbracketleftr,e;m|ρ S(t)|s,e/prime;m/angbracketright≈ρred r,s(m;t)δe,e/primed−(N−m)(51) for the matrix elements of the reduced density operator ρS(t) which strictly holds only for the equilibrium density operator[84]. Once we trace out the environment DOF e, the factor d −(N−m)is canceled and the definition of ρred r,s(m;t) introduced in Eq. ( 49) is recovered.1. Diagonal part of the reduced density matrix To evaluate the DDM, 1 =2,3=4 has to be set in Eq. ( 46) to arrive at ˙ρred l1,l1(m1;t)=/summationdisplay l2,m2/parenleftbig /Xi1l2,l1(m2,m1)ρred l2,l2(m2;t) −/Xi1l1,l2(m1,m2)ρred l1,l1(m1;t)/parenrightbig , (52) with the relaxation matrix elements /Xi1l1,l2(m1,m2)=d−(N−m1)/summationdisplay e1,e2(/Xi1+ 1,2,2,1+/Xi1− 1,2,2,1). Equation ( 51) demands that e1=e2as well as e3=e4in Eq. ( 46). Thus, only terms of the form /angbracketleftr,e;m|ρS(t)|s,e;m/angbracketright occur in Eq. ( 46), and the environment e1has been traced out on both sides of Eq. ( 52). Note that the DDM are restricted to the discarded states liof the iteration mi, since the complete basis set used to evaluate the trace of the density matixcomprises all discarded states [ 50,51] and a combination of two kept states does not contribute in Eq. ( 38). For the DDM, the relations between the different half- sided Fourier components in systems with identical chemicalpotentials in each reservoir C ν,˜m(ω2,1)+C∗ ν,˜m(ω2,1)=2f(ω2,1)/Gamma1ν,˜m(ω1,2), Cν,˜m(ω2,1)+C∗ ν,˜m(ω2,1)=2f(ω2,1)/Gamma1ν,˜m(ω2,1), are used—see also Eq. ( 48)—to derive the explicit expression of the relaxation tensor matrix elements /Xi1l1,l2(m1,m2)=2f(ω2,1) dN−m1/parenleftbig W(m1,m2) l1,l2+W(m2,m1) l2,l1/parenrightbig ,(53) W(m1,m2) l1,l2=M/summationdisplay ˜m=0/summationdisplay ν/Gamma1ν,˜m(ω1,2)X˜m l1,l2(m1,m2), (54) X˜m l1,l2(m1,m2)=/summationdisplay e1,e2/angbracketleftl1,e1;m1|f† ν,˜m|l2,e2;m2/angbracketright ×/angbracketleftl2,e2;m2|fν,˜m|l1,e1;m1/angbracketright, (55) where in general the number of reservoirs is determined by the chain length, i.e., M=N. The first term on the r.h.s of Eq, (53) describes the emission of a particle into the reservoir ˜ m and afterward a reabsorption while the second term starts withan absorption and ends with a reemission process. It is easy to check that the sum W (m1,m2) l1,l2+W(m2,m1) l2,l1is sym- metric with respect to exchanging the label pairs ( l1,m1)↔ (l2,m2). Therefore, the asymmetry in the rates /Xi1l1,l2(m1,m2) with respect to this index swap is solely caused by theprefactor. The steady-state value of the reduced density matrix is fully determined by the prefactor f(ω 2,1)dm1−N. The specific form of the remaining term W(m1,m2) l1,l2+W(m2,m1) l2,l1is irrelevant for the steady-state values and only influences the relaxationtimescales as long as all matrix elements remain coupled inthis master equation. Therefore, a decoupling of bound stateson the Wilson chain from the reservoir continuum would leadto a steady state of the system which deviates from the thermalequilibrium. We discuss two important properties of the master equa- tion, Eq. ( 52). First, the trace Tr[ ρ S]=/summationtext l,mρred l,l(m;t)i s 075149-9JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) conserved at all times t, since 0=∂tTr[ρS]=/summationdisplay l1,m1˙ρred l1,l1(m1;t) =/summationdisplay l1,m1/summationdisplay l2,m2/parenleftbig /Xi1l2,l1(m2,m1)ρred l2,l2(m2;t) −/Xi1l1,l2(m1,m2)ρred l1,l1(m1;t)/parenrightbig . (56) This can be seen by interchanging the summation indices (l1,m1) and ( l2,m2) in the second summation. Second, the steady state of the matrix elements obeys the detailed balance condition. Since f(ω2,1)e−βEl1= f(ω1,2)e−βEl2holds, and thus /Xi1l1,l2(m1,m2)e−βEl1= /Xi1l2,l1(m2,m1)e−βEl2dm1−m2, the fixed point of Eq. ( 52)i s given by ρred l,l(t→∞ ;m)=dN−m Ze−βEm l, (57) with the partition sum Z[84]: Z=N/summationdisplay m=mmin/summationdisplay ldN−me−βEm l. (58) The formalism requires that Em lis given in the absolute energy units measured relative to the ground-state energy on the lastiteration E N g, which comprises the sum of the rescaled NRG eigenenergies ¯Em land the ground-state energy shift relative to the last iteration, /Delta1EN,m g=Em g−EN g. Since the ground-state energy is reduced in each iteration step, a positive constant isadded to E m l=/Lambda1(m−1)/2¯Em l, which in combination with the low temperature 1 /βcauses an exponential suppression of the contributions for m<Neven for ¯Em l=0 on the specific iteration mafter identifying β=βN∝/Lambda1(N−1)/2¯β[58]. The steady-state fixed point stated in Eq. ( 57) is inde- pendent of the values of Xl1,l2(m1,m2) unless some matrix elements vanish. Therefore, ρred l,l(m;t) in general approaches its thermal equilibrium value. If, however, the reservoirs havedifferent chemical potentials, this statement does not hold. Inthat case, the structure of the master equation suggests the ap-proach to a steady state that differs from thermal equilibrium[41,42]. The calculation of all matrix elements of the fourth-order tensor /Xi1 l1,l2(m1,m2) for all combinations of discarded states between all iterations m1,m2is numerically very expensive since for every Wilson chain site ˜ mone needs to build X˜m l1,l2(m1,m2)[ s e eE q .( 55)] for each m1,m2∈[mmin,N], which scales as N5 SN3as outlined in the next section. Thus, this procedure appears to not be feasible. Therefore, wehereinafter propose further approximations that do not violatethe conservation of the trace as well as the thermalization ofthe density matrix but keep the approach manageable even forlarge Fock spaces. First, we restrict the summation of the reservoirs in Eqs. ( 46c) and ( 46d) and, in particular, in Eq. ( 53)t o ˜m/lessorequalslant M=min(m 1,m2). This is a consequence of the analytic prop- erties of the coupling functions /Gamma1ν,˜m(ω) discussed at the end of Sec. II D.2. Calculation of the matrix elements /Xi1l1,l2(m1,m2) The key ingredient of the master equation is the calculation of the transition rates /Xi1l1,l2(m1,m2) as defined in Eq. ( 53). While it is straightforward to evaluate the expressions form 1=m2, it is a challenge to connect different Wilson shells. Therefore, we focus on m1/negationslash=m2in the following. We make use of the NRG hierarchy, implying that f(Em2 l2− Em1 l1)≈/Theta1(m2−m1). This implies that the density matrix element ρred l1,l1(m1;t)i nE q .( 52) decays only into states with smaller energies, i.e., m2/greaterorequalslantm1. The first term on the r.h.s of this equation is a source term which increases the occupationof the state l 1via the decay of states l2from iterations m2/lessorequalslantm1. Using the properties of the coupling functions /Gamma1ν,˜m(ω) further justifies the simplification: /Gamma1ν,˜m(±/Delta1E)≈/braceleftbigg/Gamma1ν,˜m/parenleftbig ∓Em1 l1/parenrightbig for ˜m/lessorequalslantm1 0f o r ˜ m>m1.(59) To proceed, we use 1− m=m/summationdisplay m/prime=mmin/summationdisplay l,e|l,e;m/prime/angbracketright/angbracketleftl,e;m/prime| (60) and 1+ m=/summationdisplay k,e|k,e;m/angbracketright/angbracketleftk,e;m| (61) to partition the completeness relation [ 50,51] 1=1− m+1+ m, (62) of the Fock space of the Wilson chain. Since discarded states at a later iteration m2>m1only have an overlap with the kept states after the iteration m1, we need to evaluate X˜m l1,l2(m1,m2)=/summationdisplay e1,e2/summationdisplay k,e/summationdisplay k/prime,e/prime/angbracketleftl1,e1;m1|f† ν,˜m|k,e;m1/angbracketright ×/angbracketleftk,e;m1|l2,e2;m2/angbracketright/angbracketleftl2,e2;m2|k/prime,e/prime;m1/angbracketright /angbracketleftk/prime,e/prime;m1|fν,˜m|l1,e1;m1/angbracketright, (63) from which X˜m l2,l1(m2,m1) can be derived by exchanging the operators fandf†. Then the matrix elements of the creation and annihila- tion operator are diagonal in the environment variables e1,e, ande/prime, /angbracketleftl1,e1;m1|f† ν,˜m|k,e;m1/angbracketright=δe1,e(f† ν,˜m)l1,k, (64) /angbracketleftk/prime,e/prime;m1|fν,˜m|l1,e1;m1/angbracketright=δe1,e/prime(fν,˜m)k/prime,l1, (65) leaving the calculation of the general overlap matrix elements S(m1,m2) l2,l/prime 2;k,k/prime=/summationdisplay e1,e2/angbracketleftk,e1;m1|l2,e2;m2/angbracketright/angbracketleftl/prime 2,e2;m2|k/prime,e1;m1/angbracketright, (66) where we set l2=l/prime 2at the end. This can most easily be evaluated in terms of a matrix product formulation [ 57]. 075149-10RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) We recall that the NRG eigenstates at the iteration m1+1 can be expanded as |r,e;m+1/angbracketright=/summationdisplay k,αPm r,k[α]|k,α,e;m/angbracketright, (67) where kdenotes the kept states after the iteration mandα labels the DOF of the chain site m+1. The matrix Pm r,k[α] is generated during the diagonalization of HNRG m+1. Recursively applying this relation leads to the matrix product expansion |l2,e2;m2/angbracketright=/summationdisplay k/summationdisplay {αi}m2−1/productdisplay i=m1Pi[αi]|k,{αi},e2;m1/angbracketright,(68) which we insert into Eq. ( 66) to obtain the overlap tensor: S(m1,m2) l2,l/prime 2;k,k/prime=dN−m2/summationdisplay {αi}/bracketleftBiggm2−1/productdisplay i=m1Pi[αi]/bracketrightBigg l2,k/bracketleftBiggm2−1/productdisplay i=m1Pi[αi]/bracketrightBigg∗ l/prime 2,k/prime. (69) The prefactor dN−m2arises from performing the summation over the remaining diagonal environment DOF. The calculation of S(m1,m2) l2,l/prime 2;k,k/primecan be casted in the recursion relation S(m1,m2+1) l2,l/prime 2;k,k/prime=1 d/summationdisplay αm2+1/summationdisplay k1,k2Pm2+1 l2,k1[αm2+1]/bracketleftbig Pm2+1 l/prime 2,k2[αm2+1]/bracketrightbig∗S(m1,m2) k1,k2;k,k/prime. (70) Although this expression can be diagrammatically visualised in terms of matrix product states [ 57]. such a tensor with six indices is numerically not manageable and can only serve asan auxiliary quantity. The recursion relation of the tensor S (m1,m2+1) l2,l/prime 2;k,k/prime, however, allows us to derive a recursion relation for the decay rates Wl1,l2(m1,m2) defined in Eq. ( 54). For that purpose, we in- troduce the tensor Fk,k/prime(l1,m1)=m1/summationdisplay ˜m=0/summationdisplay ν/Gamma1ν,˜m/parenleftbig Em1 l1/parenrightbig (f† ν,˜m)l1,k(fν,˜m)k/prime,l1.(71) This includes all reservoir coupling functions up to ˜ m/lessorequalslantm1. Due to the analytic properties of /Gamma1ν,˜m(ω), we expect that /Gamma1ν,˜m(Em1 l1) is rapidly vanishing for ˜ m/lessmuchm1,s o ˜m=m1will be the major contribution. From the definition of Wl1,l2(m1,m2), we immediately obtain Wl1,l2(m1,m1)=dN−m1Fl2,l2(l1,m1) (72) for the Bloch-Redfield tensor elements connecting states on the same Wilson shell m1=m2. The prefactor dN−m1arises from the trace over the remaining environment DOFs andcompensates the prefactor d −(N−m1)in/Xi1l1,l2(m1,m2). Let us absorb the prefactor d−(N−m1)in the definition 1 dN−m1Wl1,l2(m1,m2)=A(m2,m1) l2,l2;l1, (73) where the tensor A(m2,m1) l2,l2;l1is given by the contraction of the overlap tensor Sand the coupling tensor F(m1): A(m2,m1) r,s;l1=1 dN−m1/summationdisplay k,k/primeS(m1,m2) r,s;k,k/primeFk,k/prime(l1,m1). (74)A2 1 =Al srP Pr sl1k k’α1(m +1,m )1 2 m +1 2(m ,m ) FIG. 6. Diagrammatic representation of the recursion relation for calculating the Atensor. Each box represents a matrix element Prm2+1,km2[αm+1] (upper row) or its complex conjugate (lower row). The state labels kandk/primeare plotted horizontally. The state label αm2+1 for the m+1 site is plotted vertically. A connected line indicates a summation over the corresponding index in analogy to Fig. 2 inRef. [ 51]. This Atensor obeys the recursion A(m2+1,m1) r,s;l1=1 d/summationdisplay αm2+1/summationdisplay k1,k2Pm2+1 r,k1[αm2+1]/bracketleftbig Pm2+1 s,k2[αm2+1]/bracketrightbig∗A(m2,m1) k1,k2;l1, (75) using the tensor Fk,k/prime(l1,m1) as the initial condition, derived from the recursion Eq. ( 70). This recursion is visualized in Fig.6. Since the coupling tensor F(m1) has been included in the definition, the Atensor has three indices for each com- bination ( m1,m2) of iterations. Note that the prefactor in S(m1,m2),dN−m2, can be combined with the overall pref- actor of /Xi1l1,l2(m1,m2),d−(N−m1), to obtain d−(m2−m1)which only depends on the relative distance between the iterations.After calculating A r,s;m2(l1,m1) for all states r,spresent at iteration m2, the diagonal matrix elements of discarded states, Al2,l2;m2(l1,m1), enter the master equation while the kept sec- tor,Ak,k/prime;m2(l1,m1), is used in the recursion Eq. ( 75). Inspecting of X˜m l2,l1(m2,m1) in the definition Eq. ( 55)r e - veals that the only difference in the calculation is the combi-nation of annihilation and creation operators. We include thisdifference into the tensor ˜F, ˜F k,k/prime(l1,m1)=m1/summationdisplay ˜m=0/summationdisplay ν/Gamma1ν,˜m/parenleftbig −Em1 l1/parenrightbig (fν,˜m)l1,k(f† ν,˜m)k/prime,l1,(76) which differs from Eq. ( 71) by the exchange of matrices for f↔f†and the sign of the energy. By adding ˜Fk,k/prime(l1,m1) andFk,k/prime(l1,m1) and using this sum as initial condition in Eq. ( 75) generates recursively the sum W(m1,m2) l1,l2+˜W(m1,m2) l1,l2after setting r,s=l2. This allows, in principle, to recursively evaluate Wl1,l2(m1,m2) and, therefore, /Xi1l1,l2(m1,m2). The calculation of all matrix elements A(m2+1,m1) r,s;l1for a fixed starting iteration m1in Eq. ( 75) requires typically Nrecursions for each m1, and we need to calculate the matrix elements for Ndifferent m2as well as Ndifferent reservoirs ˜ mso the calculation scales asN3dN5 S. This is beyond a reasonable computational effort to calculate small corrections to the weakly time-dependentreduced density matrix elements. 3. Approximations of the rates for the diagonal master equation Although the calculation of each matrix element for the diagonal parts of the Bloch-Redfield tensor is analyticallystraightforward and can be casted into the diagrammatical 075149-11JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) matrix product state recursion depicted in Fig. 6, we want to point out that one needs a third-order tensor Ar,s;m2(l1,m1) at any time of the calculations. Although the recursions forcalculating the sequence of tensors A r,s;m2(l1,m1)f o rafi x e d value m1can be independently evaluated for each start it- eration m1, running these calculations in parallel requires a large number of such tensors in the memory at any giventime. Therefore, it might be more feasible to run the recursionfor each m 1sequentially and use highly parallelized matrix multiplication libraries. However, in this paper, we have chosen a different ap- proach. Consider that the correct Boltzmann distributionis enforced by the prefactor of /Xi1 l1,l2(m1,m2),f(Em2 l2− Em1 l1)/dN−m1, which ensures that the thermodynamic state is always reached. The factor 2( W(m1,m2) l1,l2+W(m2,m1) l2,l1) only deter- mines the relaxation timescale. We recall that the deviationof the TD-NRG steady state and the NRG thermodynamicexpectation value is usually small and within 1–10%. There-fore, the main purpose of the master Eq. ( 52) is to ensure the decay of the diagonal matrix elements into the thermodynamicsteady state while maintaining the correct decay rate. Sincethe Redfield tensor decays exponentially with increasing dis-tance|m 1−m2|, we calculate /Xi1l1,l2(m1,m2) exactly only for the tridiagonal terms ( m1,m2∈{m1−1,m1,m1+1}). For |m1−m2|>1, we replace the exact value of X˜m l1,l2(m1,m2) in Eq. ( 55)b y dN−max(m 1,m2)δQ1,Q2+1where Qiis the particle number of the state |li,mi/angbracketright. This approximation includes the degeneration of states with the environment parameter eias well as the fact that only those states couple whose numbersof particles on the Wilson chain differ by one. In other words,we ignore the correct overlap matrix elements but include theproper symmetry relation between l 1andl2, which demands that transitions are only allowed if the states can be linked byan absorption or an emission of a particle from or into thereservoir. 4. Off-diagonal part of the density matrix As a consequence of the secular approximation in Eq. ( 50), only the states of the same NRG iterations mare coupled for the ODDM. As explained above, it is highly unlikely that thesame finite energy difference of the two states r,sat iteration mcan be found at any other iteration m /primegiven the energy hierarchy of the NRG approach. Then Eq. ( 46) simplifies to ˙ρred r1,r2(m;t)=−/summationdisplay r3,r4Rr1,r2;r3,r4(m)ρred r3,r4(m;t), (77) where the environment variables eihave been traced out canceling the factor d−(N−m)in Eq. ( 51). The ODDM has to vanish in the limit t→∞ to allow for the correct thermalization. This condition is met by thesolution of Eq. ( 46). By definition, 1 /negationslash=2 and 3 /negationslash=4m u s t hold: The only possible fixed point of Eq. ( 77)i sρ red r1,r2(t→ ∞;m)=0 for all r1/negationslash=r2. The calculation of the Bloch-Redfield tensor Rr1,r2;r3,r4de- fined in Eq. ( 46b) involves intermediate states which run over the complete basis set of the Wilson chain. Using Eqs. ( 60)– (62) allows us to divide the intermediate sum over index 5 in the two first terms in Eq. ( 46b) into contributions from the same Wilson shell and contributions from m/prime<mgeneratedby 1− m. Neglecting the latter contributions retains the structure of the master equation for the ODDM and only leads to aslight underestimation of the relaxation rates. 2In favor of a fast and simple implementation, we only include matrixelements of /Xi1 ± 1,2,3,4where all four indices are referring to states at the same shell and used the definitions Eqs. ( 46c) and ( 46d). 5. Combined approach In the previous sections, we derived the master equation for the reduced density matrices ρred s,r(m,t) that will replace the time-independent reduced density matrices in Eq. ( 38)b y our proposed hybrid TD-NRG approach, /angbracketleftO(t)/angbracketright=N/summationdisplay m=mmintrun/summationdisplay r,seit(Em r−Em s)Om r,sρred s,r(m,t), (78) which is the main result of this paper. For the conservation of the trace, all reduced density matrix elements of the discarded states need to be coupled and it iscrucial to maintain the symmetry of the Redfield tensor matrixelements /Xi1 l1,l2(m1,m2)i nE q .( 52). At any time, the condition N/summationdisplay m=mmin/summationdisplay lρred l,l(m,t)=1 (79) must hold, where lonly includes the discarded states at iteration m. We fulfill this requirement by solving a masterequation for the diagonal matrix elements of the reduced den-sity matrix, Eq. ( 52), as a first step. The off-diagonal dynamics only involves couplings within a single Wilson shell and isobtained in a second step. In a third step, the solutions forρ red s,r(m,t) are inserted into Eq. ( 78), and the nonequilibrium dynamics of the quantity of interest is evaluated. D. Algorithms for solving the master equations The master equations Eqs. ( 52) and ( 77) are transformed into a Lindblad-style master equation that can be solvedby diagonalizing the occurring nonsymmetric matrix. For along NRG chain with a large number N sof retained NRG eigenstates, the exact diagonalization of this nonsymmetricmatrix is not possible, and we have to rely on approximateschemes. For that purpose, the biorthogonal Lanczos algo-rithm is utilized. 1. The Lindblad master equation The DDM and the ODDM yield two separate equations that are solved separately. In both cases, the reduced densitymatrices ρ red r,s(m;t) are transformed into a supervector that contains all matrix elements. We map the diagonal matrix el-ements and off-diagonal density matrix elements onto equiv-alent vectors ρ red(t)→/vectorρDDM(t),/vectorρODDM (t)[85] and identify the corresponding relaxation matrix. For both cases, we castthe master equations into the form ˙/vectorρ(t)=−R/vectorρ(t). (80) 2Note that the fixed point ρred r1,r2(t→∞ ;m)=0f o ra l l r1/negationslash=r2 remains unaltered. 075149-12RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) For the DDM, all NRG iterations are connected, whereas in the case of the ODDM only the matrix elements of thesame shells couple to each other. However, the dimensionof the master equation of the ODDM comprises two NRGstate indices r,sof the same iteration so the dimension of the off-diagonal vector /vectorρ ODDM isd2N2 s, where Nsdenotes the number of kept states a after each iteration and dthe number of local DOF added in the next iteration step. Ris always a nonsymmetric matrix, and thus we have to distinguish left eigenvectors /vectorwkand right eigenvectors /vectorvk[44], R/vectorvk=λk/vectorvk, (81) /vectorwh kR=/vectorwhλk, (82) where λkis an eigenvalue of R. It should be stressed here that the eigenvectors {/vectorwk,/vectorvk}constitute a biorthogonal basis, which is a consequence of the fact that the matrix Ris non- symmetric. The eigenvectors obey the biorthogonality relation/angbracketleft/vectorw k,/vectorvk/prime/angbracketright=/vectorwh k/vectorvk/prime=δk,k/prime. Note that right eigenvectors are not orthogonal to each other and /angbracketleft/vectorw,/vectorv/angbracketrightdenotes the abstract scalar product. The master equations can be formally solved by /vectorρ(t)=e−Rt/vectorρ(t=0)=D/summationdisplay k=1cke−λkt/vectorvk, (83) where Dis the dimension of the density matrix vector /vectorρ(t), and the complex expansion coefficients ckare calculated by the scalar product ck=/angbracketleft/vectorwk,/vectorρ(t=0)/angbracketright. The supervector /vectorρ(t) consists either of the diagonal matrix ρred l,l(m;t) span- ning all iterations m∈[mmin,N] or the off-diagonal matrix ρred r,s(m;t)(r/negationslash=s) for each iteration mand is provided by the TD-NRG algorithm. The sum over kcomprises a full basis of eigenvectors of R, thus Eq. ( 83) is exact. 2. The biorthogonal Lanczos method Since the matrix dimension of the Redfield tensors scale asN4 sand are much too large for exact diagonalization in a typical NRG framework, we have to employ a Lanczosalgorithm to obtain approximate eigenvalues and—vectorsin a space of reduced dimension. The Lanczos method is adiagonalization scheme that yields mapproximate eigenvalues and—vectors of a given matrix, where typically m/lessmuchDholds. The biorthogonal version [ 44] is suited especially for non- Hermitian matrices. In the conventional Lanczos method, the so-called Krylov subspace K m={Rn/vectorφ0,n∈[0,m−1]}is generated by choos- ing a starting vector /vectorφ0. Then, this Krylov subspace is or- thogonalized by a Gram-Schmidt algorithm. By this pro-cedure, an m×mtridiagonal matrix T mcan be generated iteratively. From the eigenvalues and eigenvectors of Tm,t h e corresponding Ritz values/vectors of the original matrix canbe computed. For a nonsymmetric matrix R, a corresponding left Krylov subspace K L m={/vectorφn=[Rh]n/vectorφ0,n∈[0,m−1]}needs to be constructed and orthogonalization is performed betweenstates of the left and the right space similar to co- and con-travariant vectors in nonorthogonal spaces. For further detailson the algorithm, the reader is referred to Yousef Saad’s book [44] on iterative methods for sparse linear systems. /vectorφ 0=/vectorρ(t=0) is chosen as a left starting vector /vectorw0as well as a right starting vector /vectorv0for the Lanczos method while one of them needs to be normed by 1 //angbracketleft/vectorw0,/vectorv0/angbracketright.T h i s choice yields an accurate short-time solution for Eq. ( 83) which can be understood by first expanding e−Rtinto a Taylor series before inserting a complete eigenbasis/summationtextm k=1|vk/angbracketright/angbracketleftwk| spanning the Krylov subspace. The overlap matrix elementsc k=/angbracketleft/vectorwk|ρ(t=0)/angbracketrightand the approximate eigenvalues λkob- tained by the Lanczos method enter the Taylor expansion /vectorρ(t)=m/summationdisplay k=1m−1/summationdisplay n=0λn k|vk/angbracketrightck(−t)n n!+O(tm), (84) indicating that the accuracy increases with increasing Krylov subspace dimension m. 3. The eigenspectrum of the Bloch-Redfield tensor Since the Bloch-Redfield tensor Rin Eq. ( 80) is nonsym- metric, the spectrum of eigenvalues λkis generally complex. The master equation for the DDM, however, ensures that theeigenvalues as well as the eigenvectors are real to maintainthe Hermitian property of the total density matrix. For theODDM, all complex values can be ordered in complex-conjugated pairs. The Lanczos approach, however, can also be used in our context to make very accurate predictions on the long-timebehavior. In general, the true eigenvalues λ kof the tensor Rfor the ODDM are finite and Re λk>0. If the Lanczos approach maintains the condition Re λk>0 even the approximative solution in a reduced m×mspace yields a complete decay of the ODDM with possibly slightly modified relaxationtimescales. As discussed above, the tensor Rfor the DDM has one eigenvalue λ 0=0 with the corresponding right steady-state eigenvector /vectorv0. Thus, the steady-state density matrix /vectorρ(t→ ∞)=c0/vectorv0is obtained by calculating the overlap between the left eigenvector /vectorw0and the initial vector c0=/angbracketleft/vectorw0,/vectorρ(t=0)/angbracketright. As we have shown in Sec. III C 1 , this steady-state density matrix obtained via Eq. ( 52) which is given by the Boltzmann distribution for a system approaching the thermal equilibrium.As long as this thermal density matrix c 0/vectorv0has a finite overlap with the initial density matrix, c0/negationslash=0, this vector is always included in the Krylov subspace by construction. We note that the correct solution for /vectorv0with an eigenvalue λ0=0 is always found with high precision by the Lanczos approach since it is an extreme eigenvalue. Therefore, theapproximation for the DDM, /vectorρ(t)=e −Rt/vectorρ(t=0)≈m/summationdisplay k=1cke−λkt/vectorvk, (85) using the Lanczos eigenvectors /vectorvk,/vectorwkand eigenvalues λk, includes the correct limit for t→∞ . This reflects the fact that only the very large and the very small (i.e., extreme) eigen-values in the Lanczos eigenvalue spectrum [ 43] are reliable representations of the true spectrum of a matrix. Therefore,the Lanczos approach has been successfully used for thecalculation of ground states of finite size Hamiltonians. 075149-13JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) E. Summary of the employed approximations to the Bloch-Refield formalism Let us summarize the approximations used in the Bloch- Refield formalism as well as the additional approximationsperformed by us whose implications we discuss in the resultsection below. The Bloch-Redfield approach [ 24] assumes a weak cou- pling to the reservoir, ignores the back action of the systemonto the reservoirs and the effect of the reservoirs onto thesystem is treated preturbatively up to second order in thesystem-reservoir couplings. The reservoir correlation func-tions are assumed to decay fast compared to the change of thereduced density matrix leading to a Markovian approximationof the time integration. As a consequence of the Markovianapproximation, the real-time dynamics is predicted to be a su-perposition of exponential solutions, see Eq. ( 83). Therefore, the short-time dynamics always contain contributions linear inthe time t, although the exact solution for t→0 starts with a term proportional to t 2as demonstrated in the Appendixes. The secular approximation neglects fast oscillatory terms[24] and leads to coupled differential equations that are very similar to the Lindblad approach. Since the Bloch-Redfield approach leads to decoupled differential equations for the diagonal and the non-diagonalmatrix elements, we investigated the nature of the Redfieldtensor for these two cases separately. Within the secularapproximation [ 24], the dynamics of off-diagonal matrix ele- ments decouples between the different NRG iterations: we cansolve the differential equations separately for all Wilson shellsm. We also use the simplification /angbracketleftr,e;m|ρ S(t)|s,e/prime;m/angbracketright≈ ρred r,s(m;t)δe,e/primed−(N−m)that strictly holds only in equilibrium [53]. The dynamics of all Bloch-Redfield differential equa- tions are solved via a biorthogonal Lanczos approach [ 44]. However, all diagonal matrix elements of all shells are coupled via the Redfield tensor. Since we have proven thatintershell matrix elements of the Redfield tensor scale asd −|m1−m2|, we only calculate the matrix elements for |m1− m2|=0,1 exactly and replace the other intershell Redfield tensor elements by a constant times d−max( m1,m2). To benchmark the approach, we discuss the the effect of further approximations in the benchmark section, Sec IV A . Here we neglect all intershell matrix elements of the Redfieldtensor, so the diagonal part of the differential equation canbe solved for each Wilson shell separately. Alternatively, weonly include intershell matrix elements for |m 1−m2|=0,1 exactly and neglect the other intershell matrix elements. IV . BENCHMARK A. The resonant-level model Since the exact solution of the local dynamics in the RLM is known [ 51], we will use it to benchmark our hybrid NRG approach. Throughout this paper, a symmetric box density ofstates ρ(/epsilon1)=ρ 0/Theta1(D−|/epsilon1|) is used in all TD-NRG calcula- tions. The Hamiltonian of the RLM describes the hybridization of a localized level at the energy Edwith a conduction band H=Ed(t)d†d+/summationdisplay k/epsilon1kc† kck+V/summationdisplay k{d†ck+c† kd},(86)where c† kcreates a spinless conduction electron with momen- tumkand energy /epsilon1kandd†creates an electron on the localized level. We also allow for a time dependency of the single-particle energy E d(t). Here /Gamma10=πρ0V2is the hybridization width and ρ0is the conduction-electron density of states at the Fermi energy. To adapt the RLM to our hybrid approach, we set Himp= Ed(t)d†din Eq. ( 1), and HIis given by Eq. ( 3). Since the number of bath flavors M=1, we drop the index νin the following. B. Real-time dynamics Choosing ˆ nd=d†das the observable ˆOin Eq. ( 78), we consider a stepwise change in the energy of the level: Ed(t)=/Theta1(−t)Ei d+/Theta1(t)Ef d. In the wide-band limit, ( D/greatermuch /Gamma10)nd(t)=/angbracketleftˆnd(t)/angbracketrightcan be solved exactly in closed ana- lytical form using the Keldysh formalism [ 51]. For T=0, the analytic solution features an exponential decay from theinitial equilibrium occupancy of H ito the new equilibrium occupancy of Hfwith two decay rates /Gamma10and 2/Gamma10. We present data for a sudden level quench in the RLM that leads to a depletion of charge on the impurity in Fig. 7.T h e real-time dynamics of the local orbital occupancy nd(t) (solid lines) are obtained with our hybrid open chain (OC) approach,Eq. ( 78): The constant reduced density matrix ρ red s,r(m)w a s made time dependent, and its dynamics was calculated bythe Bloch-Redfield master equations. The master equationswere solved via a biorthogonal Lanczos algorithm [ 44]. The dimension of the Krylov subspace for calculating the real-timedynamics of the diagonal matrix elements was set to m= 1000, while a Krylov subspace dimension of m=100 turned out to be sufficient for obtaining the dynamics of off-diagonalmatrix elements that only require coupling matrix elementswithin a single Wilson shell. We also supplied the exactanalytic solution [ 51] as a black dashed line to the panels. n d(t) was calculated for three different Wilson chain lengths Nby varying the NRG parameter /Lambda1to ensure the same target temperature T=0.01/Gamma10. For comparison, we added the results obtained by the closed chain (CC) TD-NRG approach [ 51,51] for the same parameters as dotted lines of the same color. The arrow marksthe thermodynamic expectation value of the equilibrium NRGusing the finial Hamiltonian H f.W e zaveraged the dynamics using Nz=4 different NRG chain representations [ 50,51,86]. The zaveraging significantly reduces the finite-size oscilla- tions but the charge occupation in the CC results still doesnot converge to the thermodynamic limit as expected from theexact continuum limit. The NRG and the quench parameters are chosen close to Fig. 1(a) of Ref. [ 59] to make a connection to the literature. Usually, the averaged TD-NRG steady-state long-time limitis close to the thermodynamic NRG expectation value. Thesequench parameters, however, are deliberately chosen such thatthe deviation is large due to back reflections along the NRGchain as discussed in Ref. [ 59]. For short timescales, the TD-NRG and our OC approach track the exact result very accurately. The differences betweenthe approaches become pronounced in the long-time limitplotted in Fig. 7(b) illustrating the influence of the NRG 075149-14RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) (a) (b) FIG. 7. Real-time dynamics of the local orbital occupancy nd(t) (a) for a short timescale and (b) for long timescales obtained by the open-chain approach and shown as full lines compared to theTD-NRG approach which is added as a dotted line in the same color for the RLM. The exact analytical solution of Ref. [ 51]h a s been added as black dotted lines. Data for different discretizationparameters /Lambda1are presented for a sudden change in the energy of the level from E i d=−/Gamma10toEf d=/Gamma10. The Wilson chain length N(/Lambda1=1.59,N=50;/Lambda1=2.17,N=30;/Lambda1=3.21,N=20) was adjusted such that the same temperature T=0.01/Gamma10is reached for all curves; the corresponding values for /Lambda1are stated in the legend. NRG parameters used: D=103/Gamma10,NS=103,Nz=4. parameter /Lambda1onto the real-time dynamics. It is well under- stood [ 59,87] that the exponential decay of the tight-binding parameters of the Wilson chain leads to a tsunami effect [ 87] of a severe slowdown of charge transport along the chain: Thecharge transport velocity mismatch leads to back reflectionsthat increase with increasing /Lambda1and are the origin of the deviation between the calculated real-time dynamics and theexact analytical solution. This problem is solved by includingthe additional reservoirs perturbatively in the dynamics. Thethermal state is reproduced as a steady state in all cases withthe largest deviations at intermediate times for the largestvalue of /Lambda1. In this case, the TD-NRG shows the largest deviations as well. Furthermore, the bath couplings are thelargest in this case so the second-order perturbation theorytreating the reservoirs is insufficient to fully reproduce theexact solution. However, Fig. 7(b) clearly demonstrates theFIG. 8. Real-time dynamics of the local orbital occupancy nd(t) forN=50 with (green) and without (orange) zaveraging [ 51].Nz accounts for the number of different values used for the zaveraging. All other parameters as in Fig. 7. convergence for /Lambda1→1+: the choice of /Lambda1=1.59 already ex- cellently tracks the exact analytic solution for the continuumproblem. The plots in Fig. 7present the very good agreement of our proposed hybrid TD-NRG approach with the exact analyticalresult in the long-time limit. The OC approach provides anefficient mechanism for particle exchange with the additionalreservoirs such that charge conservation is maintained in thecoupled system but excess charge is balanced by the infinitelylarge reservoirs that couple to each chain site. The effect of the zaveraging [ 50,51,86] is illustrated in Fig. 8.T h e N z=4 result of Fig. 7for the Wilson chain of length N=50 (green) is plotted in comparison to the data without zaveraging ( Nz=1, orange curve). The discrepancy between the different data obtained from the OC approach issmall: The zaveraging evens out the finite size oscillations which are very close to the exact solutions plotted as a blackdashed line. The hybrid approach perfectly reproduces thethermal value of the occupation as indicated by the blackarrow at the right side of the figure and follows the exactsolution very accurately. An important component of our hybrid approach is the coupling of the reduced density matrix elements between allWilson shells. Since the calculation of all Bloch-Redfield ten-sor elements are in principle possible—see Sec. III C 2 —but numerically too expensive for a practicable implementation,we only calculate the shell diagonal tensor matrix elementsand those between adjacent shells m /prime=m±1 in a complete manner. For the coupling of iterations with |m1−m2|>1, we use the approximation X˜m l1,l2(m1,m2)→dN−max(m 1,m2)δQ1,Q2+1 as introduced in Sec. III C 3 . The effect of different approximations to the Bloch- Redfield tensor is depicted in Fig. 9. We augmented the OC approach data for N=50 taken from Fig. 7with the results obtained with additional approximations in calculations ofdiagonal density matrix elements. The blue curve (no shell coupling) is obtained by a tensor /Xi1 l1,l2(m1,m2) that is diagonal in the Wilson shell indices, 075149-15JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) FIG. 9. Impurity occupancy nd(t) obtained by the full open chain hybrid approach, i.e., with a coupling of all NRG iterations (green) by an adjacent approximation for /Xi1l1,l2(m1,m2) including m1=m2 andm1=m2±1 (orange) and restricting to m1=m2, i.e., with no coupling between discarded states of different NRG iterations (blue). NRG parameters as in Fig. 7forN=50. i.e.,/Xi1l1,l2(m1,m2)=δm1,m2/Xi1l1,l2(m1,m1). The coupling of the additional reservoirs generates a damping in the real-timedynamics of the orbital occupancy n d(t). Since the sum of the diagonal density matrix elements remains conserved in eachWilson shell as in the TD-NRG, the steady-state value is verysimilar to the time-averaged TD-NRG value at infinitely longtimes: The decay into the thermal steady-state is not possiblewithout coupling the discarded states of different iterations m. Note that even though the approximation /Xi1 l1,l2(m1,m2)∝ δm1,m2conserves the trace, it does not reach the correct ther- mal steady state. The reason for this is that for completelydecoupled shells, the Bloch-Redfield equations feature a non-physical conservation of each contribution to the trace withineach energy shell. As a consequence, a correct relaxation intothe thermal equilibrium characterized by the detailed balancecondition [see also Eq. ( 57)] lim t→∞ρred l1,l1(t;m1) ρred l2,l2(t;m2)=dm2−m1e−β(Em1 l1−Em2 l2)(87) not possible. This is fundamentally changed when the adjacent approx- imation which includes all tensor elements m1=m2±1i s applied (depicted as an orange line). We notice a decay ofn d(t) at intermediate times even though there is no conver- gence on the timescales plotted in Fig. 9. However, we proved analytically in Appendix (B10) as well as numerically—notshown here—that the thermal expectation value of n dwith respect to Hfis already obtained as the steady-state value in this approximation. The decay rate, however, is very low.This problem is solved by our approximate treatment of allother matrix elements /Xi1 l1,l2(m1,m2) that includes a coupling of the diagonal density matrix elements of all Wilson shellswith exponentially decaying matrix elements that are allowedby the symmetry but ignoring the precise values of the overlapmatrix elements (green curve).FIG. 10. Impurity occupancy nd(t) obtained by the OC hybrid approach but with different approximations. The green curve is obtained by the full algorithm (all discarded states are coupled) and is taken from Fig. 7. For the blue curve, we neglected the coupling between discarded states of different iterations and included the relaxation into the kept states for each iteration instead. The loss of the trace Tr[ ρ(t)] of the density matrix as a function of time is shown as a blue dashed line. Normalizing the blue curve by the time dependent trace yields the orange curve. NRG parameters as in Fig. 7 forN=50. In Fig. 10, we plot the all-coupling approach (green) versus a complete separation of the iterations (blue), comparableto the blue curve in Fig. 9. The difference lies in the fact that we now include all states, discarded and kept, at alliterations for the independent Bloch-Redfield equations. Thisimplies a realistic relaxation of the high-energy states into thelow-energy kept states for each NRG iteration. Since the keptstates of the diagonal part of the density matrix, however, arenot included in Eq. ( 78), we end up with an effective unphys- ical loss of the trace. This can easily be compensated for byartificially dividing any nonequilibrium expectation value bythe time-dependent trace and thus ensuring to keep the traceof the resulting expression constant (orange curve). That way,a correct thermalization can be realized. Even though thisapproximation is very efficient regarding computation timeand memory requirements, its motivation is unphysical. Forthat reason, we will continue this paper by using the approachthat couples all iterations and thus includes an inherent con-servation of the trace. C. Computational time considerations The TD-NRG, the CC algorithm, comprises a standard NRG run and a backward iteration for the real-time dynamicsby tracing out the reduced density matrix for each iterationthat is needed as an initial condition for the time-dependentreduced density matrix [ 50,51]. The limiting factor is the number of time points that should be evaluated for eachexpectation value. The TD-NRG run takes three to ten timeslonger than the underlying NRG itself. The first OC part is the final forward run. Here the runtime is determined by building the correlation functions C m(ω) to then build the Bloch-Redfield tensors R[see Eq. ( 46)]. 075149-16RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) This procedure scales ∝N2 SN2since for each iteration m at least the same number of individual tensors has to bebuild within our approximation. Next is the construction anddiagonalization of the tensor that connects all NRG iterationsfor the diagonal part of the density matrix. Here the Lanczosdepth m lan, diag (equivalent to the Krylov space dimension) is the limiting factor. For large values of /Lambda1and long chains, very small effective temperatures are reached, which meansthat the tensor has to cover a broad energy spectrum. Hence,for a sufficient resolution to resolve the long-time dynamics, alarger value for m lan, diag has to be chosen accordingly. For T= 0.01/Gamma10, we have chosen mlan, diag =1000. The last part of the program is the backward run where the Bloch-Redfield tensorfor the off-diagonal dynamics of each iteration is diagonalizedand then its contribution to the complete expectation valueis calculated for each individual time t. Since those tensors only contain one shell, they are significantly smaller and thusa Lanczos depth of m lan, off-diag =100 is sufficient. The latter two sections of the program roughly scale linearly with thechain length N. TheN=20 curve of Fig. 7, for instance, required a total runtime of 78 min on an Intel Xeon 12 core workstation whichis distributed roughly by 50% on the building of the tensors.Here we have parallelized with ten threads for the value ˜ m in Eq. ( 46c). The backward run and the TD-NRG portion are parallelized for the single subspaces on each iteration and theformer takes a runtime percentage of 35% while the mere TD-NRG takes less than 1%. The building and diagonalization ofthe Bloch-Redfield tensor for the diagonal part of the densitymatrix requires the remaining runtime, which lies around14%. We did not implement the parallelization of the Lanczosalgorithm. When zaveraging was utilized, an independent thread can be used for each zvalue which enables a simple parallelization of almost the entire program. V . REAL-TIME DYNAMICS FOR CORRELATED MODELS USING THE OPEN-CHAIN APPROACH After establishing the quality of the OC algorithm to the nonequilibrium dynamics by comparing the results of theapproach to the exact analytical solution of the occupancydynamics in the RLM, we apply our approach to two problemsfor which an exact analytic solution is unknown: the interact-ing RLM and the SIAM. A. Interacting resonant-level model To proceed to the first nontrivial problem of this paper, the RLM is extended by a Coulomb repulsion Ubetween the local impurity level and the band which defines the interactingresonant level model (IRLM). Here the modified impurityHamiltonian H impreads Himp=Ed(t)d†d+U/parenleftbig d†d−1 2/parenrightbig/parenleftbig f† 0f0−1 2/parenrightbig . (88) This model has been intensively studied [ 66,67] in the 1970s due to its connection to the Kondo problem [ 88]. In recent years, the interest has shifted to its nonequilibrium properties,particularly for a biased two-lead setting [ 30,31,89,90].FIG. 11. The real-time dynamics of nd(t) vs time in the IRLM for different values of Uobtained by the open-chain hybrid approach (solid lines) for a sudden change in the energy of the level from Ei d//Gamma1eff=−1t oEf d//Gamma1eff=1. The analytical U=0 result is added as a guidance (black dashed line). The thermodynamic expectation value nf dis added as a black arrow on the r.h.s of the figure for com- parison. NRG parameters: /Lambda1=1.59,N=50,D//Gamma1eff=103,NS= 103,Nz=4s oT//Gamma1eff=0.01. The IRLM shares the line of low-energy fixed points with the noninteracting RLM after renormalization of /Gamma10→/Gamma1eff≈D(/Gamma1/D)1/(1+α), (89) withα=2δ−δ2andδ=(2/π)a r c t a n( πρU/2). Neverthe- less, the nonequilibrium dynamics of both models differssignificantly [ 59,60]. While the coherent oscillations present in the analytic solution [ 51] are strongly damped in the RLM and, therefore, are only observable for extreme parameterchoices, an increasing number of coherent oscillations inn d(t) is found with increasing U[59,60] in the IRLM. The additional Coulomb repulsion Ufavors the single-electron subspace spanned by the impurity orbital and the first Wilsonchain site. The coherent oscillation frequency is given bythe energy difference between the binding and antibindingmolecular state formed by the hybridization since the initialconfiguration can be expanded into these two local states withdifferent eigenenergies. In the limit of large U, the rest of the Wilson chain is essentially decoupled from those two states,and the virtual charge fluctuations between these states and therest of the Wilson chain induces a damping of these coherentoscillations that is proportional to U −2[60]. To ensure quenches between the same initial and final equilibrium fixed points, the hybridization strength /Gamma10has been adjusted such that nd(0)=0.75 and nd(∞)=0.25 for all values of U, implying Ei d//Gamma1eff=−1 and Ef d//Gamma1eff=1 for all curves. The OC results for the local occupancy nd(t) are shown in Fig. 11. Upon increasing U, a new timescale τUemerges which is much larger than the thermodynamical relaxation timescale τ0∝1//Gamma1eff. The timescale τUcharacter- izes the decay of the amplitude of coherent oscillations. For U→∞ , the charge simply oscillates between the impurity and the first Wilson chain site, while for a finite Uthe 075149-17JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) FIG. 12. The real-time dynamics of nd(t) vs time in the IRLM for a fixed U/D=16 but different chain length Nand NRG dis- cretization parameter /Lambda1combinations. The NRG parameter /Lambda1(/Lambda1= 1.59,N=50;/Lambda1=2.17,N=30;/Lambda1=3.21,N=20) was adjusted such that the same temperature T=0.01/Gamma1effis reached for all curves andNS=300. The analytical U=0 curve is added as a dashed line for illustration purposes. oscillations are damped and the system approaches thermal equilibrium. Since the partitioning of the original continuum depends on the NRG discretiation parameter /Lambda1, we investigated the nonequilibrium dynamics of nd(t) for a fixed value of U/D= 16 and the same local quench parameters as used in Fig. 11 but for three different values of /Lambda1. The corresponding chain lengths are adjusted such that the effective temperature is thesame for all three cases. The results are plotted in Fig. 12. Remarkably little effect of /Lambda1on the oscillation frequency and the relaxation time is found, although /Lambda1strongly influences the spectral weight of the coupling to the additional reservoirs.This indicates that our OC approach is rather robust and theresults depend only weakly on the discretization parameter. The difference of our approach and the TD-NRG in the IRLM is illustrated for a few small values of Uin Fig. 13. Although the oscillation frequency is the same as reported byGüttge et al. [60], we note that the decay time τ Uof the OC approach is shorter than predicted by the CC approach. Theanalytical golden rule estimate of Ref. [ 60] is based on a CC topology, where the impurity orbital and the first Wilson chainsite ( m=0) only couple via the hopping parameter t 0to the rest of the system. The Fermi’s golden rule calculation treatsthe first two orbitals as a closed system and adds a perturbativecoupling to the rest of the chain. The long-time artifacts ofthe CC approach are suppressed in Ref. [ 60] by combining the TD-NRG with a TD-DMRG approach for a very longtight-binding chain and stopping the simulation before reflec-tions at the chain end are detectable at the impurity. In ourapproach, the additional reservoirs cause an additional decayof the coherent oscillations and ensure the thermalization tothe expectation value. In Fig. 14, we present a comparison of numerically ex- tracted parameters with their analytical predictions. In thetop panel, we show the NRG results for the ratio /Gamma1 0//Gamma1effFIG. 13. The real-time dynamics of nd(t) vs time in the IRLM for different values of Uobtained by our OC hybrid approach (solid lines) and by the TD-NRG (dotted line) in the same color as well as a fit to Fermi’s golden rule (dashed line). NRG parameters as in Fig. 7. as a solid line, /Gamma10being the bare hybridization strength of the model. The results of the perturbative RG predictionaccording to Eq. ( 89) have been added as a dotted line. Both graphs agree excellently in the limit of large U. The middle panel and the bottom panel of Fig. 14present the numerical fit to the analytical golden rule results stated in Eq. (17) of Ref. [ 60] and their analytical predictions. The oscillation frequency /Omega1of the occupation was calculated by /Omega1=/epsilon1 +− /epsilon1−=2/radicalbig (Ed/2)2+(Veff)2with Veffbeing the renormalized hybridization strength parametrizing /Gamma1eff=πV2 eff/2D. As expected, the analytical prediction agrees very well with the numerical value for the large Uregime where the golden rule result is applicable. Nevertheless, a significantdeviation between the analytical and the numerically ex-tracted relaxation time τis observed. The analytical solution FIG. 14. A comparison of analytical estimates (dotted lines) and the numerical values (stars) for three different IRLM parameters. The coupling strength U/Dhas been varied. 075149-18RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) presented in Ref. [ 60] predicts τ/Gamma1eff/parenleftbiggD U/parenrightbigg2 =π4 256/Omega1 D/Gamma1eff /Gamma1(90) and is plotted as a dotted green line in the bottom panel of Fig.14. As mentioned above regarding Fig. 13, the relaxation time extracted for the OC in the IRLM does not exactly matchthe golden rule prediction. In fact, we approximately obtainan overlay of three different decay times, the smallest onestemming from the DDM. The remaining two decay timesdamp the oscillations as exponential functions in Eq. ( 78). The largest decay time influences the long-time behavior of theoccupation and thus we have chosen this value to be plotted incomparison to the golden rule approximation for τin Fig. 14. Obviously, the long-time relaxation τ∝(U/D) 2, as predicted in the golden rule, whereas the asymptotic value for large Uis smaller, thus implying a faster relaxation, as discussed above. In the OC approach, presented here, the fundamental dif- ference to the CC approach is the direct coupling of anauxiliary reservoir to the first Wilson site m=0 as well: Even if we artificially decouple the rest of the Wilson chain fromthe first site by setting t 0=0, the oscillations remain damped for any finite Udue to the relaxation channel provided by the first bath. In the limit of large U, we expect a superposition of two damping channels: damping by the rest chain anddamping by the high-energy modes of the reservoir /Delta1 0(z). This additional damping mechanism in our OC explains thedecrease of τ Ucompared to the CC approach as demonstrated in Fig. 13. Our OC also avoids the reflections of charge waves propagating along the Wilson chain since they aredamped by the reservoirs as expected from the continuumproblem. Furthermore, the analysis of the RLM has alreadyshown that the relaxation times of our approach are slightlyexaggerated for t/Gamma1 0<10 (see, e.g., Fig. 9), which stems from the approximation in Sec. III C 3 where the matrix elements of the Bloch-Redfield tensor for |m−m/prime|>1 are still assumed slightly too large. This yields a faster relaxation for short timesfading into a smaller rate for later times. B. Single-impurity Anderson model 1. Definition of the model In the SIAM, the spin degree of freedom ν=σ, the on-site repulsion U, and an optional local magnetic field strength b(t) are added to the RLM. The SIAM impurity Hamiltonian nowreads H imp=/summationdisplay σ/bracketleftBig Ed(t)−σ 2b(t)/bracketrightBig d† σdσ+Ud† ↑d↑d† ↓d↓.(91) We choose the spin quantization axis parallel to the external magnetic field direction and absorb the prefactor gμBinto the magnetic field strength bwhich is consequently measured in the units of energy. Since we are not interested in the limitof large magnetic fields of the order of the band width [ 91], we neglect the small corrections due to the spin polarizationof the conduction band and only apply a local magneticfield for simplicity. The bath Hamiltonian and the interactionHamiltonian are given by Eqs. ( 32) and ( 34), respectively, where the spin index σis summed over M=2 values.FIG. 15. Impurity occupancy nd(t) and spin polarization Sz(t) vs time after the quench on a logarithmic timescale. The upper and lower right panels show the results of scenario (i) leavingthe hybridization strength constant. In the upper and lower left panels, the data after switching the hybridization strength on at t=0 are plotted. Parameters: /Lambda1=1.66,D=20/Gamma1 0,N=30,T= 0.01/Gamma10,Nz=4,NS=103. 2. Real-time spin and charge dynamics We apply an instantaneous quench by a change of the parameters Ei d→Ef d,bi→bf, and/Gamma1i→/Gamma1f=/Gamma10att=0. Since the hybridization strength /Gamma1f=/Gamma10is the same in all cases, all energies are given in units of /Gamma10. We investigated two different quench scenarios: We either (i) keep the impurity hybridization constant, i.e., /Gamma1i=/Gamma1f or (ii) we switch on the hybridization at t=0. The initial low-energy fixed points of both scenarios are fundamentallydifferent. The first case corresponds to the conventional low-energy fixed points of the SIAM [ 58] for the parameter choice ofU,E d, and b, while in the second scenario we start from the unstable local moment fixed point where the impurity isdecoupled from the conduction band continuum. In both cases, we leave Uconstant and only quench E dand the magnetic field b. Initially, we set bi=/Gamma10to induce a spin polarization and switch off the magnetic field at t=0. We also start with a degeneracy of the spin-up impurity state andthe unoccupied state by setting E i d−bi/2=0. For scenario (ii), the spin-down state is initially completely depopulatedson d(0)=0.5, and the local spin polarization Szis fixed to Sz(0)=1/4. For scenario (i), the initial occupation and spin polarization depend on the ratio U//Gamma10. Att=0, we quench the level position to Ef d=−U/2 and switch off the magnetic field, bf=0. Therefore, the thermodynamic low-energy fixed point of Hfis the same for all values of Uand both scenarios: the particle-hole symmetric strong coupling fixed point. In Fig. 15, the dynamics of the impurity occupancy nd(t), nd(t)=/angbracketleftd† ↑d↑+d† ↓d↓/angbracketright(t), and the dynamics of the spin polarization Sz(t), Sz(t)=1 2/angbracketleftd† ↑d↑−d† ↓d↓/angbracketright(t), 075149-19JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) FIG. 16. Comparison of NS=300 (dashed lines) to NS=1000 (solid lines) different states for the SIAM regarding the impurity occupation nd(t) and the spin polarization Sz(t). are plotted as a function of time. The data for five different values of Uare shown using our hybrid OC approach. Since the number of states increases by a factor of 4 in each NRG iteration, 3 /4 of the states are discarded at the end of each iteration in the NRG algorithm. Hence, the number ofmatrix elements of the Bloch-Redfield tensors is substantiallylarger than in the RLM case and the numerical costs of theLanczos approach for coupling the diagonal density matrixelements become very high. While the standard TD-NRG re-quires around two minutes on today’s desktop computers, theOC approach for each of the curves presented in Fig. 15took about three days on a workstation node utilizing all 16 cores. The effect of choosing different numbers of kept states N Safter each NRG iteration is demonstrated for the SIAM in Fig. 16. We supplement the data for Fig. 15shown as solid lines with NS=300 states (dashed lines) for the same quench parameters. Obviously, the differences are very small,suggesting the choice of N S=1000 states to be perfectly sufficient for our purpose. The charge relaxation and the spin relaxation occur on different timescales [ 50], as can already be seen in Fig. 15. While the charge relaxation occurs on the scale set by /Gamma1f= /Gamma10, the spin decay time shows significant Udependency. The equilibrium energy scale that governs the crossover from thelocal moment fixed point into the strong coupling fixed pointis the Kondo temperature T K. This parameter is a measure for the temperature at which the local magnetic moment is already70% screened [ 64]. To investigate the spin dynamics in more detail, we plotted theS z(t) data shown in Fig. 15versus the dimensionless times tTandtTKin Fig. 17, where Tis the system temperature. For scenario (ii)—top right panel—we find a very gooduniversality of the long-time behavior of S z(t). This scenario starts from the local moment fixed point with a decoupledimpurity and approaches the symmetric strong coupling fixedpoint and, therefore, partially tracks a thermodynamic flow.The dynamics is clearly governed by the Kondo scale forlarge Kondo temperatures where T/lessmuchT K. The long-timeFIG. 17. Sz(t) data taken from Fig. 15for/Gamma1i=0 at the top and /Gamma1i=/Gamma10at the bottom. The time is scaled by the system temperature Ton the left and by the respective Kondo temperature TK(which depends on U) on the right. tails of the U//Gamma10=2,4,5,8 curves show universality. Since TK(U//Gamma10=8)=0.046/Gamma10, we start to see deviations since the temperature is T//Gamma10=0.01 in all simulations. For U=16/Gamma10, the system temperature T≈4.2TKis clearly above the Kondo temperature. The top left panel of Fig. 15suggests that the relevant decay scale is set by the thermal fluctuations forT>T K,a sSz(t) decays on the scale of 1 /T. For scenario (i), depicted in the two lower panels of Fig. 15, the Kondo temperature does not provide such a universalscaling. The characteristic decay time is of the order of T Kfor temperatures T/lessmuchTKbut it depends on the initial preparation of the system. Upon increasing the relative temperature T/TK, the thermal fluctuations start to dominate the decay time as inscenario (ii). As a further indication for the correctness of the TD-NRG results, an analytic solution will be used for case (i). Thedynamics of the density operator is calculated up to secondorder in the impurity coupling function.This solution is onlyvalid on short timescales and becomes asymptotically exactin the limit t→0. The calculation requires a numerical evaluation at finite temperature but in the limit of T→0w e arrive at the compact analytical expression n d(t)=1 2+2B(t,U/2),Sz(t)=1 4−B(t,−U/2),(92) with B(t,/epsilon1)=/Gamma10t πSi(ωt)+/Gamma10 ωcos(ωt)−1 π/vextendsingle/vextendsingle/vextendsingle/vextendsingleD−/epsilon1 ω=−/epsilon1, (93) and Si( /epsilon1) being the sine integral. The full calculations can be found in Appendix C. The OC (solid line) and CC (dotted line) numerical data for the change of the time-dependent spin (orange) and charge(blue) expectation values are compared to the analyticalcurves (dashed lines) for U//Gamma1 0=2 and D//Gamma10=20 in Fig. 18. The CC (TD-NRG) agrees perfectly with the analytics fortimes t/Gamma1 0<0.1. Here, the deviation of the OC solution from both curves is clearly visible, but this effect is exaggerated 075149-20RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) FIG. 18. Impurity occupation nd(t)a n ds p i n Sz(t) of the open chain (OC) compared to the closed chain (CC, dotted) and theanalytical solution (a.s., dashed) according to Eq. ( 92) plotted on a double logarithmic scale. by the double logarithmic plot. The OC and the CC approach merge on timescales 0 .1<t/Gamma10. The initial deviations are a generic feature of the Bloch-Redfield formalism where short-time quantum correlations are ignored due to the factorisationunder the integral. VI. CONCLUSION We presented a hybrid approach to the nonequilibrium dynamics in QISs which combines the strength of the NRGand the strength of weak coupling approaches for openquantum systems to restore the original continuum problem.The continuous fraction expansion of the coupling functionbetween the quantum impurity and the environment yields aHamiltonian representation of the original problem decom-posed into a discrete Wilson chain and a set of high-energyadditional reservoirs, each coupled to a single Wilson chainsite. These reservoirs represent the high-energy modes of theoriginal coupling function that only couple indirectly to thequantum impurity. Therefore, the standard NRG is definedas an approximation which neglects the coupling to the ad-ditional reservoirs. A different discretized representation of a QIS augmented with a Lindblad dynamics was previously considered [ 41,42] in the context of nonequilibrium quantum transport. Theirapproach treats the Lindblad coupling tensor elements as fit-ting parameters that are determined by a variational approach.In our method, we are able to analytically construct theexact coupling functions to the additional reservoirs that arerequired to recover the original continuous coupling functionof the problem. Since the NRG has been established as an excellent tool [58] for the equilibrium problem, we propose to augment the TD-NRG with a perturbative Bloch-Redfield treatment of thecoupling to the auxiliary reservoirs. We modified the standardBloch-Redfield approach [ 24] derived for the full density ma- trix of a finite size system: The approach is applied to the setof reduced density matrices that are required for the dynamicsof local observables at and around the quantum impurity to handle the huge amount of discarded states generated by theNRG truncation. The fourth-rank Bloch-Redfield tensor isevaluated exactly from the analytically constructed couplingfunctions to the additional reservoirs within a Wilson shelland for the coupling between the diagonal matrix elementsof the density matrix of adjacent shells. We used the genericscaling properties of the matrix elements to substitute thecumbersome exact enumeration by a simplified analyticalform for the tensor elements connecting states of Wilsonshells that are far apart from each other. This is justifiedsince the matrix elements decay exponentially with the shelldistance |m 1−m2|and their precise value does not affect the steady state solution of the master equation. It turns out to be crucial that all diagonal matrix elements of the reduced density matrices of all energy shells are coupled.We have proven that the steady state of the approach is theNRG thermal equilibrium value for a hybrid system coupledto reservoirs that share a common chemical potential. Adifferent current-carrying steady state can be achieved in atwo-lead setup with different chemical potentials [ 41,42]. This will be the subject of a further publication. We used the known analytic solution of the RLM [ 51] as a benchmark for the proposed hybrid approach and foundexcellent agreement between the analytical and the numericalcurves. A comparison of real-time dynamics between the TD-NRG and the OC hybrid approach was presented for two non-trivial strongly correlated models: the IRLM and the SIAM.In all cases, our hybrid approach significantly reduced thefinite-size oscillations as well as removing the slight deviationbetween the nonequilibrium steady-state expectation valuesand the NRG thermal equilibrium values. Our hybrid approach has the potential to be extended in two ways: (i) adding leads with different chemical potentialsand numerically calculating a current carrying steady state inthe strong coupling limit and (ii) deriving a similar approachfor the NRG spectral functions to remove the necessity foran artificial broadening [ 58] and replacing it by the physical processes included in the original continuum model prior tothe discretization. APPENDIX A: RELATION BETWEEN THE FIRST WILSON-CHAIN PARAMETER t0AND THE CONTINUOUS FRACTION COUPLING PARAMETER V0 Below we will show, that the zeroth reservoir of the OC, which represents the start of our reservoir algorithm, is suffi-cient for a Wilson chain parameter t 0of any /Lambda1> 1. Inserting Eq. ( 16) into Eq. ( 17) yields πV2 0=V2/integraldisplay∞ −∞dωIm/Delta1(ω) Re/Delta1(ω)2+Im/Delta1(ω)2 =2D π/integraldisplayD −Ddωπ2 4artanh2(ω/D)+π2=π 3D2.(A1) Since t2 0=D2 4(1−/Lambda1−1)(1+/Lambda1−1)2 1−/Lambda1−3, (A2) the inequality V0>t0follows for any /Lambda1> 1. 075149-21JAN BÖKER AND FRITHJOF B. ANDERS PHYSICAL REVIEW B 102, 075149 (2020) APPENDIX B: DERIV ATION OF THE BLOCH REDFIELD APPROACH The dynamics of the density operator ρI(t) is governed by the differential equation ∂tρI(t)=i[ρI(t),VI(t)] (B1) in the interaction picture, where the system-reservoir coupling takes the form VI(t)=eiH0tHI(N)e−iH0t(B2) and ρI(t)=eiH0tρ(t)e−iH0t. (B3) Here the operators are transformed by H0=HNRG N+Hres(N). For expectation values of local operators, it is sufficient to know the local density operator ρS(t)=TrR[ρI(t)] where we have traced out all the reservoir DOFs. This operator is actingonly on the Wilson chain or system S, respectively. Now Eq. ( B1) can be adapted to derive a Bloch-Redfield equation for the reduced density matrix ρ S(t) by integrating the equation ρI(t)=ρI(0)+i/integraldisplayt 0dt/prime[ρI(t/prime),VI(t/prime)] (B4) and substituting the resulting ρI(t) back into the differential equation. The expression ∂tρI(t)=i[ρI(0),VI(t)]−/integraldisplayt 0dt/prime[[ρI(t/prime),VI(t/prime)],VI(t)] (B5) is obtained which is used to derive the dynamics of the local density operator ∂tρS(t)=−/integraldisplayt 0dt/primeTrR[[[ρS(t/prime)ρR,VI(t/prime)],VI(t)]] (B6)after tracing out all reservoir DOFs. This operator is acting only on the DOF of the Wilson chain. The first term of ther.h.s of Eq. ( B5) vanishes due to particle number conservation. To derive the dynamics of the reduced density operator, the weak coupling approximation [ 24] is employed and the full density operator ρ I(τ)≈ρS(τ)ρRis factorized, where ρRdenotes the equilibrium density operator of the reservoir which remains unaltered by the coupling to the Wilson chain. The bath coupling functions /Gamma1νm(/epsilon1) derived in Sec. II C enter the expression for the greater and lesser reservoir GF foreach reservoir [ 92]. The lesser or particle GF, G < ν,˜m(t,t/prime)=i|t/prime ν˜m|2TrR[ρRc† 0ν˜m(t)c0ν˜m(t/prime)] =i/integraldisplay∞ −∞d/epsilon1/Gamma1H ν˜m(/epsilon1) πf(/epsilon1)ei/epsilon1τ =G< ν,˜m(τ)=G<∗ ν,˜m(−τ), (B7a) and the greater or hole GF, G> ν,˜m(t,t/prime)=−i|t/prime ν˜m|2TrR[ρRc0ν˜m(t)c† 0ν˜m(t/prime)] =−i/integraldisplay∞ −∞d/epsilon1/Gamma1H ν˜m(/epsilon1) πf(−/epsilon1)e−i/epsilon1τ =G> ν,˜m(τ)=G>∗ ν,˜m(−τ), (B7b) only depend on the time difference τ=t−t/primein equilibrium and fully determine the effect of the reservoirs onto thedynamics on the Wilson chain. Their Fourier transformationsare defined as G >(<)(ω)=/integraldisplay∞ −∞dte−iωtG>(<) ν,˜m(t). (B8) The reduced density operator ρS(t) obeys the time-local differential equation ∂tρS(t)=−iN/summationdisplay ˜m=0/summationdisplay ν/integraldisplayt 0dτρS(t)[f† ν˜m(t−τ)fν˜m(t)G> ν,˜m(−τ)−fν˜m(t−τ)f† ν˜m(t)G< ν,˜m(−τ)] +iN/summationdisplay ˜m=0/summationdisplay ν/integraldisplayt 0dτ[f† ν˜m(t−τ)ρS(t)fν˜m(t)G> ν,˜m(−τ)−fν˜m(t−τ)ρS(t)f† ν˜m(t)G< ν,˜m(−τ)] +iN/summationdisplay ˜m=0/summationdisplay ν/integraldisplayt 0dτ[f† ν˜m(t)ρS(t)fν˜m(t−τ)G> ν,˜m(τ)−fν˜m(t)ρS(t)f† ν˜m(t−τ)G< ν,˜m(τ)] −iN/summationdisplay ˜m=0/summationdisplay ν/integraldisplayt 0dτ[f† ν˜m(t)fν˜m(t−τ)G> ν,˜m(τ)−fν˜m(t)f† ν˜m(t−τ)G< ν,˜m(τ)]ρS(t)( B 9 ) after substituting the explicit form of VI(t) into Eq. ( B6) and making use of the Markov approximation [ 24]: For fast decaying correlation functions G> ν,m(τ),G< ν,m(τ) relative to the change of ρS(t), one can replace ρS(t−τ)→ρS(t) under the integral, converting the integro-differential equation intoa master equation for ρ S(t) and neglecting retardation effects. This approximation is the origin of the deviation between theanalytical solution and the OC approach in Fig. 18for very short times. By calculating the trace on both sides of Eq. ( B9), one obtains ∂tTr[ρS(t)]=0, since for each reservoir GF a pair of terms can be found which cancel each other out. Thus, thederived differential equation conserves the trace of the densityoperator at all times. 075149-22RESTORING THE CONTINUUM LIMIT IN THE TIME- … PHYSICAL REVIEW B 102, 075149 (2020) Conservation of the trace under the restriction m2∈{m1−1,m1,m1+1}(Eq. ( 52) has been used): N/summationdisplay m1=mmin/summationdisplay l1˙ρred l1,l1(m1;t)=N/summationdisplay m1=mminm1+1/lessorequalslantN/summationdisplay m2=m1−1/greaterorequalslantmmin/summationdisplay l1,l2/parenleftbig /Xi1l2,l1(m2,m1)ρred l2,l2(m2;t)−/Xi1l1,l2(m1,m2)ρred l1,l1(m1;t)/parenrightbig . (B10) The two sums are interconvertible, so the trace is conserved. APPENDIX C: ANALYTICAL SOLUTION TO SHORT-TIME DYNAMICS IN THE SIAM When Eq. ( B5) is integrated over time and then inserted into the time-dependent expectation value of any local operator O, we obtain /angbracketleftO(t)/angbracketright=Tr{ρ0OI(t)}+/angbracketleft O/prime(t)/angbracketright,/angbracketleftO/prime(t)/angbracketright≈−/integraldisplayt 0dτ1/integraldisplayτ1 0dτ2Tr{ρ0[HI(τ2),[HI(τ1),OI(t)]]} (C1) after replacing the full dynamics of the density operator by its initial values in the step from line one to line two. This is asymptotically exact for t→0 and defines a second-order approximation in the impurity bath coupling function. Here HI(τ)=/summationdisplay k,σVk(c† kσ(τ)dσ(τ)+ckσ(τ)d† σ(τ)) (C2) is the term for the interaction of the impurity level and the bath excitations. The operators in the interaction representation read dσ(t)=|0/angbracketright/angbracketleftσ|e−i/epsilon1dt−σ|−σ/angbracketright/angbracketleft2|e−i(/epsilon1d+U)t, (C3) ckσ(t)=ckσe−i/epsilon1kt, (C4) where |0/angbracketrightis the vacuum state on the impurity, |2/angbracketrightthe double occupied state, and |σ/angbracketrightaccounts for either spin state ↑or↓.T h e density matrix ρ0factorizes for the interaction quench. We chose the parameter E0 d=b0/2=/Gamma10/2 in the Hamiltonian Eq. ( 91) fort<0. Inserting Eq. ( C2) into Eq. ( C1) and evaluating the double commutators using the initial density matrix ρ0, we obtain /angbracketleftO/prime(t)/angbracketright=2 Z/summationdisplay k,σV2 kA(/epsilon1k−/epsilon1d,t)·[f(/epsilon1k)e−βE0(/angbracketleftσ/angbracketright−/angbracketleft 0/angbracketright)+f(−/epsilon1k)e−βEσ(/angbracketleft0/angbracketright−/angbracketleftσ/angbracketright)] +2 Z/summationdisplay k,σV2 kA(/epsilon1k−/epsilon1d−U,t)·[f(/epsilon1k)e−βEσ(/angbracketleft2/angbracketright−/angbracketleftσ/angbracketright)+f(−/epsilon1k)e−βE2(/angbracketleftσ/angbracketright−/angbracketleft 2/angbracketright)], (C5) where we have used the shortcut notations /angbracketlefts/angbracketright=/angbracketleft s|O|s/angbracketright,s∈{0,↑,↓,2}andA(/epsilon1,t)=1−cos(/epsilon1t) /epsilon12. For a constant hybridization function [see Eq. ( 35)] and applying the low-temperature limit, Eq. ( C5) can be transformed to O/prime(t)=(/angbracketleft↑/angbracketright + /angbracketleft↓/angbracketright − 2/angbracketleft0/angbracketright)B−D,0(t,/epsilon1d)+(/angbracketleft0/angbracketright−/angbracketleft ↑ /angbracketright )B0,D(t,/epsilon1d)+(/angbracketleft2/angbracketright−/angbracketleft ↑ /angbracketright )B−D,0(t,/epsilon1d+U). (C6) The integration can be done in an exact manner with Ba,b(t,/epsilon1/prime)=/Gamma10 π/integraldisplayb ad/epsilon1A(/epsilon1−/epsilon1/prime,t)=/Gamma10t πSi((/epsilon1−/epsilon1/prime)t)+/Gamma10 /epsilon1−/epsilon1/primecos((/epsilon1−/epsilon1/prime)t)−1 π/vextendsingle/vextendsingle/vextendsingle/vextendsingleb /epsilon1=a, (C7) where Si( /epsilon1) is the sine integral. 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PhysRevB.82.035317.pdf

Finite-momentum condensation in a pumped microcavity R. T. Brierley1and P. R. Eastham1,2 1Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom 2School of Physics, Trinity College, Dublin 2, Ireland /H20849Received 7 December 2009; revised manuscript received 5 July 2010; published 28 July 2010 /H20850 We calculate the absorption spectra of a semiconductor microcavity into which a nonequilibrium exciton population has been pumped. We predict strong peaks in the spectrum corresponding to collective modesanalogous to the Cooper modes in superconductors and fermionic atomic gases. These modes can becomeunstable, leading to the formation of off-equilibrium quantum condensates. We calculate a phase diagram forcondensation and show that the dominant instabilities can be at a finite momentum. Thus we predict theformation of inhomogeneous condensates, similar to Fulde-Ferrel-Larkin-Ovchinnikov states. DOI: 10.1103/PhysRevB.82.035317 PACS number /H20849s/H20850: 03.75.Kk, 71.36. /H11001c, 71.35.Lk, 78.67.Hc I. INTRODUCTION The appearance of order at an equilibrium phase transition is a central concept in many areas of physics, from con-densed matter to the physics of the early universe. Recentlythere has been considerable interest in the more generalproblem of ordering far from thermal equilibrium, motivatedby the possibility of quantum quench experiments in coldatomic gases. 1In a quench the parameters of the system are rapidly switched from a disordered to an ordered phase andthe disordered state forms the initial conditions for a dynam-ics with the new parameters. An interesting regime is that ofcoherent relaxationless dynamics, which can lead to the for-mation of nonequilibrium order including crystallization, 2 condensation, and ferromagnetism.3 Among condensed matter systems, semiconductor micro- cavities are promising candidates for studying such quenchdynamics. The nonequilibrium dynamics of microcavitieshas attracted considerable interest both experimentally 4–9and theoretically10–14with recent experiments demonstrating re- gimes where the low-energy quasiparticles, polaritons, forma condensate. More recently, an experiment has been pro-posed to implement a quantum quench, 15by rapidly prepar- ing a microcavity in a noncondensed initial state. The coher-ent dynamics of this noncondensed state is predicted to leadto a form of nonequilibrium condensation, similar to thatpredicted in a quenched Fermi gas. 16–18We show here that, as in the Fermi gas, such condensation is due to the appear-ance of a new collective mode. Moreover, we show that inthe microcavity the dominant instability occurs at a finitewave vector. Thus we predict that microcavities could beused to realize inhomogeneous condensates, 19i.e., those characterized by a spatially varying phase. These conden-sates are similar, in this essential respect, to those predictedby Fulde, Ferrel, Larkin, and Ovchinnikov /H20849FFLO /H20850in unbal- anced Fermi systems. 19 In this paper, we first calculate the optical spectra of a microcavity a short time after it has been prepared in a non-condensed state, i.e., immediately after the “quench.” Wefind that the collective mode responsible for condensation isdirectly observable in these spectra. We use this analysis tocalculate a phase diagram for the nonequilibrium condensa-tion and show that the condensation generally occurs at afinite momentum. While we focus on a microcavity contain- ing quantum dots, our analysis is based on the Maxwell-Bloch equations. These describe a wide variety of coupledlight-matter systems, implying a broad relevance of ourwork. The remainder of this paper is structured as follows. In Sec. IIwe briefly review the proposed quench experiment and outline our model. In Sec. IIIwe present absorption spectra of the system. In Sec. IVwe discuss the phase dia- gram and the possibility of finite momentum condensation,and in Sec. Vwe discuss the connections to FFLO and the role of nonlinear terms. Section VIsummarizes our con- clusions. Finally, the appendix contains a brief treatment ofthe preparation of noncondensed initial states by opticalpumping. II. MODEL We consider an experiment, proposed in Ref. 15,o nas e t of localized exciton states in a planar semiconductor micro-cavity. Such excitonic states could be realized in practiceusing either highly disordered quantum wells /H20849where exci- tons are localized by disorder /H20850or quantum dots. The pro- posed experiment involves two stages which are separated intime and can be regarded as independent. In the first stage,the localized states are driven by a chirped laser pulse. Thispulse creates an energy-dependent population in the inhomo-geneously broadened exciton line by adiabatic rapid passage/H20849ARP /H20850. For certain populations a second stage may then oc- cur, where the population evolves into a nonequilibrium con-densate due to the photon-mediated interactions between theexcitons. As in Ref. 15we describe the system using a generaliza- tion of the Dicke model. 20The localized exciton states are treated as two-level systems with the standard dipole cou-pling to the electromagnetic field. The state localized at site i is specified by the Bloch vector /H9268i=/H20855/H9268ˆi/H20856, where the inversion /H9268iz=1/H20849−1/H20850for an occupied /H20849unoccupied /H20850state, and /H9268ˆi−is the exciton annihilation operator. Angle brackets /H20855/H20856denote ex- pectation values in the quantum state of the system. We consider time scales short compared with the exciton lifetime, which is at least 100 ps,21and treat the electromag- netic field using a mean-field approximation. In this approxi-PHYSICAL REVIEW B 82, 035317 /H208492010 /H20850 1098-0121/2010/82 /H208493/H20850/035317 /H208497/H20850 ©2010 The American Physical Society 035317-1mation the photon creation and annihilation operators are replaced with their expectation values and hence become c numbers. The resulting equations of motion are linear in theremaining operators so that we may take their expectationvalues without further approximation. The resulting dynam-ics obeys the generalized Maxwell-Bloch equations i /H9274˙k=/H9275k/H9274k+g/H20885PkdE+fk+F/H9254k−p, /H208491/H20850 iP˙k=EPk−g/H20858 k/H11032Dk−k/H11032/H9274k/H11032, /H208492/H20850 iD˙k=2g/H20858 k/H11032/H20849Pk/H11032−k/H11569/H9274k/H11032−Pk/H11032+k/H9274k/H11032/H11569/H20850. /H208493/H20850 Here/H9274kis the complex amplitude of the microcavity mode with in-plane wave vector kand energy /H9275k/H20849/H6036=1/H20850.I ti sr e - lated to the expectation value of the photon annihilation op- erator by /H9274k=/H20855/H9274ˆk/H20856//H20881N, where Nis the total number of local- ized states. This normalization is convenient when dealingwith condensation since macroscopic occupation corre-sponds to a finite /H9274kin the thermodynamic limit N→/H11009.W e allow for the finite lifetime of the photon modes by takingI /H9275k=−/H9253.fkis introduced to allow us to calculate the linear response. Fis an externally applied pump field, with wave vector p, that is used to create the nonequilibrium popula- tion. The coupling g=gi/H20881nin Eqs. /H208491/H20850–/H208493/H20850is related to the di- pole coupling strengths of the localized states, gi, and their area density n. To simplify the notation we have taken gito be the same for all states; the extension to a distribution isstraightforward. In the dipole gauge g i=d/H20881Ei 2/H92800/H9280w, /H208494/H20850 where dis the matrix element of the dipole operator erˆbe- tween the zero-exciton and one-exciton states, and Eitheir energy difference. wis the effective width of the cavity, which arises from the normalization of the cavity mode func-tions. P k/H20849E/H20850is the collective polarization of the ensemble at wave vector kdue to states with energy in a small interval near E Pk/H20849E/H20850/H9254E=1 N/H20858 i/H11032/H20855/H9268ˆi−/H20856e−ik·ri, /H208495/H20850 where the prime indicates that the sum runs over states with exciton energies between Eand/H9254E.Dk/H20849E/H20850is the collective inversion, defined in a similar way with /H9268ˆzreplacing /H9268ˆ−. In the following we shall be concerned with large Nand wave vectors which are small compared with the inversespacing of the localized states. In these limits we may ap-proximate sums over dot positions, such as those in Eq. /H208495/H20850, by1 N/H20858 i/H11032e−ik·ri/H11015/H9263/H20849E/H20850/H9254k,0/H9254E+O/H208731 /H20881N/H20874. /H208496/H20850 Here/H9263/H20849E/H20850is the distribution of localized states in energy, normalized to one. Thus N/H9263/H20849E/H20850/H9254Eis the number of terms in the primed sum, Eq. /H208496/H20850. For k/HS110050 the phasor sum is a two- dimensional random walk, producing the O/H208491//H20881N/H20850 corrections.22 The approximation of Eq. /H208496/H20850corresponds to replacing the response of the disordered dielectric with its homogeneousaverage response so that the wave vector is well-defined.This is similar to the linear dispersion model which has been extensively used for inorganic microcavities. 23TheO/H208491//H20881N/H20850 corrections describe Rayleigh scattering from density fluc-tuations in the dielectric. They are generally small correc-tions because the scatterers are dense so that on long wave-lengths the medium appears homogeneous. The correctionscan become important at very small or large wavevectors, 24,25for modes whose group velocity becomes very small. In this case a long lifetime is required for the wavevector to be well-defined so that even weak scattering orabsorption destroys the quasipropagating modes. Here, how-ever, we are concerned with modes that have a significantdispersion due to their photon component. Furthermore, thelifetime of these modes is massively enhanced by resonantgain from the populated excitons. Thus the leading approxi-mation of Eq. /H208496/H20850will capture the physics at the experimen- tally relevant wave vectors. III. ABSORPTION SPECTRA In the experiment Fis a chirped pulse, which creates a nonequilibrium population of excitons using ARP. In the Ap-pendix we demonstrate this explicitly, using a model pulse,Eq. /H20849A5/H20850, for which analytical solutions to the dynamics ex- ist. Following the pulse the exciton states are populated witha distribution given by Eq. /H20849A6/H20850and the fields and polariza- tions are negligible /H9274k/H110150,Pk/H110150. To establish the optical properties of the microcavity im- mediately after the pump pulse we find the response to aweak probe f k. The susceptibility can then be found from the induced electromagnetic field /H9254/H9274k/H11013/H20858 k/H11032/H20848/H9273k,k/H11032/H20849t −t/H11032/H20850fk/H11032/H20849t/H11032/H20850dt/H11032. If the system is stable then /H9274kand Pkare small /H20849of order fk/H20850for all times whereas if it is unstable they are only small soon after the pump pulse. In both regimes wemay neglect terms above first order in /H9274kandPk. Equation /H208493/H20850then gives D˙k=0 so the nonequilibrium population is constant. Fourier transforming the linearized Eqs. /H208491/H20850and /H208492/H20850 gives /H9275/H9254/H9274k=/H9275k/H9254/H9274k−g2/H20858 k/H11032/H20885Dk−k/H11032/H9254/H9274k/H11032 /H9275−EdE+fk. /H208497/H20850 The pumping populates the states independently of their position, within the pump spot. Thus the sum in Dk−k/H11032is strongly peaked near the forward scattering direction k−k/H11032 =0, as discussed above /H20851Eq. /H208496/H20850/H20852. Neglecting the smaller off- diagonal scattering terms we obtain a diagonal responsefunctionR. T. BRIERLEY AND P. R. EASTHAM PHYSICAL REVIEW B 82, 035317 /H208492010 /H20850 035317-2/H9273k/H20849/H9275/H20850=1 /H9275−/H9275k+g2/H20885D0/H20849E/H20850 /H9275−EdE. /H208498/H20850 The absorption coefficient of the microcavity follows from the susceptibility26,27 A/H20849/H9275/H20850=−2 l i m /H9280→0Im/H9273/H20849/H9275+i/H9280/H20850, /H208499/H20850 where the infinitesimal /H9280appears due to causality and can be physically understood as a small damping constant for theexcitons. The sign is such that A/H20849 /H9275/H20850/H110220 corresponds to ab- sorption of energy by the system. Thus from Eq. /H208498/H20850we obtain A/H20849/H9275/H20850=2/H9253−g2/H9266D0/H20849/H9275/H20850 /H20873/H9275−/H9275c+g2P/H20885D0 /H9275−EdE/H208742 +/H20851/H9253−g2/H9266D0/H20849/H9275/H20850/H208522 =2H/H20849/H9275/H20850 G/H20849/H9275/H208502+H/H20849/H9275/H208502. /H2084910/H20850 When the dots are unoccupied D0/H20849/H9275/H20850/H110210 and the empty ex- citon states contribute to absorption. For energies wherethere are occupied exciton states D 0/H20849/H9275/H20850/H110220, describing gain due to the population. In general, the response, Eq. /H2084910/H20850, peaks near the zeroes of G/H20849/H9275/H20850, which are at the energies of the normal modes. These energies differ from the energy of the cavity resonance dueto the coupling to the exciton states. For an unpopulated statethe condition G/H20849 /H9275/H20850=0 recovers the usual polariton energies of the Lorentz oscillator model28but, in general, the spec- trum differs due to the presence of the nonequilibrium popu-lation. The modes have a lifetime determined by the secondfactor in the denominator with contributions from the cavitylosses and the resonant mixing with the band of excitonstates. As expected it is damping which controls the overallstrength of the absorption so that the damping factor H/H20849 /H9275/H20850 also appears in the numerator. The only dependence of the spectra, Eqs. /H208498/H20850and /H2084910/H20850,o n the wave vector is in the energy of the cavity mode, /H9275c =R/H20849/H9275k/H20850/H11015/H92750+/H20841k/H208412//H208492m/H20850. We therefore show results as func- tions of /H9275c, which corresponds experimentally to both the incident probe angle and the cavity width. Figure 1illustrates absorption spectra obtained from Eq. /H2084910/H20850for both a pumped and unpumped exciton line. These spectra are valid at all times if condensation does not occur/H20849see later /H20850but only soon after the pump pulse if it does. We have taken a Gaussian model for the inhomogeneouslybroadened exciton line with standard deviation /H9268, and mea- sure energies relative to the center of the line. We choose theduration /H9270of the pump pulse as our unit of energy and have taken g=13 //H9270,/H9253=1.5 //H9270,/H9268=15 //H9270. These parameters, with /H9270 =3 ps, are reasonable for a microcavity containing strongly disordered quantum wells.29As discussed in the appendix the pump creates a population equivalent to a Fermi functionwith temperature 1 //H20849 /H9266kB/H9270/H20850, and Fermi energy /H9262dictated by the chirp and center frequency of the pulse; we choose apulse for which /H9262=−12.5 //H9270.The lower panel of Fig. 1shows the expected result for an unpopulated microcavity. There is a pronounced peak in theabsorption at the cavity mode energy, which broadens as thecavity mode is tuned through the excitons. There is somesuggestion of an anticrossing near resonance, i.e., a polaritonsplitting but since the inhomogeneous broadening is rela-tively large compared with the coupling this is a weak effect.The top panel shows that the population dramaticallychanges the absorption spectrum. For these parameters itleads to a range of probe frequencies for which /H9253 /H11021g2/H9266D0/H20849/H9275/H20850, and the absorption coefficient, Eq. /H2084910/H20850, be- comes negative. This occurs when the gain from the popu-lated exciton states exceeds the losses so that there is a netgain for the probe beam. Moreover, we see a pronouncedadditional peak in the absorption spectrum, which first ap-pears near the upper edge of the population as the cavitymode energy is decreased. As the cavity energy is decreased−80 −40 0 40ωprobeτA(ω) −70−45−20530 ωcτ (a) −80 −40 0 40ωprobeτA(ω) −70−45−20530 ωcτ (b) FIG. 1. Absorption spectra /H20851Eq. /H2084910/H20850/H20852in arbitrary units, for the parameters described in the text. Top panel: spectra with the excitonpopulation created by the pump pulse. Bottom panel: spectra of theunpopulated microcavity. Each curve corresponds to a differentvalue of the cavity mode energy /H9275c, vertically offset as indicated by the right-hand axes. Note the peak developing in the populatedsystem at /H9275probe /H11015−13 //H9270, indicating the presence of a collective mode. There are regions of probe gain /H20851A/H20849/H9275/H20850/H110210/H20852between −31 /H11021/H9275probe/H9270/H11021−13, where gain from the populated excitons over- comes the cavity losses. Dotted lines indicate spectra with unstablenormal modes. Instabilities occur when a normal mode lies in theregion of gain and in these cases the absorption spectra have nega-tive peaks, which are hidden from view in this figure.FINITE-MOMENTUM CONDENSATION IN A PUMPED … PHYSICAL REVIEW B 82, 035317 /H208492010 /H20850 035317-3still further this peak moves down through the region of gain, before the spectrum finally reverts to one dominated by theunperturbed cavity mode. This additional peak in the absorption spectrum is analo- gous to the Cooper pairing mode in a superconductor orFermi gas, that gives rise to the Cooper instability. The anal-ogy can be seen by noting that the normal-mode conditionG/H20849 /H9275/H20850=0 contained in Eq. /H2084910/H20850is the Cooper equation, as discussed for this system in Ref. 15. The nonequilibrium exciton population corresponds to the Fermi distributionwhile the photon-mediated interaction between excitons cor-responds to the pairing interaction between the electrons. Asin a superconductor the sharp step in the population leads tocollective modes generated by the pairing interaction. Figure1shows that, for reasonable parameters, these collective modes give rise to strong features in the spectra. It is interesting to compare the spectra of Fig. 1with the predictions for an equilibrium condensate in the samemodel. 27In that case the condensation opens a gap in the single-particle spectrum, which is the analog of the Coopergap of the superconductor. Inside this gap is a collectivemode, which is the analog of the Cooper mode or phasemode of the superconductor. The features visible in Fig. 1 arise from the nonequilibrium generalization of the collectivemode /H20849which is a different spectral feature than the gap /H20850.I ti s clear from Fig. 1that it is the collective mode which domi- nates the spectrum. Thus, although the single-particle fea-tures may be affected by condensation, this would have littleeffect in practice. It may be possible to isolate the singleparticle spectrum in a Rayleigh scattering experiment, as hasbeen proposed for equilibrium condensates. 30 IV. PHASE DIAGRAM The normal modes of the system, with frequencies deter- mined by G/H20849/H9275/H20850=0, have decay rates H/H20849/H9275/H20850. If a normal mode frequency lies in the H/H20849/H9275/H20850/H110210 region produced by the non- equilibrium population it will be unstable, growing exponen-tially to give a state with a highly populated mode, i.e., acondensate. The condition for the onset of such an instabilitygives a nonequilibrium phase diagram, which is shown forour chosen parameters in Figs. 2and3. Figure 2shows the phase diagram assuming that only a single cavity mode, of energy /H9275c, is relevant. The dotted line shows the phase boundary for equilibrium condensation inthe same model 27with a temperature and chemical potential corresponding to the pumped population. We see that onesheet of the nonequilibrium phase boundary extends theequilibrium result to allow for the cavity damping. Whereasin equilibrium the presence of the collective mode is suffi-cient to create an instability, in the open system condensationonly occurs if the gain at the energy of the collective modeovercomes the cavity loss. Thus the collective mode can ex-ist even in the normal state /H20849see Fig. 1/H20850and the damping pushes the transition to larger couplings. In addition, we seethat there is a lower limit on /H9275cin Fig. 2. This lower thresh- old is a purely dynamical effect, not present in the equilib-rium case. Below it there is a bosonic collective mode at anenergy well below that of the populated states /H20849see Fig. 1/H20850.Although this mode would be occupied in equilibrium it is far out of resonance with the excitons. As a result, it is notoccupied dynamically and the uncondensed state is meta-stable. A similar metastable region has been predicted inquenched atomic gases. 17 Figures 1and2show that, for a given gand/H9253, the con- densation instability occurs over a range of /H9275c. In a micro- cavity different values of /H9275c=/H92750+/H20841k/H208412/2mcorrespond to ei- ther changing the cavity width, which varies the detuning /H92750, or considering modes at a different wave vector k. As such, a range of unstable /H9275cimplies that for a fixed cavity detuning there can be instabilities at many wave vectors with differentgrowth exponents /H20841H/H20849 /H9275/H20850/H20841. At short times after the population has been created the mode with the highest growth exponentwill dominate. Figure 4shows that as /H92750is lowered this dominant mode occurs at k/HS110050, implying a condensate with finite momentum, and a spatially inhomogeneous order pa-rameter. Thus the full phase diagram, allowing for the con-tinuum of in-plane modes, takes the form shown in Fig. 3. The phase diagram of Fig. 3can be understood physically by noting that the condensation is a result of the exciton-Condensed γNormal 0 10 20 30 40 50 60-40-20020406080100ωc gττ FIG. 2. Nonequilibrium phase diagram for a populated micro- cavity with a single photon mode as a function of coupling strengthgand cavity detuning /H9275cwith cavity damping /H9253/H9270 =1, 15, 30, 45. Arrow indicates curves of increasing /H9253. Dotted line indicates the location of the equilibrium phase boundary /H20849Ref. 27/H20850with temperature and chemical potential corresponding to that of the pumped population /H20851T=1 //H20849kB/H9266/H9270/H20850,/H9262=−12.5 //H9270/H20852. Normal Homogeneous Inhomogeneous 0 5 10 15 20/Minus20/Minus1001020 gΤΩ0Τ FIG. 3. Nonequilibrium phase diagram for a populated micro- cavity with a continuum of in-plane photon modes as a function ofthe coupling constant gand detuning at k=0, /H92750. The cavity damp- ing/H9253=1.5 //H9270. In the inhomogeneous region modes with k/HS110050 have the highest growth exponent. The vertical boundary is a result of thegain-loss criterion, H/H20849 /H9275/H20850/H110220.R. T. BRIERLEY AND P. R. EASTHAM PHYSICAL REVIEW B 82, 035317 /H208492010 /H20850 035317-4photon interactions. If the cavity mode is detuned well below the excitons then the quasiparticles at k=0 are essentially photons, uncoupled from the excitons. Thus the condensationshifts to the higher momentum states, where the photons andexcitons are closer to resonance, and there are strong cou-pling effects. V. DISCUSSION There has been extensive theoretical work on states with finite momentum Cooper pairing in the context of equilib-rium superconductors, atomic gases, and quark matter. 19 These FFLO states, which have been sought in a wide rangeof systems, may be the ground states where there is an im-balance in the populations of the two pairing species. How-ever, they involve increasing the kinetic energy in order togain pairing energy and in practice this restricts them tosmall regions of parameter space. Here, however, the stateachieved is determined by the Cooper equation and a gain-loss criterion with the energetics playing a subsidiary role.Thus, as indicated by Fig. 3, condensation at a finite momen- tum may be achieved without fine tuning of parameters. The connection to FFLO may be made more explicit us- ing a representation for the exciton operator /H9268ˆi−=cˆi,↑cˆi,↓, where cˆis a fermionic annihilation operator. In the simplest case of a plane wave condensate at wave vector kthe mean- field order parameter is Pk, the macroscopic component of the exciton polarization. This becomes Pk=/H208491/N/H20850/H20858 i/H20855cˆi,↑cˆi,↓/H20856e−ik.ri =1 /N2/H20858 p,q/H20855cˆp,↑cˆq,↓/H20856/H20858 iei/H20849p+q/H20850.ri−ik.ri /H11011/H208491/N/H20850/H20858 p/H20855cˆp,↑cˆ−p−k,↓/H20856. /H2084911/H20850 Thus we see that the condensate can be formally represented as a coherent state of fermions, pairing with a finite totalmomentum. While in this respect the state is similar toFFLO, there are other differences. For example, in Eq. /H2084911/H20850the relative wave function of the pair is independent of mo- mentum and the pairing is entirely local. In a general FFLOstate there is a momentum-dependent pairing function, de-scribing Cooper pairs of finite size. Because the growth exponent depends only on /H20841k/H20841the condensate emission at short times will cover a circle ofin-plane wave vectors, giving a cone of emitted light. How-ever at later times the nonlinear terms neglected in Eq. /H208497/H20850 will break the degeneracy, selecting a spatial form for thecondensate. In equilibrium such interactions favor conden-sate structures consisting of a pair of antipodal wave vectors/H20849k,−k/H20850, or more complex structures such as face-centered cubes. 19Here the nonlinearity corresponds to the depletion of the exciton population by the growth of the condensate.This will reduce the gain 13for collective modes of similar energies, suggesting that a single plane wave /H20849Fulde-Ferrel /H20850 state may be favored. Although these nonlinearities deter-mine a particular form for the condensate it is unlikely theywill lead to a homogeneous state so we do not treat them indetail here. It is interesting to note that finite-momentum polariton condensates have been observed 4,5,9though in a different ex- perimental protocol to that considered here. In these casesthere is continuous pumping and relaxation and a spatialstructure imposed by a pump and trap. The mechanisms lead-ing to this finite momentum condensate have yet to be estab-lished and are likely different from those discussed here.Nonetheless, these experiments demonstrate that microcavi-ties could support exotic ordered states that have proved elu-sive in equilibrium. VI. SUMMARY We have calculated the linear response of a microcavity with a nonequilibrium population of excitons. The popula-tion produces new collective modes, which are analogs ofthe Cooper pairing mode in superconductors. We have shownthat these modes are visible as peaks in the optical spectra.By considering the growth exponents of these collectivemodes we have found a phase diagram for the dynamicalcondensation. In a microcavity with a continuum of in-planewave vectors there can be multiple unstable modes of differ-ent wave vectors. For some parameters the dominant /H20849and, for sufficiently negative detuning, only /H20850instabilities can oc- cur at a nonzero wave vector. In these regimes the microcav-ity will develop a condensate with spatial structure, signaledby coherent emission at an angle to the cavity normal. ACKNOWLEDGMENTS This work was supported by Science Foundation Ireland under Grant No. 09/SIRG/I1592 and EPSRC under GrantNo. EP/F040075/1. We thank P. B. Littlewood, J. Keeling,and M. Parish for discussions and comments on the manu-script. APPENDIX: ANALYTICAL PUMP SOLUTION Reference 15gives the results of numerical simulations of Eqs. /H208491/H20850–/H208493/H20850, driven by a linearly chirped Gaussian pump-100 -80 -60 -40 -20 0 20 ω0τ024681012k/(2mph/τ)1/2 012345678 FIG. 4. Growth exponents − H/H20849/H9275/H20850of the unstable modes with wave vector modulus kin a populated cavity with detuning /H92750, coupling strength g=13 //H9270, and decay constant /H9253=1.5 //H9270. The wave vector of the most unstable mode increases as /H92750is reduced. For /H9275c/H11351−30 //H9270thek=0 mode is stable and there are only finite mo- mentum instabilities.FINITE-MOMENTUM CONDENSATION IN A PUMPED … PHYSICAL REVIEW B 82, 035317 /H208492010 /H20850 035317-5pulse. These simulations show that there are parameter re- gimes in which the dynamics separates into a fast pumpingstage, followed by a slower condensation stage. In this ap-pendix we present an approximate analytical solution to Eqs./H208491/H20850–/H208493/H20850which gives the population and /H20849negligible /H20850polariza- tion at the end of the pump pulse. This solution forms thestarting point for the dynamics discussed in the body of thepaper. The full numerical solutions in Ref. 15show that the only significant polarization during the pumping is at the pumpwave vector p. Moreover, this polarization can be seen to be small compared with the applied pump field F. Thus during the pumping we may neglect the second term in Eq. /H208491/H20850for all wave vectors. With this approximation, Eqs. /H208491/H20850–/H208493/H20850re- duce to an ensemble of independent two-level systems, driven by a field /H9274p0which is the externally applied field F filtered by the cavity response. For pumping at high angles, outside the stop band of the mirrors, /H9274p0is proportional to the pump pulse. Thus Eqs. /H208491/H20850–/H208493/H20850become the Bloch equations /H20898P˙ x/H11032 P˙ y/H11032 D˙0/H20899=/H208980 E−/H9004/H20849t/H20850 0 −E+/H9004/H20849t/H20850 0 g/H9024/H20849t/H20850 0 −g/H9024/H20849t/H20850 0/H20899/H20898Px/H11032 Py/H11032 D0/H20899, /H20849A1/H20850 where we have defined /H9274p0=/H9024/H20849t/H20850e−i/H20848t/H9004/H20849t/H20850dt/H11032, /H20849A2/H20850 P/H11032=Ppei/H20848t/H9004/H20849t/H20850dt/H11032, /H20849A3/H20850 P/H11032=1 2/H20849Px−iPy/H20850. /H20849A4/H20850 Note that the collective polarizations are at the pump wave vector while the collective inversion is spatially uniform. For a model pump pulse of the form g/H9024/H20849t/H20850=/H90240 /H9270secht−t0 /H9270,/H9004/H20849t/H20850=/H9251 /H9270tanht−t0 /H9270+/H92630, /H20849A5/H20850 Equation /H20849A1/H20850has an analytical solution.31The form of the population D0at times t/H11271/H9270after the pulse is D0/H20849E/H20850 /H9263/H20849E/H20850=2cosh2/H9266/H9251 2− cos2/H9266/H20881/H902402−/H92512 2 cosh/H20877/H9266 2/H20851/H20849E−/H92630/H20850/H9270−/H9251/H20852/H20878cosh/H20877/H9266 2/H20851/H20849E−/H92630/H20850/H9270+/H9251/H20852/H20878 −1 . /H20849A6/H20850 In the limit /H90240/H11022/H9251/H112711 the distribution becomes D0/H20849E/H20850 /H9263/H20849E/H20850=2nF/H20849E−/H9262+−/H92630/H20850/H208511−nF/H20849E−/H9262−−/H92630/H20850/H20852−1 , /H20849A7/H20850 where nF/H20849E/H20850is a Fermi distribution with temperature 1//H20849kB/H9266/H9270/H20850and the chemical potentials /H9262/H11006=/H92630/H11006/H9251 /H9270. If the den- sity of states /H9263/H20849E/H20850is sufficiently small at energies below /H9262− then this lower edge is irrelevant. The occupation function D0/H20849E/H20850is then equivalent to an equilibrium Fermi distribution with/H9262=/H9262+. In this paper we consider parameters where this applies, choosing /H92630=−30 //H9270,/H90240=18, and /H9251=17.5. Since the dynamics during the pumping, Eq. /H20849A1/H20850, in- volves only Pp, the polarization at any other wave vector Pk/HS11005premains zero. For the polarization at the pump wave vector, the analytical solution gives a window of energies /H11011/H9270 in which there is a nonzero polarization after pumping. How- ever, in the absence of an external field and with /H9274p/H110150, as is the case after pumping, the subsequent evolution of the po-larization is free. As a result, the total polarization P p =/H20848Pp/H20849E/H20850dEdecays by free induction decay and so may be neglected after a time of order /H9270. 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PhysRevB.78.092506.pdf

Muon spin rotation studies of SmFeAsO 0.85and NdFeAsO 0.85superconductors Rustem Khasanov,1,*Hubertus Luetkens,1Alex Amato,1Hans-Henning Klauss,2Zhi-An Ren,3Jie Yang,3Wei Lu,3and Zhong-Xian Zhao3 1Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 2IFP , TU Dresden, D-01069 Dresden, Germany 3National Laboratory for Superconductivity, Institute of Physics and Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, P .O. Box 603, Beijing 100190, People’ s Republic of China /H20849Received 13 August 2008; published 29 September 2008 /H20850 Measurements of the in-plane magnetic-field penetration depth /H9261abin Fe-based superconductors with the nominal composition SmFeAsO 0.85/H20849Tc/H1122952 K /H20850and NdFeAsO 0.85/H20849Tc/H1122951 K /H20850were carried out by means of muon-spin rotation. The absolute values of /H9261abatT=0 were found to be 189 /H208495/H20850and 195 /H208495/H20850nm for Sm and Nd substituted samples, respectively. The analysis of the magnetic penetration depth data within the Uemuraclassification scheme, which considers the correlation between the superconducting transition temperature T c and the effective Fermi temperature TF, reveals that both families of Fe-based superconductors /H20849with and without fluorine /H20850fall to the same class of unconventional superconductors. DOI: 10.1103/PhysRevB.78.092506 PACS number /H20849s/H20850: 74.70. /H11002b, 76.75. /H11001i The recent discovery of the Fe-based layered supercon- ductor LaO 1−xFxFeAs /H20849Ref.1/H20850with the transition temperature Tc=26 K has triggered an intense research in the oxypnic- tides. In its wake, a series of superconductors with Tconset of up to 55 K have been synthesized successively by substi-tuting La with other rare-earth /H20849Re/H20850ions such as Sm, Ce, Nd, Pr, and Gd. 2,3Recently the family of the oxypnictide super- conductors ReFeAsO 1−xwith the doping induced by oxygen vacancies instead of fluorine substitution were synthesized.4,5 One of the questions, which awaits to be explored, is to which class of the superconducting materials the discoveredFe-based superconductors belong. The search for relationsbetween the various physical variables such as transitiontemperature, magnetic-field penetration depth, electrical con-ductivity, energy gap, Fermi temperature, etc. may help toanswer this question. Among others, there is a correlationbetween T cand the zero-temperature inverse squared magnetic-field penetration depth /H20851/H9261−2/H208490/H20850/H20852that generally re- lates to the zero-temperature superfluid density /H20849/H9267s/H20850in terms of/H9267s/H11008/H9261−2/H208490/H20850. In various families of underdoped high- temperature cuprate superconductors /H20849HTSs /H20850, there is the empirical relation Tc/H11008/H9267s/H11008/H9261−2/H208490/H20850, first identified by Uemura and co-workers.6,7In this respect it is rather remarkable that the magnetic-field penetration measurementson LaO 1−xFxFeAs /H20849x=0.1 and 0.075 /H20850/H20849Ref. 8/H20850and SmO 0.82F0.18FeAs /H20849Ref. 9/H20850result in values of the superfluid density, which are close to the Uemura line for hole dopedcuprates, indicating that the superfluid is also very dilute inthe oxypnictides. In this Brief Reports we focus on the different classifica- tion scheme proposed by Uemura and co-workers, 7,10which considers the correlation between Tcand the effective Fermi temperature TFdetermined from measurements of the in- plane magnetic penetration depth /H9261ab. Within this scheme strongly correlated unconventional superconductors, such asHTSs, heavy fermions, Chevrel phases, or organic supercon-ductors, form a common but distinct group, characterized bya universal scaling of T cwith TFsuch that 1 /10/H11022Tc/TF /H110221/100. We show that within the Uemura classification scheme both families of oxypnictide superconductors /H20849withand without fluorine /H20850fall to the same class of unconventional superconductors. Details on the sample preparation for SmFeAsO 0.85/H20849Tc /H1122952 K /H20850and NdFeAsO 0.85 /H20849Tc/H1122951 K /H20850can be found elsewhere.5Both samples studied in the present work were in the form of sintered powders. Zero-field /H20849ZF/H20850, longitudinal- field /H20849LF/H20850, and transverse-field /H20849TF/H20850muon-spin rotation /H20849/H9262SR/H20850experiments were performed at the /H9266M3 beam line at the Paul Scherrer Institute /H20849Villigen, Switzerland /H20850. Here LF and TF denote the cases when the magnetic field is appliedparallel /H20849LF/H20850and perpendicular /H20849TF/H20850to the initial muon-spin polarization. ZF and LF /H9262SR measurements are used to probe the intrinsic magnetic properties of a sample. WhileZF /H9262SR provides information on the internal magnetic-field distribution, the complementary application of LF measure-ments allows discrimination between static and fluctuatingfields. In TF /H9262SR measurements, the additional relaxation due to the inhomogeneous internal field distribution in thevortex phase of the type-II superconductors allows to extract,e.g., the absolute value of the magnetic penetration depth.For detailed description of ZF, LF, and TF /H9262SR techniques, see Ref. 11. During the experiments we were mostly concen- trated on SmFeAsO 0.85, which shows the highest Tcamong other oxypnictide superconductors discovered until now. ForNdFeAsO 0.85we studied only the temperature dependence of the superfluid density in a field of 0.2 T. First we present the results of ZF and LF /H9262SR experi- ments on SmFeAsO 0.85. The recent ZF /H9262SR studies of the parent LaOFeAs compound reveal that there are two intersti-tial lattice sites where muons come to the rest, namely, closeto the Fe magnetic moments within the FeAs layers and nearthe LaO planes. 12Therefore, the ZF and the LF muon-time spectra for T/H1135180 K were analyzed by using the following depolarization function: PZF,LF/H20849t/H20850=Aslowexp/H20849−/H9011slowt/H20850+Afastexp/H20849−/H9011fastt/H20850./H208491/H20850 Here Aslow/H20849Afast/H20850and/H9011slow/H20849/H9011fast/H20850are the asymmetry and the depolarization rate of the slow /H20849fast/H20850relaxing component, respectively. The whole set of ZF /H20849LF/H20850data was fitted simul-PHYSICAL REVIEW B 78, 092506 /H208492008/H20850 1098-0121/2008/78 /H208499/H20850/092506 /H208494/H20850 ©2008 The American Physical Society 092506-1taneously with the ratio Aslow /Afastas a common parameter and the relaxations /H20849/H9011slowand/H9011fast/H20850as individual parameters for each particular data point. The total asymmetry Aslow +Afastwas kept constant within each set of the data /H20849ZF or LF/H20850. Above 80 K the fit is statistically compatible with a single exponential component only. The results of the analy-sis and the representative ZF and LF muon-time spectra areshown in Fig. 1. From the data presented in Fig. 1the following important points emerge: /H20849i/H20850Both/H9011 fast/H20849T/H20850and/H9011slow/H20849T/H20850measured in the zero and the longitudinal /H20849up to 0.6 T /H20850fields coincide within almost the whole temperature region. This, together with theexponential character of the muon polarization decay, revealsthe existence of fast electronic fluctuations measurablewithin the /H9262SR time window. Since no sign of static or dy- namic magnetism has been observed in optimally dopedLaO 0.9F0.1FeAs,8we attribute the observed relaxation to be due to fluctuations of the Sm magnetic moments. Assumingthe fluctuation rate with a temperature dependence /H9263/H11011exp/H20849E0/kBT/H20850/H20849E0is the activation energy /H20850and accounting for the saturation of /H9011/H11229/H90110at 10 K /H11351T/H1135135 K, the relax- ation is expected to follow:13 1 /H9011=1 /H90110+1 Cexp/H20849E0/kBT/H20850. /H208492/H20850 The fit of Eq. /H208492/H20850to the experimental ZF /H9011fastdata yields /H90110,fast=2.38 /H208496/H20850/H9262s−1,C=0.012 /H208492/H20850/H9262s, and E0=23/H208492/H20850meV.This kind of activated process can be anticipated for a ther- mal population of Sm crystal-field levels. /H20849ii/H20850The fact that both the slow and the fast relaxation rates exhibit similartemperature dependences /H20851see Fig. 1/H20849b/H20850/H20852strongly suggests that there is a common source for both relaxations. The mag-nitudes of /H9011 fastand/H9011sloware thus related to the different couplings between muons and Sm moments at the distinctmuon stopping sites. /H20849iii/H20850There are no features appearing in the vicinity of the superconducting transition. Figure 1/H20849b/H20850 implies that /H9011 fastincreases continuously with decreasing temperature. This may suggest that the magnetic fluctuationsresponsible for the effects seen in Fig. 1are not related to superconductivity. It is worth mentioning that, in systemsexhibiting an interplay between the superconductivity andmagnetism, the slowing down of the spin fluctuations /H20849in- crease in /H9011/H20850correlates with T c/H20849see, e.g., Ref. 14/H20850. Another argument comes from the comparison of /H9011fast/H20849T/H20850for SmFeAsO 0.85/H20849Tc/H1122952 K /H20850, studied here, with that reported by Drew et al.9for SmO 0.82F0.18FeAs /H20849Tc/H1122945 K /H20850/H20851see Fig. 1/H20849b/H20850/H20852. Apparently the Sm spin fluctuations are independent of Tc./H20849iv/H20850The fast increase in both /H9011fastand/H9011slowbelow 5 K is most probably associated with additional local-field broaden-ing due to the ordering of the Sm moments. 9,15 The superconducting properties of SmFeAsO 0.85and NdFeAsO 0.85were studied in the TF /H9262SR experiments. The temperature scans were made after cooling the samples fromabove T cdown to 1.7 K in /H92620H=0.2 T. Following Hayano et al.16it can be shown that the effect of fast fluctuations on the longitudinal and transverse depolarizations becomessimilar. By taking this into account the TF /H9262SR data were analyzed by using the following functional form: PTF/H20849t/H20850=PLF/H20849t/H20850exp/H20849−/H92682t2/2/H20850cos/H20849/H9253/H9262Bt+/H9278/H20850. /H208493/H20850 Here Bis the average field inside the sample, /H9253/H9262=2/H9266 /H11003135.5342 MHz /T is the muon gyromagnetic ratio, /H9278is the initial phase, /H9268is the Gaussian relaxation rate, and PLF/H20849t/H20850 is described by Eq. /H208491/H20850./H9268vsTdependences for SmFeAsO 0.85 and NdFeAsO 0.85are shown in the inset of Fig. 2. As it statesFIG. 1. /H20849Color online /H20850/H20849a/H20850Temperature dependences of the fast /H20849/H9011fast/H20850and the slow /H20849/H9011slow/H20850components of the ZF and the LF muon depolarization rates of SmFeAsO 0.85./H20849b/H20850/H9011fast/H20849T/H20850and/H9011slow/H20849T/H20850nor- malized to their values at T=17 K. Open triangles represents the normalized data of SmO 0.82F0.18FeAs from Ref. 9. The inset in /H20849a/H20850 shows the ZF and the LF /H9262SR time spectra of SmFeAsO 0.85mea- sured at T=15 K. The solid lines in /H20849a/H20850and/H20849b/H20850are the fits of Eq./H208492/H20850to ZF/H9011fast/H20849T/H20850. σ µ σµ FIG. 2. /H20849Color online /H20850Temperature dependences of /H9268sc/H11008/H9261ab−2for SmFeAsO 0.85and NdFeAsO 0.85measured after field cooling the samples in /H92620H=0.2 T. The inset shows the Gaussian depolariza- tion rate /H9268.BRIEF REPORTS PHYSICAL REVIEW B 78, 092506 /H208492008/H20850 092506-2in Ref. 17the Gaussian relaxation rate /H9268for a highly aniso- tropic type-II superconductor obeying, in addition, low-temperature magnetic ordering consists of a superconducting /H20849 /H9268sc/H11008/H9261ab−2/H20850, a magnetic /H20849/H9268m/H20850, and a small nuclear magnetic- dipole /H20849/H9268nm/H20850contribution. We may conclude that, therefore, a continuous increase in /H9268below Tcrelates to appearance of superconductivity while the sharp rise of /H9268at low tempera- tures is due to extra Sm /H20849Refs. 9and15/H20850and Nd /H20849Ref. 18/H20850 orderings /H20851see inset of Fig. 2and ZF /H9011fast/H20849T/H20850data in Fig. 1/H20849a/H20850/H20852. Because /H9268mis only present at low temperatures, data points belo w 5 K for SmFeAsO 0.85 and 10 K for NdFeAsO 0.85were excluded in the analysis. The supercon- ducting contribution /H20849/H9268sc/H11008/H9261ab−2/H20850was then determined by sub- tracting in quadrature /H9268nmmeasured above Tcfrom/H9268and is presented in the main panel of Fig. 2. The data in Fig. 2were fitted with the power law /H9268sc/H20849T/H20850//H9268sc/H208490/H20850=1−/H20849T/Tc/H20850nwith/H9268sc/H208490/H20850,n, and Tcas free pa- rameters. The fit yields /H9268sc/H208490/H20850=1.73 /H208495/H20850/H9262s−1,Tc =52.0 /H208492/H20850K, and n=3.74 /H2084916/H20850, and/H9268sc/H208490/H20850=1.63 /H208495/H20850/H9262s−1,Tc =51.0 /H208493/H20850K, and n=1.98 /H2084914/H20850for SmFeAsO 0.85 and NdFeAsO 0.85, respectively. At the present stage we are not going to discuss the temperature dependences of /H9268sc.W e would only mention that the power-law exponentn=3.74 /H2084916/H20850is close to the universal two-fluid value n/H110134 while n=1.98 /H2084914/H20850is close to n=2, observed in a case of dirty d-wave superconductors. As is shown by Brandt, 19/H9268scfor anisotropic powder su- perconductor may be directly related to the in-plane mag-netic penetration depth /H9261 abvia /H9268sc2 /H9253/H92622= 0.00371/H902102 /H9261eff4= 0.00126/H902102 /H9261ab4, /H208494/H20850 where /H90210=2.068 /H1100310−15Wb is the magnetic-flux quantum. Here we also take into account that in anisotropic supercon-ductors, such as Fe-based oxypnictides, 20the effective mag- netic penetration depth /H9261eff, measured in /H9262SR experiments, is solely determined by the in-plane penetration depth as/H9261 eff=1.31/H9261ab.21From measured /H9268sc/H208490/H20850’s the absolute values of the in-plane magnetic penetration were found to be/H9261 ab/H208490/H20850=189 /H208495/H20850nm for SmFeAsO 0.85 and /H9261ab/H208490/H20850 =195 /H208495/H20850nm for NdFeAsO 0.85. The magnetic-field dependence of /H20849/H9268sc/H11008/H9261ab−2/H20850for SmFeAsO 0.85measured at T=15 K is shown in Fig. 3.A t low fields a maximum in /H9268sc/H20849H/H20850is observed followed by a decrease in the relaxation rate up to the highest fields. Con-sideration of the ideal triangular vortex lattice of an isotropics-wave superconductor within the Ginsburg-Landau ap- proach leads to the following expression for the magnetic-field dependence of the second moment of the magnetic-fielddistribution: 22 /H9268sc/H20851/H9262s−1/H20852= 4.83 /H11003104/H208491−B/Bc2/H20850 /H11003/H208511 + 1.21 /H208491−/H20881B/Bc2/H208503/H20852/H9261−2/H20851nm/H20852./H208495/H20850 Here Bis the magnetic induction, which for applied field in the region Hc1/H11270H/H11270Hc2isB/H11229/H92620H/H20849Hc1is the first critical field, and Bc2=/H92620Hc2is the upper critical field /H20850. According to Ref. 22, Eq. /H208495/H20850describes with less than 5% error the field variation of /H9268scfor an ideal triangular vortex lattice andholds for type-II superconductors with the value of the Ginzburg-Landau parameter /H9260/H113505 in the range of fields 0.25 //H92601.3/H11351B/Bc2/H113491. Since we are not aware of any re- ported values of the second critical field for SmFeAsO 0.85, we used Bc2/H2084915 K /H20850/H1122980 T obtained from Bc2/H208490/H20850/H11229100 T reported by Senatore et al.23for SmO 0.85F0.15FeAs. The black dotted line, derived by using Eq. /H208495/H20850, corresponds to /H9261ab=232 nm. Figure 3implies that the experimental /H9268sc/H20849H/H20850 depends stronger on the magnetic field than it is expected ina case of fully gaped s-wave superconductor. We have two possible explanations. /H20849i/H20850As shown by Amin et al. 24the field dependent correction to /H9267smay arise from the nonlocal and nonlinear responses of a superconductor with nodes in theenergy gap to the applied magnetic field. The solid line rep-resents the result of the fit by means of the relation: /H9267s/H20849H/H20850 /H9267s/H20849H=0/H20850=/H9268sc/H20849H/H20850 /H9268sc/H20849H=0/H20850=1− K·/H20881H, /H208496/H20850 which takes into account the nonlinear correction to /H9267sfor a superconductor with a d-wave energy gap.25Here the param- eter Kdepends on the strength of nonlinear effect. Since Eq./H208496/H20850is valid for the intermediate fields /H20849Hc1/H11270H/H11270Hc2/H20850, only the points above 0.02 T were considered in the analysis./H20849ii/H20850Strong dependence of /H9268scon magnetic field is observed in two-gap superconductors such as MgB 2/H20849Ref. 26/H20850and NbSe 2,27and is explained by the faster suppression of the contribution of the smaller gap to the total superfluid densitywith increasing field. Now we focus on the Uemura classification scheme, which considers the correlation between the superconductingtransition temperature T cand the effective Fermi temperature TFdetermined from measurements of the penetration depth. Using this parametrization, Uemura et al.7confirmed a close correlation between Tcand TF. HTSs, heavy fermion, or- ganic, fullerene, and Chevrel phase superconductors all fol-low a similar linear trend with 1 /100/H11021T c/TF/H110211/10, in con- trast to the conventional BCS superconductors /H20849Nb, Sn, Al,σ µ µ FIG. 3. /H20849Color online /H20850/H20849/H9268sc/H11008/H9261ab−2/H20850as a function of applied field H for SmFeAsO 0.85. Each point was obtained after field cooling the sample from above TctoT=15 K in the corresponding magnetic field. The solid and the dashed lines represent the results of analysisby means of Eqs. /H208495/H20850and/H208496/H20850.BRIEF REPORTS PHYSICAL REVIEW B 78, 092506 /H208492008/H20850 092506-3etc/H20850for which Tc/TF/H110211/1000. The “Uemura plot“ of log/H20849Tc/H20850vs log /H20849TF/H20850, shown in Fig. 4, thus appears to discrimi- nate between the “unconventional“ and “conventional“ su-perconductors. The T Brepresents the Bose-Einstein conden- sation temperature for a noninteracting three-dimensionalBose gas having the boson density n B=ns/2 and mass mB =2m/H11569. Intriguingly, all the unconventional superconductorsare found to have values of Tc/TBin the range of 1/3–1/30, thereby emphasizing the proximity of these systems to Bose-Einstein-like condensation. Following suggestions of Uemura et al. , 7the effective Fermi temperatures of the Fe-based superconductors werecalculated as k BTF=/H6036/H9266cintns mab/H11569/H11008cint/H9268sc. /H208497/H20850 Here ns/mab/H11569=/H9267sis the in-plane superfluid density, which within the London approach is proportional to /H9261ab−2and, thus, to/H9268sc/H20849ns/mab/H11569/H11008/H9261ab−2/H11008/H9268sc/H20850,nsis the charge-carrier concentra- tion, mab/H11569is the in-plane charge-carrier mass, and cintis the distance between the conducting planes. The points forSmFeAsO 0.85and NdFeAsO 0.85, and that obtained from /H9261ab/H208490/H20850values of LaO 1−xFxFeAs /H20849x=0.1 and 0.075 /H20850/H20849Ref. 8/H20850 and SmO 0.82F0.18FeAs /H20849Ref. 9/H20850are shown in Fig. 4by solid red stars. As is seen the Fe-based superconductors follow thesame linear trend as is established for various unconven-tional materials, suggesting that they all probably share thecommon condensation mechanism. To conclude, measurements of the in-plane magnetic-field penetration depth /H9261 abin superconductors SmFeAsO 0.85/H20849Tc /H1122952 K /H20850and NdFeAsO 0.85/H20849Tc/H1122951 K /H20850were carried out by means of muon-spin rotation. The absolute values of /H9261abat T=0 were estimated to be 189 /H208495/H20850and 195 /H208495/H20850nm for Sm and Nd substituted samples, respectively. The analysis within theUemura classification scheme, considering the correlationbetween the superconducting transition temperature T cand the effective Fermi temperature TF, reveal that both families of Fe-based superconductors /H20849with and without fluorine /H20850fall to the same class of unconventional superconductors. This work was performed at the Swiss Muon Source /H20849S/H9262S/H20850, Paul Scherrer Institute /H20849PSI, Switzerland /H20850. *rustem.khasanov@psi.ch 1Y. Kamihara et al. , J. Am. Chem. Soc. 130, 3296 /H208492008/H20850. 2Z.-A. Ren et al. , Chin. Phys. Lett. 25, 2215 /H208492008/H20850. 3Z.-A. Ren et al. , Mater. Res. Innovations 12,1/H208492008/H20850. 4J. Yang et al. , Supercond. Sci. Technol. 21, 082001 /H208492008/H20850. 5Z.-A. Ren et al. , Europhys. Lett. 83, 17002 /H208492008/H20850. 6Y. J. Uemura et al. , Phys. Rev. Lett. 62, 2317 /H208491989/H20850. 7Y. J. Uemura et al. , Phys. Rev. Lett. 66, 2665 /H208491991/H20850. 8H. Luetkens et al. , Phys. Rev. Lett. 101, 097009 /H208492008/H20850. 9A. J. Drew et al. , Phys. Rev. Lett. 101, 097010 /H208492008/H20850. 10A. D. Hillier et al. , Appl. Magn. Reson. 13,9 5/H208491997/H20850. 11A. Amato, Rev. Mod. Phys. 69, 1119 /H208491997/H20850. 12H. H. Klauss et al. , Phys. Rev. Lett. 101, 077005 /H208492008/H20850. 13T. Lancaster et al. , J. Phys.: Condens. Matter 16, S4563 /H208492004/H20850. 14A. Amato et al. , Physica B /H20849Amsterdam /H20850326, 369 /H208492003/H20850.15L. Ding et al. , Phys. Rev. B 77, 180510 /H20849R/H20850/H208492008/H20850. 16R. S. Hayano et al. , Phys. Rev. B 20, 850 /H208491979/H20850. 17R. Khasanov et al. , Phys. Rev. B 76, 094505 /H208492007/H20850. 18Y. Qiu et al. , arXiv:0806.2195 /H20849unpublished /H20850. 19E. H. Brandt, Phys. Rev. B 37, 2349 /H208491988/H20850. 20J. Jaroszynski et al. , Phys. Rev. B 78, 064511 /H208492008/H20850;S . Weyeneth et al. , arXiv:0806.1024 /H20849unpublished /H20850. 21V. I. Fesenko et al. , Physica C 176, 551 /H208491991/H20850. 22E. H. Brandt, Phys. Rev. B 68, 054506 /H208492003/H20850. 23C. Senatore et al. , Phys. Rev. B 78, 054514 /H208492008/H20850. 24M. H. S. Amin et al. , Phys. Rev. Lett. 84, 5864 /H208492000/H20850. 25I. Vekhter et al. , Phys. Rev. B 59, 1417 /H208491999/H20850. 26S. Serventi et al. , Phys. Rev. Lett. 93, 217003 /H208492004/H20850. 27J. Sonier et al. , Rev. Mod. Phys. 72, 769 /H208492000/H20850.FIG. 4. /H20849Color online /H20850The superconducting transition tempera- tureTcvs the effective Fermi temperature TF. The unconventional superconductors fall within a common band for which 1 /100 /H11021Tc/TF/H110211/10, as indicated by the gray region in the figure. The dashed line corresponds to the Bose-Einstein condensation tempera-ture T B/H20849see Ref. 7for details /H20850. The points for SmFeAsO 0.85and NdFeAsO 0.85calculated from /H9261ab/H208490/H20850values obtained in the present study and that for LaO 1−xFxFeAs /H20849x=0.1, 0.075 /H20850/H20849Ref. 8/H20850and SmO 0.82F0.18FeAs /H20849Ref. 9/H20850are shown by solid red stars.BRIEF REPORTS PHYSICAL REVIEW B 78, 092506 /H208492008/H20850 092506-4

PhysRevB.98.075135.pdf

PHYSICAL REVIEW B 98, 075135 (2018) Bond disproportionation, charge self-regulation, and ligand holes in s-p and in d-electron ABX 3perovskites by density functional theory G. M. Dalpian,1,2Q. Liu,1,*J. Varignon,3M. Bibes,3and Alex Zunger1 1Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA 2Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, 09210-580, Santo André, SP , Brazil 3Unité Mixte de Physique, CNRS, Thales, Université Paris Sud, Université Paris-Saclay, 91767 Palaiseau, France (Received 30 April 2018; revised manuscript received 2 July 2018; published 20 August 2018) Some ABX 3perovskites exhibit different local environments (DLE) for the same Batoms in the lattice, an effect referred to as disproportionation, distinguishing such compounds from common perovskites that havesingle local environments (SLE). The basic phenomenology associated with such disproportionation involves theabsence of B-atom charge ordering, the creation of different B-Xbond length (“bond alternation”) for different local environments, the appearance of metal (in SLE) to insulator (in DLE) transitions, and the formation of ligandholes. We point out that this phenomenology is common to a broad range of chemical bonding patterns in ABX 3 compounds, either with s-pelectron B-metal cations (BaBiO 3, CsTlF 3) or with noble-metal cations (CsAuCl 3), as well as with d-electron cations (SmNiO 3, CaFeO 3). We show that underlying much of this phenomenology is the “self-regulating response,” whereby in strongly bonded metal-ligand systems with high-lying ligand orbitals,the system protects itself from creating highly charged cations by transferring ligand electrons to the metal, thuspreserving a nearly constant metal charge in different local environments, while creating B-ligand bond alternation and ligand-like conduction band (“ligand hole” states). We are asking what are the minimal theory ingredientsneeded to explain the main features of this SLE-to-DLE phenomenology, such as its energetic driving force,bond length changes, possible modifications in charge density, and density of state changes. Using as a guidethe lowering of the total energy in DLE relative to SLE, we show that density functional calculations describethis phenomenology across the whole chemical bonding range without resort to special strong correlation effects,beyond what DFT naturally contains. In particular, lower total energy configurations (DLE) naturally developbond alternation, gapping of the metallic SLE state, and absence of charge ordering with ligand hole formation. DOI: 10.1103/PhysRevB.98.075135 I. INTRODUCTION: SINGLE VERSUS MULTIPLE LOCAL BONDING MOTIFS FOR THE SAME ELEMENT I NAC R Y S T A L Single repeated structural motifs—be they the AX 4tetra- hedron, AX 6octahedron, or A3B3Xtrigonal prism—have established the basis of our understanding of structure andbonding in solids and molecules [ 1–3]. Furthermore, the tradition of using in electronic structure calculations theeconomically smallest possible unit cell naturally forced insimple models the situation where each bonded element wasdescribed via a single local environment (SLE), the so-called monomorphous representation. Ionic solids were generally modeled by the NaCl structure, intermetallic compounds bythe L1 0CuAu–type structure, and ternary ABO3oxides via the cubic perovskite ( Pm¯3m) structure. This view also underlies the description of disordered AxB1−xalloys via the popular single-site coherent potential approximation approach (CPA)[4,5] where all Aatoms (and separately all Batoms) are assumed to see the same potential. At the same time, the existence of more than one in- equivalent Wyckoff position for identical elements in a latticeis no foreigner to crystallography. The classic example of *Present address: Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China.polymorphous structures manifesting different local envi- ronments for the same chemical element involves elements capable of existing in multiple valences. For example, columnIII elements with the configuration s 2p1have, at low atomic number ( Z), and for the top of the periodic table column (B, Al, Ga), a formal oxidation state (FOS) of 3, whereas for high atomic numbers, at the bottom of the column, the FOS might be 1 (e.g., Tl). The reason is that the relativistic mass-Darwineffect [ 6] is sufficiently large to localize the s 2electrons and make them quasi-core-like orbitals, leaving a single p1electron at the high- Zbottom of the periodic table column as chemically active. The intermediate- Zelements—In and in part Tl—have two stable valences. An analogous transition occurs in columnIV elements, where the light elements (Si, Ge) utilize all theirfour (s 2p2) valence electrons, whereas the high- Zelements (Pb) are mostly divalent ( p2) with the intermediate one (Sn) having two stable valences. In such dual-valence atoms a singlevalence would disproportionate into two different valences, asillustrated by the “negative- Ucenter” of In in PbTe [ 7], and by the dual valence of Sn in Cu 2ZnSnS 4[8]. A particularly interesting case of different local environ- ments to the same element is the disproportionation of apair of identical 2 /angbracketleftBX/angbracketrightsingle local environments (SLE) in anABX 3perovskite into a structure with two different local environments (DLE) associated with the same element B, i.e., /angbracketleftBX/angbracketright(1)+/angbracketleftBX/angbracketright(2)observed in A2[B(1)B(2)]X6perovskites 2469-9950/2018/98(7)/075135(13) 075135-1 ©2018 American Physical SocietyDALPIAN, LIU, V ARIGNON, BIBES, AND ZUNGER PHYSICAL REVIEW B 98, 075135 (2018) FIG. 1. Schematic representation of the total energy of a fixed- composition compound ABX 3appearing in a few hypothetical phases (α, β,γ, δ ) of different local environments versus the “local environ- ment descriptor” which can, for example, be the formal oxidationstate (FOS), or the coordination number, or the magnetic moment of the electronically active element. A phase below (a) or above (b) the tie line connecting the nearest-neighbor phases (dashed blue line) willmanifest in SLE and DLE behavior, respectively. (such as BaBiO 3[9,10], YNiO 3[11], CaFeO 3[12,13]). Whereas the appearance of two different local environments isubiquitous in double perovskites , when the disproportionating sites are different chemical elements as in A 2[BC]X6(for example Sr 2[FeMo]O 6[14] and Cs 2[AgIn]Cl6[15]), in the current paper we discuss the unusual case of disproportionationwith the same chemical element B. The existence of stable SLE ora stable DLE can be represented in a generalized “convex Hull” plot of total (free)energy of different phases versus some descriptor of the localenvironment (such as FOS or coordination number), as shownin Fig. 1. In the first case [Fig. 1(a)], atom Bin either the βor theγstate will not disproportionate because this would raise its energy relative to the “tie line” represented by the straightblue line and will stay as an SLE, whereas in the second case [Fig. 1(b)]a t o m Bin the γstate will disproportionate into β+δbecause this lowers its energy relative to the tie line. We will follow in the present work this total energy guide for thetendency of various systems to manifest SLE or DLE behavior. II. THE MAIN QUESTIONS ADDRESSED WITH RESPECT TO SLE VERSUS DLE IN ABX 3 We phrase below a number of questions posed regarding disproportionation in ABX 3compounds, and will address them in this paper by considering six compounds showing DLEbehavior, including BaBiO 3[9,10,16–20], CsTlF 3[21,22], SmNiO 3[11,23–26], CsAuCl 3[27], CaFeO 3[12,13,28–30], and CsTe 2O6[31]. Although the last compound is not a perovskite, we also studied it to show that the conclusionsdrawn here can also be expanded to other types of structures.This selection represents a rather broad range of perovskitecompounds with the Batom being an s-pelement (Bi, Tl, Te), or a transition metal (Ni, Fe) or noble metal (Au). We willdemonstrate common behavior for all such cases, highlightingthe broad appeal of SLE versus DLE selectivity.A. What level of electronic structure theory is sufficient to predict the energetic tendency (Fig. 1)o fa c t u a l ABX 3 compounds to be SLE or DLE? Previous studies implied that disproportionation is a cor- relation effect that may require an explicitly dynamicallycorrelated approach. For example, Park et al. [32] presented density functional plus dynamical mean field theory (DFT-DMFT) calculations “which show that the bond-length dispro-portionation and associated insulating behavior are signaturesof a novel correlation effect.” Cammarata et al. [33] proposed a “spin-assisted covalent bond formation” as a mechanism for DLE. However, thismodel cannot be general, since compounds like BaBiO 3, CsTlF 3,C s A u C l 3, and CsTe 2O6are not spin-polarized yet they have a DLE phase. Also, the spin-assisted mechanismcannot provide an understanding of the metal-insulator phasetransition in CaFeO 3and SmNiO 3since (i) they transit from a paramagnetic metal to a paramagnetic insulator not showingnet spin polarization and (ii) the Néel temperature is well belowthe metal-insulator transition (MIT). The magnetic interactionscannot therefore account for the MIT for these two compounds.In our case, DFT was able to predict both phenomena: wealways observe a decrease in energy when going from SLEto DLE, and we also concomitantly observe the opening ofa band gap for both spin-polarized and non-spin-polarizedconfigurations. We note another DFT explanation by Mercyet al. [26] who have recently provided compelling evidence that the octahedra rotations are triggering the MIT in nickelatesand CaFeO 3[29], reproducing the experimental observations. In the present study we find that SLE-DLE selectivity exists in s-pas well as 3 delectron compound alike, and that density functional theory (DFT) suffices to correctlydescribe the energetic selection between SLE and DLE inall such compounds. This establishes such single-determinant,mean field band theory as an adequate tool for describing thephenomenology related to bond disproportionation, includingmagnetism [ 25] and defects in disproportionated structures. B. Is the FOS a physically meaningful “local environment descriptor” for predicting within the convex hull construct of Fig. 1the tendency of actual ABX 3to be SLE or DLE? The most basic understanding of the disproportionation problem suggests that when DLE occurs, the elements locatedat theBsite in ABX 3perovskites will have different FOS. In this view the meaningful descriptor of the local environmentof the disproportionating Batom is the FOS. For BaBiO 3, for instance, the Bi site was said [ 10] to disproportionate to represent the two valences of the Bi atom, resulting inBa 2[Bi3+Bi5+]O6. The same view of charge ordering is often used for the other compounds, such as Cs 2[Au+1Au+3]Cl6, Ca2[Fe3+Fe5+]O6, and Cs 2[Te4+Te6+ 3]O12. In 1951, Frost [ 34] proposed a way to determine whether a specific FOS of a certain element is stable or not by plotting thefree energy versus the FOS [ 35,36] (a specific choice of a “local environment descriptor” in Fig. 1), and learning what would be expected for this specific compound. These graphs wereconstructed by using solutions and electrode potential freeenergies. This view that integer oxidation states are physicallyrealizable (as opposed to being formal labels) led to the 075135-2BOND DISPROPORTIONATION, CHARGE SELF- … PHYSICAL REVIEW B 98, 075135 (2018) picture equating such disproportionation to “charge ordering” [19,37,38] whereby the formal oxidation states correspond to physical charges, alternating on the different chemicallyidentical elements throughout the lattice. To establish whether a structural change such as SLE-to- DLE is associated with a change in charge distribution wecompute the quantity most directly related to our question,namely the variational charge density ρ(r) calculated self- consistently by DFT for DLE and for SLE geometries. Wetherefore calculated the charge accumulation function, i.e., thecharge enclosed in a sphere of radius Raround the Batom, as a function of R. Contrary to other methods of estimating the charge around an atom, such as Bader analysis, where afixed boundary is chosen, the charge accumulation functionis a direct measure of the charge density, providing directevidence of the charge distribution around a certain atom. Fromthese plots, it is also straightforward to clearly see the chargedensity difference around a certain atom. We clearly see thatthe physical charge density is essentially unchanged around theBatom as a result of the structural change. If one considers instead an indirect measure such as formal oxidation states, onededuces that it changes very significantly by the SLE-to-DLEtransformation. We conclude that the FOS has little or nothingto do with the physical charge density. The reason for thiswas discussed in detail in Ref. [ 39] in terms of the “charge self-regulating response,” whereby charge rearrangement onthe cation is offset by opposite rearrangement on the ligands,resulting in a minimal net change in physical charge density.In contrast, the FOS concept focuses just on the atom whosecharge is counted (namely, the Bcation in the present case), seeing therefore just a piece of the picture. Similar conclusionswere reached for the case of transition metal impurities insemiconductors [ 39], LiCoO 2vs CoO 2[40], and for Sn atoms in perovskites such as CsSnI 3and Cs 2SnI 6[41]. C. Is the bond geometry a physically meaningful descriptor for predicting within the convex hull construct of Fig. 1the tendency of actual ABX 3to be SLE or DLE? Once it is understood that the charge residing in a certain B atom is basically constant for different local environments, itis important to look for a more relevant descriptor for this dis-proportionation. X-ray techniques can precisely determine thedifference in bond lengths between the Batoms and the ligands. They can clearly differentiate the large and small octahedra inperovskites such as those studied in this paper [ 10]. While it is possible to assign different bond distances to different FOS[9], the fact that the physical charge residing on different B atoms is nearly identical suggests that this assignment doesnot reflect a causal mechanism. For example, PbCoO 3is said [42] to have both A-site and B-site charge ordering, leading to a formal description as [Pb2+Pb4+ 3][Co2+ 2Co3+ 2]O12. The char- acterization as a charge-ordering compound in this case camefrom x-ray measurements that show two groups of Pb-O bondsand two groups of Co-O bonds and not the direct measurementof any quantity related to charge. Our analysis indicates that itis best to use the bond geometries to differentiate these DLE,since this is the property that is usually measured, and not theFOS. We will show that DFT can predict the observed bonddisproportionation in all DLE compounds studied. Sawatzkyand collaborators reached a similar conclusion for BaBiO 3[17] and Varignon et al. for rare-earth nickelates [ 25]. We therefore use the term “bond disproportionation” rather than “chargeordering/disproportionation.” D. How is the SLE versus DLE selection related to metallic versus insulating character of the compound? It has been often observed that structural disproportionation comes with a simultaneous metal-insulator transition, e.g.,in rare-earth nickelates [ 11,26,32] and CaFeO 3[29]. For transition metal compounds, correlation effects have been usedto explain the metal-insulator transition [ 32]. We find in the standard DFT description for both s-pABX 3systems and ABX 3d-electron systems that whenever the SLE phase is metallic, the formation of the DLE configuration lowers thetotal energy [viz., Fig. 1(b)] and becomes automatically insu- lating. Specifically, in RNiO 3, the metal-insulator transition is developed by lattice mode couplings between rotations in DLErather than by pure correlation [ 26]. Thus the metal-insulator transition is an energetic consequence of disproportionation inthese systems. E. How is disproportionation related to ligand hole? The basic electronic structure [ 43,44] of metal oxides involves a valence band maximum (VBM) made either ofoxygen porbitals (in late transition metal oxides such as NiO) or from metal atom dorbitals (in early 3 doxide compounds such as VO 2[45]). The conduction band minimum (CBM) of such metal oxides is generally composed of transition metaldorbitals (in early transition atom oxides such as YTiO 3) or metal sorbitals (in late 3 doxides, CdO). A special case is when the CBM is made of ligand orbitals, called ligand hole states [ 17,18,25,29,43]. Ligand holes have been shown to exist in disproportionated systems [ 46] but there seems to be significant lack of clarity on whether they are intimatelyrelated to d-electron systems and whether they are specific to disproportionated systems. We demonstrate by DFT calculations that the conduction band wave function of both s-pelectron and d-electron dispro- portionated ABX 3compounds discussed here represents lig- and hole states. As to the mechanism of LH formation, we notethat this requires that the relevant metal states be deeper thanligand orbitals so electrons can be transferred to the metal (asin late but not early transition metal oxides, or Bi compoundswith low- selectron valence states) and that a sufficiently strong metal-to-ligand coupling exist so as to split the ligand VBinto occupied and unoccupied parts, the latter being LH. Thus,LH does not require disproportionation (indeed we find it toexist in SLE configurations), but in disproportionated statesthere exists a short enough B-ligand bond that could facilitate splitting of the valence band and LH formation. The basicdriving force for LH formation is the self-regulating response [39]: Total energy lowering favors the formation of LH when without such a LH, the charge on the metal would be verylarge (such as Ni 4+in RNiO 3or Bi5+in BaBiO 3), which is not favored energetically. Consequently, the ligand transferselectrons (thus, forming a hole) to the metal cation so as toself-regulate its charge (creating the [Ni 2+−O1−] complex in 075135-3DALPIAN, LIU, V ARIGNON, BIBES, AND ZUNGER PHYSICAL REVIEW B 98, 075135 (2018) FIG. 2. Geometrical representations of ABX 3crystal structures with SLE and DLE. (a) Orthorhombic structure with SLE, (b) cubicstructure with SLE, (c) monoclinic structure with DLE. In this case, the different colors of the octahedra indicate different local environments. RNiO 3and the [Bi3+−O1−] complex in BaBiO 3, where the hole is localized on the oxygen octahedra). In addition, ligandholes were recently discussed in organometallic systems whenstrong π−πinteraction splits the ligand πband so the upper π ∗band is unoccupied [ 47,48]. We next provide a detailed discussion of the five questions above, leading to the conclusion that the phenomenology ofdisproportionation—absence of charge ordering, formation ofbond length disproportionation, gap formation, as well asligand hole formation—is derived and detected by total energylowering within standard DFT, and is common to both s-p electron and d-electron ABX 3compounds. III. METHOD Density functional description of the SLE to DLE transfor- mation. With the advent of accurate first-principles exchange and correlation functionals [ 49] and effective energy mini- mization strategies (local gradients [ 50], minima hopping [ 51], global space group optimization [ 52], etc.), the possibility of affording larger-than-minimal supercells, which provide anopportunity for chemically identical atoms to develop theirown unique local environments, has arisen. Consequently, ithas become possible to simulate these kinds of SLE/DLEmaterials from a computational perspective, getting insightsinto the origin of these different configurations for the sameatom. Electronic Hamiltonian and its solver. Calculations were performed using the plane wave pseudopotential total energyDFT approach as implemented in the V ASP [ 53] code within the projected augmented wave (PAW) approach and the gener-alized gradient approximation (GGA-PBESol [ 54] for CaFeO 3 and SmNiO 3and GGA-PBE [ 55] for the other materials). We have also performed hybrid functional calculations [ 56]o n GGA-converged structures, in order to obtain total energies andband gaps. An on-site self-interaction correcting “DFT +U” [57] term was added to dorbitals of Ni ( U=2.0 eV) and Fe (U=3.8 eV). These values of Uwere chosen based on extensive tests from previous studies [ 25]. Basis set cutoff en- ergies were set to 600 eV for CaFeO 3and SmNiO 3and 400 eV for the other compounds. The Brillouin zones were sampledwithk-point meshes up to 8 ×8×6 for orthorhombic phases (20 atoms) and 6 ×6×6 for cubic ones (5 atoms).TABLE I. Calculated space group for the SLE and DLE com- pounds after DFT optimization. The experimentally observed config- urations are marked in bold. Material Space Group SLE Space Group DLE Reference BaBiO 3 Pnma P21/c [10] CsTlF 3 Pm¯3m Fm¯3m [21] CsAuCl 3 Pm¯3m I4/mmm [27] CsTe 2O6 Pnma R¯3m [31] SmNiO 3 Pnma P21/c [23] CaFeO 3 Pnma P21/c [13] Input crystal structures for relaxation. When available, we have used crystal structures reported in the ICSD [ 58] for our calculations, and optimized both lattice vectors andinternal atomic coordinates to minimize total energies untilthe forces on each atom for each Cartesian coordinate aresmaller than 0.001 eV /˚A. Such a relaxation scheme allows the system to change the symmetry of the trial structure.The most common structures observed in our SLE and DLEconfigurations are shown in Fig. 2. In the SLE case [Figs. 2(a) and2(b)], all octahedra have the same shape and size. In the DLE configuration [Fig. 2(c)] there are two different octahedra, arranged in such a way that a large octahedron is surroundedby six small octahedra and vice versa. Table Ireports the space group symmetries of our optimized structures for both SLEand DLE phases, together with the references for experimentalpapers reporting these structures. For CaFeO 3, both SLE and DLE configurations have been observed experimentally (so theconvex hull illustrated in Fig. 1must be rather shallow), the DLE being the low-temperature structure [ 13]. For this case we use as a trial structure the experimental crystal structure, witha ferromagnetic configuration for the spin arrangement, thenrelax the structure. For SmNiO 3only the SLE configuration has been observed [ 23], although our theoretical calculations show that the DLE configuration should be more stable. Inthis case, the initial DLE structure was copied from CaFeO 3, then fully relaxed. For other compounds (BaBiO 3,C s T l F 3, CsAuCl 3, and CsTe 2O6) only the DLE configuration has been experimentally identified so the trial SLE configuration wasbuilt by using either a cubic or an orthorhombic phase, inspiredin other perovskites, followed by full relaxation [ 59]. TABLE II. Energy difference between DLE and SLE phases (/Delta1E(DLE−SLE)) for the studied materials. The band gaps have been calculated within both the GGA and HSE (in parentheses) functionals for the DLE phase. Material /Delta1E(DLE−SLE) DLE Band Gaps (meV/f.u.) (GGA (HSE) eV) BaBiO 3 −107 0.00 (0.63) CsTlF 3 −40 0.91 (1.78) CsAuCl 3 −732 0.70 (1.51) CsTe 2O6 −136 0.38 (1.59) SmNiO 3 −82 0.05 (1.50) CaFeO 3 −4 0.09 (0.77) 075135-4BOND DISPROPORTIONATION, CHARGE SELF- … PHYSICAL REVIEW B 98, 075135 (2018) FIG. 3. Charge density profiles for the DLE phase of (a) BaBiO 3, (b) CsTlF 3, (c) CsAuCl 3, (d) CsTe 2O6,( e )C a F e O 3, and (f) SmNiO 3.T h e images on the left represent a 2D plot of the total charge density in a plane containing the Batoms in the small ( BS)a n dl a r g e( BL) octahedra (in units of e/˚A3). The central figures show a 1D plot of the total charge density along the line shown in the figures on the left. The graphs on the right represent the total charge density integrated in a sphere of radius Rcentered on the BSandBLatoms, as a function of R(charge accumulation function). In all these figures, it is possible to observe that the charge around both Batoms ( BSandBL) is basically the same. IV . RESULTS A. Energy lowering due to disproportionation An interesting as well as pragmatic question is, what is the minimum electronic structure theory framework neededto systematically predict spontaneous SLE/DLE symmetrybreaking when it occurs? Although one can always go tothe highest level methods to study certain materials (such as quantum Monte Carlo [ 60]) a reasonable question is what is the minimal set of physical ingredients that provide sucha prediction. Recent publications claimed that the band gapopening for the DLE phase, for instance, is a strongly correlatedeffect [ 32], and that methodologies such as dynamic mean field theory should be used to describe it. It turns out that, 075135-5DALPIAN, LIU, V ARIGNON, BIBES, AND ZUNGER PHYSICAL REVIEW B 98, 075135 (2018) FIG. 4. Bond lengths for the studied materials in both SLE and DLE configurations. For the DLE phase, distances are depicted for both the large (blue) and small (green) octahedra. For SLE, all octahedra are the same (red). White circles indicate experimental measurements. The calculated space groups and the average difference between experiment and theory are indicated. Note different scales on different graphs. as Table IIshows, the single-determinant, mean field Bloch periodic DFT band theory with an appropriately flexible unitcell that permits symmetry breaking if it lowers the energy isessentially sufficient, at least for the (rather chemically broad)set of representative compounds used here. The energy lowering from SLE to DLE for all compounds in Table IIindicates that the DLE phase is lower in energy than the SLE one. This is a clear indication that DFT can predictDLE configurations, and should be a good choice for studyingthis kind of phenomena. In Sec. IV D , we will show that DFT is also enough to describe band gap opening in these compounds,leading to a complete description of them.B. The physical charge density around the different disproportionated atoms is nearly constant, thus no “charge ordering” It was once thought [ 12,13,21,28,61] that disproportion- ation leads to different physical charges on the different B atoms in the lattice, an effect referred to as “charge ordering.”This kind of phenomenon has been labeled in several differentways such as charge ordering [ 62], charge disproportionation [13], valence skipping [ 20], missing valence states [ 63], or mixed valence compounds [ 22,64]. This view resulted from confusing formal charges assigned on the basis of the extreme 075135-6BOND DISPROPORTIONATION, CHARGE SELF- … PHYSICAL REVIEW B 98, 075135 (2018) ionic view, with physical charge observed in the variational charge density ρ(r). A few recent works have challenged the existence of charge ordering [ 17,65], suggesting that bond disproportionation, or different local environments, should bea better description of the physical reality. We have analyzed the variational DFT calculated charge density profiles around the Batoms for six structurally relaxed compounds. In Fig. 3we plot the total valence charge density of these compounds in a few different ways. First, in the leftpanels, we present a 2D representation of the total chargedensity in a plane containing both Batoms [inside small ( B S) and large ( BL) octahedral] of the DLE phase. The figures in the center panel show the charge density along a line containingbothBatoms, as indicated in the left panels, reporting a very small difference in the absolute values for the charge residing in both. The “charge accumulation function” Q(R)=/integraltext R 0ρ(r)dr integrated in a sphere of radius Raround each Batom is shown in the figures on the right panels as a function of the sphere radius R. We note again a minimal difference between Q(R) of SLE and DLE phases, and a larger difference is seen as R approaches the ligands, far from the Batom. For all the studied cases we find that the charge around the Batoms in different local environments, supposedly designated by widely different oxidation states, is rather similar. This principle of conservation of cation charge under differ- ent bonding conditions in strongly coupled metal-ligand com-pounds has been discussed in the context of the self-regulatingresponse [ 39–41,66] for the case of transition metal impurities in semiconductors [ 39], for Co in LiCoO 2vs delithiated CoO 2 [40], and for Sn atoms in perovskites such as CsSnI 3and for its reduced form where 50% of the Sn is removed as inCs 2SnI 6. The description above is a clear confirmation that the use of “valence” or “charge” for differentiating both atoms indifferent local environments is not a good choice, since thecharge in both is basically the same. Although there mightexist a very small difference between the charge density onthe two atoms, this difference is far from the two electronsargued by the valence skipping proposals. Such behaviorswere explained earlier [ 25,39–41,66] by the cooperation of the ligand orbitals that rehybridize in response to a changein total charge (reduction, delithiation, charging a sample) soas to minimize the perturbation—a manifestation of the LeChatellier principle. Given that the physical charge on the disproportionated atoms is rather similar, the next obvious question is, what is aphysically meaningful descriptor of the local environment ofthe disproportionating Batom? This will explain, via Fig. 1, which compound disproportionates and which stays as an SLE.The answer, as discussed next, is the bond geometry aroundeach of the disproportionated atoms. C. The different B-Xbond lengths in DLE and SLE octahedra form good markers for disproportionation A good way to differentiate the Batoms at different local environments in DLE compounds is through a directlymeasurable quantity such as the bond distance between the B andXatoms. Figure 4shows the calculated B-Xbond lengths in SLE and DLE phases of each of the studied materials,TABLE III. Amplitudes of key distortions (in ˚A) appearing in the ground state of each material (both SLE and DLE) obtainedon the basis of a symmetry-adapted mode analysis with respect to a high-symmetry cubic phase. The analysis is performed with AM- PLIMODES from the Bilbao Crystallographic Server. Only octahedrarotations (OR and OT), the breathing mode (BM), and antipolar (O-AP) displacements are reported. DLE SLE BaBiO 3 P21/c Expt. Theory BM (R− 2) 0.308 0.245 OR (R− 5) 1.226 1.370 O-AP ( X− 5) 0.178 0.229 OT (M+ 2) 0.384 0.414 CsTlF 3 Fm¯3mP 21/c Expt. Theory Theory BM (R− 2) 0.648 0.373 0.441 OR (R− 5) – – 1.484 O-AP ( X− 5) – – 0.615 OT (M+ 2) – – 0.938 CsAuCl 3 I4/mmm Expt. Theory BM (R− 2) 0.256 0.125 O-AP ( R− 3) 0.949 0.839 CaFeO 3 P21/c Pnma Expt. Theory Theory BM (R− 2) 0.180 0.081 – OR (R− 5) 1.078 1.091 1.084 O-AP ( X− 5) 0.406 0.461 0.456 OT (M+ 2) 0.833 0.798 0.938 O-AP ( R− 4) 0.176 0.081 0.091 SmNiO 3 P21/c Pnma Theory Expt. Theory BM (R− 2) 0.166 – OR (R− 5) 1.277 1.244 1.175 O-AP ( X− 5) 0.616 0.525 0.526 OT (M+ 2) 0.914 0.983 0.802 O-AP ( R− 4) 0.140 0.188 0.116 and a comparison with the available experimental results. For the SLE phase (red bars in Fig. 4), the equal octahedra can either have six equal bonds, as in the case of BaBiO 3,C s T l F 3, CsAuCl 3, and CsTe 2O6, or be distorted with two different bond lengths, such as in the case of CaFeO 3and SmNiO 3. For the DLE compounds, the small octahedra are representedby green bars, and the large octahedra are represented by bluebars. For the DLE cases, owing to the lower symmetry of thecompounds, there can be different groupings of bond lengthsin each octahedra. Experimental bond distances are shown inFig.4as white circles. Although there are claims stating that standard DFT “strongly underestimates the breathing distortion parameters”in DLE configurations [ 16], it is clear from Fig. 4that DFT provides a good description (see percent deviation listed inFig.4) of the trends on bond lengths upon disproportionation across the different bonding groups, whether the active B 075135-7DALPIAN, LIU, V ARIGNON, BIBES, AND ZUNGER PHYSICAL REVIEW B 98, 075135 (2018) cation is delectron, s-pelectron, or noble metal. An outlier is CsTlF 3with a deviation of 6% which might be related to sample stability/quality concerns reported in Ref. [ 21]. While future improved DFT functionals could hopefully improve thequantitative agreement with experiment, there is little doubtthat even current DFT functionals provide already a reasonablepicture of SLE-to-DLE spontaneous symmetry breaking. Besides the bond distance analysis shown in Fig. 4,w e have also performed a more detailed comparative analysisof the magnitude of the different symmetry-allowed normalmodes [ 67] including octahedral tilting (OT), octahedral rota- tion (OR), breathing modes (BM) producing the rocksalt-likepattern of compressed and extended oxygen cage octahedrals,and oxygen antipolar (O-AP) displacements X − 5andR− 4for our compounds. The results presented in Table IIIcorrespond to the actual amplitude of each lattice distortion appearing inthe ground state with respect to a high-symmetry cubic phase.In other words, atomic displacements are decomposed on thebasis of the eigenvectors of the dynamical matrix (phononmodes) of the perfectly symmetric structure of perovskites(Pm¯3m) phase, therefore providing the amount of each pure lattice distortion appearing in the material. The total amount ofatomic displacements is reported in angstroms. This analysisshows that the DFT calculated structures are, in most cases, invery good agreement with experiment. D. Energy lowering upon DLE formation is accompanied by gapping and metal-to-insulator transition In some compounds like CaFeO 3[29] and in some rare- earth nickelates such as YNiO 3[26] the structural transforma- tion from the DLE (an insulator) to SLE (a metal) configurationwith increasing temperature is accompanied by an insulator-to-metal transition. The fundamental origins of the band gapopening as well as the transition from SLE to DLE are still amatter of debate. Table IIshows the calculated band gaps both within the GGA approximation (or GGA +Ufor transition metal compounds) and also using HSE hybrid functionals for thestudied compounds. HSE calculations consistently give largerband gaps for the compounds, as expected. For BaBiO 3,a s previously discussed in the literature, the GGA band gap iszero. As the VBM and CBM are in different points of theBrillouin zone [ 68], and there are no levels crossing the Fermi energy, this zero gap should not be a major problem for theanalysis reported below. Figures 5and6show the density of states of the selected compounds in both SLE and DLE configurations. Figure 5 reports results for compounds that are not spin polarized,whereas in Fig. 6we report the density of states for the magnetic compounds (only spin-up), where the ferromagnetic configu-ration was assumed. Other complex magnetic configurationsmight exist in these compounds [ 25], but they will not be discussed in the present paper. Our test calculations with AFMconfigurations show that our main conclusions will not changewith a different magnetic configuration. As can be observed on the left panels of Figs. 5and6,a l l the SLE configurations are metallic. The red curve representsthe ligand (oxygen, fluorine, or chlorine) states, whereas thegreen curves represent the B-atom orbitals. For the DLE FIG. 5. Projected density of states for (a) BaBiO 3, (b) CsTlF 3,( c ) CsAuCl 3, and (d) CsTe 2O6. The figures on the left are for the metallic SLE configuration, where the green curve represents the Batom and red is related to the ligands (oxygen or fluorine or chlorine). The figures on the right are for the insulating DLE configuration, where green refers to Batoms inside the large octahedron, and purple to B atoms inside the small octahedron, and the yellow region indicates the band gap. All these PDOS were calculated using GGA. The inset shows details of the region around the band gap. 075135-8BOND DISPROPORTIONATION, CHARGE SELF- … PHYSICAL REVIEW B 98, 075135 (2018) FIG. 6. Projected density of states for (a) CaFeO 3and (b) SmNiO 3. The figures on the left are for the metallic SLE configuration, where the red curve represents the oxygen levels and green is for theBatom. Only spin-up components are plotted. The figures on the right are for the insulating DLE configuration, where green refers to the B atoms inside the large octahedron, purple to the Batom inside the small octahedron, and the yellow region indicates the band gap. The insets show details of the region around the band gap. configuration, the right panels of Figs. 5and6, we separate the contribution from the Batom inside the large octahedra (painted green) and from the Batom inside the small octahedra (painted purple). Red curves again represent the ligands. Thelevels related to the Aatom do not appear in the selected energy range. We can clearly see a specific qualitative behaviorin these compounds: the levels related to the Batom in the large octahedra are mostly localized in the valence band,whereas those related to the Batom on the small octahedra are in the conduction band. As the coupling between the Batom and the ligands is larger in the small octahedra, these levelsare pushed to higher energies (purple curves), whereas thosefrom the large octahedra are mostly filled in lower energies(green curves). The different coupling between large andsmall octahedra is a clear and straightforward explanation ofwhy the DLE phase is insulating. The hybridization between Batom and ligands also leads to a large DOS contribution from ligand (oxygen, fluorine,and chlorine) atoms around the Fermi energy. As shown inFigs. 5and6, for BaBiO 3, CaFeO 3, and SmNiO 3, the ligand contribution to the CBM is much larger than that from theBatom, consistent with the previous observation/prediction[17,29,43], while for CsTlF 3and CsAuCl 3, the CBM has almost similar contributions from the ligand and the Batom. We will further discuss such ligand hole states in Sec. IV F. E. Model for energy lowering and band gap opening in SLE-to-DLE conversion The fact that for all studied compounds there is energy lowering and band gap opening when going from SLE to DLEis a clear indication that there is a universal behavior in thesephenomena and, as such, it should be possible to develop aunified model to explain such properties. This can be donethrough an energy level diagram, as shown in Fig. 7. In a first approximation, considering only the electronic contribution tothe total energies, the energy lowering and band gap openingin DLE configurations can be understood through the differentstrengths of coupling between the BandXatoms in the BX 6 octahedra. In Fig. 7we will use BaBiO 3as an example, al- though similar trends can be extended to all other compounds. SLE bonding. In Fig. 7(a), the coupling between oxygen 2pand bismuth 6 sand 6plevels is depicted for the SLE configuration. As can be observed, this will lead to a metallicconfiguration, basically owing to electron counting. As alloctahedra are similar, there will be only one strength for thecoupling between the Batom and the Xoctahedra. DLE bonding. In Fig. 7(b)we show the same diagram for the DLE case. As there are now two different octahedra, one largeand one small, we have to separately consider both of them.The coupling between Bi and O levels in the small octahedrais stronger, owing to the shorter bond distance. This will pushthe hybrid levels of the small octahedra upwards, emptying ones-phybrid orbital. This empty hybrid orbital is usually called a ligand hole orbital, as will be discussed in the next section. For the large octahedra the coupling is weaker, leading to weakerrepulsion. By considering both large and small octahedra, thefinal effect is that holes are pushed to higher energies andelectrons are pushed to lower energies, leading in first orderto an electronic energy gain. If the difference in the strengthof the coupling for small and large octahedra is large enough,this will also open a band gap in this material. The coupling diagram described in Fig. 7can also be transported to all other compounds, as can be clearly observedin their projected density of states in Figs. 5and6:the VBM of the DLE phase is always localized in the large octahedra,whereas the CBM is related to the small octahedra. Forcompounds containing delectrons, each material can have a different order of eandtlevels, and the coupling is slightly different, although it leads to similar conclusions of energylowering and band gap opening. F. Ligand holes in DLE compounds with strong metal-ligand bonds signal a self-regulating response The conduction band of metal oxides [e.g., Fig. 8(a) for NiO] is generally composed of either transition metal dorbitals (in early transition atom oxides such as YTiO 3) or from metal s orbitals (in late 3doxides). A special case is when the CBM is made of ligand orbitals, called “ligand hole” states [ 17,24,29]. Ligand holes have been shown to exist in disproportionatedsystems [ 46] but there seems to be significant lack of clarity 075135-9DALPIAN, LIU, V ARIGNON, BIBES, AND ZUNGER PHYSICAL REVIEW B 98, 075135 (2018) FIG. 7. Schematic diagram showing the electronic coupling between BandXatoms in ABX 3. Here we use BaBiO 3as an example. (a) In the SLE configuration all octahedra have the same size. The coupling between Bi and O levels leads to a metallic configuration and the creation of a ligand hole ( L). Filled (empty) boxes represent filled (empty) hybrid orbitals. (b) In the DLE configuration the octahedra have two different sizes leading to different strengths in the coupling between Bi and O. Small octahedra have a stronger coupling (green) and large octahedra have a weaker coupling (blue). This different coupling pushes the hybrid levels to higher energies, opening a band gap and lowering the total energy of the system. on whether they are intimately related to d-electron systems and whether they are specific to disproportionated systems.The existence of ligand holes has been often associated withsuperconductivity in oxides [ 69–71], making its understanding even more interesting. DFT evidence for ligand holes in s-p and d electron ABX 3 with strong metal-ligand bonding . Figures 5and6show the density of states of the empty conduction band indicatinga clear ligand (oxygen) component. Figure 8shows a 2D representation of the CBM charge density in a plane containingfourXligands and the Batom in the small octahedra for the studied compounds. For guiding the eye, we first show inFig.8(a)the conduction band wave function square of the NiO system that lacks ligand holes. It is very clear from this figure that the charge is strongly localized on the Ni atom (center ofthe figure), with no contribution from the ligands (oxygen).This is a clear case of a positive charge transfer compound;i.e., charge is transferred from the metallic atom towards theligand. The difference with respect to ligand hole systems isapparent. The existence of ligand holes can be verified in thesecompounds by looking at the figures shown in Figs. 8(b)–8(g): we observe a strong signal on the ligand atoms, indicating anegative charge transfer compound, or the presence of ligandholes. In some cases, mainly for the compounds containingtransition metal atoms, there is still a metal-atom component,but the picture is very different from NiO, where absolutely nocontribution was observed on the ligands. This is true acrossdifferent bonding patterns for both s-pelectron and d-electron disproportionated ABX 3compounds. In fact, there is a LH state even in the SLE cases as well, as can be clearly observedby their projected density of states. Energy level model for ligand hole formation . Figure 7 gives the essential features of ligand hole formation .L Hformation requires that the relevant metal states should be deeper than ligand orbitals (so electrons can be transferredto the metal) and that a sufficiently strong metal-to-ligandcoupling exists so as to create unoccupied hybrid levels witha large fraction of ligand character. Thus, LH does not requiredisproportionation, but in disproportionated states there existsa short enough B-ligand bond that increases hybridization and consequently increases the ligand character on empty states.These empty levels will have strong oxygen- p(or fluorine or chlorine) character, showing that the holes are localized onthe ligands. This character of the CBM is different from mostsemiconductor compounds, and is clearly increased by shortcation-ligand bonds. The driving force for LH formation is the self-regulating response [39]. Total energy lowering favors the formation of LH when without such LH, the charge on the metalwould be highly positive (such as Ni 3+in RNiO 3or Bi4+ in BaBiO 3). This is not favored energetically, so the ligand transfers electrons (thus, forming a hole) to the metal cationso as to self-regulate its charge, creating the [Ni 2+−O1−] complex in RNiO 3and the [Bi3+−O1−] complex in BaBiO 3 where the hole is on the oxygen octahedra. V . CONCLUSIONS Quantum materials such as transition metal oxide per- ovskites present a wide range of interesting properties, such asmetal-insulator transitions, high-temperature superconductiv-ity, and a variety of magnetic orders, and can exhibit differentlocal environments (DLE) for the same atoms manifested bybond disproportionation. The basic phenomenology associ-ated with such disproportionation involves the absence ofB-atom charge ordering, the creation of different B-X bond 075135-10BOND DISPROPORTIONATION, CHARGE SELF- … PHYSICAL REVIEW B 98, 075135 (2018) FIG. 8. Square of the wave function of the ligand hole levels (lowest unoccupied states) for (a) NiO, (b) BaBiO 3, (c) CsTlF 3, (d) CsAuCl 3, (e) CsTe 2O6, (f) CaFeO 3, and (g) SmNiO 3in a plane containing four ligands and the central Batom in the small octahedra of the DLE phase. The white circles indicate the position of the B atom, and the black/yellow dots indicate the position of the ligands. length (bond alternation) for different local environments, the appearance of a metal (in SLE) to insulator (in DLE)transition, and the formation of ligand holes. We point out thefollowing: (i) The broad phenomenology associated with dispropor- tionation is common to a range of chemical bonding patternsinABX 3compounds, either with s-pelectron B-metal cations (BaBiO 3,C s T l F 3) or noble-metal cations (CsAuCl 3), as well asd-electron cations (SmNiO 3, CaFeO 3).(ii) Using as a guide the lowering of the total energy in DLE relative to SLE, we show that density functional calculations describe this phenomenology across the chemical bonding range without resort to special correlation effects. Inparticular, lower (DLE) total energy configurations naturallydevelop bond alternation, gapping of the metallic SLE state,and absence of charge ordering with ligand hole formation. (iii) Underlying much of this phenomenology is the “self- regulating response” (SRR), whereby in strongly bondedmetal-ligand systems with high-lying ligand orbitals, the sys-tem protects itself from creating highly charged cations bytransferring ligand electrons to the metal, thus preserving anearly constant metal charge in different local environments,while creating B-ligand bond alternation and ligand-like con- duction band (“ligand hole” states). We address the five questions posed in the Introduction as follows: (a) DFT provides an adequate level of theory of interelec- tronic interactions for predicting the tendency of actual ABX 3 to be SLE or DLE. (b) The formal oxidation state is not a physically mean- ingful “local environment descriptor” for predicting within theconvex hull construct of Fig. 1the tendency of actual ABX 3 to be SLE or DLE. (c) Bond geometry is a physically meaningful descriptor for predicting within the convex hull construct of Fig. 1the tendency of actual ABX 3to be SLE or DLE. (d) The SLE vs DLE selection is directly related to metallic vs insulating character of the compound. (e) Disproportionation per se is not related to ligand hole formation which is a more general phenomenon associatedwith strong metal-ligand bonding for the orbital order ofligand orbital energy being above metal orbital energies.However, in creating a compressed octahedron with short bondlengths, disproportionation provides a platform for ligand holeformation. ACKNOWLEDGMENTS The work at the University of Colorado at Boulder was supported by the US Department of Energy, Office of Science,Basic Energy Sciences, Materials Sciences and EngineeringDivision, under Grant No. DE-SC0010467 to the Universityof Colorado. G.M.D. also acknowledges financial supportfrom the Brazilian agencies FAPESP and CNPq. 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PhysRevB.84.014517.pdf

PHYSICAL REVIEW B 84, 014517 (2011) Universal quenching of the superconducting state of two-dimensional nanosize Pb-island structures Jungdae Kim, Gregory A. Fiete, Hyoungdo Nam, A. H. MacDonald, and Chih-Kang Shih* Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA (Received 24 October 2010; revised manuscript received 25 May 2011; published 27 July 2011) We systematically address superconductivity of Pb nano-islands with different thicknesses and lateral sizes via a scanning tunneling microscopy/spectroscopy (STM/STS). Reduction of the superconducting gap ( /Delta1)i s observed even when the island is larger than the bulk coherence length ( ξ) and becomes very fast below ∼50-nm lateral size. The suppression of /Delta1with size depends to a good approximation only on the volume of the island and is independent of its shape. Theoretical analysis indicates that the universal quenching behavior is primarilymanifested by the mean number of electronic orbitals within the pairing energy window. DOI: 10.1103/PhysRevB.84.014517 PACS number(s): 74 .78.−w, 73.21.Fg, 74 .55.+v The remarkable properties of a superconductor are at- tributable to its Cooper pair condensate (formed from a macro-scopic number of electrons), which can be described by a single quantum wave function. According to the celebrated Bardeen- Cooper-Schrieffer (BCS) theory of superconductivity, 1there is a minimum length scale (the coherence length ξ) on which the condensate-order parameter can vary. The fate of super-conductivity in systems with spatial dimensions smaller thanthe coherence length ξhas been the subject of intense interest for decades because it is dependent on quantum confinement, interaction, and quantum-coherence effects in intrinsically many-particle context–ingredients that drive much modern re-search in quantum many-body physics. 2–21Attention has often focused on two-dimensional (2-d) systems which can havefragile order because of quantum and thermal fluctuations. Inthis context the recent discovery of robust superconductivity in epitaxial thin Pb films with thicknesses that are orders of magnitude smaller than ξseems surprising. 4,10,11Even at a thickness of only two atomic layers ( ∼0.6 nm), the superconducting transition temperature ( TC∼5K )o fP b films is only slightly smaller than the bulk value 7.2 K.4 Nevertheless, there remain disputes regarding the thicknessdependence of T Cat thickness below 10 monolayers (MLs). Although this observation is already interesting, much more telling information on the fate of superconductivity at small length scales emerges when the lateral dimensions of theultrathin films are also reduced. What happens if all dimensions are reduced? Ultimately at some length scale, superconductivity should cease to exist.What determines this scale? Here we systematically addressthis fundamental question via a detailed scanning tunnelingmicroscopy/spectroscopy (STM/STS) study of thin-film su-perconducting islands with different thicknesses and lateralsizes. By controlling the lateral size of ultrathin 2-d islands, wediscovered that as the lateral dimension is reduced, suppressionof the superconducting order occurs. The reduction of thesuperconducting order parameter starts slowly but then accel-erates dramatically when the level spacing starts to approachthe value of the gap. While the values of the superconductinggap of the nano-islands show thickness dependence at alllateral sizes, when normalized to the extended film limit theycollapse to a universal curve that depends only on the volumeof the nano-island. Most surprisingly, our results for the size ofthe gap /Delta1versus island volume show quantitative agreementwith recent STS studies of hemisphere-shaped Pb droplets despite their dramatic difference in geometry. 5,9 The experiments were conducted in a home-built low- temperature (LT) STM system with an in situ sample prepara- tion chamber. The striped incommensurate (SIC) phase of the Pb-Si surface shown in Fig. 1(a) was prepared by deposition of∼1 ML of Pb onto the Si(111) 7 ×7 surface at room temperature, followed by sample annealing at 400 ∼450◦C for 4 min to form the surface template.22–24Depending on the detailed kinetic control, either flat film or 2-d nano-islandsof Pb can be obtained. Flat films can be grown by holding the sample temperature at ∼100 K during Pb deposition. 4 Figure 1(b) shows a 5-ML film grown on the Pb-on-Si (111) SIC surface. In order to get 2-d islands with a variety oflateral sizes and thickness, Pb was deposited on the templateat∼200 K with a deposition rate of 0.5 ML per minute. Figure 1(c) shows 2-d islands of different sizes and shapes on the same substrate. Also shown in Fig. 1(d) is a close-up view of two 3-ML 2-d islands, one with an effective diameter (d eff) of 74 nm and one with an effective diameter of 15 nm. The effective diameter ( deff) of each island is calculated by usingdeff=√ ab, where aand bare lengths along the major and minor axes, respectively. Figure 2(a)shows dI/dVtunneling spectra acquired at 4.3 K for a 5-ML film and 5-ML 2-d islands of various effectivediameters. Interestingly even at a diameter of 235 nm, whichis much larger than the bulk coherence length ξ(∼80 nm), one already observes a small reduction of the superconducting gapin comparison to that of the extended film. As one can observedirectly from the data, the trend toward gap reduction continuesas the diameter of the 2-d islands decreases further. Figure 2(b) shows dI/dVtunneling spectra acquired at 4.3 K for 3-ML 2-d islands with diameters ranging from 15 nm to 74 nm. The sizedependence of the superconducting gap of 4-ML 2-d islandsfollows a similar trend to that of 3-ML islands. It is importantto recognize that as long as the individual island is isolated,the measured superconducting gap is very uniform throughoutthe island until its very edge. Recently, Brun et al. 15reported a study of thickness dependence of the superconducting gap of Pb 2-d islands withlateral size larger than ξ, which they interpreted to represent the superconducting properties of extended films. They foundthat the superconducting gap scales with the inverse of islandthickness (1/d) and superconductivity should vanish at a 014517-1 1098-0121/2011/84(1)/014517(7) ©2011 American Physical SocietyKIM, FIETE, NAM, MACDONALD, AND SHIH PHYSICAL REVIEW B 84, 014517 (2011) FIG. 1. (Color online) STM topography image of (a) the SIC phase of Pb-Si surface, (b) a globally uniform 5-ML Pb film, and(c) 2-d Pb islands with various lateral sizes and thicknesses. The numbers labeled on the image (c) indicate thickness of islands in ML. (d) Two 3-ML Pb islands with effective lateral size ( d eff)o f7 4n m and 15 nm (inset) are shown in the same length scale (sample bias Vs=(a,d) 0.3 V , (b,c) 2 V , tunneling current I=(a) 20 pA, (b,c,d) 10 pA). thickness of 2 ML. While the recent observation of strong superconductivity at 2 ML already implied the breakdownof the 1 /d scaling, 4the current result that gap reduction occurs in 2-d islands with lateral dimension much less thanξshould settle the true behavior of superconductivity in these nano-systems. In order to more quantitatively characterize the suppression of superconductivity, we measured the transition temperatureT Cfor different lateral sizes and thicknesses of 2-d islands. To extract an energy-gap value for different temperatures /Delta1(T)w e fit the normalized dI/dVspectrum measured by STS to BCS- like density of states (DOS), as shown in Fig. 2(c).A l lS T M signals were carefully shielded against radio frequency (RF)noise, and a Gaussian-broadening parameter was implementedin the fitting to model the effect of the remaining noise. Astandard deviation of 0.3 mV rmswas used for the Gaussian broadening (see Appendix Afor more details). To determine TC, extracted values of /Delta1(T) are fitted to the BCS-gap equation [see Fig. 2(d)]; 1 N(0)V=/integraldisplayED 01/radicalbig ε2+/Delta1(T)2tanh/parenleftBigg/radicalbig ε2+/Delta1(T)2 2kBT/parenrightBigg dε, (1) where N(0) is the DOS at the Fermi level, Vis the electron- phonon coupling, and EDis the Debye energy. Strictly speaking the universal curve in /Delta1vsTis valid for a weak- coupling superconductor ( ED/kBTC/greatermuch1), but it is a good approximation in most cases, including Pb.5,25 Figure 3(a) demonstrates that TCof 3-ML, 4-ML, and 5-ML islands depends on their lateral size deff. Interestingly FIG. 2. (Color online) Lateral size dependence of differential conductance spectra ( dI/dV) measured at 4.3 K is shown for (a) 5-ML islands and (b) 3-ML islands. All differential conductance spectra were taken with the same tunneling parameter with the junction stabilized at Vs=20 mV and It=30 pA tunneling current. (c) Normalized conductance spectra (blue) measured as a function oftemperature from a 3-ML Pb island with a 62-nm-lateral size were fitted using the BCS DOS for the tunneling conductance (red). Each normalized spectrum is offset successively by 0.5 for clarity. (d) Thesuperconducting energy gaps ( /Delta1) for each temperature were obtained from (c) and plotted as red circles. The blue curve is a fitting of these energy gap data using a BCS gap equation to obtain a T Cof∼5.6±0.1 K for a 3-ML island with 62-nm-lateral size. All TCvalues determined from such fitting are estimated to have an error bar of ±0.1 K. there is a transition region (somewhere between 40 to 60 nm) below which electrons rapidly lose the strength of supercon-ducting coherence. We also observe a variation of T Cas a function of island thickness for a given deff:TC(3 ML) > TC(4 ML) >T C(5 ML), attributable to the quantum oscilla- tions of TCfrom the vertical electronic confinement, a topic that has been intensively investigated recently4,19(3-ML data was previously unavailable because of difficulty in preparing 3-MLfilms). The transition region from slow to rapid T Creduction also shows thickness dependence: it occurs at ∼40 nm for 5-ML islands, at ∼50 nm for 4-ML islands, and slightly above 50 nm for 3-ML islands. On the other hand if wenormalize the T Cof 2-d islands to the thin-film value at a given thickness and plot it as a function of island volume,the data collapse onto a single curve, revealing a universalbehavior of superconductivity suppression with respect to theisland volume [Fig. 3(b)]. Remarkably this universal curve is in quantitative agreement with earlier STS studies for Pb droplets,despite the dramatic difference in geometry. 5,9This collapse demonstrates that superconductivity in regularly shaped nano-islands is sensitive to a large degree only to the average levelspacing of a nano-structure, affirming to a surprisingly degreetheories 7,8,26of finite-size superconductivity suppression in which the mean number of electronic orbitals within thepairing energy window plays the central role. 014517-2UNIVERSAL QUENCHING OF THE SUPERCONDUCTING ... PHYSICAL REVIEW B 84, 014517 (2011) FIG. 3. (Color online) (a) TCas a function of island lateral size for 3-ML, 4-ML, and 5-ML islands. A transition region from slow to dramatic reduction of superconductivity is shaded in green. (b)T Cof islands was normalized to the globally flat film TCfor each thickness and plotted as a function of cube root of island volume. For theTCnormalization, 5-ML film TCwas measured to be 6.1 K, and estimated values of 6.9 K and 6.3 K from the TCtrend in (a) were used for 3-ML and 4-ML film, respectively. The data collapse of Fig. 3(b) can be understood within BCS theory in the following way. Starting with the mean-fieldgap equation at zero temperature , 1 V=εn=ED/summationdisplay n1/radicalbig /Delta12+ε2n, (2) where Vis the electron-phonon coupling as before, εnare the discrete single-particle energy levels on the island, and ED is the Debye (phonon) energy as before, we can determine the dependence of /Delta1on the size of the island, which sets the spacing between the energy levels, εn. Since the lateral size of the islands is finite, one should in general use the gapequation with a discrete sum Eq. ( 2) rather than an integral Eq. ( 1). However, if the islands are large enough and one is looking at finite temperatures, the integral form Eq. ( 1)i sa good approximation. That is the reason we have used it to fitour data earlier to obtain /Delta1(T). Here we wish to understand the fundamental reason that the suppression of superconductivitywith lateral size follows a universal curve. For that purposewe begin with zero temperature considerations and thengeneralize to finite temperature to obtain the universal curvetheoretically [see the black curve in Fig. 3(b)]. To understand weakened superconductivity in small islands, one must retainthe discrete nature of the energy levels as their increasedspacing with shrinking lateral size plays an important rolein the suppression of superconductivity. 7,8,26We first note that our data show an absence of Coulomb blockade effects onthe islands, which would have a charging scale of E C∼ e2/C, which can be estimated as EC∼10–40 meV . Here eis the charge of the electron and the island capacitance C=2πκL, where Lis the size of the dot and κis the dielectric constant. This indicates some electrical contact withthe substrate (also needed for the STM measurement itself),though we are not able to determine if the coupling is in theintermediate or strong regime. Because the contact of the SICis only a “ring” around the Pb island (i.e., the Pb island isnot “sitting” on the SIC), even a highly conducting SIC-Pbinterface would only weakly affect the basic superconductingproperties, which are primarily determined by the “bulk”island DOS and electron-phonon coupling. Thus, the essentialphysics of the universal suppression in our experiments iscontained in Eq. ( 2), as we now show. The typical scale of the variation in the DOS (number of energy levels on the island per unit energy) in Eq. ( 2)i ss e t by the energy scale of the electronic degrees of freedom E F (Fermi energy), which is 9.5 eV for Pb. On the other hand ED is typically tens or hundreds of Kelvin (88 K in Pb), or roughly 10−2electron volts. Because ED/lessmuchEF, the DOS (i.e., average level spacing) will be constant to a good approximation overthe energy range of the sum in Eq. ( 2), even for islands that are somewhat irregularly shaped. First consider an island with ashape that supports perfectly even space levels. (We will latershow that when the energy levels are not perfectly spaced,there is very little change in the physics expressed througha self-consistency condition in the BCS-gap Eq. ( 2)—this is ultimately what is responsible for the universal suppression ofthe superconductivity.) When fluctuations in level separationsare neglected ε n=nδ, where δis the level spacing. The corresponding DOS for such an island is thus 1 /δ. From the well-known formula for the bulk gap in the ther- modynamic limit, /Delta10=2EDexp{−1/λ}=2EDexp{−δ/V}, and the bulk value of the transition temperature TC= /Delta10/2.2=7.2 K, we find δ/V=2.5 for Pb under the equal-level spacing assumption εn=δn.H e r e Vis the strength of the electron-phonon coupling in units of energy.Thus, for the critical island size where the gap /Delta1=0, we have 2.5=δ V=nmax,0=ED/δ/summationdisplay n=11 n, (3) which implies that nmax,0=7 when the island becomes so small that the transition temperature vanishes, or that δ=ED/ 7∼13 K is roughly of the order of the bulk gap /Delta10=12 K. The key analytical feature of Eq. ( 3) is the “1 /n” contribution from the different energy levels: Even for large n, there are apprecia- ble contributions to the sum (1 /nis logarithmically divergent). This means that any random (not too large) fluctuationsfrom the equal-level spacing approximation attributable to anirregular-shaped island will be approximately averaged out.(We have verified this in numerical checks with random energyspacings constrained to have the same average.) Moreover, inislands larger than the critical size (with smaller level spacingδ) there will be even more effective averaging because the larger “small” ncontributions will carry less weight because 014517-3KIM, FIETE, NAM, MACDONALD, AND SHIH PHYSICAL REVIEW B 84, 014517 (2011) of the presence of the gap in the denominator of Eqs. ( 2) and (4), and there will be more terms in the sum to be averaged, nmax>n max,0: 2.5=δ V=nmax=ED/δ/summationdisplay n=11/radicalBig/parenleftbig/Delta1 δ/parenrightbig2+n2. (4) From Eq. ( 4) the gap dependence on level spacing /Delta1(δ) is determined. We emphasize this analysis is valid even inthe presence of tunneling coupling of the island states to thesubstrate, with only a small numerical change in the results.The previous arguments are not simply hand waving but ratherrely on the properties of the summation of 1 /nfor a finite number of terms. As we now show, similar considerations apply to the level- spacing dependence of the critical temperature ,T C, which is proportional to the gap. Formally, the temperature dependenceof the gap in a small island is given by the discrete version ofEq. ( 1), 2.5=δ V=nmax=ED/δ/summationdisplay n=1Tanh[/radicalbig (/Delta1(T))2+(nδ)2/(2kBT)]/radicalBig/parenleftbig/Delta1(T) δ/parenrightbig2+n2, (5) which will exhibit the same insensitivity to level-spacing fluctuation as Eq. ( 4). The critical temperature TCis given by Eq. ( 5) with /Delta1(TC)=0, 2.5=δ V=nmax=ED/δ/summationdisplay n=1Tanh[nδ/(2kBTC)] n. (6) For islands of equal volume but different shape the value ofnmaxin Eq. ( 6) will be roughly the same, since that only counts the total number of states in the energy window withinE Dof the Fermi energy, and that number is expected to be a weak function of island shape [distorting a regularly shapedisland into an irregular shape mainly rearranges energy levels,which is unimportant because of the “averaging” effect fromthe slow decay of the summand in Eq. ( 5)]. We believe the effective averaging of level-spacing fluctuations attributableto a roughly constant average DOS over the relevant energy range E Dis the reason we find that the normalized critical temperature curves in Fig. 3(b) collapse to a universal form that is to a good approximation independent of island shape andgiven by Eq. ( 6). We remark that first principles calculations have predicted some changes in the phonon spectrum andelectron-phonon coupling, but this makes a quantitativelysmall change in the transition temperature in the thin-filmgeometry. 19We believe our argument based on the BCS mean-field gap equation simply explains all the trends in ourdata and related experiments. 5,9 In summary we have performed a STM/STS study of 2-d superconducting islands with different thicknesses andlateral sizes. As the lateral dimension is reduced, the strengthof the superconducting order parameter is also reduced,first slowly at a dimension larger than the bulk coherencelength, then dramatically at a critical length scale of 40 ∼ 50 nm, which corresponds to level spacing of order thebulk gap /Delta1. Interestingly the systematic suppression ofsuperconductivity with size depends to a good approximation only on the volume of the island and is independent ofits shape. We have explained this feature with theoreticalarguments based on BCS theory. We expect this work to havebroad implications for device implementation that dependson detailed knowledge of size-dependent superconductivityand to stimulate further fundamental studies on nanoscalesuperconductivity. ACKNOWLEDGMENTS This work was supported by NSF Grants No. DMR- 0906025, CMMI-0928664, Welch Foundation F-1672, andTexas Advanced Research Program 003658-0037-2007. GAFacknowledges support by ARO W911NF-09-1-0527 and NSFDMR-0955778. APPENDIX A: DETERMINATION OF THE SUPERCONDUCTING GAP 1. Normalization of tunneling spectra Because of the existence of quantum-well states, the DOS of the sample is often not a constant within the relevant energywindow of E F±20 meV . Consequently the raw dI/dVdata contains an asymmetric background. Moreover, as will bediscussed in Appendix B, there exist “pseudogap” features (the depression in dI/dVin the range between −10 mV to 10 mV) at temperature above T C.16These two factors—the nonconstant DOS near EFand the pseudogap—need to be considered in the normalization procedure. In most cases one can normalizethe spectra by dividing them with the spectra acquired aboveT C, which should represent the normal-state DOS. However, because of the existence of the pseudogap such a normalizationprocedure will artificially raise the dI/dVvalue in the vicinity of the superconducting gap, thus distorting the lineshape [seeFigs. 4(a) and4(b)]. We found that the spectra normalized in this manner would not fit well with a BCS-like gap function(discussed subsequently). Another possibility is to use spectraacquired at a temperature much above T C(say,>80 K) to represent the normal-state DOS. However, the thermal driftencountered with such a large temperature change makes itdifficult to guarantee that the tip will stay at the same locationwith respect to the sample. Here we use a modified normalization procedure by using a high-order (in this case 5th-order) polynomial to fit the spectra FIG. 4. (Color online) (a) dI/dVtunneling spectra obtained from 9-ML Pb films at 4.3 K and 8.5 K. (b) The result of normalization with the spectrum acquired at 8.5 K (blue line) and the 5th-order polynomial fitting (red line). 014517-4UNIVERSAL QUENCHING OF THE SUPERCONDUCTING ... PHYSICAL REVIEW B 84, 014517 (2011) outside the gap region to represent the normal DOS [see the dashed line in Fig. 4(a)]. The resulting normalized spectra are s h o w ni nt h er e dl i n eo fF i g . 4(b). This normalization procedure works quite well for a general shape even when the normalDOS contains a dip or peak. Moreover, it can be incorporatedinto a computer program that can automatically generatethe normalized spectra without human bias. We recognize,however, the existence of pseudogap degrades the precisionof fitting a small gap for spectra acquired very close to T C. Consequently we regard those gap values below 0.2 meV asunreliable data and they are excluded from the T Cfitting. 2. Gap fitting While the BCS-gap function1is not strictly applicable to the case of strong coupling like Pb, it remains a goodapproximation. Within this approximation, the differentialconductance ( dI/dV)based on the tunneling current between a normal metal (STM tip) and a superconductor (sample) canbe expressed as dI dV∝/integraldisplay∞ /Delta1|E|√ E2−/Delta12/bracketleftbigg −df dE(E+eV)/bracketrightbigg dE, (A1) where f(E) is the Fermi distribution function, Vis an applied voltage bias, and /Delta1is the superconducting energy gap. In addition a Gaussian broadening of finite width (calledthe broadening parameter) is applied. This broadening isattributable to the incomplete shielding of the RF interferencepresent at the tunneling junction, as we will subsequentlydiscuss. For each RF-shielding configuration there is only one broadening parameter. This parameter changes when theconfiguration of the RF shielding changes. Shown in Figs. 5(a) and 5(b) are results of STS measurement of 9 ML without the RF filter and with our best filter configuration we haveachieved so far, respectively. For the no-filter configuration,a 0.8 meV broadening parameter is needed to fit the spectra, FIG. 5. (Color online) Temperature dependence of superconduct- ing gap spectra taken from 9-ML Pb films (a) without RF filter and(b) with RF filter. 0.8-meV and 0.2-meV broadening were applied in the BCS DOS (red lines) of (a) and (b), respectively. Each normalized spectrum is offset successively by 0.3 for clarity. FIG. 6. (Color online) (a) Direct comparison between experimen- tal spectra measured from 9-ML Pb films with our best RF shielding configuration and theoretical BCS DOS without any broadening. Each spectrum is offset successively by 0.4 for clarity. (b) Thesuperconducting energy gaps ( /Delta1) obtained from Figs. 5(a) and5(b) were plotted as empty and solid blue squares for each temperature, respectively (error bar for each /Delta1value is smaller than the size of square mark). The red curve is a fitting of these energy gap data using a BCS gap equation. Those gap values from two shielding configurations align very well with a single BCS curve within a T C of∼6.6±0.1 K. (c) Temperature dependent gap spectra measured from 5-ML Pb films. corresponding to a RF noise with rms amplitude of 0.8 mV . On the other hand with a good RF-shielding configuration,only 0.2 meV broadening is needed. For a comparison wealso show the theoretical spectra at different temperatureswithout any broadening in Fig. 6(a). One can see that the spectra acquired with our best RF-shielding configurationare reasonably close to the theoretical curves. It shouldbe noted that in the temperature range where the spectrawere acquired, the width of the Fermi-Dirac distributionalready exceed the broadening parameter typically used in ourexperiments. Most importantly while the raw data of the two config- urations differ significantly in their line shape, they yielda very similar T C(to within ±0.1 K). In Fig. 6(b) we fit the gap values as a function of temperature from these twodifferent configurations (shown as empty and solid squares).As one can see from this plot, the data from differentfilter configurations fall quite well onto a single BCS-like/Delta1(T) curve. The best RF-shielding configuration used in the experiment unfortunately compromises the operation of the LT walker,which is an important component of LT-STM. Thus, for mostof the experimental results reported in the main text, thespectra were acquired in a different RF-filter configuration,corresponding to a 0.3-meV broadening parameter. This sec-ond RF-filter configuration allows us to simultaneous operatethe ultra-high-vacuum LT walker. Since we have shown thatthe fitted gap and the resulting T Care independent of the 014517-5KIM, FIETE, NAM, MACDONALD, AND SHIH PHYSICAL REVIEW B 84, 014517 (2011) broadening parameter being used to within ±0.1 K, we believe the scientific conclusions would be consistent. We should alsoemphasize that the temperature dependence of the gap nearT Cprovides the most reliable determination of TCbecause the gap varies most rapidly close to TC. This can be clearly seen by observing the raw data of a 5-ML film in Fig. 6(c); while at 5.8 K, the superconducting gap is still clearly observable,it disappears at 6.3 K, consistent with the fitted T Cvalue of 6.1 K. APPENDIX B: THE PSEUDOGAP STRUCTURE AT VOLTAGES ABOVE THE GAP The majority of this paper is focused on the supercon- ducting properties of nano-islands of different thicknesses andshapes. However, our data also show interesting features atvoltage biases above the gap when the temperature is below T C and for all voltages we probed when the temperature is above TC. As seen in Figs. 7(a) and7(b) (where the temperature is above TC), there is a “pseudogap” feature that appears (even for very large islands). The pseudogap does not measurablychange when the temperature drops below T C, suggesting that it is likely an intrinsic property of the normal state ofthe Pb islands and not related to superconductivity in anyway. It has been suggested in an earlier study 16on films that such features may be attributable to phonon effects.We show here that we obtain a very good fit to a theorybased on a pseudogap feature arising only from the combinedinfluences of electron-electron interactions and disorder. 27 Moreover, we are able to extract a quantitative estimate of thenormal state conductivity of the Pb nano-islands used in our experiments. FIG. 7. (Color online) (a,b) Comparison of dI/dVspectrum measured at 6 K (above TC) for 3-ML islands and theoretical tunneling DOS based on electron-electron interaction and disorder [Eq. ( B1)]. Because T=0Ki sa s s u m e di nE q .( B1), 1.2-meV broadening effect is applied to the fitting curves to take finite temperature and instrument noise into account. (c,d) 2-d conductivity ( σ) values for Pb islands obtained from the fitting result in (a,b).According to Ref. 26, the tunneling DOS should follow a form dI dV=c1+c2|V|α+mV, (B1) where c1,c2,α, and mall depend on details of the system, and Vis the voltage. The final term mVaccounts for a weakly energy-dependent tunneling-matrix element between the STMtip and the Pb substrate rather than intrinsic island propertiesand does not enter our estimates of island conductivity. Forour purposes we are most interested in αas it is inversely proportional to the 2-d conductance of the nano-island/thinfilm substrate α=e 2 hσln(2πae2dn/dμ ) 2π, (B2) where eis the charge of the electron, his Planck’s constant, σ is the 2-d conductivity of the island or film, ais the tip-sample spacing, and dn/dμis the compressibility of the island or film. (Here nis the 2-d density and μthe chemical potential.) Because the expression is only logarithmically dependent onthe compressibility, we can estimate it by using a free-electronapproximation. For 2-d electrons dn/dμ=4πm ∗/h2, where m∗=1.14meis the effective mass of Pb, and meis the bare mass of the electron.28 The plots of conductance vs island size are given in Figs. 7(c) and 7(d) for 3-ML- and 4-ML-film thicknesses. The values are consistent with those measured in earlyfilm samples, 29,30but here we provide the first conductance measurements of islands . The general trend is for thicker and larger islands to be more conducting and for the conductanceto be higher as the temperature is lowered. These findingsare all consistent with expectations. Figure 8shows that the pseudogap feature at energies above the gap does not changeas the temperature drops below the superconducting transitiontemperature. FIG. 8. 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PhysRevB.82.155429.pdf

Atomistic engineering in the control of the electronic properties of CdSe nanotubes F. Mercuri,1,*S. Leoni,2J. C. Green,3and M. Wilson4 1ISTM-CNR, c/o Department of Chemistry, University of Perugia, I-06123 Perugia, Italy 2Max Planck Institute for the Chemical Physics of Solids, D-01187 Dresden, Germany 3Department of Chemistry, Inorganic Chemistry Laboratory, University of Oxford, Oxford OX13QR, United Kingdom 4Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX13QZ, United Kingdom /H20849Received 31 August 2010; published 18 October 2010 /H20850 The electronic band structure of inorganic nanotubes /H20849INTs/H20850formed both from percolating hexagonal- and square-net motifs are obtained by density-functional-based methods for a key technological material, CdSe. Anenergetic crossover from hexagonal- to square-net-based structures is observed at low INT radius indicative ofa potential synthetic pathway. Molecular-dynamics simulations, using an existing potential, demonstrate theisolated INTs to be thermally stable. Electronic structure calculations indicate remarkable differences betweenINTs of different morphology. The results demonstrate that the electronic properties of CdSe nanotubes may beeffectively engineered. DOI: 10.1103/PhysRevB.82.155429 PACS number /H20849s/H20850: 73.22. /H11002f, 61.48. /H11002c, 61.66.Fn, 71.20.Nr The exploitation of nanomaterials is becoming an increas- ingly important economic priority. A major factor is the po-tential for these materials to display unique physicochemicalproperties which may be controlled through the effectivecommand of the size and morphologies of the nanoparticles.These nanostructures adopt a range of low-dimensionalstructures including pseudozero-dimensional nanoparticles,pseudo-one-dimensional nanowires and complex mixturesthereof. These broad classifications may be further subdi-vided. For example, the pseudo-one-dimensional structuresmay consist of space-filling bulklike sections or may benanotubular. Full exploitation of the potentially unique prop-erties afforded by such nanoscale materials relies, therefore,on the ability to understand and control their formation on anatomistic level. To date work in this area has focused largely on carbon nanotubes /H20849CNTs /H20850. 1However, significant progress has re- cently been made in obtaining low-dimensional nanotubularsystems based on more general inorganic compounds /H20851inor- ganic nanotubes /H20849INTs/H20850, for example, see Ref. 2/H20852. To gener- ate nanotubular structures of inorganic compounds a varietyof synthetic techniques, such as filling of CNTs from moltensalts, 3template synthesis,4or hydrothermal treatment5have been adopted. The characteristics of the INTs, which usuallydiffer from those of their bulk /H20849macroscopic /H20850counterparts, have been the subject of intensive research efforts, targetedto the elucidation of their structural, electronic and reactiveproperties. 6 The focus of this paper is to consider INT structures for a key target system, CdSe. Our aim is to explore the possibilityof engineering the electronic properties of single-walled cad-mium selenide nanotubes /H20849CSNTs /H20850by means of theoretical calculations. CdSe has attracted significant attention, in par-ticular, for its behavior as a semiconductor and the ability toadopt nanoparticular forms 7including both single-crystal and polycrystalline8,9nanotubes. Significantly, the electronic band structure appears highly sensitive to the size, dimen-sionality, and geometry of the nanoparticles. 10,11These prop- erties underlie key applications, for example, as photolumi-nescent materials, 12in lasers,13as biodetectors,14and in solid-state devices.15,16In this work we compare the elec-tronic structures of a range of possible INT morphologies based upon both percolating hexagonal- and square-net mo-tifs. Our methodology is to employ an existing potentialmodel 17to generate relaxed INT structures whose coordi- nates can be used as a starting point for subsequent density- functional-theory /H20849DFT/H20850optimizations and electronic struc- ture calculations. Previous theoretical work /H20849see Ref. 18and references therein /H20850has indicated that two general classes of INT may be identified for the MX /H20849M/H11005metal, X /H11005halogen /H20850stoichiometry, considered as formed by folding sections of hexagonal- andsquare-net sheets, respectively /H20849Fig.1/H20850. The relative stability of these two classes of INT is correlated with the relativestabilities of the four- and six-coordinate bulk crystal struc-tures. CSNTs are generated by folding the sheets around achiral vector C h=/H20849na1+ma2/H20850, where a1anda2are the unit- cell vectors /H20849as indicated in the figure /H20850. In order to distin- guish between the two tube morphologies we adopt a nota-tion/H20849n,m/H20850 X, where nandmare the chiral vector indices and FIG. 1. /H20849Color online /H20850Two-dimensional hexagonal- /H20849left/H20850and square- /H20849right/H20850net structures shown for the MX stoichiometry. In both cases the two atom unit cells are highlighted by a dashed box,as are the respective unit cell vectors, a 1anda2. Example chiral, Ch, and translational, T, vectors are also shown.PHYSICAL REVIEW B 82, 155429 /H208492010/H20850 1098-0121/2010/82 /H2084915/H20850/155429 /H208495/H20850 ©2010 The American Physical Society 155429-1X=hexorsqcorresponding to INTs formed from hexagonal or square nets, respectively. The initial CSNT structures were generated by foldin g the appropriate sheets using the Cd-Se bond lengths ob-tained from fully relaxed infinite hexagonal and square-netsheets /H208492.57 Å and 2.72 Å, respectively /H20850. Each INT was relaxed to a local energy minima using a steepest descentalgorithm utilizing the effective pair potential /H20849EPP/H20850model developed by Rabani. 17This potential model predicts a wurtzite /H20849four-coordinate /H20850ground state with the energy mini- mum /H1101112 kJ mol−1below that of the rocksalt structure /H20849six coordinate /H20850. The closeness of the two energy minima indi- cates that both hexagonal- and square-net-based INTs may bestabilized in the confining CNT environment. Subsequently,CSNT geometries relaxed using the EPP were used as start-ing structures for DFT optimizations. DFT calculations werecarried out within the local density approximation for theexchange correlation functional. 19Electronic states were ex- panded by a numerical basis set of double-zeta plus polariza-tion quality with a norm-conserving pseudopotential descrip-tion of core levels. Charge density was described by a plane-wave representation with an energy cutoff of 200 Ry.Periodic boundary conditions in the dimension parallel to thenanotube axis were used with a converged number of K points along the periodic axis and sufficiently large vacuumsize/H20849around 15 Å /H20850in the nonperiodic dimensions. Geom- etries were optimized by application of the conjugated gra-dient algorithm, until a maximum force on atoms smallerthan 0.01 eV /Å was reached. Density of states /H20849DOS/H20850and projected density of states were computed on the DFT opti-mized structures with an increased number of Kpoints along the periodic direction. DFT calculations were performed withthe SIESTA program package.20 The optimized geometry of selected /H20849n,0/H20850and/H20849n,n/H20850 hexagonal- and square-net CSNTs are shown in Figs. 2and 3, respectively. The structures of optimized /H20849n,m/H20850hexCSNTs /H20849with n/HS11005m/H20850exhibit similar features. The relaxation of all considered CSNTs results in tubular /H20849hollow /H20850structures, with the exception of the /H208494,0/H20850square-net INT, which re- laxes to a core-shell structure. For all structures consideredDFT optimization leads to significant relaxation in the ioncoordinates with the evolution of significant “corrugation”with Cd and Se atoms tilting inward and outward with re-spect to their starting position on the ideal cylindrical sur-face. Such relaxations have been observed in other inorganiclow-dimensional nanostructures 21and are often related to “rumpling” on blende or wurtzite crystal surfaces.22The dif- ference between the diameter of the resulting outer and innernanotubes lies within 1.0 and 1.5 Å for the CSNTs consid-ered /H20849with outer diameters ranging from 2.8 to 10.9 Å /H20850and is slightly more pronounced for hexagonal-net nanotubes.Accordingly, the Cd-Se bond lengths are uniform and prac-tically unchanged with the diameter in hexagonal-net nano-tubes, ranging from 2.51 to 2.57 Å, thus around 1–4 %shorter than in the optimized bulk geometry. Conversely, thedistribution of Cd-Se bond length is much more spread out insquare-net nanotubes ranging from 2.50 to 2.98 Å. Figure 4 shows the INT energetics as a function of their effective radiishown relative to the energy of the bulk wurtzite structure.The INTs are /H110110.5 eV /H20849/H1122950 kJ mol −1/H20850higher in energythan the bulk ground state. This difference is comparable to that observed between graphite and C 60/H20849Ref. 23/H20850and represents only /H110115% of the overall lattice energy. The ki- netic stability of these structures, therefore, appears highlyplausible. FIG. 2. /H20849Color online /H20850Optimized structures of /H20849n,0/H20850and/H20849n,n/H20850 /H20849n=4,5,6 /H20850hexagonal-net CSNTs. Cd and Se atoms are in dark gray /H20849light blue online /H20850and light gray /H20849yellow online /H20850, respectively. FIG. 3. /H20849Color online /H20850Optimized structures of /H20849n,0/H20850and/H20849n,n/H20850 /H20849n=4,5,6 /H20850square-net CSNTs. Cd and Se atoms are in dark gray /H20849light blue online /H20850and light gray /H20849yellow online /H20850, respectively.MERCURI et al. PHYSICAL REVIEW B 82, 155429 /H208492010/H20850 155429-2The energy of the DFT-optimized hexagonal- and square- net CSNTs is generally more favorable than that of theirplanar limits, as shown in Fig. 4, and tends toward their respective high- Rplanar limits. This behavior is the opposite to that expected from purely mechanical considerations. 24 Moreover, the hexagonal-net INT energetics appears to beinvariant to the chiral vector. Conversely, the square-net INTenergetics shows a strong dependence upon the chiralvector 24with the formation of INTs of /H20849n,n/H20850sqmorphology strongly favored over those of /H20849n,0/H20850sq. Furthermore, the /H20849n,n/H20850sqINT energetics show a much stronger Rdependence than either the /H20849n,0/H20850sqor/H20849n,m/H20850hexsystems. A significant con- sequence of this difference is that there is an energetic cross-over from favoring hexagonal-net-based INTs to square-net-based structures at low R/H20851corresponding to /H20849n,n/H20850 sqCSNTs with n/H113494/H20852and thus indicating a possible synthetic pathway to the manufacture of CdSe nanostructures based on squarenets. These energetic considerations provide evidence for a potentially complex polymorphism for CdSe nanotubularstructures. In turn, the electronic properties of CSNTs can beexpected to be strongly influenced by the details of theatomic arrangement. This observation suggests a potentialroute to control the electronic properties of CSNTs based onthe selection of the appropriated family of polymorphs, ac-cording to the energetic criteria of Fig. 4. Molecular- dynamics simulations, performed using the EPP and startingfrom both ideal INTs periodically repeated along one direc-tion, show the hexagonal-net-based INTs to be highly stableup to at least T/H110111000 K independent of morphology. The square-net-based INT stability shows a strong morphologicaldependence but with clear “islands” of stability reflecting theunderlying energetics shown in Fig. 4. Similarly to CNTs, the electronic properties of CSNTs can be derived from those of the generating planar sheets interms of band-folding arguments. 25The band structure of hexagonal- and square-net planar arrangements of CdSe andthe relative DOS are reported in Fig. 5and compared with the properties of bulk /H20849wurtzite /H20850CdSe. The band structure of planar hexagonal-net CdSe /H20851see Fig. 5/H20849b/H20850/H20852displays features similar to other layered systems based on the hexagonal net-work, such as graphene. 26Namely, the DOS around the Fermi energy /H20849EF/H20850can be partitioned into /H9268and/H9266compo- nents with the two bonding /H9268and/H9268/H11032bands degenerate at /H9003. However, in contrast to graphene, the band structure ofhexagonal-net CdSe planar sheets exhibits a wide band gapatKbetween the /H9266and the /H9266/H11569bands and a direct band gap /H208492.00 eV /H20850originates at /H9003between the /H9268and/H9268/H11569bands. There- fore, hexagonal-net CdSe sheets rolled to form nanotubes areexpected to exhibit semiconducting behavior irrespective ofthe folding direction, in contrast to the behavior observed forCNTs. Conversely, the planar square-net CdSe sheets exhibit me- tallic character, with two bands /H20851labeled as /H9268/H11032and/H9268/H11033/H11569in Fig. 5/H20849c/H20850/H20852crossing EFin regions close to the edges of the Bril- louin zone and composed mainly of mixed Cd-Se in-plane p andsstates, respectively. To assess the effect of curvature on the electronic proper- ties of planar sheets, band-structure and DOS calculationswere also performed on models of hexagonal- and square-net5 10 15 R’0.40.450.50.550.60.65E [eV] FIG. 4. Relative energies of CSNTs /H20849in eV/atom /H20850calculated at the DFT level for different square- and hexagonal-net INT mor-phologies, as a function of the effective radii expressed in terms of the identifying indices, R /H11032=2/H9266 a0R. Key: circles: /H20849n,0/H20850hex; squares: /H20849n,n/H20850hex; diamonds: /H20849n,m/H20850hex;/H11612:/H20849n,0/H20850sq; and ⊲:/H20849n,n/H20850sq. The hori- zontal lines indicate the limiting energies for the hexagonal /H20849solid/H20850 and square /H20849dashed /H20850planes, respectively. Model energies are ex- pressed relative to the bulk /H20849wurtzite /H20850energy. The arrow highlights the crossover from hexagonal-to square-net based INTs as the INTradius is reduced. FIG. 5. Band structure and DOS for bulk CdSe /H20849wurtzite /H20850/H20849left/H20850hexagonal-net /H20849center /H20850and square-net /H20849right/H20850planar sheets of CdSe. Dashed and dotted lines represent the DOS projected on states of Cd and Se atoms, respectively.ATOMISTIC ENGINEERING IN THE CONTROL OF THE … PHYSICAL REVIEW B 82, 155429 /H208492010/H20850 155429-3CSNTs. The electronic structures of /H208495,0/H20850hex,/H208495,5/H20850hex, /H208495,0/H20850sq, and /H208495,5/H20850sqnanotubes are reported in Figs. 6and7 as representatives of more general cases. The electronicstructure of hexagonal-net CSNTs can be derived from that of the parent planar sheet, analogously to CNTs, 25and indi- cates a clear semiconducting behavior and similar atomiccomponents for /H9268and/H9266bands close to EF. Moreover, the main features of the band structure are qualitatively similarfor/H20849n,0/H20850 hexand/H20849n,n/H20850hexnanotubes. The situation is different for square-net CSNTs: here, rolling the generating planarsheet induces different strains for /H20849n,0/H20850and/H20849n,n/H20850nanotubes, as also evidenced by a different formation energy /H20849see Fig. 4/H20850. The change in the geometry of the CdSe square-net upon rolling affects also the degree of Cd-Se hybridization, with aconsequent modification of the /H9268bands around EF, leading to the opening of a finite band gap /H20849see of Fig. 7/H20850. The band- gap opening is slightly more pronounced in /H20849n,0/H20850sqCSNTs due to a larger structural distortion, upon relaxation, withrespect to the arrangement of atoms on the ideal rolled planarsheet. Conversely, the electronic structure of /H20849n,n/H20850 sqCSNTs maintains some of the features of the square-net planar sheet,exhibiting dispersed bands around E Fand a relatively smallband gap. The dispersion is particularly pronounced for the conduction bands, constituted mainly by hybrid Cd/Se emptystates, indicating a marked one-dimensional character for/H20849n,n/H20850 sqCSNTs and thus suggesting their use as potential building blocks for nanoscale electronic and optoelectronicdevices. 27Remarkably, the details of the electronic structure of CSNTs depend strongly upon their morphology. In par-ticular, the band gap of hexagonal- and square-net CSNTscan be correlated with their linear density /H9267, defined as the number of atoms per unit length, which is directly propor-tional to the diameter of ideal /H20849unrelaxed /H20850nanotubes. The linear density /H9267has been found to provide a reasonable cor- relation of the electronic properties with the diameter of bothhexagonal- and square-net nanotubes, taking also into ac-count the effect of deformations described above. The rela-tionship between /H9267and the band gap of CSNTs is shown in Fig.8. Similarly to other INTs,28the band gap of hexagonal- net CSNTs decreases with linear density /H9267, irrespective of the chiral vector, and converging to the limiting value for theplanar sheet. As a result, as evidenced by the electronic struc-ture of the planar CdSe sheet, the change in the band gapwith /H9267can be ascribed to the degree of /H9268-/H9266hybridization FIG. 6. Band structure and DOS for /H208495,0/H20850and/H208495,5/H20850hexagonal- net CSNTs. Dashed and dotted lines represent the DOS projected onstates of Cd and Se atoms, respectively. FIG. 7. Band structure and DOS for /H208495,0/H20850and/H208495,5/H20850square-net CSNTs. Dashed and dotted lines represent the DOS projected onstates of Cd and Se atoms, respectively.MERCURI et al. PHYSICAL REVIEW B 82, 155429 /H208492010/H20850 155429-4induced by mechanical strain,29which, in turn, is a function of the diameter only. The band gap of the square-net CSNTs, however, is also related to the rolling vector, with values for the /H20849n,n/H20850sq morphologies generally smaller than for /H20849n,0/H20850sqCSNTs for linear densities larger than 2 atoms /Å/H20849corresponding to anapproximate outer diameter of 5.6 Å /H20850, reflecting the differ- ences in the band structure discussed above. In particular, theband gap of the /H20849n,0/H20850 sqCSNTs reaches a maximum value /H208491.80 eV /H20850atn=6 and then decreases for nanotubes of larger diameter. Conversely, the band gap of /H20849n,n/H20850sqCSNTs de- creases monotonically with nand exhibits a slight oscilla- tion, due to the different symmetry of CSNTs with odd or even values of n, respectively. Remarkably, the smallest di- ameter square-net /H20849n,0/H20850CSNTs considered /H20849/H9267 /H110152.1 atoms /Å/H20850exhibits metallic character, due to relax- ation to a core-shell structure with formation of intermetallicCd-Cd bonds. 21 In conclusion, the electronic properties of single-walled CSNTs are found to be strongly dependent on the underlyingplanar morphology /H20849square or hexagonal net /H20850. In addition, for the square-net-based nanotubes, both electronic proper-ties and stability also depend upon the chiral vector associ-ated with the nanotube. These observations suggest that theband gap of such nanotubes may be effectively engineeredby selecting their diameter. The authors thank the Oxford Supercomputing Center and the Center for Information Services and High-PerformanceComputing /H20849ZIH/H20850Dresden for providing computing time. We also thank G. Seifert for helpful discussions. F.M. thanks theAccademia dei Lincei for the “A. Vaciago” research grant. *Corresponding author; merc@thch.unipg.it 1R. Saito, G. Dresselhaus, and M. Dresselhaus, Physical Proper- ties of Carbon Nanotubes /H20849Imperial College Press, London, 1998/H20850. 2M. Bar-Sadan, I. Kaplan-Ashiri, and R. Tenne, Eur. Phys. J. Spec. Top. 149,7 1/H208492007/H20850. 3J. Sloan, A. Kirkland, J. Hutchison, and M. Green, C. R. Phys. 4, 1063 /H208492003/H20850. 4J. Hulteen and C. Martin, J. Mater. Chem. 7, 1075 /H208491997/H20850. 5B. Yao, Y. Chan, X. Zhang, W. Zhang, Z. Yang, and N. Wang, Appl. Phys. Lett. 82, 281 /H208492003/H20850. 6R. Meyer, J. Sloan, R. Dunin-Borkowski, A. Kirkland, M. No- votny, S. Bailey, J. Hutchison, and M. Green, Science 289, 1324 /H208492000/H20850. 7V. Colvin, M. Schlamp, and A. Alivisatos, Nature /H20849London /H20850370, 354/H208491994/H20850. 8H.-S. Shim, V. Shinde, J. Kim, T. Gujar, O.-S. Joo, H. Kim, and W. Kim, Chem. Mater. 21, 1875 /H208492009/H20850. 9X. Jiang, B. Mayers, T. Herricks, and Y. Xia, Adv. Mater. 15, 1740 /H208492003/H20850. 10L.-S. Li, J. Hu, W. Yang, and A. Alivisatos, Nano Lett. 1, 349 /H208492001/H20850. 11E. Rabani, B. Hetényi, and B. Berne, J. Chem. Phys. 110, 5355 /H208491999/H20850. 12B. Saunders and M. Turner, Adv. Colloid Interface Sci. 138,1 /H208492008/H20850. 13T.-J. Lin, H.-L. Chen, Y.-F. Chen, and S. Cheng, Appl. Phys. Lett. 93, 223903 /H208492008/H20850.14W. Chan and S. Nie, Science 281, 2016 /H208491998/H20850. 15D. L. Klein, R. Roth, A. K. L. Lim, A. P. Alivisatos, and P. L. McEuen, Nature /H20849London /H20850389, 699 /H208491997/H20850. 16W.-K. Woo, K. Shimizu, M. Jarosz, R. Neuhauser, C. Leatherdale, M. Rubner, and R. Bawwnsi, Adv. Mater. 14, 1068 /H208492002/H20850. 17E. Rabani, J. Chem. Phys. 116, 258 /H208492002/H20850. 18C. Bishop and M. Wilson, J. Mater. Chem. 19, 2929 /H208492009/H20850. 19J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 /H208491981/H20850. 20D. Sanchez-Portal, P. Ordejon, and E. Canadell, Principles and Applications of Density in Inorganic Chemistry II , Structure and Bonding Vol. 113 /H20849Springer, Berlin, 2004 /H20850, pp. 103–170. 21N. Kuganathan and J. C. Green, Chem. Commun. /H20849Cambridge /H20850 2008 , 2432. 22J. LaFemina, Surf. Sci. Rep. 16, 137 /H208491992/H20850. 23B. V. Lebedev, L. Ya. Tsvetkova, and K. B. Zhogova, Thermo- chim. Acta 299, 127 /H208491997/H20850. 24C. Bishop and M. Wilson, Mol. Phys. 106, 1665 /H208492008/H20850. 25N. Hamada, S. I. Sawada, and A. Oshiyama, Phys. Rev. Lett. 68, 1579 /H208491992/H20850. 26A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. 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PhysRevB.91.115125.pdf

PHYSICAL REVIEW B 91, 115125 (2015) Dilute magnetic topological semiconductors Kyoung-Min Kim,1 Yong-Soo Jho,1 and Ki-Seok Kim1 ,2 1 Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea 2 Institute o f Edge o f Theoretical Science (1ES), Hogil Kim Memorial Building, 5th Floor, POSTECH, Pohang, Gyeongbuk 790-784, Korea (Received 11 September 2014; revised manuscript received 10 February 2015; published 13 March 2015) Replacing semiconductors with topological insulators, we propose the problem of dilute magnetic topological semiconductors. Performing the renormalization group analysis for an effective field theory, where doped magnetic impurities give rise to a spatially modulated random axion term, we find a novel insulator-metal transition from either a topological or band insulating phase to an inhomogeneously distributed Weyl metallic state with such insulating islands, where extremely broad distributions of ferromagnetic clusters combined with strong spin-orbit interactions are responsible for the emergence of randomly distributed Weyl metallic islands. Since electromagnetic properties in a Weyl metal are described by axion electrodynamics, the role of random axion electrodynamics in transport phenomena casts an interesting problem beyond the physics of percolation in conventional disorder-driven metal-insulator transitions. DOI: 10.1103/PhysRevB.91.115125 PACS number(s): 71.10.Hf. 71.23.- k , 71.30.+h, 75.30.-m I. INTRODUCTION The role of localized magnetic moments in metal-insulator transitions [1] lies at the heart of modern condensed matter physics, for example, the mechanism of high-Tc superconduc tivity [2], the nature of non-Fermi-liquid physics near heavy- fermion quantum criticality [3], and the problem of metal- insulator transitions in doped semiconductors [4— 6]. Dilute magnetic semiconductors had been investigated for more than twenty years [7], where such spin-polarized electric currents have been realized but at low temperatures much below room temperature, prohibiting us from device applications. How ever, interactions between doped magnetic ions and a small number of charge carriers raised interesting and fundamental physics problems, for example, the nature of the RKKY (Ruderman-Kittel-Kasuya-Yosida) interaction [8] away from good metals, the mechanism of ferromagnetic ordering in randomly distributed magnetic ions, and anomalous transport properties in the presence of scattering with random magnetic impurities. In this study, we propose the problem of dilute magnetic topological semiconductors, replacing nontopological semi conductors with topological semiconductors [9,10]. Recently, it has been reported that the evolution of average mag netic correlations from ferromagnetic to antiferromagnetic in Fe, Bi 2Te3 gives rise to changes in transport properties of magnetoresistivity and Hall effect, identified with topological “phase transitions” driven by dynamics of doped magnetic impurities, where the paramagnetic topological “semicon ductor” of Bi2Te 3 turns into a normal semiconductor with ferromagnetic-cluster glassy-like behaviors around x ~ 0.025, and it further evolves into a topological “semiconductor” with valence-bond glassy-like behaviors, which spans over the region between x ~ 0.03 up to x ~ 0.1 [11], Although these experiments could not reach the semiconducting regime, interactions between randomly distributed magnetic ions and itinerant electrons with topological properties cast a novel physics problem beyond the problem of dilute magnetic semiconductors, that is, interplay between the evolution of magnetic correlations in localized magnetic moments and anomalous transport phenomena in itinerant electrons with topological properties.Performing the renormalization group analysis for an effective field theory to describe the first occurring “phase transition” within the “ferromagnetic” regime, we find that the variance of the distribution for randomly quenched effective magnetic fields due to ferromagnetic clusters goes toward an infinite fixed point as the concentration of magnetic ions increases. Recalling that time reversal symmetry breaking in this strong spin-orbit coupled system gives rise to the Weyl metallic state [12-15], the infinite variance fixed point implies the emergence of randomly distributed Weyl metallic islands which coexist with topological semiconducting phases inhomogeneously, where local breaking of time reversal symmetry due to ferromagnetic clusters with large effective magnetic fields is responsible. See Fig. 1. Based on this physical picture, we propose a schematic phase diagram of Fig. 2 in (A .so,r ,7 ’), where A .so is the spin-orbit coupling constant, T is the variance of the distribution for randomly quenched effective magnetic fields given by ferromagnetic clusters, and T is temperature. First of all, we find an unstable fixed point with ( k [ :0,re ) at T = 0, where k£0 corresponds to the quantum critical point of a topological phase transition between a topological semiconductor (kso— > o o ,r = 0) and a normal semiconductor (ks o = 0,F = 0 ) [16], and Tc identifies a novel disordered quantum critical point between one fixed point of (k s o ,r = 0) and the other of ( k j : 0 ,r -* ■ oo) at T = 0 . Although the nature of this infinite variance fixed point is not fully clarified within our perturbative renormalization group analysis, we conjecture to identify such a fixed point with the inhomogeneously distributed Weyl metallic state which coexists with insulating islands, as discussed before. The appearance of inhomogeneously distributed Weyl metallic islands suggests a novel disorder-driven insulator- metal transition, regarded to be counterintuitive since the metallic state results from increasing the strength of magnetic disorders. Frankly speaking, it is not clear at all whether or not the finite variance fixed point corresponds to this insulator-metal transition exactly because a percolation-type transition must be involved in order to have a genuine metallic state, where Weyl metallic islands should be connected to each other. However, this metal-insulator transition is beyond the percolation physics [17] since electromagnetic properties 1098-0121/2015/91(11)/115125(24) 115125-1 ©2015 American Physical Society KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) FIG. 1. (Color online) Band structure of the Weyl metallic state. Each band structure corresponds to H = 0, H < m(\k\), H 2> m(\k\), and H 3> m(\k\) = 0, respectively, where H = Hz is an effective magnetic field, which originates from ferromagnetic clusters conjectured to appear from RKKY interactions. The last two cases are identified with a Weyl metallic phase. in a Weyl metal are described not by conventional Maxwell dynamics but by axion electrodynamics [10,18-20]. See Appendix B for axion electrodynamics in a Weyl metallic phase. The role of random axion electrodynamics in transport phenomena of the disordered metallic state implies that the present metal-insulator transition does not fall into the class of either Anderson-type [5] or Mott-type [1] metal-insulator transitions [6], regarded as a novel class of metal-insulator transitions. The problem of dilute magnetic topological semiconductors differs from that of randomly doped magnetic impurities on the surface state of a topological insulator. One may speculate that half-quantized Hall conductance appears with Anderson localization if doped magnetic ions exhibit ferromagnetic ordering. On the other hand, an anomalous metallic phase can emerge to fall into the universality class of the quantum Hall plateau-plateau transition in the paramagnetic phase although an actual transition occurs between the quantum Hall plateau and the gapless surface state, where Anderson localization may not exist due to the presence of time reversal symmetry on average [21]. Although self-consistency must be incorporated to determine both the magnetic structure and Anderson localization at the same time, this surface-state problem should be distinguished from the problem of dilute magnetic topological semiconductors in the respect that axion electrodynamics does not appear. Randomly arising axion electrodynamics is the characteristic feature of dilute magnetic topological semiconductors. II. MODEL HAMILTONIAN We start from an effective-model free energy [20] F =/O O dIr r > P (Ir r > ) In -OO /d3k ( j y — y ^ a(*-T )|(9 T - F)Iaa' ® I aa'J Dxj/^DSrx exp + vk ■ o aa'FIG. 2. (Color online) A schematic phase diagram based on the renormalization group analysis for an effective field theory Eq. (3). The y axis represents a spin-orbit coupling constant A s o or a mass parameter m, and the x axis denotes a variance of the distribution for randomly quenched effective magnetic fields T, given by ferromagnetic clusters. T is temperature. Arrows mean renormalization group flows. There exist three stable fixed points, where (As o = 0,F = 0) and (As o — > oo,T = 0) correspond to a band insulating phase (BI) and a topological semiconducting state (TS), respectively, while the infinite variance fixed point of (A ^T — > • oo) is interpreted to be an inhomogeneously distributed Weyl metallic phase (disordered WM) which coexists with randomly distributed insulating states. Two unstable fixed points imply two kinds of phase transitions. (A^.F = 0) is a quantum critical point between the topological insulating and normal semiconducting phases in the absence of magnetic impurities. (A [;0,r c) identifies a novel quantum critical point, conjectured to be associated with an insulator-metal transition, where the metallic state is not a conventional diffusive Fermi-liquid phase but quite an unconventional inhomogeneous Weyl metallic state. However, it is not clear at all whether or not this critical point coincides with this novel metal-insulator transition since a percolation-type transition must be involved in order to have a genuine metallic state, where Weyl metallic islands should be connected to each other. An important point is that electromagnetic properties of a Weyl metal are described by axion electrodynamics, where they are unknown transport properties in this disordered metallic state and at the disordered quantum critical point. rL ' + m(\k\)I oo yy «* d(x''J fa'a>{k,T)-J dr x J d3rJ\l/la(r,r)(< ja a ' < g > I aa') f a'a'{r,x) ■ S ( r ,r ) H d?r'Irr'S(r,x) ■ S(r',r) — S, (1) Here, t/rC T a (A:,r) represents a four-component Dirac spinor, where a and a are spin and chiral indexes, respectively. o aai and xaa> are Pauli matrices acting on spin and “orbital” spaces. The relativistic dispersion is represented in the chiral basis, where each eigenvalue of r z aa, expresses either + or — chirality, respectively. The mass term can be formulated as m(\k\) = m — p\k\2, where sgn(m)sgn(p) > 0 corresponds to a topological insulating state while sgn(m)sgn(p) < 0 115125-2 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) corresponds to a normal band insulating phase. The topological insulating state identified with positivity of the mass of Dirac electrons corresponds to the case that the spin-orbit coupling constant is larger than its critical value. On the other hand, when the spin-orbit coupling constant is smaller than its critical value, the sample resides in a band insulating phase, where the sign of the mass is negative. /z is a chemical potential, controlled by doping. Magnetic impurities S (r,r) experience RKKY interactions [22], denoted by /rr< . Since they are doped at random positions, we take the coupling constant as a random variable, described by the Gaussian distribution function P(lr r ' ) - Such magnetic impurities also interact with conduction electrons, described by the Kondo-type interaction J.SR is a Berry phase term in the spin coherent-state representation [23]. Hinted from the recent experiment [11], we take into account randomly distributed ferromagnetic “cluster” ordering. Introducing a neutral fermion field to describe an impurity spin as S (r,r) = \fl(r,z)oao'fa'{r,z) with a single-occupancy constraint f}(r,z)fa(r,z) = 1, and performing the standard decomposition [2] in this effective model, we construct •7r M F [^> r>kr;/r, 7’] = -T J dIr r P (Irr')\n J D ij/a c t (r ,z)Dfa(r ,z) exp \r f l^ T f cP kiT j ( 2 r f v'“ “ ’T)K8' " '*)'a e x yy R act I aa' r\,r r + vk ■ oa a ' ® T z a a , + m{\k\)la a ' ® T * a ,} ilr a' a i(k,z) — J dz J d3 r J fla(r,z)(oaa'® Ia a ' ) \ l r < T > a ir,z)-Q r - J^,T dz J d3 r J d3 r' fl(r,z)[(dz + X r)8 aa, & ( 3 ) (r - r') + IrA^r’ ■ < J )a o '}fa'{r' ,r) - j j d3 r J d3 r'[kr< 5 (3)(r - r') - I r r < J > r ■ 4 > r- (2) where < I > r is a ferromagnetic order parameter and kr is a Lagrange multiplier field to impose the single-occupancy constraint, given by a functional of I rrr in the self-consistent mean-field analysis. This magnetic evolution gives rise to the variation in transport properties as discussed in the introduction. In particular, normal metallic transport properties in magnetoresistivity and Hall effect appear from topological semiconducting transport behaviors in the ferromagnetic-on-average region before the antiferromagnetic-on-average region. According to the above physical picture, randomly frozen magnetic clusters described by 4 > r generate effective magnetic fields to itinerant electrons. As a result, the “local” spectrum of itinerant electrons becomes modified into Er(k) = - n ± Jv2 (k2 x + k2 y) + [ J \ < t > r\± ^UH\k\) + V 2 k2 z]2 . This local spectrum implies that the gap of a topological semiconductor vanishes at position r in the case of 7 |O r | > |m(|&|)|, splitting the Dirac spectrum into a pair of Weyl points locally. See Fig. 1. Then, a Weyl metallic island arises from a topological semiconducting island at position r, regarded as being an insulator-metal “transition” driven by random magnetic moments. Inhomogeneously distributed topological semiconductor and Weyl metal islands are thecharacteristic feature of the dilute magnetic topological semi conductor in the ferromagnetic regime. IT T . RENORMALIZATION GROUP ANALYSIS In order to understand the nature of such inhomogeneous mixtures, we construct an effective field theory based on the above physical picture. It is straightforward to show that randomly quenched effective magnetic fields (J$r) corre spond to random chiral gauge fields (c), rewriting the effective Hamiltonian of the ferromagnetic regime into the standard representation of the Dirac theory with the introduction of Dirac gamma matrices as follows, S = / d4 x \j/(i y M 3M — m + K 'V gJVg where the distribution function of cf i is assumed to be Gaussian with its variance T, meaning that the distribution of effective magnetic moments of ferromagnetic clusters is Gaussian [24]. See Appendix A. Applying the replica trick and performing the Gaussian integral for random chiral gauge fields, we find that effective nonlocal-in-time “interactions” of chiral currents arise between different replicas [5]. Rewriting the bare action in terms of renormalized fields and renormal ized coupling constants with the introduction of counterterms, i.e., Sb — S r + Scz, we construct an effective field theory for renormalization group analysis: Sr - Mdd ~'r /o +'l'R(r,z)(iyT d T + ivR y ■ V - mR )i/a R (r,z) y- dz'<ira R{r,z)Y^Y 5VAr,r)ira R(r,zl )YR .Y 5VR(J,z''^, Scz = J dz J dd lr ^ x /r a R (r,z)(8pYT 9 r + S^ivR Y ■ V - S m m R )f%(r ,z) - M r — [ d z’ira R {r,z)Ylx Y 5VR {r,z)^ a R(r,z')Ylx Y 5VR(r,z') J - Jo(3) 115125-3 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) Here, i/ ^ ( r ,r ) is a renormalized electron field with a replica index a — 1......... R, and vR,mR,TR are renormalized velocity, renormalized mass, renormalized variance, respectively. 5^, <5*, Sm, and < 5 r are introduced to absorb infinities resulting from quantum corrections. These renormalized field and parameters are related with the bare field and parameters as follows: rB (r, r) = zf*rR (r, r), vR = Zk f Z^~x vR , m B = zm z% ~ x m R , rB = iiszrz“ ~ 2 r R , (4) where renormalization constants are given by z ; = 1+8%, z* = i+«*, Zm = 1 + 8 m, Zy = 1 + S p . (5) Here, // is a scale of momentum, distinguished from the chemical potential before and s — d — 3 . Before going further, we would like to clarify that the present study focuses on insulating phases as the first step while the previous experiment [11] has been performed in the metallic region. In this respect it needs some care to apply our renormalization group analysis to the experiment directly. We come back to this point below. Performing the standard procedure for the renormalization group analysis, we find renormalization group equations, where both vertex and self-energy corrections are introduced self-consistently. See Fig. 3, where all quantum corrections are shown as Feynman diagrams up to the one-loop order for vertex corrections and the two-loop order for self-energycorrections. All details are shown in Appendixes C, D, E, and F. Here, we point out that the renorm alization constant o f the “interaction” vertex remains to be Z r = 1, where the divergence of the particle-hole ladder diagram is canceled by that of the particle-particle channel while other vertex corrections do not give rise to divergences. On the other hand, the Fock diagram in the one-loop order and both the rainbow diagram and the crossed diagram with a vertex correction in the two-loop order for self-energy corrections contribute to the wave-function renormalization constant while others do not cause divergences. In particular, the role of the rainbow diagram turns out to be crucial in the emergence of a novel metallic fixed point of -* oo and mR -»• 0, identified with a disordered Weyl metallic phase. As a result, we find d In F/f d in //- arT/? - byT2 R, d\nmR d In //F/? T bm r R, (6) where positive numerical constants are given by ar = hr = am = f , and b m — ZL. We em phasize that the chemical potential lies between the band gap. The renor m alization group flow of these equations is shown in Fig. 4, which confirms our proposed phase diagram (Fig. 2). First, we focus on the quantum critical point of the topological phase transition, identified with mR — 0. Then, it is easy to see that there exists an unstable disorder fixed point = Tc, which FIG. 3. Feynman diagrams up to the one-loop order for vertex corrections and the two-loop order for self-energy corrections. First of all, we point out that quantum corrections including fermion loops vanish in the replica limit of NR - > ■ 0. Three types of quantum corrections contribute to the vertex renormalization. It turns out that the 1/e divergence in the ladder diagram of the particle-hole channel is canceled by that of the particle-particle channel, where the diagram with a vertex correction does not cause the divergence. As a result, the vertex renormalization constant remains to be Zr = 1 . The Fock diagram results in the 1 /e divergence for the fermion self-energy in the one-loop order, and both the rainbow diagram and the crossed diagram with a vertex correction also cause that in the two-loop order. The wave-function renormalization constant Z£ is given by these three contributions. In particular, the role of the rainbow diagram turns out to be essential, meaning that we cannot reach the disorder-driven novel metallic fixed point without it, where the sign of b m changes from negative to positive in the renormalization group equation for the mass parameter. 115125-4 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) FIG. 4. (Color online) Renormalization group flow as the solution of the coupled renormalization group equations (6). A characteristic feature is the emergence of a novel stable fixed point ( mR = 0 ,r R — > oo), identified with an inhomogeneously distributed Weyl metallic phase which coexists with insulating islands (right). This metallic fixed point originates from random fluctuations of chiral currents due to effective random magnetic fields of ferromagnetic clusters. There exists a quantum phase transition of the second order between the Dirac semimetallic state ( mR = 0,TR = 0) and the disordered Weyl metallic phase ( mR = 0,1^ -* oo), identified with a disorder-driven quantum critical point (mR = 0,TS = Tc) (left). On the other hand, it would be the first-order quantum phase transition between an insulating phase of either (mR — > oo,TR = 0) (topological insulator) or (mR — ► — oo,T R = 0) (normal semiconductor) and the disordered Weyl metal state. See the text for more details. separates two stable fixed points of VR = 0 and TR -> oo. This means that the Dirac semimetallic state, arising at the critical point, remains to be stable in the case of weak randomness, expected since the density of states vanishes. However, it is quite interesting that antiscreening appears for random fluctuations in chiral currents in contrast with those in charge currents. Recall that the electric charge is screened to decrease at low energies [2], Second, we start from an insulating phase, increasing the variance of random chiral gauge fields. Then, we reach a novel stable fixed point o f (mR -» 0 ,r ^ oo), separated from two insulating fixed points of (m*-^ oo, Tfl — > ■ 0) (topological insulator) and — oo,Tff — > 0) (band insulator). The appearance of this fixed point is quite surprising since the mass param eter renormalizes to vanish, which originates from random fluctuations of chiral currents. Although metallicity can be enhanced by the interplay between disorders and interactions [25], the present metallicity results from the interplay between randomness and topology of a band structure in the approach of an effective field theory. One may understand the emergence o f this novel metallic fixed point as follows. First of all, T/e — * ■ oo is difficult to be com patible with m R -* ± o o since T R -» oo implies that most regions becom e Weyl metallic with mR, giving rise to gap closing inevitably. The only consistent way for the existence of (mR -+ ± o o ,F R — * oo) is that the mass param eter increases faster than the variance of effective magnetic fields. Actually, we find that the mass gap increases faster than the variance if we neglect the renormalization given by the rainbow diagram. This means that the Weyl metallic island does not occur and the insulating phase survives although the variance goes toward an infinite fixed point. We believe that this does not make any sense because wefail to figure out the nature of a quantum phase transition between two insulating phases given by (mR — » oo,T r = 0 ) and (mR -* oo, > oo), focusing on the semiconducting (either topological or normal) side. Suppose that the sample lies deep inside in the band or topological insulating region, i.e., either m — * ■ —oo or m — » • oo, respectively. It is certainly true that there exists a quantum critical point to distinguish T« -» 0 from Fr — ► oo, which resides in the weak-coupling regime, thus justified within our perturbative renormalization group analysis. Then, we have two stable fixed points along the line of either m ^ - o o or « oo. W hat does the quantum critical point mean? Rem em ber that we are in an insulating phase. It is difficult to imagine the existence of such a quantum critical point in this situation. If we believe in the existence of the quantum critical point, certainly justified within our renorm alization group analysis, we conclude that the infinite randomness fixed point or a strong-coupling variance fixed point beyond the perturbative regime at least should exist. Incorporating the contribution of the rainbow diagram, we observe that the sign of bm in the renormalization group equation for the mass param eter changes from negative to positive, giving rise to the disorder-driven metallic fixed point. We interpret this infinite variance fixed point with zero mass gap as an inhom ogeneously distributed Weyl metallic state, where transport properties are described by axion electro dynamics, which should be distinguished from the diffusive Ferm i-liquid state [5], The nature of the quantum phase transi tion between (m« — > ± o o ,T R = 0) and (mR = 0 ,F S -» oo) is expected to be the first order, where an insulating phase persists just before the disordered Weyl metallic state. How ever, we cannot exclude the possibility of an additional phase transition associated with percolation, which may be 115125-5 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM responsible for a genuine insulator-metal transition beyond the present description. We speculate what would happen when the chemical potential lies above the band gap, resulting in a Fermi surface. First of all, the presence of the Fermi surface changes the engineering dimension of the variance F« from +1 to -1 , making it relevant. Keeping the physics of antiscreening, we write down the renormalization group equation for the “interaction” vertex where c is a positive numerical constant. This allows the -h> - oo fixed point only in the low-energy limit. Considering the emergence of randomly distributed Weyl metallic islands in the case of zero chemical potential, we expect that the Dirac point is separated into a pair of Weyl points locally due to local time reversal symmetry breaking if the critical point of m R = 0 is taken into account for example, and thus the single Fermi surface with degeneracy in the Dirac spectrum splits into a pair of chiral Fermi surfaces locally, which encloses each Weyl point with definite chirality. The nature of a pair of chiral Fermi surfaces turns out to differ from a normal Fermi surface in the respect that both the Berry curvature, which originates from the Weyl point identified with a magnetic monopole in momentum space, and chiral anomaly, which means that this pair of Weyl points are not independent but connected to each other, change electromagnetic properties seriously, described by the axion electrodynamics [12,15,19,20], as discussed before. The emergence of a randomly distributed pair of chiral Fermi surfaces is an essential feature when the chemical potential lies above the band gap, regarded to be an extended physical picture of the case of zero chemical potential. IV. DISCUSSION AND SUMMARY A. Role of the time component of the random chiral gauge field One may criticize that the effective field theory [Eq. (3)] contains an additional ingredient, compared with our “mi croscopic” lattice model [Eq. (2)], which corresponds to the time component of the random chiral gauge field. It plays the role of a random chiral chemical potential, given by /x5 = /x+ — /i_, where ± denotes an index of chirality. Actually, we repeated performing our renormalization group analysis in the absence of the random chiral chemical potential. The one-loop self-energy correction given in the first diagram of Fig. 3 is modified as follows: £ (1)|eo=0 = - ^ - ( - 3 p o T ° + 3m) + 0(1). Recall ^ ^ lc 0 5 iO = — f(— 2poy® + 4m) -I- 0(1) in Appendix Dlb, where the time component of the random chiral gauge field is introduced. This modification does not change our renormalization group result, where some coefficients are modified in the following way: a T ^ 2^T ) &m ^ 4^m*PHYSICAL REVIEW B 91, 115125 (2015) The two-loop rainbow self-energy correction (the second diagram of Fig. 3) can be obtained as r 2 i £ (2 )’ r|co=° = ^ - ( 2 p o y ° + 6m )+ 0(1), compared with £ (2)’ r = ^ ± ( 3 p o / + 10m) + 0(1) of Appendix Dlb. On the other hand, it is not easy to evaluate the two-loop crossed self-energy diagram (the third diagram of Fig. 3), where loop momenta are highly entangled. Although we cannot give a definite solution for this correction, we spec ulate that this correction is not essential since the coefficient of the 1/e divergent term is expected to be much smaller than that of the two-loop rainbow self-energy correction. Recall + £ >PmYm + 2 m ) + 0(1) when the time component of the chiral gauge field is introduced. In this respect we conclude that our renormalization group result will not be modified regardless of whether the time component of the random chiral gauge field is introduced or not. Physically speaking, the time component of the random chiral gauge field allows random electric bulk currents between a pair of Weyl points in a puddle of a Weyl metallic state since there exists a difference of the chemical potential, usually referred to as chiral magnetic effect [27], Although it is not easy to give any definite intuitive claims, we are expecting that the presence of such random electric bulk currents within the Weyl metallic puddle makes the system unstable, which may cause the chiral chemical potential to vanish. B. Large-A extension of the effective field theory One may doubt the existence of the infinite randomness fixed point since it lies well outside the regime of validity of our renormalization group analysis. Moreover, whenever the renormalization group equations [Eq. (6)] are valid, the renormalized mass m R is highly relevant; from Fig. 4 it seems that this behavior changes only for » 1, again, well beyond the range where Eq. (6) applies. In this respect one may claim that in reality mR is relevant for all FR . A standard way to lead such a strong-coupling fixed point toward a weak-coupling one is to extend the spin degeneracy of electrons from a = f,4 to a = 1,..., A, referred to as multi flavors. Here, we consider the following type of a multiflavor model, Sd is oc Yla,p=i where both gamma matrices and space-time integrals are omitted for simplicity. Based on this large-A generalization and following exactly the same procedure as the above, we evaluate self-energy corrections, where vertex corrections turn out not to appear as in the case of the finite-A model. As a result, we obtain d In r R _ i _ 2NFR _ 3 A 2F 2 din/X 7T 7T2 dlnmR = i 3NFR | 7 A 2r 2 din/x it 2 ji2 Compare these renormalization group equations with Eq. (6). It is straightforward to see that the quantum critical point is shifted from = — ^ ~ 2 ) ~ 0.81 of Eq. (6) to T^ = More importantly, the large-A generalization causes the T* value, at which the sign of the renormalization group 115125-6 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) equation for the mass parameter is altered or the relevance of the mass parameter turns into irrelevance, to lie in the weak-coupling regime, i.e., from T* = 3jr+^ ^ ~ 7.20 of Eq. (6)toT^ — 1 3 ^+^ a/2 3 ^ M2. As a result, our perturbative renormalization group analysis can access the regime which shows the nature of the infinite randomness fixed point. It is noticeable that the large-A model does not take into account the crossed diagram for the self-energy correction while our previous calculations contain the crossed diagram, where its contribution is small. We do not claim that this large-iV generalized model is physical. However, we would like to point out that there must be an extended model which leads the strong-coupling fixed point to be within the weak-coupling regime. C. Experimental signatures Although we expect that the present infinite variance fixed point should exhibit the strong inhomogeneity, its thermo dynamic nature looks very complicated, where randomly distributed ferromagnetic clusters would interact with each other beyond our present description. Then, quantum Griffiths phenomena [26] are expected to appear, implying that the nontrivial power-law exponent of the spin susceptibility at low temperatures should show nonuniversal continuous evolutions within the ferromagnetic-on-average region. Such quantum Griffiths effects may be incorporated, resorting to a power- law distribution function instead of the Gaussian distribution function for random chiral gauge fluctuations. We believe that this infinite randomness fixed point can be verified by atomic force microscopy. Although the local electronic spectrum will not show strong inhomogeneity around zero bias in the metallic regime, it should be observed deep inside the spectrum around — /z, where /z is the chemical potential. In a certain region a gap feature appears while such a gap does not exist in the vicinity of the same energy scale at a different position, where a Weyl metallic island exists. See Fig. 5. FIG. 5. (Color online) A schematic picture for the local density of states probed by atomic force microscopy Consider the case when the chemical potential lies above the band gap, which corresponds to a metallic state. Recalling that the infinite variance fixed point is identified with inhomogeneous mixtures between a normal Fermi surface with degeneracy in a Dirac spectrum and a pair of chiral Fermi surfaces without degeneracy in a pair of Weyl spectrum, we predict that a gap feature appears in a certain region which corresponds to the region of small ferromagnetic clusters while a V-shaped pseudogap feature results at a different position which coincides with the region of large ferromagnetic clusters exceeding the band gap.An important unsolved question is how to evaluate transport coefficients at this infinite variance fixed point. As discussed before, both Berry curvature and axion electrodynamics play a central role in electromagnetic properties of Weyl metallic islands. Recalling that each Weyl point can be identified with a magnetic monopole in momentum space, inhomogeneous mixtures of topological semiconductors and Weyl metals are interpreted as randomly distributed monopole-antimonopole pairs of momentum space. This physical picture leads us to speculate that effects of such random monopole-antimonopole pairs may be canceled on transport coefficients, giving rise to normal “metallic” behaviors in magnetoresistivity and Hall effect. However, this speculation focuses on only the aspect of Berry curvature, missing the role of axion electrodynamics in transport coefficients. It is the problem of dilute magnetic topo logical semiconductors to develop how to calculate transport coefficients at this infinite variance fixed point beyond the spec ulation, incorporating both effects of Berry curvature and axion electrodynamics. D. Summary In summary, we proposed the problem of dilute magnetic topological semiconductors, the novel physics of which be yond that of dilute magnetic semiconductors is the emergence of randomly distributed Weyl metallic islands. Performing the renormalization group analysis for an effective Dirac theory with random chiral gauge fluctuations, expected to encode the information of randomly quenched magnetic moments, we find that the variance of random chiral gauge fields reaches an infinite fixed point as long as average magnetic correlations remain to be ferromagnetic, which enforces the mass gap to vanish. As a result, we find a disorder-driven novel metallic phase and an associated insulator-metal phase transition beyond either the Anderson or the Mott metal- insulator transition, where this metallic state appears to be identified with the infinite variance fixed point. Recalling that quantum Griffiths phenomena may arise in the vicinity of this infinite variance fixed point, we predicted continuous nonuniversal changes in the temperature exponent of the uniform spin susceptibility. In addition, we claimed that this picture of inhomogeneous mixtures can be verified by atomic force microscopy. However, a difficult fundamental problem remains, that is, how to understand transport coefficients near this infinite variance fixed point, where random axion electrodynamics arises to govern electromagnetic properties, identified with the problem of dilute magnetic topological semiconductors. ACKNOWLEDGMENTS This study was supported by the Ministry of Education, Science, and Technology (No. 2012R1A1B3000550 and No. 2011-0030785) of the National Research Foundation of Korea (NRF) and by TJ Park Science Fellowship of the POSCO TJ Park Foundation. We would like to thank V. Dobrosavljevic for insightful discussions. K.S. also appreciates discussions with Jeongwoo Kim. 115125-7 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) We show all technical details for our renormalization group analysis. In Appendix A we discuss our effective field theory, referred to as a random chiral-gauge-field model. In Appendix B we prove that axion electrodynam ics works in a Weyl metallic state instead of normal M axwell dynamics. In Appendix C we construct a replicated field theory for our renorm alization group analysis, where random chiral-current fluctuations play a central role. In Appendix D we perform the perturbative renormalization group analysis up to the one-loop order for vertex corrections and the two-loop order for self-energy corrections, respectively. In Appendix E we evaluate renormalization constants, and find renormalization group equations in Appendix F. We believe that this supplementary material will be helpful even to beginners for the weak-coupling renormalization group approach. APPENDIX A: FROM EFFECTIVE MAGNETIC FIELDS TO CHIRAL GAUGE FIELDS The kinetic-energy sector for dynamics of bulk electrons can be rewritten as the standard representation o f the Dirac theory in the following way: rP f d3 k SW \1r]= J dr J j ^ r aa(k,T){dT Iaa,® Ia a , + vF k-< Taa,® Tt a a ,+m(\k\)Ia ' T ,®Tx a a ,} ^ A k ,T ) = f d r J = J dr J d3 x\lr\x,x) - f2 0 hx2 0 hxl h x l 03r+ V f(-lV )■ hx2 0 o hx2a 0 0 — a0 h x i hx2 ( T j r * ’T) Q^+m/4 x 4 JtK*,r) d4 xxl/(x){iy°dT -I-vFiV ■ y -\-m}f(x), where Dirac gamma matrices are given by y° = ( Q ') and yk = ( ° * ) . Next, we consider an effective Zeeman coupling term, He f f = H0 - J $ ■ S = H0 + Hmt, where H0 is a free Dirac Hamiltonian and 0 is an effective magnetic moment given by a ferrom agnetic cluster with the Kondo coupling J. S = ^ ( I ® a )f represents a spin of itinerant electrons o f the bulk sample. Then, it is easy to show that an effective magnetic field is equal to a chiral gauge field in this Dirac theory, given by flint = ^ ■(I®o))ir = f^p ( - i y $ ■ pi ®o)ir = $ ( i y $ ■ yy5 ) ^ = ^r{C ■ yy5 )ir. In the last equality we used the identity of pi ® a = (° ‘)(£ °) = (° * ) = -yy5 . Generally, we introduce a time com ponent of the chiral gauge field and represent the Zeeman coupling term as Hm l = if(Cl , yI L y5 ) i j / . We reach the following expression for an effective field theory in the ferrom agnetic-on-average regime: S[$,M = J d4 xir(x)(iyI J - d tl+m)f(x) + J V l H * ) = + Sm t[ ijf,ijr , C „ ]. (A l) Then, an effective free energy becomes F = ~ T J DCnMP[C^(x)]\n J D({fr(x),\lf(x))exp(-S0 [ is ,ilr ] - 5int[ ^ , ^ ; C ^j), (A2) where B [C M (ar)] = is the distribution function for chiral gauge fields with their variance T, originating from randomly quenched ferromagnetic clusters. The coefficient JV is determined from the normalization condition of Af f = l . APPENDIX B: AXION ELECTRODYNAMICS IN THE WEYL METALLIC PHASE We start from QED4 (quantum electrodynamics in one time and three spatial dimensions) with the topological-in-origin E ■ B term, i QED4= J Di,(x) ex p l - f dr jd3 r\4r(x)(iyI M [ d l x + ieAM ] + m)f(x) - - F tlv F> J j V + 8{r) 16 T C 2fivps p p- c 1 flV 1 pd , (B l) where i j/{x) with x = (.r , r ) is a four-com ponent Dirac spinor and the coefficient (H r) is spatially modulated. Resorting to the anomaly equation d ^ y ^ y 5 ^)\6n2p p 1 pLV1 pd> (B2) 115125-8 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) one may rewrite the above expression as follow s: Zwm = J Df(x)exp J dr Jdir\i/r(x)(iy1 1 [ 3 M + ieAp] + m + c M y ' * y 5 ) V r ( * ) (B3) where the chiral gauge field = (c r ,c) is given by cT = 0 and c = V r6(r). In the previous section w e have shown that topological insulators under magnetic fields can be described by Eq. (A l), identical to Eq. (B 3). Effective magnetic fields are identified with V r0(r). It is straightforward to integrate over gapped fermion excitations, resulting in an effective field theory for electrom agnetic fields: 1 e2 £axio„ = F • llvF > tv + (B4) where tim e dependence in 9{r,t) has been introduced for generality. Applying the least-action principle to Eq. (B 4), w e reach M axw ell equations to describe the axion electrodynamics: 1 3D An ( 1 V D = 4 np -f 2a (V P 3 ■ B), V x H----------- = — j — 2a I (V P 3 x E) + -(d/P^B c dt c \ c 1 9 j BV X E + ------- = 0 , V • B = 0,c dt where w e follow the standard cgs notation with Pj(r,t) oc 0(r,t) and the fine-structure constant a [18].(B5) APPENDIX C: EFFECTIVE FIELD THEORY FOR RENORMALIZATION GROUP ANALYSIS IN THE REPLICA TRICK A physical observable is defined as follow s: (0(i/nA)) j DCp.P[Cp.]- j- D ^ ^ e-s 0[Y,Y]e-s MiY,r,CA(C l) w hich can be formulated from {0{f•*»-/DCpP[CpiSJIn Z[Cn,J], Z[Ca,J ] y=o /D (ij/ ^■)e - s o[^'V,le - sini['A.'/';Q]+/aGyo(^,i(r)> ^ 2) where J is a source coupled to an operator 0(^,x//) locally. Since the averaging procedure for disorder is not straightforward within this formulation, w e take the replica trick o f In Z = l i m ^ o Z ^ , where the replicated partition function is given by ZR = f D( i/ra ,T/fa) e x p [ — S [\[r a ,\l/a ■ , C p\ + f dA xJ 0({fra , x l r a)] with a replica index “a”. Then, the above expression is reformulated as follows: {OW,f-,Cp.)) = l i mJ — lim — fR ^ O R JDCpP[Cp'h ' /(ZR - 1) y=o a= 1 R = l i m - T [ R ^ O R ^ Ja= 1D C p P [C p ] I D (x[r\xlsa) Y / 0 ( i ' a , i ' a)e~ j : °= 'S['fa'r ''C" ] // D (fa ,ira )0 ($ a ^e-'ZLiSoir.We-T.Litf z lim — R - >0 R- ^ 2 J D (\J ra ,ljfa) 0(\l/a , f a)e~^‘= l So[^°^°]e^*x=i S o dzf o d x ’ S d ix\Wb T Y ‘ L Y $ ' l ’ b A W c - l ' Y ' lY 5V ,,) a = 1 J = lim - V f .R ^ O R 4-/ JD (i/r, \ J/ ) O (\ J ra, y j f a ) ea s 0ir,r]-T.a.b = i si ] s ir.r,tb,t bi (C3) where the average for disorder has been performed first to result in S dis[itb,tltb,if c = /o dr /q d r' f d2 x^[i!/b x {x)Ydy 2 isx(.x )][ijic X '(,x)Y pY 5 ’ i'x'tx)}. We point out the positive sign, arising from lowering the index from y M to Y p . Averaging for random chiral gauge fluctuations gives rise to effective interactions between chiral currents with all replicas, where effective all-tim e interactions allow momentum exchange only (not energy exchange). An effective field theory is given by Sb= / A ■ x ^ fiO y ° 9 o + vB iyK d k + m B ) f a B +JdTfd’fdd (C4) 115125-9 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) in the replica trick, where Einstein convention has been used. B (R) stands for “bare” (“renormalized”). Performing the dimensional analysis, where space and time coordinates have -1 in mass dimension, we observe dim[i/r] = dim|/w] = 1, and dim[T] = 3 - d . In this respect we perform the renormalization group analysis in d = 3 + e dimensions, where e is a small parameter. In the end of the calculation the dimension is analytically continued to the physical dimension (d = 4), setting 6 = 1. Taking into account quantum corrections, divergences are generated, which can be absorbed by renormalization constants, redefining fields and parameters. Rewriting the effective field theory in terms of renormalized fields and parameters, we obtainSB = J dd x ( Z ^ i Y 0 M a R+Z)lvR ^liYk d k ^ + Zm mR ^ a R ) - { - J dr J dr' J dd~lxZr~- with Yn = (Z^)1 /2 isa R , mB = Zm( Z p - 1 m«, vB = Z*{Z$)~'vR , and VB = Zr ( Z ; r 2r ff, where ZJJ is a wave-function renormalization constant, Zm , mass renormalization, z£, velocity renormalization, and Zr, vertex renormalization. It is more customary to rewrite this field theory, separating the renormalized part from counterterms that absorb divergences, in the following way: SB = S r + S qt, S R = J dd xYR (iy°d0 + vR iyk d k+m R )YR + j dr j dr’ j dd ~lx - V «)r„ (C5) Sct = J ddx8^YR (SpY°do + 8 kvR iyk d k + S m mR ) Yr + J dr j dr' J dd~1 x 8 r ^ ( f Ry lly 5irR)t (ijrc Ryp ,y5YR ) r,, where Z" = 1 + Z* = 1 + S*, Zm = 1 + 8 m, and Zr = 1 + < 5 r - APPENDIX D: EVALUATION OF FEYNMAN DIAGRAMS 1. Self-energy corrections a. Feynman diagrams Within the replica trick, we are allowed to perform the perturbative analysis. The Green’s function of G(x,y) = 77 Hp,q e~,p' x+iq’ yG(p,q) with G(p,q) — (ir(p),ir(q)) is evaluated as follows: G(p,q ) 1 ^ c = lint — / D {{jf , Y ^ a {p)^ra {q)e^^R ^' a= 1 limRi m ^ J D(xj/,Ye~^'S o l ' l '°'r] a= 1 J xS(3 \p i - P 2 + P 3 - P4)8po vpo8po po + (~y) I] Hifa(p )$ a(q ) - - Y ^ ,^ a^ p ^ a {~ ( i ^ b^ y , 1 'y i ^ b^P2)V{pi)Yk iY 5V {pA ) b ,c = 1 Pi V(p)'l'a(q)V(p\)Yd 'Y5'l'b(P2)V(P?,)YpY5V(P4) b .c .d .e P i,q , x Y ( q l)Yvy 5x lid(q2 )Y(q3)YvY5^ e(q^80)(Pi - p2 + Pi ~ p4)Spo po Spo po /3 \ qi - q 2 + q 3 - q4)8q oq o 8 qoq o + O(T^) 1 ^ A 1 ^ ' / F \ = + ^ { - Y ) T , ^ a^ a^ b^ y ' ly 5^ b(P2)r(P2)YpY5r(P4))o8(4 )(Pi) a = \ a , b , c = \ ' ' p i J _^ / p \ 2 _^ + r J 2 W a(PW a(q)'i'b(Pi)YI X Y5'l'b(P2)i'c(P3)YpY5V (P 4 )i'd(q\)YvY5'l'd(q2We(q3)yvy 5'l'e(q4))on h r - ' ' f i t . O sa ,b ,c d ,e x8< - 4 \ p i)8w (qi) + 0 ( T j), where we introduced a shorthand notation of 5 ^ 4)(p,) = Si3 )(p\ — p2 + p 2 — p4)8p opo 8pop o with the four-vector notation of x = (r,x) and p = (co„,p). The first term is just the bare propagator. From now on, we omit momentum arguments and summations for the moment in order to focus on replica indices. 115125-10 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) a , ab cQ.-.p < 3 T P P. a a “ * ~ T l * 3 “ a c c b b aa b b a FIG. 6. All possible quantum corrections in the first order without the replica limit. First-order corrections are given by (Fig. 6) l * lim - J2 { f ? t j f b k(Y*YS )kltlfC m (YliY5 )m nfnlo a,b,c=\ I * = 1™ £ I ] [{K f])0{ f i f b k ) 0{Vntfc m ) 0(Y*Y\l(Y»Y5 )mn+{Wfj)0[ K f b k ) 0{ ' l ' ! ’i'C m )o (Y 'X Y5 MY»Y5 ) n a,b,c= 1 “ 2 {fi^k ) 0 {^ntm)o (yM y5 )kl(YnY5 ) m n + 2 (P>*)0 (Y^Y^kliYuY^mn] 1 * lim - 5 3 [ G ^ . G ^ y + G * - G ^ G p y a,b,c= 1 -2Gft G?jGSm( y V 5)H(K,y5k ^ ^ 5 cc + 2G* G 't G?m( y V k ^ P U P A / A c ] lim — £ Gfltr[G W M G ^ y ^ A i A c + 5 3 Gatr[Gc(y V ^ ^ y ^ A A / A c a,b,c= 1 a,/7,c=l /? /? - 2 ^ G V V ) G f c tr[Gc(yM y 5) A A A c + 2 5 3 Gfl(yM y 5)Ge(y V ) G * A A A a,b,c= 1 fl,/?,c= l where “2” results from identical contributions and — comes from the odd number of fermion loops (one loop). Since all G reen’s functions with different replica indices are identical, the first term is proportional to R3, the second, R 2 , the third, R 2 . and the fourth, R . Taking the replica limit of lin i/^ o p only the fourth term survives. As a result, we find G (1) = G(p) (- - ) 53 y»y5 G(p - q)Y^G {p) = G{p)Y,mG{p) (D l) in the one-loop order (Fig. 7). Here, we point out that Feynman diagrams whose internal propagators are not connected to external lines (the third diagram in Fig. 6) always vanish in the replica limit. In other words, contributions with fermion loops vanish identically in the replica limit. FIG. 7. The Fock correction, which contributes to the wave-function renormalization constant only in the first order. 115125-11 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) FIG. 8. All possible second-order self-energy diagrams without the replica limit, omitting vacuum and one-particle reducible diagrams. Om itting vacuum and one-particle reducible diagrams, we have self-energy corrections in the second order: X W af a'irbY'i Y5'l'b'i'cY vl.Y5V i r dYvY5i ' b'i'cYvY5V ) o \iP i a,b,c,d,e l = R X [8GaGb y^y5 Gb Ga tT [G cy y 5 Gdy^y5 M G eyvy5 ] S a b S b a 8 c d S d c S e e - 4Gay»y5 Gbyvy5 GdY l x y5 a ,b , c ,d , e x Gc tT [G eyvy 5 ]8 a b S b d S d c S c a 8 ee-4 G ayl x y5 Gb tx[G cy vy5 Gdyvy5 Gey^y5 ] S a b S b a S c d S d e S e c + (-1)16Gflx V G V v K 5 x Ge tT [G cy vy5 GdY l x y5 ]8 a b 8 b e S e a 8 c d 8 d c + 8Gay, 1 y5 Gh y vy 5 Gdyvy5 Geyl ,y 5 Gc S a b 8 b d S d e 8 e c S c a + 8Gay»y5 Gby'’ y5 Gdylly5 Gcyvy5 Ge 8 a b 8 b d 8 d c 8 c e 8 e a ] . See Fig. 8. The first term is proportional to R3, the second, third, and fourth, R2 , the fifth and last, R . As a result, only the fifth and last term s survive in the replica limit. Therefore, the relevant self-energy correction is given by (Fig. 9) G[ 2 ) - r(p) = G(p) ^ X - q)yvyS G {p -q - l)yvy5 G(p - q)y^y5 G(p) = G (p)S (2)’ r G(p), G(2U (P ) = G(p) ^ X YtlY5 G(p - q)yv y5 G(p - q - l)y^y5 G(p - l)yvy5 G(p ) = G{p)T.( - 2 ) ' cG{p), T‘{2 )'r = ( — y) Y l ylMy 5G('P ~ ^ y vy 5G(' P ~ ^ ~ l'> y 'iy 5G^ p ~ ^ y ^ y 5’ ( ° 2 ) e'2u = ( — y) X ylly5G (p ~ q)yv y5 Gip - q - i)Y n Y 5 G(p - i)Y v y5 . (D3) «(«o = 0)q(g0 = 0) 1 (1o = 0) ........ ......... . ......... up, o ; ...—• * " —... V / P-<J / p - q - l \ p - 9 \ V v . / p-q ip-q-l \ p - 1 \ P 7V 7"75 7.7s 7„75Yl° 7/y5 > 7 5 FIG. 9. Relevant second-order self-energy corrections in the replica limit. 115125-12 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) b . Evaluation of relevant Feynman diagrams The first-order Fock diagram (Fig. 7) is -> (i)ddq , — — 27tS(q0)Y ,M Y \ ,2 2(2 n)d (p — q)2 — m-- - m u 5 r R y V = ~ yfdd lq y ¥ p o y ° + PkYk - qtyk + m)yl x (2jt)4-1-Po -(P - ?)2 r«r (2 — d)qky k — 2poy° + 4m VR -2 p 0y° +4m T (l - ^-l) 1 2 J (2 7t)d~l q2 + p l+ m 2 2 ( 4 * ¥ n i ) (p2 + m2)i= ¥ _ 2(47r) Then, a relevant part for renormalization is( - 2 p 0 y ° + 4 - y - I n ( p o +«2 ) +ln47r + O(e)^. p 1 S (1 ) = - - * - ( - 2 p o y 0 + 4m) + 0(1). 4n €(D4) The second-order rainbow diagram (the first diagram in Fig. 9) is "(2),r _ ’ 2 = £« f dd~'q f 4 J (2jr)rf- ' J PoY + pmy "2itS(q0 ) Jd~\ l/r2 ^ 5 (/0) y M y/4 5 _ ~ m v 5— 2 -------- t tV— (j — l — m (p — q)2 — m2' ' (p — q — l)2 — m2^v^ (p - q)2 — m2r IVK (W ° + Fry* - qkYk + m v poy 0 + piy 1 - qty l - hyl - m (2jt)d~l ’ qmy m +m-Po r i p - q )2 - m 1 ~P2 o — ip — q — l)2 — m2-Yv -Po - i p - q )2-Yu r «f ^ ,uPoY° + pky k - q ky k +m v ’ f dd ll poY° + PiY1 ~ qiY1 ~ hyl ~ m~ 4 J (2 7 T ) d~ '} {p-q)2+pl + m 2 } J (2jt)d~] (p - q - l)2 + pi + m2Yv PoY + pmy m - qmy m + mx-------------- r------ ------- r-----yM ip - q ) + P o + m 1 if ■ “ ifdd 'q u-qkYk + poY° + m (2tt) ^ - i dd~lq (2 7 T > 4 -iy^9" + Po + m - g t y f c + p o y 0 + iw or + P o + m2/-Z /y / + /z0k° - - (2jt>d-ll2 + pl+,- q my m + poy + m Yv .7.7.7 Yu 1 r(l - ¥ ) y v(p0y° - m)yv « * ) * r (l) {p2 +m2 )i-¥q- + Po + m2 -<?».y”' + p o y u + m q2 + P o + m2 r 21 R r (¥) 4 ( 4 jr ) V (pg + m 2) Rearranging the numerator as follows,f dd~lq J i2jt)d~lq Yfli-qkYk + poy0 + m)(-2p0y° - 4rn)(-qmy m + p0y° + m)y.l{ q2 + P2 o + ™ 2 ) N = y di-q ky k + p0y° + m )(-2 p 0y° - 4 rn)(-qmym + p0y° + m)yl x = -qkqmy > l y ki2poY0 +4m)ymyll - y M (p0y° + m)(2p0y° + 4m)(p0y° + m)yM = -qkqmYlx Yki2poY° + 4m)ymyn + (-4 p ly0 + 32 mpl + 20m2p0y° - 16m3) = -qkqmYflYki2p0y 0 + 4 m)ymyll + fip) with /( p ) = - 4 pq/ ° + 32mpo + 20m2p0y° — 16m3, we obtain F 2£(2),r = _ - s r(¥) ( 4 ^ ) ‘ ,21 ( pq + m 2) "/^ -qkqmy liYki2poy0 + 4m)ymyl x + fip) ¥ J {2n)d-1 (?2 + P2 o + wi2) r2 r(¥) (4;r) V (pq + m 2)y # iy*:(2p0y° + 4m)y^y|[ i r(2d- 1i) 2(47r)VT(2)(po + m 2)2 -fc i-l+/¥) F ( 2 - ¥ ) (4jr) 1 T(2)(p2 + m2)9_d — l^ 2 115125-13 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) r 21 R r(¥) 4 d- 1(47T) 2 (p^ + m2) 22 (< * - 2 )(< Z - l)poy 0 + 4d(d - l)m T ( ^ ) /( p ) F ( ^ ) 2(4t t )_+ ( . P l + m 2) 2 (4?r) 2 ( pl+ m 2 )' r 21 D 4(47 t)2 r 2“ - Y + In(4 t t) - In (p2 + m2) + 0 (e )j [(1 + e)(2 + e)p0K° + 2(3 + e)(2 + e)m ] YY + ln (4 ;r) - In (p^ + m2) + 0(e)) (e\ ~4ply° + 32m pi + 20m2 p0 y° — 16m3 pl + m2 We note that the second term vanishes when we use the on-shell condition, given by p0 y° + m = 0 and pi = m2. As a result, we obtain £ (2)’ ’r 2 l 16 7 T 2-(3 /Jo r + 10m) + 0(1) + 0 (< r2). (D5) The second-order crossed diagram (the second diagram in Fig. 9) is E (2 )- c = -r* dd q (2jt)d2nS(qo)fdd l , , c p — d — m w 2 , , m y v i-i-i (p-q -l)2 - m 2 Y l x Y (p - l)— I — m - A2 . m2yv y 5 « [ d“ -li Po? 4 J(2jt)d ~1 J(2n)d - ' Y P o Y° + Pm /m - lm ym +m Yv -Po - (P - l)2 — m2 n r dd ~lif d“ - ,~ 41 (2n)d -' J (2n)d ~l ' r£r dd ~x i [ " [ dd ~lq N ~ 41 ( 2 7 r ) d~l lJ (27r)d_1 Dpoy0 + PkYk - qkyk + m v p0 y° + piy' - qtyl - Uyl - m y 7 , .S T o Y j J .~Po - (p - qf - m2 ~Po - (P ~ q - l )2 - m 2 dd ] q ,^-qkYk + PoY°+ r n v-qiy'-hyl ~ P lY 1 + PoY°-m -lm ym + P oy0 + m Y , , . , . 1 . 1 Y u ,7,7,7 Y vq1 + pI + ml -lm ym + poy0 - l2 + P o + m2(q + 1 + p)2 + pl+ m2 l2 + pl+ m2 -Yv, where the denominator D 1 and the numerator N are given by D~x = [ dx[q2 + x ( l -x)(l + p f + pi + m2]-2 = [ Jo Jodx[q + A (Z,p)]- 2 and N = yd[-qkyk + x(lk + pk)yk + p0 y° + rnty^-qiy' - (1 - x)(l, + pi)y‘ + p0y° - m]yM = qkqiY^YkYvYlY i d + Y^Udk + Pk)yk + P oY ° + m]yv[-{ 1 - x)(l, + pi)y' + p0 y° + m]yM = qk qi[-2y‘yvyk + (4 - d)ykyvyl] + /(Z,p), respectively. Performing the momentum integral with these expressions, we obtain f dd~x q N = rxdx r dd ~x q qk q,[-2y‘ 'y1 v yk + (4 - d)yky' V ] + /(Z,p)J (2n)d ~ ' D J0 X J( 2jt)d -' [ q2 + A(l,p)]2 = / L J 1 ( 2- < V y > ; r ( i - ^ i ) 1 /(z,p) r( 2- ^ i ) 1 J o [ (4t t¥ 2 T(2) A (Z ,p )'-¥ (4 n)^ r(2) A (/,p)2~ ^ (2 — d ){3 — d)yv T ( ^ ) ^ / ( / , p ) T ( ^ ) ~ 2 (4 rr)^ A (Z ,p)¥ (4 t t) ¥ A (Z ,p)¥_ ‘ Then, we have S (2),c f dd ~l i r ' {2-d)(i-d)y7T (^ ) f(i ,P ) r ( ^ ) J (2n)d ~y J o X 2(4 n)d ~ r - A ¥ ( 4 ^ ) ^ A t-L ym + P o y0 + m Z 2 + Po + w2 ^ (2 — d)(3 — d) r ( 2 ^ ) 2{4n)d -r - A ¥/^ !/ — (2 — d)lm ym — 2ppy° + 4m ( 2 * ) '- 1 Z 2 + p 2 + m2 r(^) r dd ~ 'l f(l,p)-lm Y ’ "+p 0y0+m (4n ) ^ J (2n)d~l Z 2 + po+m 2 ^ 115125-14 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) r 2 f 1 s / ./r'(2 -d )(3 -d )r(2 = * ) f — (2 — d)lm ym — 2poy° + 4w , r ( ¥ ) f dd ~ll1 L i a 4 Jo 2(4t t) ¥ a¥ J(2jr)d _ 1 Z 2 + Pq + m2 (4;r)¥ J(2n)d ~l y^[x(lk + pk)yk + p0 y° + m]yv[-{ 1 - x)(Z; + pi)y‘ + p0y° + m]yM - lm ym + p0 y° + m a¥ l2 + pl + m2 y' r 2 f1R dx■(2 - <0(3 - r f ) r(3=^) /* ¥ '* / - ( 2 - - d)lm ym — 2p0y° + 4m, r ( ¥ )r dd ~'l N ~1 C l J C 4 Jo 2(4t t) ¥ a¥ JW - i Z 2 + Po + m2 (4n-)¥ J(2n)d ~l D where D~l = [x(l -x)(l + p)2 + p2 + m2 ] ^ [ l2 + p2 + m2 ]~ ' _ d ____________________ [ * ( 1 - X ) ] ¥ y ¥ - 1 _____________________r ( ^ + !) J o ' (y{(Z + p) 2 + [x(1 - x)Yx (pl +m2 )) + (1 -y)(l2 + pl + m2) ) ¥ r ( ^ ) = £ ( ¥ ) ['dy __________________ [x(i - x ) ] ¥ y ¥ __________________ r (¥) Jo ' (l2 + 2yp l + yp2 + {[*(1 - x)]"1 + 1 - y}(pl + m2 ))¥ _ r ( ¥ ) f ' d _____________ [*(1 - x ) ] ¥ y ¥ _____________ r ( ¥ ) J o ( Z 2 + t( 1 - y)p2 - ( M i - * ) ] _ 1 + i - y}p2 )1 ^L r ( ¥ ) [*(1 - x y p y * rm J o y [/2 + A (p )]¥ for the fourth equality, l — > l — yp and — = p2 + m2 , and N = yd{x [lk + (1 - y)pk\yk + Poy° + m}yv{ -(l - x)[Z/ + (1 - y)pi]y‘ + poY° ~ m )y,l{-lm ym + ypm ym + p0 y° + m)yv = [— jc(1 - x)lk llydykyv ylyl l + xlky»ykyv[-{ 1 - x)(l - y)p,yl + p0 y° - m]Y l l + yd[x( 1 - y) + pkYk +PoY° + m]yv[-( 1 - x)/,y']yM + y M [x(l - y)pkyk + p0 y° + m]yv[-( 1 - x)(l - y)piyl + p0 y° - m]yM ] x[-im ym + ypm ym + poy° + w]yv] = x( 1 - x)lk l,y^ykyvylyn(yPmym + Poy° + m)yv - xlk lm y'xykyv[-{ 1 - x)(l - y)p,y' + p0 y° - m]yM ym yv +(1 - x)///my M [x(l - y)pkyk + p0 y° + m]yvy,ytlym yv + y M [x(l - y)pkyk + p0 y° + m ] x y l’[— (1 x)(l - y)piyl + poy° - m]Y l x (ypm ym + p0 y° + m)yv = lk l,[-x( 1 - x)ydykyvylytl(ypm ym + P oy° + /n)y, - x y V V [-(1 - *)(1 - ? W " + ZW° - m]yM yVv +(1 - x)yd[x( 1 - y)/?m y m + p0y° + m]y V V ¥ > ¥ + y ¥ x (l - y)pkyk + A>y° + m ] x y v[ - ( l - x)(l - y)piy' + pQ y° - m]yli(ypm ym + p0 y° + m)yv = hh ■ f(p) + g(p)- As a result, the above expression becomes more simplified in the following way: y(2),c _ _ H [ \ , ( 2 - < * ) ( 3 - r f ) l W ) f dd ~'l — (2 — d)lm ym — 2p0 y° + A m 4 Jo 2(47 t) ¥ a¥ J (2*)d ~l l2 + pl + m2 n r(¥) f ' r [ dd-H r(¥) r , /d /- /( p ) + g ( p ) 4 (4 t t) ¥ J o J ( 2 ^ - 1r(^ )J o ^[x(l-x)]¥ [ Z 2 + A(/7)]¥- The first line is easy to perform integrals. It seems to have a double pole, but the presence of (2 — d)(3 — J) in the numerator gives only 2 as a leading term as follows: _ r | f 1 dx(2-d)0 -d)r (3 -Y) -2p0 y° + 4m T ( ^ ) 4 Jo 2(4 t t) ¥ a¥ (47r ) ¥ ( p 2 + m 2) ¥ = ~ Y - I n A +ln4jr + 0 (e)j - y -ln(po + m2) +ln47r + O(e)^ (~2p0 y° +4m) r 2 l= - r r — 2 - ( - 2 p 0y° + 4m)+ 0(1).32t t z € 115125-15 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) The evaluation of the second line is quite complicated especially because of the contribution from Dirac algebra in the anomalous dimension and potential appearance of poles by the integration for Feynman parameters of (x,y). We need to analyze both f(p) and g(p) carefully, separating them from each other: r 2 1■ 2 /•]t Ldxdy3 - d y 2 [ x ( i - x ) ] ¥T(3 - d) gk , ■ f(p) + T(4 - d) g{p) _2{4n)d 1 A (p)3 d (4^)rf-i A(p)4 ~ d First, let us analyze the second term with g{p). At a glance, it does not have a pole. However, it is possible to diverge if the x integral contains the contribution of fQ ’ dx[x^( 1 - x ) ^ ] - for example. We rearrange g(p) and -t t-t as follows:A (/7 ) g(p) = yM [x( 1 - y)pk Yk + PoY ° + m]yv[-( 1 - x)(l - y)PlYl + p0 y° - m]Y l J ,{ypm ym + P oY ° + m)yv = *(i - * )y(i - yf x ( • ■ •) + x(i - x)(i - y)2 x ( • • •) + ■ • • + 1 x ( • ■ ■ ) , l {p)d -\x{\ - x) - [x(l - x)]2(l - y)2 + [x(l - x)]3(l - y)4 + • • • ], A (p ) W-^siLi+a-^r' where (• ■ •) indicates quantities which depend on p but not on x and y. We note that the expansion of A (p) is justified because of 0 < x,y < 1, which cannot be done in a reciprocal way, i.e, H -----]• Then, we obtain /'1 , , i = S 3 = i g(p)dxdyx 2 ( 1 — x) 2 y 2A - d s d - 4= (P) = (pf~fJ oA (p) dxdy[x( 1 -x)]T y¥[x(l - x ) y ( lf - y)2 x (••■ ) + ••• + 1 x (• • - )]{jc(1 - x ) - [x(l - x)]2(l - y)2 + • • •} = (/>)1 dxdy[x^ ( 1 — xjVy Y x (...) h-----] J o ,.-4rr(¥)r(¥)r(¥)r(i) , 1 d —>3 r w - i ) r ( ^ )+ V finite, where we used the formula of B(x,y) = / 0 ' dt[tx '(1 — t)y '] = . The leading term is already finite, which leads us to conclude that the second term gives only a finite value. Next, let us focus on the first term. The first term contains T(3 - d), which gives rise to a divergence as d ->• 3. Also, it may cause a divergence in the x integral by the same reason as before. In this respect it is more plausible to show a divergent behavior. Performing the Dirac algebra repeatedly, g k i ■ f(p) = - x ( l - x)yllyky vykyll(ypm ym + PoY° + m)yv - xy V k y v[-(l - x)(l - y )pm ym + p0 y° - m]yliykyv + (1 - x)y*[x( 1 - y)pm ym + p0 y° + m]yvykyl x ykyv = ~ x)[-2ykyvyk + (4 - d)yky vyk](ypm ym + p0y° + m)yv + [x(l - x)(l - y)pm [-2ym y vy k + (4 - d)yky vym ] - xp0 [-2y0 y vyk + (4 - d)yky vy°] + xm[4gk v + ( d - 4)yky v ]]y kyv + [*(1 - *)(1 - y)pm[~2.yky vym + (4 - d)ym y vy k] + (1 - x)pQ [-2ykyvy° + (4 - d)y°yvyk] + (1 - x)m[4gv k + (d - 4)yvyk]]y kyv = x(l - x)ypm yk [4g? + ( d - 4)ykym ] (2 - d) - x(l - x)p0 yk [4g ° k + (d - 4)y^y°] (2 - d) -x (l - x)mykyk{ 2 - d)2 - 2x(l - x)(l - y)Pm ym (d - 1 )d + x(l - x)(l - y)pm yk\4g * + ( d - 4)ym yk] x (4 -d ) + 2 xp0 y°(d - 1 )d - xp0 yk[ 4 g ° k + ( d - 4)y°yk](4 - d ) + 4xm(d - 1) + xm(d - 4)(2 - d)(d - 1) - 2x(l - x)( 1 - y)pm yk[4g^ + (d - 4)ym yk] + x(l - x)(l - y)pm ym { 4 - d)(d - 1 )d - 2 ( l- x ) p 0 yk[4g0 k + (d -4 )y 0 yk \ + ( l - x ) p 0y0 (4 -d )(l-d )d + 4 (l-x )m (d -l) + (1 - x)m(d - 4)(d - l)d = — 4x(l — x)ypm ym (2 — d) — x(l — x)ypm ym { 2 — d)(d — 4)(d — 1) — x(l — x)poy°(d — 4)(2 — d)(d — 1) -x (l - x)m(2 - d)2 (d - l)-2x(l -x)(l -y )p m ym ( d - l)d+4x(\ -x)(l - y)pm ym { 4 - d) + x(l - x){\ - y)pm ym (d - 4)2 (d - 3 ) + 2 xp0 y°(d - 1 )d + xp0 y°(4 - d)(d - 4 )(d - 1 ) + 4 xm(d - 1 ) 115125-16 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) + xm(d - 4)(2 - d)(d - 1) - 8x(l - x)(l - y)pm ym - 2x(l - x)(l - y)pm ym (d - 4)(3 - d) + x(\ — jc)(1 - y)pm ym { 4 - d){d - \)d + 2{\ - x)p0 y°(d - 4)(d - 1) + (1 - x)p0 y°(4 - d)(d - 1 )d + 4(1 — x)m(d — 1) + (1 — x)m(d — 4 )(d — 1 )d = P ,nym [x( 1 - jc)(1 - y)[-2d{d - 1) + 4(4 -d ) + ( d - 4)\d - 3) - 8 + 2 (d - 4){d - 3) - {d - 4)(d - 1 )d ] + x(l - x)y[4(d - 2) + (d - 4)(d - 2)(d - 1)]] + p0 y°[x( 1 - x)[(d - 4)(d - 2)(d - 1)] + (1 - x)[2(d - 4)(d - 1) - (d - 4)(d - l)d] + x[2d(d - 1) - (d - 4)2 (d - 1)]] + m[x(l -x )[ -(d - 2 f( d - 1)] + (1 -x)[4(d- l) + d (d -4 )(d - l)] + x[4(d- l) - (d - 4)(d - 2)(d - 1)]] = pm y"‘ [x( 1 - x)(l - y)[-6d2 + 2U - 16] + x(l - x)y[d 3 - Id2 + \M - 16]] + p0 y°[x( 1 - x)[d3 - Id2 + 14 d - 8] + (1 - x)[-d 3 + Id2 - 14 d + 8] + x[-d3 + 1 Id2- 26 d + 16]] + m[x( 1 — x)[-d3 + 5 d2 — % d + 4] + (1 - x)[d3 - 5d2 + & d - 4] + x [-r/3 + Id2 — 10cf + 4]], we simplify the above expression as follows: T2 P1 — / dxdy 4 J o [x(l - x ) ] ¥T (3 - d ) g k r f( Py 2 (4 7 t)d ~1 A (p)3-d 8(4 n)d2 P1 Ldxdy3 - d y 2 T(3 - d) [x(l - x ) ] ¥ Mp)3 ~d■zj(PmYm [x(\ - x)(l - y)[-6d2 + 2U - 16] + x(l - x)y[d 3 - Id2 + 18c? - 16]] + p0 y°[x( 1 - x)[d3 - Id2 + 14 d - 8] + (1 - x)[-d 3 + Id2 - 14 d + 8] + x[-d3 + 1 Id2 - 26 d + 16]] + m[x(l - x)[-d3 + 5d2 - 8d + 4] + (1 - x)[d3 - 5 d2 + M - 4] + x [ - J 3 + Id2 - 10 d + 4]]). There appears a simple pole already due to T(3 — d), but another pole can be made by the x integration. The y integration turns out not to cause a divergence. Since we are interested in contributions with a simple pole, we are allowed to take two kinds of terms only: One has a double pole with e in a numerator and the other contains a simple pole only. Performing integrals carefully, we find F21 P r(3 - d) 8(4T x ) d ~' A(p)3-dP m yr (d- 1 ■ )r( W [ ^r id -i, r(3=!)28d - 16] +I W ) 2 r ( ¥ ) r(d-i>r(5^)[d3 - I d 2+\%d - 16] + poyr ( ¥ ) r ( ^ ) r ( d - D r ( ^ )[d3 - Id2 + 14 d - 8] + r(¥)r(¥)r(¥) r(d -2 ) r ( ¥ )[ - d 3 + Id2 - 14d + 8] + +r (¥ )r (¥ )r (¥ )r ,3[-d3 + 1 Id2 — 26d + 16]T(d-2) r ( ^ ) i-A ^ [ d 3 - 5d2 + 8rf - 4] ++ m> (¥ ) 2r(¥) r y - D r ( ^ )- d 3 + 5d2 - M + 4 ) r(¥ )r(¥ )r(¥ )r ,3 rj2 , , r(¥)r( ¥ ) r(¥ )r ,3 , , j2 r21 Rr(d - 2 ) r(¥) 1P(d - 2 ) r(¥)[-d3 + Id2 - 1 O d + 4 ] 128t t2 + mP m ym [8 + 0(e)] + p0 y° — 2 + 0(e) + T ( - )(12+14e) — 2+ 0(e) + f ( - )(12 + lOe) r2 1 r ^ - ( - 8 p m ym - 26P oy° - 18m) + 0 (e “2) + 0(1).1287T2 e As a result, the self-energy correction from the crossed diagram is given by E ( 2 ) ’c = ~ ^ l { ~ 2poy° + 4 m ) ~ r n ^ \ { ~*P m Y m ~ 2 6 p o Y ° “ 1 8 m ) + 0(e_2) + 0 (1 } 1 :(— 34 p0 yu - % P mYm - 2m) + 0 (e -2) + 0(1)128t t2(D6) 115125-17 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) It is important to notice that the sign of the self-energy correction in the two-loop order differs from that of the Fock diagram in the one-loop order, which turns out to play a central role in the renormalization group equation for the mass parameter. 2. Vertex corrections a. Feynman diagrams The vertex renormalization can be found from the four-point function of G(x,x\y,y') = j^J 2P P 'q q ' e - i P x+ip x -iq-y+ iq -y G(p,p',q,q') with G(p,p’ ,q,q') = {T [\l/(p )\j;(p ')-^-(q )ilf(q ')]). Performing the perturbative analysis up to the order, we obtain G(p,p',q,q') 1 ^ „ c lint — ^ 2 / D ( 4 r,\lp ) \lr a ( p ) \ ir a ( p ') \ lr a ( q ) \ l/ a ( q ' ) e ~ ^ ' % “.« •"] e Z *c = i fo dTIo d r' f d dx r - f a= 1 R= lim-^^ J a= 1 JV(p)V(p')'l'a(q)V(.q') + E E i,c = 1 PiTr ' 2\ J / a (p )ip -a (p')\j/a (q )\lra {q ') b,c,d,e=\ Pi,qix[^b(p\)y^y5V(P2)][i'c(P3)yf J -y5^c(P4)]S{ 3 \pi - p2 + pi - p4)8po< p o8poi P o + J2 J2 (—y ) Vip)V(p') xil/a(q)$a(q'Mb(p\)yf l y5' l ' b(P2)Mc(P3)Yl x y5V(P4)Md(q\)y, 1 y5' l s < 1 (q2)Me(q3)yf i y5V(q4)]80\p\ - p2 + p3 ~ P4)8po ,p o S p o tP o 80)(qi - q 2+q3- q4)8q ^q «8q aq o + O (f^ ) = Iim n 4 ^ ( l H p ) ^ a( p V a(9)iA< V )} o + lim ~ a=\ a,b,c=\ ' ' Pi x^b(p2)^c(P3)yI J 'y5V(P4))o8w(pi)+ Hmi '%2(i'a(p)^a(p')i'a(q)'lfa(q')i'b(pi)y'My5 a,b,c,d,e=\ ' ' pi,qi x xjrb (p2 )^c(p3 )y^y5 x l / c (p4 )^d(qi)yfly5 ird(q2)^e(q3)y,ly5 ire (q 4 ))0 8w (pi)8w (qi) + 0(V 3 R ), FIG. 10. All possible first-order corrections for the four-point function without the replica limit. 115125-18 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) = lim R-where we introduced a shorthand notation of 5(4 )(p, ) = & (3> (p\ - pi + pi — p4)8p op o 8p op o . All possible first-order contributions are given by (Fig. 10) 1 R ^ [ i ' i ^ y k ^ l ^ bm(.YIX Y 5)mn'lfb n $ C o< <Yl l Y \ p V p )Q a,b,c=\ l R U m - £ [ 4 ( ^ ^ U ^ ^ 1 o ( ^ V \ ) o ( ^ r 0 ) 0( y ^ 5 U ( y V 5 )O P - 4 ( ^ fl^ ) 0(^vr)0 a,b,c= 1 x [fb n r o ) 0 {rP r m ) 0(YR Y5 )n ,n (Y ^ Y 5 ) o P - 4 (w r X W ^ a)0 X r m ) 0 [rP r a \ (k V u *0^y5 )o p+4 [ r i r j W b nfi)0 {rP r m ) 0 {rkr o ) 0 (y V w y V w + ^ b r x X X X x[V p'4ra i\{K'ifC o ) 0(YR 'Y 5 )m n {Y R 'Y\p\ 1 R■ [ X ~ R E [ ^ GtiGL G C p o ( Y ^ Y X ( Y ^ Y X S aaSaa8bbScc- 2 G ‘ lj Ga klGb noGc pJ y ^ Y 5)mn(Y^Y5)oPSaa8aaSbcScb a,b,c= 1 AGa ijGb n lG a k m Gc p o {YI X Y5 )rnn{Y^Y\P ^aa^ab&cc+^Ga ijGb n lG C p m Ga k o {Y> l Y5 )m n ^Y \P ^a 8 b a 8 c b & a c + 4 Gb njG“ mGc plGa ko8ba8ab8ca8ac(y^Y5)n,n(Y^Y5)op] 1 r ri= lim —R^>0 RK R 2 [Ga ]2 ® tr[G V M T5M G cyM K5] a,b,c= lK Y n Y ^ S a a S a a S b b S c c + 2 [G“]2 < g > tr[Gfty M y 5 Gcypy5 ] S a a S a a S b c S c b a,b,c= l -4 G “ ® Gay^y5 )Gb tT [G cy^y5 ]8 a a S b a 8 a b S c e + 4Ga ® Gay^y5 Gcy^y5 Gb 8 a a 8 b a 8 c b 8 a c + 4G“yfly5Gc ® Gayl x y5 Gb 8 b a 8 a b 8 c a 8 a where 4 results from identical contributions in the first equality and — comes from the odd number of fermion loops (one loop). The first term is proportional to R3, the second and third, R2 , the fourth and fifth, R . As a result, only the fourth and fifth terms survive in the replica limit. But, the fourth term is just the product of a bare propagator and a propagator with a Fock self-energy. Therefore, the four-point function and the scattering matrix element are (Fig. 11) Gm(p,p'\q ) = 4 ( - v r ) G(p)y, 1 y5 G(p + q) ® G(p')y^y5 G(p' - q), M( ]\p,p-,q) = 4 y^y5 ® y R " y5- FIG. 11. Tree level vertex. I 15125-19 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) FIG. 12. Second-order vertex corrections without the replica limit. All possible quantum corrections in the second order are given by (Fig. 12) 1 R R E [ { f i ^ j f k ' l ' l f bm (y,ly 5) m n ^ C o(ytly 5)opVp ^ dq(Y'’y 5)qr ^ f es(YvY 5) s t ^ ) 0 ] a,b,c,d,e=\ Rconnected = 1™ j E [16 GaY^y5 Gb < g > GayvysG‘ti[Geylly5 Gdy vy5 ]8 a b 8 b a 8 ai8 d e 8 a e 8 e a + 16Gay»y5 Gb y vy5Gd a.b,c,d,e=\ Za*, 5n.es,®Gay^y5 Gcyvy5 Ge 8 a b 8 b d 8 d a 8 a c 8 c e 8 e a + 1 6Gayliy5 Gb y vy5 Gd ® Gayvy5 GeY fly5 Gc 8 a b 8 b d 8 d a 8 a e 8 e c 8 C l + 32 Gay^y5 Gb yvy5 Gdy^y5 Gc ® Gayvy5 Ge 8 a b 8 b d 8 d c 8 c a 8 a e 8 e a 1 , where only diagrams fully connected with the external lines have been taken into account. The first term is proportional to R2 while all other terms are ~ A 1 . As a result, the four-point function and the scattering matrix element in the second order are given by (Fig. 13) G( 2 \p,p';q) = Y^[l6G(p)y, 1 y5G(p - l)yvy5 G(p + q) < g > G (//)yM y 5 G{p' + l)yvy5 G(p' - q) + 16 G(p)yI J -y5 G(p - l)yvy5 G(p + q) ® G(p')yvy5 G(p' - l - q)y^y5 G(p' - q) +32G(p)y^y5 G(p - l)yvy5 G(p + q - Z)yM y 5G(/? + q) ® G(p')yvy5 G(p’ - q )}, Mu\p,p',q) = 4Ap p + 4AP h + 8Aver, App ~ G(p - l)yvy5 ® Y f iY5 G(p' + l)yvy 5, (D7) (D8) (D9) where pp and ph represent “particle-particle” and “particle-hole”, respectively, and ver means “vertex”. Y V p - l p + q 1 ( 1 o = 0 ) l + q(qo = 0) P 1 -Q p ' - l - q P ' FIG. 13. Second-order vertex corrections in the replica limit. 115125-20 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) b. Evaluation of relevant Feynman diagrams First, we focus on the particle-hole diagram (the first diagram in Fig. 13), given by 2 ' dd l ........... 5 p - t - •^ph — I? ) /;2 n 8 (lo )y ^ y y v y5®Y uYj r 2 ~ 4I f -V /(2 it)d-1-Po ~ ( P - t r ip — l)2 — m 2 yv ® K/x' -<f -1 -i ip' — q — l)2 — m2^'^ dd ~[ l poy° + pk yk - lk yk + m v p ' 0 y° + p\yl - q0 y° - my1 - hyl + m -ip' 0 + <7o)2 -ip' - q - l)2 - r n 2Y v dd ll y^ipoy0 + PkYk ~ k yl + m)yv y ^ [ip '0 - qo)y° + p',yl ~ qiY1 - hyl + rn]yv i2n)d ~l dd ~ll N 4 J (2it)d~l D ’(l - p)2 + p2 + i (/-/?' + q)2 + (p'0 - q0 )2 + m2 with D ' = f dx[l2 + *(1 -x)ip - p’ + q)2 +xpl + (1 - x)ip’ 0 - q0 )2 + m2 ] 2 = f dx[l2 + A(p,p’- q)] 2, J o J o N = y^[-lkyk + (1 - x)ipk - p 'k + qk)yk + P oY° + m]yv < g > y^-liy1 - x(p, - p\ + q,)yl + ( p '0 - q0 )y° + m]yv = hhy^Yky v®YnYlYv+y^li 1 - x)(pk - p 'k + qk)yk + poY° + m]yv < g > y^[-x(p, - p [ + q{)yl + ( p '0 - q0 )y° + m]yv = lkliY^YkYv ® YuY'Yv + fip,p' - q ). Then, we obtain d“ - ll lk hYl x YkYv ® YuYlY v + fip,p' ~ q)* p h p 2 rl r = -*- / dx 4 Jo(2jt)d - 1 [l2 + A ]-2 -1 y^YkYv ® Y u Y kY v F (l - ^f1) 1 ( ■4 jr)d-l r 2 f 1-tL r 2 c 1--tLdx dxr(2) A1 - n-d\1 Y^ykYv < g > Y u Y kY v r ( V ) + fiP’P ' - q) r ( V )fiP ’P '-q ) r (2 ~ V ) i (4 T t) d ^ r(2) A2- ^ 5 -d (4 7 t)d ydyky v ® Y u Y kY vA T (4n)d-l A ^A 2 2(4;r)Y — In A (p,p' -q ) + ln4;r + 0(e)) + f(P'P e J ( 47t) 2 A 2(DIO) Next, we evaluate the particle-particle diagram (the second diagram in Fig. 13). It is almost identical with the way for the particle-hole channel to perform the integral of the particle-particle channel. We find ~ r i)f F2 P1= — I d 4 Joddl (2n)d2 7 i8 (l0 )YdyM .,5_— / — m (p - l)2 — m2YV Y5 ® YuYJp' + / - m (p' + l)2 — r n 2YvY = Aph ip - q -> p’ ,l -» - l2 ) Yd Y Y v ® YuYkYv f 2 2(4jt) V e Lastly, we evaluate the third diagram in Fig. 13, given byy — In A ip,p') + ln47r + 0(e) ) +fiPiP') r(V ) (47T)2 ? A(p,p')1 T(Dll) ( r ff\ 2 r dd i , V 2 J J (2n Y ‘ ( dd - li 4 J1 (2 7 r)r f ->/ n f dd ~ll y^( 4 J’ (2 T X ) d ~X rl r dd ~H n 4 J' (2 7 t)d~l D '2 n 8 ( la ) y ^ yu 5 _l — mv,,5 (p — l)2 — m2 - lkyk + m YY Y■ t + 4 — m (p — l + q)2 — m2YuY ® YvY -P o ~(p - l ) 2 ~m < P o + q o )2 - i p + q - l ) 2 - m 2-YuY ®YvY [(/ - p)2 + Pq+ m2 ] [ (l - p - q ) 2 + ipo + q o )2 + m2 115125-21 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM PHYSICAL REVIEW B 91, 115125 (2015) with D ' — [ dx[l2+x(l -x )q 2+xql + 2xp0 q0 + + m2 ] 2 = f dx[l2 + A'(p,q)] 2 J o Jo N = y^(-lkyk - xqkyk + p0 y° + m)yv[-liyl - (1 - x)q,y‘ + (p0 + q0 )y° - m]y^y5 < g > yvy5 = khy ^K V / V mK5 ® YvY5 + Y^(-xqkYk + PoY° + m)yv[ ( 1 - x)q,yl + (p0 + q0 )y° - m]yM y 5 ® yvy5 = l kli[-2yly vyk + (4 - d)yky vy‘ ]ys ® yvy 5 + g(p,q). Following a similar procedure as the above, we obtain j1 d x J dd~H ljM-2yly vyk + (4 - d)ykyvyl]y5 < g > yvy5 + g(p,q) r 21 R (27T)d-1[l2 + A '(p,q)]2 f z el fi r 2 cx = T lr 2 ~ Adx dx dx1 [— 2y'yvyk + (4 — d)yky vy']y5 ■ K vK5r(¥ ) (4w)V 2 (2-J) y W 8 y ^ 5r ( ¥ )+#Gm) r (M) .(47T) — g(€ + l) 4 7 T+r (2) a'(Pi9) ¥ (47 t)d~l r ( 2) A \P,q) ^ g(P,q) r(¥) i 2 r(2) A>(p,q)^ (4t t)rf— 1 r(2) A '(p,9 ) ¥ ----- y - In A'(p,q) + In4^ + 0 ( f ) V y 5 0 y„y5 + 8^ P’ ^ ^ 2 ^ (4?r)2 ? A'{p,q)2 J(D12) APPENDIX E: EVALUATION OF RENORMALIZATION CONSTANTS Combining Eq. (D4), Eq. (D5), and Eq. (D6) in the following way, 2 x E<‘> + 8 x S (2 > 'r + 8 x E (2 > 'e + [8 % p 0 y° + 8 kPkyk + 8 m m ] = 2 x . R - ( — 2Poy° + Am ) + 8 x — ^ Kn/i — o4rr € o c o _ „.016jt2 €(3poy + 10m) + 8 x1287T2 e-OApoy + 8pm ym + 2m) = P o Y we find counterterms+S$PoY'> + 8^pk yK + 8 mr n + 0(1) _ 29F2 1 + + * ; ) + pm ym ( 0 - + « ; i +m7 T e 8 t t2 e r R l 2 9 r2 l n e 8:r2 e ’ which give rise to renormalization constants of¥ = 81 = -r 2 l 27T 2 68 m =2rR 1 4 i r 2 1 7 T € 87T 2 6 2r« l 4ir2 l n € 8rr2 z% = 1 “ ¥ ln^ _ l ? lnM - e x p [ - ^ lnM - ln /z z * = 1 _ & ln ^ - exP [ - 0 lnd] Zm = 1 + lnM- ^ l n ^ ~ e x p [ ^ l n M-(El) where 2 is replaced with a cutoff scale, In /x. Similarly, one can find the renormalization constant for vertex corrections. It turns out that the particle-hole contribution Eq. (DIO) cancels the particle-particle correction Eq. (Dll) while the vertex part Eq. (D12) is finite, given by r r 4 x Aph + 4 x Ap p + 8 x Av e r + 8 r ~ ( y vy 5 0 yvy5 ) = 0(1) + 8r~^-{yvy5 < 8 > yvy5 ) . As a result, we find Zr = 1 up to the order. APPENDIX F: DERIVATION OF RENORMALIZATION GROUP EQUATIONS Recall the relation between the bare and renormalized coupling constant: FB = p?~d(Z^)~2 Zrr R . It is straightforward to find the renormalization group equation for the variance parameter: d In r « ^^^t/lnZ^ dlnZr dlnp, d I n / x dlnp,(FI) 115125-22 DILUTE MAGNETIC TOPOLOGICAL SEMICONDUCTORS PHYSICAL REVIEW B 91, 115125 (2015) /?r(o fun FIG. 14. (Color online) Beta functions. Similarly, mR = /U,(Zp x Zm mR results in d\nmR _ t | d\nZ“ d\nZm dln/j, dln/j, d In /tt Substituting Eq. (El) and Zp = 1 into Eq. (FI) and Eq. (F2), we obtain the renormalization group equations for the variance and mass parameter (see Fig. 14) dlnT^ = j _ 2T« _ 2 9 T | dlnfi 7 t 4n2(F3) d\nmR _ _ j _ 3T R 3 T | d ln/t, t c 2jt2(F4) [1] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996). [2] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006). [3] H. V . Lohneysen, A. Rosch, M. Vojta, and P. Wolfle, Fermi- liquid instabilities at magnetic quantum phase transitions, Rev. Mod. Phys. 79, 1015 (2007). [4] H. V. Lohneysen, Electron-electron interactions and the metal- insulator transition in heavily doped silicon, Ann. Phys. (Berlin) 523, 599 (2011). [5] P . A. Lee, and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985). [6] V . Dobrosavljevic, Introduction to metal-insulator transitions, in Conductor-Insulator Quantum Phase Transitions , edited by V . Dobrosavljevic, N. Trivedi, and J. M. Valles Jr. (Oxford University Press, Oxford, 2012). [7] T. Dietl, and H. Ohno, Dilute ferromagnetic semiconductors: Physics and spintronic structures, Rev. Mod. Phys. 86, 187 (2014). [8] M. A. Ruderman and C. Kittel, Indirect exchange coupling of nuclear magnetic moments by conduction electrons, Phys. Rev. 96, 99 (1954); T. A. Kasuya, Theory of metallic ferro- and antiferromagnetism on Zener’s model. Prog. Theor. Phys. 16, 45 (1956); K. Yosida, Magnetic properties of Cu-Mn alloys, Phys. Rev. 106, 893 (1957). [9] M. Z. Hasan and C. L. Kane, Colloquium: Topological insula tors, Rev. Mod. Phys. 82. 3045 (2010).[10] X.-L. Qi, and S.-C. Zhang, Topological insulators and supercon ductors, Rev. Mod. Phys. 83, 1057 (2011). [11] H.-J. Kim, K.-S. Kim, J.-F. Wang, V . A. Kulbachinskii, K. Ogawa, M. Sasaki, A. Ohnishi, M. Kitaura, Y.-Y. Wu, L. Li, I. Yamamoto, J. Azuma, M. Kamada, and V. Dobrosavljevic, Topological phase transitions driven by magnetic phase transitions in Fe^Bi2 Te 3 ( O ^ x ^ O .l) single crystals, Phys. Rev. Lett. 110, 136601 (2013). [12] F. D. M. Haldane, Berry curvature on the Fermi surface: Anomalous Hall effect as a topological Fermi-liquid property, Phys. Rev. Lett. 93, 206602 (2004). [13] S. Murakami, Phase transition between the quantum spin Hall and insulator phases in 3D: Emergence of a topological gapless phase, New J. Phys. 9, 356 (2007). [14] A. A. Burkov and L. Balents, Weyl semimetal in a topological insulator multilayer, Phys. Rev. Lett. 107, 127205 (2011). [15] H. B. Nielsen, and M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal, Phys. Lett. B 130, 389(1983). [16] L. Fu and C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76, 045302 (2007). [17] S. Kirkpatrick, Percolation and conduction, Rev. Mod. Phys. 45, 574(1973). [18] F. Wilczek, Two Applications of axion electrodynamics, Phys. Rev. Lett. 58, 1799 (1987). [19] H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, and L. Li, Dirac versus Weyl fermions in topological insulators: Adler-Bell-Jackiw anomaly in transport phenomena, Phys. Rev. Lett. Ill, 246603 (2013). 115125-23 KYOUNG-MIN KIM, YONG-SOO JHO, AND KI-SEOK KIM [20] K.-S. Kim, H.-J. Kim, and M. Sasaki, Boltzmann equation approach to anomalous transport in a Weyl metal, Phys. Rev. B 89, 195137 (2014). [21] L. Fu and C. L. Kane, Topology, Delocalization via average symmetry and the symplectic Anderson transition, Phys. Rev. Lett. 109, 246605 (2012). [22] The nature of effective interactions between doped magnetic ions is not clearly specified here. Actually, this issue is not settled completely yet as far as we know. There exist several systematic studies on the nature of effective interactions between doped magnetic ions. When magnetic ions were doped into (Bi, Sb)2Te3 (BST), Bi2Se3 did not give any signatures on ferromagnetism while Bi2Te3 and Sb2Te3 showed hysteresis in magnetic behaviors [Phys. Rev. Lett. 112, 056801 (2014); Science 339, 1582 (2013); Phys. Rev. B 74, 224418 (2006)]. In addition, these samples still showed ferromagnetic behaviors even in their insulating phases [Adv. Mater. 25, 1065 (2013)], regarded as being beyond either the Ruderman-Kittel-Kasuya- Yosida (RKKY) or Van Vleck type mechanism, proposed previously [Science 329, 61 (2010)]. In order to uncover this issue, Jeongwoo Kim and Seung-Hoon Jhi (POSTECH) have investigated Cr doping to BST materials, based on the first- principles approach. They found that superexchange interactions due to wave-function overlaps between Cr d orbitals and BST p orbitals are responsible for ferromagnetic behaviors in insulating phases of BST (private communications). Of course, RKKY interactions play their roles in the dynamics of doped magnetic ions in metals, certainly.PHYSICAL REVIEW B 91, 115125 (2015) [23] A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, New York, 1994). [24] We point out that this distribution function should be deter mined self-consistently. For example, one can transform the distribution function of P (/rr< ) into P(4>r), resorting to the solution 4 > r [/rr/] of the saddle-point analysis in Eq. (2). If one starts from a log-normal distribution function for the RKKY interaction, he/she may get a power-law distribution function for the effective magnetic moment [S. Kettemann, E. R. Mucciolo, I. Varga, and K. Slevin, Phys. Rev. B 85, 115112 (2012)], which gives rise to quantum Griffiths phenomena [26]. Unfortunately, we do not touch this difficult issue in the present problem. Instead, we consider a Gaussian distribution function for random chiral gauge fields, where ferromagnetic clusters are assumed to be independent of each other. [25] K. Byczuk, W. Hofstetter, and D. Vollhardt, Anderson localiza tion vs. Mott-Hubbard metal-insulator transition in disordered, interacting lattice fermion systems, Int. J. Mod. Phys. B 24,1727 (2010 ). [26] E. Miranda and V . Dobrosavljevic, Disorder-driven non-Fermi liquid behavior of correlated electrons, Rep. Prog. Phys. 68, 2337 (2005). [27] For recent reviews on Weyl metals, see either Pavan Hosur and Xiaoliang Qi, C. R. Phys. 14, 857 (2013) ; or Ki-Seok Kim, Heon-Jung Kim, M. Sasaki, J.-F. Wang, and L. Li, Sci. Technol. Adv. Mater. 15, 064401 (2014) . 115125-24 Copyright of Physical Review B: Condensed Matter & Materials Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.

PhysRevB.81.155439.pdf

Relativistic embedding method: The transfer matrix, complex band structures, transport, and surface calculations M. James and S. Crampin * Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom /H20849Received 9 November 2009; revised manuscript received 1 March 2010; published 20 April 2010 /H20850 We develop the relativistic embedding method for electronic-structure studies. An expression for the transfer matrix is derived in terms of the Green’s function of the Dirac equation, and we outline its evaluation withinthe relativistic embedding framework. The transfer matrix is used to find the complex band structure and anembedding potential that can replace a semi-infinite substrate in ab initio electronic-structure calculations. We show that this embedding potential may be used to define an operator that gives the current flowing across asurface; the eigenstates of which define channel functions for conductance studies, and which enable thederivation of a relativistic generalization of the known expression for the conductance across a nanodeviceconnected to leads. Finally, the application of the embedding potential in relativistic electronic-structure studiesis illustrated using an electronlike basis to solve the surface-embedded Dirac equation for Au /H20849111/H20850. A calcu- lation with a single layer of atoms within the embedded volume correctly predicts the magnitude of theRashba-type splitting of the zone center surface state. DOI: 10.1103/PhysRevB.81.155439 PACS number /H20849s/H20850: 71.15.Rf, 73.20.At, 71.70.Ej I. INTRODUCTION The embedding method of Inglesfield1,2has developed into a powerful tool for performing electronic-structure cal-culations for extended systems that may be naturally dividedinto two or more distinct regions. The key advantage of themethod is that it enables properties of the entire system to bedetermined by finding explicit solutions only in a smallerembedded region, with an additional surface operator, theembedding potential, added to the Hamiltonian to ensure thecorrect matching of wave functions on the surface separatingthe embedded region from the rest. A particular class ofproblems where the embedding method is especially usefulare semi-infinite systems such as surfaces and interfaces. Forexample, in contrast to slab and supercell methods, whichrequire large numbers of layers to limit spurious interactionsbetween states localized on opposite surfaces, 3in embedding calculations surface states, resonances and the bulk con-tinuum are faithfully reproduced and easily distinguishableeven in calculations that include just a single surface layer. 4 This can reduce computational demands. Furthermore, be-cause the embedding potential furnishes the surface calcula-tion with details of the bulk band structure, scattering prop-erties needed for photoemission, or tunneling spectroscopycalculations are correctly described, and Fermi surface ef-fects such as surface resistivity can be studied. 5Exploiting this, recent developments have extended the embedding ap-proach to transport problems and field emission. 6–10The em- bedding approach has also been applied to surfaces and in-terfaces in strongly correlated materials. 11 Most surface and transport calculations to date using the embedding method have been performed using the scalar-relativistic full potential linearized augmented plane wavemethod /H20849FLAPW /H20850. 4,6,7,12–14Scalar-relativistic methods and those based on the Schrödinger equation do not include thespin-orbit interaction which is responsible for important ef-fects in many systems, including significant energy shifts andsplitting in materials containing heavy elements, determiningthe spin orientation in crystals /H20849magnetocrystalline aniso- tropy /H20850and providing mechanisms for spin manipulation in the field of spintronics. Recently an embedding scheme hasbeen developed that is based upon the Dirac equation 15and which therefore naturally includes the spin-orbit interaction as well as other relativistic effects. In that work the embed-ding potential used was based upon the reflection matrix ob-tained from a layer multiple-scattering calculation—an ap-proach that is generally insufficient for complex and openstructures. Here we describe how fully relativistic, full-potential em- bedding calculations may be performed in the FLAPWframework for both surface and transport problems. The de-velopments are based upon the transfer matrix that describesthe relationship between boundary values of the solutions ofthe Dirac equation across an embedded region and parallelsimilar advances in the nonrelativistic framework. 6,16The transfer matrix enables accurate embedding potentials to beobtained for general systems, as well as providing the basisfor the calculation of the complex band structure, an impor-tant concept in understanding states at surfaces and inter-faces. We also derive a generalization of the expression forthe conductance across an interface in terms of the embed-ding potential and the Green’s function. The outline of our work is as follows. First, in Sec. IIwe summarize the key ideas of the relativistic embeddingmethod and the basis set used in our calculations. In Sec. III we derive an expression for the relativistic transfer matrixand discuss the nonrelativistic limit. We illustrate how thecomplex band structure may be obtained from the transfermatrix and how the transfer matrix may use to determine anembedding potential to replace a semi-infinite substrate. InSec. IVwe discuss conductance within the embedding framework, identifying surface-orthogonal channel functionsand using them to derive an expression for the transmissionthrough an interface in terms of the relativistic embeddedGreen’s function and embedding potentials. Finally in Sec. V we perform a demonstration fully relativistic surfacePHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 1098-0121/2010/81 /H2084915/H20850/155439 /H2084911/H20850 ©2010 The American Physical Society 155439-1electronic-structure calculation, using an embedding poten- tial derived from the relativistic transfer matrix, reproducingthe known dispersion and spin-orbit splitting of the surfacestate on Au /H20849111/H20850in a one layer calculation. II. RELATIVISTIC EMBEDDING METHOD In relativistic embedding problems we are interested in understanding systems described by the single-particle Diracequation /H20849H−w/H20850 /H9274=/H20849c/H9251·p+/H9252mc2+V−w/H20850/H9274=0 , /H208491/H20850 where the Hamiltonian describes an extended system /H20849re- gions I+II /H20850but the focus of our interest is a small region of space, region I—see Fig. 1. Examples include investigating surface or defect physics or transport properties across aninterface. It is also possible to study systems in which amagnetic field is present—for clarity of presentation we onlyconsider the scalar potential in Eq. /H208491/H20850. Instead of solving Eq. /H208491/H20850we solve the embedded Dirac equation 15 /H20849H+HS−w/H20850/H9274=0 /H208492/H20850 in region I alone, where HSis a surface operator acting on S, the surface dividing I and II, that ensures solutions of Eq. /H208492/H20850 coincide with solutions of Eq. /H208491/H20850inside region I HS/H9274/H20849r/H20850=ic/H6036/H9268·nˆS/H20849rS/H20850/H9254/H20849r−rS/H20850 /H11003/H20900/H9274s/H20849rS/H20850−ic/H6036/H20885 Sd2rS/H11032·/H9003/H20849rS,rS/H11032;w/H20850/H9268/H9274l/H20849rS/H11032/H20850 0 /H20901. /H208493/H20850 Here/H9274land/H9274srefer to the upper and lower two-spinors that make up the Dirac four-spinor /H9274, which we refer to as the large /H20849l/H20850and small /H20849s/H20850components on account of their typical relative magnitudes for electron states. The position vector rS is on S, and nˆS/H20849rS/H20850is the surface normal /H20849f r o mIt oI I /H20850atrS. /H9003, which contains all information about region II, is the rela- tivistic embedding potential . This 2 /H110032 matrix function re- lates the amplitudes of small and large components of /H9273/H20849w/H20850, the solution of the Dirac equation for region II at energy w, on the surface S/H9273s/H20849rS;w/H20850=ic/H6036/H20885 Sd2rS/H11032·/H9003/H20849rS,rS/H11032;w/H20850/H9268/H9273l/H20849rS/H11032;w/H20850. /H208494/H20850 Solutions of Eq. /H208492/H20850satisfy the Dirac equation in I, and also satisfy Eq. /H208494/H20850onS, thereby correctly matching on to solu- tions in II. They therefore describe solutions of the Diracequation for I+II in region I. 15 Basis The embedded Dirac Eq. /H208492/H20850is typically solved by ex- panding /H9274in a suitable basis set and solving the resulting matrix problem. For extended systems it is usually advanta-geous to actually solve for the corresponding Green’s func-tion /H20849H+H S−w/H20850G/H20849r,r/H11032;w/H20850=−/H9254/H20849r−r/H11032/H20850. /H208495/H20850 The choice of basis set is motivated by the need to efficiently describe the spatial variations of the solutions and the bound-ary values. 8,12,13,17–19In the present work where we use an all-electron description and our focus is on relativistic sys-tems with underlying two-dimensional translational symme-try, we use a fully relativistic, electronlike linearized aug-mented plane wave basis set 20/H20849RLAPW /H20850so that the wave function is expanded in plane waves in the interstitial regionand atomiclike solutions in the atomic spheres. In the inter-stitial region /H9273gn/H9268/H20849r/H20850=/H20873/H9278/H9268 /H9253k/H9268·k+/H9278/H9268/H20874eik+·r/H11006/H20873/H9278/H9268 /H9253k/H9268·k−/H9278/H9268/H20874eik−·r,/H208496/H20850 where the /H11006sign determines whether the zdependence of the large component is sin-like or cos-like, as in correspond-ing nonrelativistic implementations of the embeddingmethod. 12,13,19The wave vectors in Eq. /H208496/H20850arek/H11006=k/H20648 +g/H11006knzˆwhere k/H20648is a wave vector in the two-dimensional Brillouin zone, gis a two-dimensional reciprocal lattice vec- tor, and the components in the zˆdirection are defined over the interval /H208510:L/H20852,kn=2n/H9266/L.Lis greater than the length of the embedded region so that the basis functions have a rangeof amplitudes on S/H20849see Sec. V/H20850. The /H9278/H9268are the usual Pauli two-spinors and /H9253k=c/H6036//H20849Wk+mc2/H20850with Wk=/H20881c2/H60362k2+m2c4. These basis functions are extended into the atomic sphere aslinear combinations of the radial solutions of the Dirac equa-tion and their energy derivatives for the spherical componentof the atomic potential at that site, found at some pivot en- ergy W /H9260/H9251. In sphere /H9251atR/H9251 /H9273gn/H9268/H20849r/H20850=/H20858 /H9011/H20851Agn/H9268/H9251/H9011u/H9011/H9251/H20849r/H9251/H20850+Bgn/H9268/H9251/H9011u˙/H9011/H9251/H20849r/H9251/H20850/H20852, /H208497/H20850 where /H9011=/H20849/H9260,/H9262/H20850,r/H9251=r−R/H9251,u˙=/H11509u//H11509W, and u/H9011/H9251/H20849r/H20850=/H20875g/H9260/H9251/H20849r/H20850/H9024/H9260,/H9262/H20849rˆ/H20850 if/H9260/H9251/H20849r/H20850/H9024−/H9260,/H9262/H20849rˆ/H20850/H20876 /H208498/H20850 with/H9024/H9260,/H9262/H20849rˆ/H20850the usual linear combination of spherical har- monics. The coefficients Agn/H9268/H9251/H9011andBgn/H9268/H9251/H9011may be chosen by matching amplitudes of large and small components at thesurface of the spheres, or alternatively by matching both am-plitude and derivative of the large component. This “elec-I II S FIG. 1. Geometry for a general embedding problem. The region of interest, region I, is separated from extended region II by surfaceS.M. JAMES AND S. CRAMPIN PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-2tronlike” basis prevents variational collapse21to negative en- ergy solutions of the Dirac equation while enabling anaccurate solution of Eq. /H208492/H20850or Eq. /H208495/H20850for layered systems. The additional spin index in the basis functions means basissets used in this relativistic formulation are twice the size ofthose used in conventional embedding studies. III. TRANSFER MATRIX The transfer matrix is a powerful tool in nonrelativistic calculations, playing a key role in studies of complex bandstructures, 16transport properties,6and, of particular signifi- cance here, in deriving embedding potentials for use inelectronic-structure calculations involving semi-infinitesubstrates. 13In this section we show how the idea of the transfer matrix can be implemented in a fully relativisticframework. To define the transfer matrix we consider a system in which a region of space /H9024extends infinitely in two dimen- sions and is finite in one dimension, bounded by surfaces S L andSR—see Fig. 2. The transfer matrix T/H9024relates the am- plitude of the wave function on SL,/H9023L, to the amplitude on SR,/H9023R /H9023R=T/H9024/H9023L. /H208499/H20850 Starting from the Dirac equation, it can be shown15that the wave function at position rin/H9024may be related to its ampli- tude on the bounding surface through the Green’s function G /H9023/H20849r/H20850=−ic/H6036/H20885 Sd2rS·G/H20849r,rS/H20850/H9251/H9023/H20849rS/H20850. /H2084910/H20850 Here S=SL/H33371SRand the surface normal is out of /H9024. This result is independent of the boundary conditions satisfied byGso choosing G lsto vanish on SLandSRgives /H9274Ll=FLR/H9274Rs+FLL/H9274Ls, /H9274Rl=FRR/H9274Rs+FRL/H9274Ls, /H2084911/H20850 where position vectors are suppressed and multiplication im- plies integration over the appropriate surface. /H9274Llis the large component of the wave function on SL,FLR= −ic/H6036Gll/H20849rL,rR/H11032/H20850/H9268·nˆR/H20849rR/H11032/H20850etc. Note that in obtaining these re- lations we allow rin II to approach the surface from within /H9024, establishing the distinct ordering when Green function arguments rSandrS/H11032are on the same surface. Rearranging Eq./H2084911/H20850, we may therefore construct the transfer matrix as T/H9024=/H20873FRRFLR−1FRL−FRRFLR−1FLL FLR−1−FLR−1FLL/H20874. /H2084912/H20850 The transfer matrix only involves the large-large component of the Green’s function for region /H9024. In the following section we illustrate this for a problem that can be solved analyti-cally. More generally we calculate T /H9024with the embedding formalism. In this case in Eq. /H208495/H20850we put HS=HL+HR, and then set to zero the left and right embedding potentials. Thisvariationally imposes the boundary condition that the smallcomponent of the Green’s function vanishes on S Land SR that was assumed in deriving the transfer-matrix expression. At this point it is interesting to consider the nonrelativistic limit of the transfer matrix in Eq. /H2084912/H20850by letting c→/H11009.I n this limit the large component of the wave function reducesto a nonrelativistic wave function /H20849with appropriate Pauli two-spinor /H20850and the small component, being of order 1 /c, vanishes. In general, G llbecomes the nonrelativistic Green’s function and all other components vanish. It is clear that FLR etc. are of order cand a nonrelativistic limit of these quan- tities is ill defined. Regarding the transfer matrix, TllandTss persist as c→/H11009,Tslis of order 1 /cand vanishes, while Tls, which is order c, is ill defined. To regularize the limit we introduce the transformation /H9023˜S=CS/H9023S,T˜/H9024=CRT/H9024CL−1with CS/H20849rS,rS/H11032/H20850=/H208981 0 0iw−v/H20849rS/H20850+mc2 c/H6036/H9268S/H20849rS/H20850/H20899/H9254/H20849rS−rS/H11032/H20850. /H2084913/H20850 The transfer matrix becomes as c→/H11009 T˜/H9024→/H20898GRRGLR−1−/H60362 2m/H20851GRL−GRRGLR−1GLL/H20852 −2m /H60362GLR−1−GLR−1GLL/H20899,/H2084914/H20850 which is formally identical to the expression for the nonrel- ativistic transfer matrix in terms of the Green’s function16 although here the entries are 2 /H110032 quantities. The reason we must make the transformation in Eq. /H2084913/H20850 in order to recover the nonrelativistic limit is as follows. TheSchrödinger equation is second order in space, meaning thatboth the amplitude and first derivative of the wave functionmust be specified on S LandSRto properly define a transfer matrix. In contrast, the Dirac equation is first order in spaceand only the amplitude of the wave function must be speci-fied in the bounding surfaces. However, the small componentmay be expressed in terms of the first derivative of the largecomponent but a prefactor which varies like 1 /casc→/H11009 means it vanishes and only the large component survives inthe limit. The transformation in Eq. /H2084913/H20850ensures that it per- sists. Despite the identical appearance of the c→/H11009limit of T ˜ and the nonrelativistic transfer matrix,16there are significant differences. In practice the 2 /H110032 blocks in Eq. /H2084914/H20850are not diagonal when the Green’s function is required to satisfy theΨL Ω SLΨR SR FIG. 2. Geometry for a general definition of the transfer matrix, T/H9024, which relates the amplitude of a wave function on surface SL, /H9274L, to that on surface SR,/H9274R.RELATIVISTIC EMBEDDING METHOD: THE TRANSFER … PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-3boundary conditions leading to Eq. /H2084911/H20850, and indeed the transformation in Eq. /H2084913/H20850gives /H9023˜=/H20873/H9274l /H11509nS/H9274l+i/H9268·/H20851nˆS/H11003/H11633/H9274l/H20852/H20874 /H2084915/H20850 the second term in the lower component being an additional term not present in the nonrelativistic theory. A. Analytic example An illustrative example which may be treated analytically is a region /H9024of constant potential v, between parallel planes SLandSRatz=Landz=Rrespectively, with R−L=d. The resulting transfer matrix can be used, e.g., to study scatteringby piecewise constant potentials. Translational invariance parallel to the planes reduces the problem to finding the wave-vector-resolved transfer matrixT /H9024,k/H20648. At energy wand wave-vector k/H20648solutions of the Dirac equation in /H9024with zero-amplitude small component at z=0 are/H9274k/H20648,/H9268/H20849z/H20850=/H20898c/H20849/H6036/H9268/H20648·k/H20648+/H9268zpz/H20850 w−v−mc2sinkzz/H9278/H9268 sinkzz/H9278/H9268/H20899, /H2084916/H20850 where kz=/H20881/H20849w−v/H208502−m2c4−c2/H60362k/H206482/c/H6036. From these, the Green’s function which satisfies the zero-amplitude smallcomponent boundary conditions on z=Land z=Rmay be constructed by the direct method; for z/H11021z /H11032 Gk/H20648,/H9268/H20849z,z/H11032/H20850=w−v−mc2 c2/H60362kzsinkz/H20849R−L/H20850/H11003/H9274k/H20648,/H9268/H20849z−L/H20850/H9274k/H20648,/H9268/H11003/H20849z/H11032−R/H20850 /H2084917/H20850 with Land Rin the wave function interchanged when z /H11022z/H11032. Placing z/H11032and then zon the surface planes from within /H9024one finds, e.g., FLL=ic/H6036 w−v−mc2/H20875kzcotkzd−i/H20849kx−iky/H20850 −i/H20849kx+iky/H20850−kzcotkzd/H20876 /H2084918/H20850 and eventually the transfer matrix T/H9024,k/H20648=/H20898coskzd −i/H20849kx−iky/H20850 kzsinkzdi/H20849w−v+mc2/H20850 c/H6036kzsinkzd 0 i/H20849kx+iky/H20850 kzsinkzd coskzd 0 −i/H20849w−v+mc2/H20850 c/H6036kzsinkzd i/H20849w−v−mc2/H20850 c/H6036kzsinkzd 0 cos kzd −i/H20849kx−iky/H20850 kzsinkzd 0 −i/H20849w−v−mc2/H20850 c/H6036kzsinkzdi/H20849kx+iky/H20850 kzsinkzd coskzd/H20899. /H2084919/H20850 The llblock, which is unaltered by the transformation T→T˜introduced in the previous section and which therefore survives unchanged in the c→/H11009limit, is not diagonal, dem- onstrating the difference with the nonrelativistic transfer ma-trix obtained in Ref. 16. Only for k /H20648=0, when the momentum is parallel to the spin-quantization axis, do we recover anidentical form T˜/H9024,k/H20648=0=/H20873coskzd1 −kz−1sinkzd1 −kzsinkzd1 − cos kzd1/H20874. /H2084920/H20850 B. Complex band structure Assume that region /H9024in the transfer matrix problem is a representative layer /H20849containing one or more planes of atoms /H20850 of a bulk crystal which has three-dimensional translationalsymmetry. Such a layer can reproduce the bulk crystal by repetition in directions normal /H20849zˆ/H20850to the layers with an asso- ciated layer-layer translation vector dthat reflects the trans- lational periodicity of the solid and also takes surface S LtoSR—see Fig. 3. It follows that Bloch states of the crystal satisfy /H9023R=eik·d/H9023L /H2084921/H20850 and comparing with Eq. /H208499/H20850we see that solutions of the eigenvalue problem16 /H20885 SLd2rL/H11032T/H9024/H20849rL+d,rL/H11032/H20850/H9274/H20849rL/H11032/H20850=/H9261/H9274/H20849rL/H20850/H20849 22/H20850 yields eigenvectors that are Bloch states on SLcorresponding to eigenvalues /H9261=eik·d. At real energies, eigenstates of T/H9024with eigenvalues that satisfy /H20841/H9261/H20841=1 have real wave vectors kthat are part of the conventional band structure of the solid. Those with kz/H110220 are right propagating Bloch states and those with kz/H110210 are left propagating Bloch states. Alongside these solutions areothers for which /H20841/H9261/H20841/HS110051, corresponding to solutions with complex wave vectors with imaginary values of k z. These form the complex band structure . The associated wave func- tions have an exponentially varying envelope, decaying toM. JAMES AND S. CRAMPIN PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-4the left /H20849right /H20850if/H20841/H9261/H20841/H110221/H20849/H110211/H20850. Such evanescent waves do not satisfy physical boundary conditions for an infinite bulkcrystal but may persist if this translational symmetry is bro-ken, for example, by a surface or interface. In these situa-tions screening confines modification of the potential to afew layers close to the surface/interface, beyond which thepotential is bulklike and where the wave function may in-clude contributions that decay into the crystal. Indeed inband gaps there are no states with real kand it is the mag- nitudes of the imaginary allowed k zthat determine the decay of localized states away from interfaces,22and tunneling across metal/insulator/metal junctions.23In general the eva- nescent Bloch states are essential for a complete descriptionof scattering. The transfer matrix given in Eq. /H2084912/H20850enables the complex band structure to be determined accurately ingeneral systems when relativistic effects are significant. As an example, for the analytic case studied in Sec. III A it is straightforward to show that the four eigenvalues ofT /H9024,k/H20648are the twofold degenerate pair /H9261/H11006=e/H11006ikzdand that the corresponding /H20849unnormalized /H20850eigenvectors may be chosen to be Uk/H20648,/H9268/H11006=/H20898/H9278/H9268 c/H6036/H9268·/H20849k/H20648/H11006kzzˆ/H20850 w−v+mc2/H9278/H9268/H20899, /H2084923/H20850 which as expected describe the amplitudes on SLof Bloch states with wave vector k/H20648/H11006kzzˆ. For realistic systems the complex band structure can be obtained by diagonalizing the numerically determined trans-fer matrix. To do so it is first necessary to identify the region/H9024, or equivalently the surfaces S Land SR. The nature of electron wave functions /H20849and Green’s functions /H20850in solids is such that it is not normally possible to identify suitable sur-faces for this purpose that enable a numerical representationthat is both efficient and straightforward. A surface on whichthe wave function is relatively smooth must weave betweenatomic cores, complicating the identification of suitable sur-face expansion functions while a planar surface that affords aroutine surface expansion will normally cut through coresand require a large number of expansion functions to accu-rately describe the rapid spatial variations. Solutions to this problem involve transforming the prob- lem on curvy surfaces S L,SRto an equivalent problem statedon planar surfaces.12,13,19,24Here we adapt the approach of Ref. 16to the relativistic case. Buffer regions /H9004Land/H9004Rare introduced to either side of /H9024as in Fig. 4so that the new volume /H9024/H11032=/H9004L/H33371/H9024/H33371/H9004Rhas planar surfaces PLandPR.A subsidiary problem is introduced containing the space /H9024/H11033 =/H9004L/H33371/H9004R/H11032, where /H9004R/H11032is region /H9004Rtranslated by − d. The po- tential in the buffer regions is arbitrary but for practical pur-poses is chosen to be smoothly varying, for example, a con-stant or the smooth extension of the interstitial expansioninto these regions. The transfer matrices T /H9024/H11032and T/H9024/H11033are found using Eq. /H2084912/H20850where the surface integrals that are now required are taken over planar surfaces well removed fromatomic cores, which are straightforward to perform. In termsof the transfer matrices for the individual regions we have T /H9024/H11032=T/H9004RT/H9024T/H9004L,T/H9024/H11033=T/H9004R/H11032T/H9004L/H2084924/H20850 and one can show that T=T/H9024/H11032T/H9024/H11033−1/H11013T/H9004RT/H9024T/H9004R/H11032−1/H2084925/H20850 satisfies an eigenvalue equation similar to Eq. /H2084922/H20850but with position vectors on PR/H11032notSL, and with the same eigenvalues asT/H9024. Hence the complex band structure may be obtained by constructing and diagonalizing T, which only involves ex- pansions on, and integrals over, planar surfaces. Using the RLAPW basis, contraction of the Green’s func- tion expansions onto the planar surfaces leads to surface ex-pansion in basis functions /H9273/H9263/H20849r/H20648/H20850=1 /H20881Aei/H20849g+k/H20648/H20850·r/H20648/H9278/H9268 /H2084926/H20850 at wave vector k/H20648, where /H9263=/H20849g,/H9268/H20850is a composite index and the complex band structure is obtained from the explicit ei-genvalue problem /H20858 /H9263/H11032eig·d/H20648T/H9263/H9263/H11032/H9020/H9263/H11032=eikzdz/H9020/H9263. /H2084927/H20850 Figure 5shows results obtained for Au /H20849111/H20850. In this calcula- tion the planes PLandPR/H20849Fig. 4/H20850have been placed 2.7 a.u. to either side of the plane of atoms and the potential withinthe buffer region is the continuous extension of the planewave expansion of the interstitial potential into these regions.We use a basis set containing 19 gvectors. The real bandsd Ω SL SR FIG. 3. /H20849Color online /H20850Geometry for calculating the transfer ma- trix in the bulk. Adjacent layers are related by a translation d. The transfer matrix connects states on surface SLacross region /H9024to states on surface SR.Ω∆L ∆R SL SR PL PR∆L ∆/prime R PLSL P/prime R FIG. 4. /H20849Color online /H20850Geometry of embedding regions used in practical embedding calculations. Left: buffer regions /H9004L,/H9004Rex- tend/H9024so that the combined region has planar bounding surfaces, PL,PR. Right: the auxiliary problem involving regions /H9004Land/H9004R/H11032, bounded by surfaces PLandPR/H11032.RELATIVISTIC EMBEDDING METHOD: THE TRANSFER … PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-5are in excellent agreement with previous results and the com- plex bands show their continuation into the complex plane.25 We see loops connect extrema of the real bands with /H90116 symmetry. One connects the L6−level near −7.6 eV with a band minimum with kz/H110150.3/H9003Land smaller loops connecting extrema away from the high symmetry points are found cen-tered on k z/H110150.5/H9003Land 0.7 /H9003L. The L6−andL6+levels below and above EFare also connected by a complex band with Re/H20849kz/H20850=L.I nt h e /H20849E,kz/H20850range show in Fig. 5two loops con- nect extrema of the real bands with extrema away from thereal kaxis. The /H9011 4+5band extrema at E=−5.3 eV is con- nected to a minimum of the complex bands with E= −3.0 eV and Re /H20849kz/H20850=/H9003, and the /H90116band extreme at E= −5.4 eV connects with the complex band with E=−4.2 eV and Re /H20849kz/H20850=/H9003. Finally, we mention that Dal Corso et al.26 have recently reported a scheme for calculating relativistic complex bands that employs two-component wave functionsand relativistic pseudopotentials—here the bands are foundfrom an all-electron four-component solution of the fullDirac equation. C. Embedding potential Another application of the transfer matrix is to generate the embedding potential for a semi-infinite bulk crystal, forsubsequent surface or interface studies. In Eq. /H208494/H20850the embed- ding potential at energy wrelates the amplitudes on Sof small and large components of /H9273, a solution of the Dirac equation at energy win II. If II corresponds to a semi-infinite left substrate then /H9273can be expressed as a linear combination of outgoing Bloch states, i.e., either decaying to the left orwith k z/H110210. The surface values on SLof these Bloch states are given by the eigenvectors of the transfer matrix and itfollows that the embedding potential may be found by invert-ing a matrix of eigenvectors of the transfer matrix. An em-bedding potential for a semi-infinite right substrate is simi-larly obtained from the Bloch states that decay or carry fluxto the right. 16 It is once again beneficial to work on planar surfaces. An eigenstate /H9020jofTin Eq. /H2084925/H20850is defined on surface PR/H11032/H20849Fig. 4/H20850and is related to the eigenstate /H9274jofT/H9024with the sameeigenvalue by /H9274j=T/H9004R/H11032−1/H9020j; hence /H9023j=T/H9024/H11033−1/H9020jis the amplitude on plane PLof the Bloch state /H9274jback propagated from SL, and can be used to derive an embedding potential for a leftsubstrate on the proviso that the buffer region is also in-cluded in the subsequent embedded system calculation. Us-ing surface expansion functions in Eq. /H2084926/H20850we obtain /H9003 /H9263/H9263/H11032=i c/H6036/H20858 j/H20851/H9023s/H20852/H9263j/H20851/H9023l−1/H20852j/H9263/H11032/H9268z,/H9268/H11032/H9268/H11032, /H2084928/H20850 where /H20851/H9023s/H20852is a matrix formed of the small components of transferred eigenvectors /H9023jcorresponding to outgoing Bloch states of the left substrate and /H20851/H9023l−1/H20852is the inverse of the corresponding matrix of large components. IV. CHANNEL FUNCTIONS AND TRANSPORT The ability to handle extended substrates makes the em- bedding scheme attractive for ballistic transport studies of“nanodevices.” Embedding potentials may be used to replacecurrent carrying leads so that only the device region, region/H9024in Fig. 6, needs to be explicitly treated. Conduction through the device region /H9024results from electrons being transmitted from open channels in the left-hand lead, through/H9024and into the open channels in the right hand lead. Calcu- lations of the conduction therefore reduce to calculation ofthe transmission probability between open channels in theleads. Here we show how this can be found in the relativisticformulation. In relativistic theory the probability current J Scarried across surface Sby state /H9274is JS=c/H20885 Sd2rS·/H9274†/H20849rS/H20850/H9251/H9274/H20849rS/H20850. /H2084929/H20850 Expanding /H9274into its large and small components, and using Eq. /H208494/H20850to replace /H9274s, we obtain for the current JS=c2/H6036/H20885 Sd2rS/H20885 Sd2rS/H11032/H9274l†/H20849rS/H20850/H9018/H20849rS,rS/H11032/H20850/H9274l/H20849rS/H11032/H20850, /H2084930/H20850 where /H9018/H20849rS,rS/H11032/H20850=i/H9268S/H20849rS/H20850/H20851/H9003/H20849rS,rS/H11032/H20850−/H9003†/H20849rS/H11032,rS/H20850/H20852/H9268S/H20849rS/H11032/H20850. /H2084931/H20850 Thus the current may be obtained from the large component of the wave function and the embedding potential evaluatedRe(kz)Γ L -22-9-6-303 Im(kz) [a.u.]E-EF(eV) FIG. 5. /H20849Color online /H20850Complex band structure for Au /H20849111/H20850 found from the transfer matrix evaluated at k/H20648=0. Only the bands with /H20841Im/H20849kz/H20850/H20841/H11349/H9266are shown.SL SR φ χ χΩ R L FIG. 6. /H20849Color online /H20850Geometry for studying conductance across a nanodevice in region /H9024, connecting left /H20849L/H20850and right /H20849R/H20850 leads. SLandSRare dividing surfaces. In the left lead /H9278describes a state carrying current toward SL; this extends into /H9024+Ras/H9273,i nR carrying current away from SRto the right.M. JAMES AND S. CRAMPIN PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-6at the energy of the state /H9274. Our convention for the surface normals when embedding region /H9024means that, when applied to the geometry of Fig. 6,JSis the current flowing in the direction into the left /H20849right /H20850lead when SisL/H20849R/H20850. A. Channel functions The eigenfunctions of /H9018, which satisfy /H20885 Sd2rS/H9018/H20849rS,rS/H11032/H20850ui/H20849rS/H11032/H20850=/H9261iui/H20849rS/H20850/H20849 32/H20850 can be used to define channel functions of the lead associated with S. Since /H9018is Hermitian, its eigenvalues /H20853/H9261i/H20854are real and its eigenfunctions /H20853ui/H20854may be chosen to be orthogonal over S. The 2 /H110032 nature of /H9018means that the functions uiare 2 /H110031 spinors; we can identify these as the surface values of the large components of an extended function /H9274i, whose small component on Smay be found from the embedding potential, Eq. /H208494/H20850, and whose value within the lead may be found from the Green function for the lead using Eq. /H2084910/H20850. Thus uifully determines a channel function /H9274i. With uinor- malized over Sit follows from Eqs. /H2084930/H20850and /H2084932/H20850that the current across Sassociated with channel function /H9274iisJi =c2/H6036/H9261i, and hence /H9261i/H113500 as a result of the outgoing bound- ary conditions implicit in the embedding potential. Nonzero/H9261 icorrespond to current-carrying or open channels and zero eigenvalues correspond to closed channels /H20849evanescent /H9274i/H20850. Expanding /H9018in terms of its eigenfunctions, we then have /H9018/H20849rS,rS/H11032/H20850=/H20858 i/H9261iui/H20849rS/H20850ui†/H20849rS/H11032/H20850, /H2084933/H20850 where the sum need only be taken over open channels. Figure 7shows the eigenvalues of /H9018atk/H20648=/H208490,0/H20850for a semi-infinite Au /H20849001 /H20850substrate, alongside the conventional band structure. In this calculation the embedding plane ispositioned midway between atomic planes. Bloch states, which are commonly used as channel func- tions in conductance studies are orthogonal functions withinthe volume of the lead but not in general orthogonal on S;i n contrast, the embedding channel functions are orthogonal onSbut not in general in the volume of the leads. The surface orthogonality of the embedding channel functions is attrac-tive as this is precisely where the channel functions are re- quired in calculating the conductance and their orthogonalityprovides new opportunity for interpreting how the nanode-vice in /H9024conducts between left and right leads. B. Conductance We now consider the conductance through a nanostructure with device region /H9024sandwiched between leads, as in Fig. 6. Anincoming state from the left lead L,/H9278, is transmitted across /H9024into outgoing states in the right lead R. These leads will be replaced by embedding potentials acting on SLand SR, respectively, and the problem will be solved in /H9024alone. The embedding potential introduced in Sec. IIrelates small and large components of outgoing states, being evalu- ated at energy w+i/H9254, where /H9254is a positive infinitesimal. In the conductance problem we have an incoming state/H9278in the left lead, which is a time-reversed outgoing state. In relativ- istic theory the time reversal operator is Kˆ=−i/H9268y/H208494/H20850Kˆ0where Kˆ0is the complex conjugation operator27so that /H9278=Kˆ/H9274 where /H9274is an outgoing state. Using Eq. /H2084910/H20850this enables us to express the large component of the wave function in /H9024 +Rin terms of the large-large component of the Green’s function and /H9018for the left lead /H9273l/H20849r/H20850=ic2/H60362/H20885 SLd2rL/H20885 SLd2rL/H11032Gll/H20849r,rL/H20850/H11003Kˆ/H9018L/H20849rL,rL/H11032/H20850Kˆ†/H9278l/H20849rL/H11032/H20850. /H2084934/H20850 We now assume that the incoming state corresponds to an open channel function of the left lead, KuL,i, and expand the transmitted state on SRas a sum over channel functions of the right lead, /H9273l/H20849rR/H20850=/H20858jtijuR,j/H20849rR/H20850, where tijis a transmission coefficient. If now the channel functions are normalized tocarry unit current /H20849c 2/H6036/H9261i/H20848Sd2rS/H20841uS,i/H20849rS/H20850/H208412=1/H20850then tij=ic4/H60363/H9261i/H9261j/H20885 SLd2rL/H20885 SRd2rRuj†/H20849rR/H20850/H11003Gll/H20849rR,rL/H20850vi/H20849rL/H20850. /H2084935/H20850 The current through the device region is proportional to the transmission probability Tij=/H20841tij/H208412summed over all open channels, where each channel function carries the same fluxas may be shown by demonstrating a unitary transformationrelates them to the corresponding set of Bloch states. 9Mak- ing use of the property of the time-reversed Green’s function, KˆG/H20849r,r/H11032/H20850Kˆ†=G†/H20849r/H11032,r/H20850, along with Eq. /H2084932/H20850, we find that the transmission is given by /H20858 ijTij=/H20849c2/H60362/H208502TrKˆ/H9018LGLRKˆ†/H9018RGRL, /H2084936/H20850 where position arguments have been suppressed for brevity and where the trace is taken over both spin and positionvariables. Transmission through the device region, /H9024, may therefore be determined fully relativistically, using the large-large part of the Green’s function calculated only in /H9024 /H20849which may be found via embedding /H20850, and the embedding potentials for the leads. This result generalizes corresponding−8−6−4−202 Γ XE−EF(eV) k⊥5 10 15 20 λi[a.u.] FIG. 7. Au /H20849001 /H20850interface at k/H20648=0. Left: band structure in the /H9003Xdirection; right: eigenvalues of /H9018k/H20648=0evaluated on an embed- ding plane midway between atomic planes.RELATIVISTIC EMBEDDING METHOD: THE TRANSFER … PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-7nonrelativistic expressions.6,9The appearance of the time- reversal operation in Eq. /H2084936/H20850may be explained by recogniz- ing the Green’s function as a propagator. Beginning at the right-hand side, /H9018/H20849rR/H11032,rR/H20850Gll/H20849rR,rL/H20850describes propagation from SLtoSR, the direction of the current and /H9018/H20849rL,rL/H11032/H20850Gll/H20849rL/H11032,rR/H11032/H20850describes propagation from SRtoSLbut is time reversed and so is also in the direction of the current. V. EMBEDDED SURFACE CALCULATIONS A class of problems to which the embedding method is particularly well suited is calculation of the electronic struc-ture of surfaces and we demonstrate here a relativistic sur-face embedding calculation with Au /H20849111/H20850. We choose this system because it has a well-characterized surface state 28 which through a combination of the broken inversion sym-metry at the surface and the relativistic spin-orbit interactionexhibits a spin-split Rashba-type dispersion E/H20849k/H20850/H11229E 0+/H60362/H20849k/H11006k0/H208502 2m/H11569. /H2084937/H20850 Furthermore, this state has particular properties that provide a stringent test of the extent to which the embedding poten-tial correctly reproduces the influence of an extended sub-strate when added to the Hamiltonian of the surface region ofa crystal. First, it is known from previous studies /H20849e.g., Ref. 29/H20850that the Au /H20849111/H20850surface state extends a significant dis- tance into the crystal, making it especially sensitive to thecrystal potential beneath the surface layers. An illustration ofthis comes from the magnitude of the interaction seen in thinfilm or periodic supercell calculations for Au /H20849111/H20850, where surface states that form on one crystal face are sensitive tothe presence of the second, resulting in an energy splitting. 3 find the magnitude of this is in excess of 500 meV for seven-layer slabs and only becomes less that 10 meV for slab thick-nesses greater than 23 layers. Second, 30have exploited the fact that the dominant contribution to the spin-orbit interac-tion originates from the potential in a small volume sur-rounding each nucleus to decompose the spin-orbit splittinginto layer by layer contributions. They find only /H1122958% of the splitting may be attributed to the spin-orbit interaction inthe surface layer with successively deeper layers accountingfor 25%, 11% and 4%. Thus the subsurface region makes animportant contribution to the total relativistic effect on theAu/H20849111/H20850surface state. In our calculations we use an embedded region containing the outermost or the three outermost atomic layers of thesurface. The semi-infinite substrate is incorporated via anembedding potential that is found from the transfer matrix asoutlined in Sec. III C . This embedding potential provides a formally exact replacement of the influence of the semi-infinite substrate on the electron wave functions in the em-bedded region. 15In the surface calculation, by calculating the embedding potential from the transfer matrix there is madethe implicit assumption that the Hamiltonian for the substrateis “bulklike” up to the embedding surface, i.e., the effectivepotential is unchanged from that deep in the bulk. In metallicsystems screening is efficient so that for close-packed sur-faces this condition holds to a sufficient degree even afterjust one or two layers, whereas more would be required for open surfaces or nonmetallic systems. The geometry isshown in Fig. 8. The buffer region, /H9004 L, introduced when transferring the bulk embedding potential /H9003Lfrom the curvy surface SLto the plane PL, is also included as is required in order to properly compensate and ensure that the Green’sfunction in the surface region correctly matches to that in thebulk. On the vacuum side the space beyond which the poten-tial is assumed to be uniform /H20849in our calculations chosen to be/H110118 Å beyond the outermost atomic plane /H20850is incorpo- rated via another embedding potential, for which we use ananalytic expression. This is derived from Eq. /H208494/H20850using for /H9273 a general outgoing wave function for this constant potential. This gives /H9003R/H20849rS,rS/H11032;w/H20850=/H20885d2k/H20648 /H208492/H9266/H208502/H9003k/H20648/H20849w/H20850eik/H20648·/H20849rS−rS/H11032/H20850/H2084938/H20850 with /H9003k/H20648/H20849w/H20850=−i w−vvac+mc2/H20875kz/H20849k/H20648,w/H20850−kx+iky +kx+ikykz/H20849k/H20648,w/H20850/H20876,/H2084939/H20850 where kz/H20849k/H20648,w/H20850=/H20881/H20849w−vvac/H208502−m2c4−/H60362k/H206482c2//H6036cand vvacis the vacuum level measured relative to the zero of energy.Our attention is focused on the surface state near E F, for which the constant vacuum potential beyond PRis a suitable approximation. It is also possible to derive an analytic em-bedding potential for an imagelike potential, which would benecessary for studying relativistic effects on states nearer thevacuum level. 31 The embedded Green’s function at energy w,G/H20849w/H20850,i s found by solving Eq. /H208495/H20850with surface Hamiltonian HS=HL +HR, and where Gis expanded in a double basis of RLAPWs. From the Green’s function we obtain the localdensity of states n/H20849r;w/H20850=− /H9266−1Im Tr G/H20849r,r;w/H20850, charge den- sity, and other electronic properties. Self-consistent calcula-tions are performed within the local density approximation/H20849LDA /H20850to density-functional theory, using the Perdew- Zunger parametrization of the exchange-correlationpotential. 32We assume an ideal unrelaxed structure with lat- tice parameter 4.08 Å. Our basis sets contain interstitialplane waves up to 220 eV , spin-angular functions withinatomic spheres /H20849radii 1.395 Å /H20850corresponding to /H5129 max=9.←bulk vacuum→ ∆L 0PLSL PRLz FIG. 8. /H20849Color online /H20850The geometry for an embedded surface calculation. Embedding potentials act on planes PL/H20849substrate em- bedding potential /H20850and PR/H20849vacuum embedding potential /H20850; buffer region /H9004Lextends from PLtoSL, beyond which the actual surface potential applies.M. JAMES AND S. CRAMPIN PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-8The valence density is found by integrating the Green’s func- tions around a semicircular contour extending from extend-ing from just below the valence band up to the Fermi level,evaluated using 31 Gauss-Chebyshev points, and using theequivalent of 486 kpoints in the full two-dimensional Bril- louin zone. 33 In Fig. 9/H20849a/H20850we illustrate results obtained for the disper- sion of the zone-center surface state found in a calculation inwhich just the outermost layer of atoms is included in theembedded region. The dispersion is obtained from calcula-tions of the energy and wave-vector-resolved local density ofstates, integrated over the surface region, and which is evalu-ated with a small imaginary component in the energy tobroadens spectral features and aid their identification. Thesurface state clearly exhibits the characteristic spin-orbitsplitting and is easily distinguished from bulk states andresonances even in this one layer calculation as the embed-ding potential ensures that the bulk continuum is faithfullyreproduced, Fig. 9/H20849b/H20850. Calculating the expectation value of the spin operator in the surface state we find, in agreementwith both experiment 29,34,35and previous theoretical calculations,29,31,35that the electron spin vectors lie in the surface plane and perpendicular to the electron momentum. In Table Iwe compare the dispersion parameters and the wave-vector splitting at the Fermi energy found in our cal-culations /H20851Fig. 9/H20849c/H20850/H20852with those obtained in previous calcula- tions and those observed in angle-resolved photoemissionexperiments on Au /H20849111/H20850surfaces and a vicinal surface with /H20849111/H20850terraces. The wave-vector splitting that we find agrees well with previous calculated values and those from experi- TABLE I. Compilation of theoretically determined Au /H20849111/H20850surface state dispersion parameters and re- sults from angle-resolved photoelectron spectroscopy experiments. Some key calculation details are noted.LDA and generalized gradient approximation /H20849GGA /H20850refer to different treatments of exchange-correlation effects in the density-functional theory. FLAPW indicates the full-potential calculations using two-componentlinearized augmented plane waves with variational inclusion of spin-orbit interaction. Further details may befound in the cited articles. E 0 /H20849eV/H20850 m/H11569/mekf /H20849Å−1/H20850/H9004kf /H20849Å−1/H20850 Notes Theory −0.39 0.025 23 layer slab, LDA, FLAPW, Ref. 36 −0.50 0.23 0.023Semi-infinite crystal, LDA+image barrier, atomic sphere, Ref. 31 −0.51 0.20 /H110060.149, 0.172 0.023Semi-infinite crystal, LDA, muffin-tin, Refs. 29 and35 −0.484 0.22 0.031 23 layer slab, LDA, FLAPW, Ref. 3 −0.326 0.25 0.031 23 layer slab, GGA, FLAPW, Ref. 3 −0.52 0.24 /H110060.159, 0.191 0.032 24 layer slab, LDA, pseudopotential, Ref. 37 0.028 23 layer slab, FLAPW, Ref. 38 −0.52 0.25 /H110060.155, 0.184 0.029 1 embedded layer, LDA, RLAPW, this work −0.52 0.25 /H110060.155, 0.184 0.029 3 embedded layer, LDA, RLAPW, this work Experiment −0.417 0.25 /H110060.153, 0.176 0.023 Ref. 28 −0.487 0.255 /H110060.172, 0.197 0.025 Ref. 36 −0.487 0.255 /H110060.167, 0.192 0.025 Ref. 39 −0.439 0.254 /H110060.157, 0.184 0.026 Au /H2084923,23,21 /H20850, Ref. 40 −0.47 0.25 /H110060.160, 0.186 0.026 Ref. 35 −0.46 0.25 0.026 Ref. 34 −0.479 0.26 /H110060.172, 0.197 0.024 Ref. 3−0.2 −0.1 0 0.1 0.2(a) kx[Å−1]−0.4−0.200.2E−EF[eV]−2.0 −1.5 −1.0 −0.5 0.0(b) n(E,kx) [arb. units] E−EF[eV]kx=0.16 Å−1 0.15 0.17 0.19(c) n(EF,kx) [arb. units] kx[Å−1]∆k=0.029 Å−1 E=EF FIG. 9. /H20849Color online /H20850Results for the Au /H20849111/H20850surface state ob- tained using the relativistic embedding method including a singleatomic layer within the embedded volume. /H20849a/H20850Intensity plot of the local density of states /H20849LDOS /H20850n/H20849E,k x/H20850forkxalong /H9003K, calculated with Im E=10−4Ha. The shaded area is the bulk continuum. /H20849b/H20850 LDOS for fixed kx=0.16 Å−1, showing the bulk continuum and spin-orbit split surface state. /H20849c/H20850LDOS at E=EFshowing the wave- vector splitting of the surface state.RELATIVISTIC EMBEDDING METHOD: THE TRANSFER … PHYSICAL REVIEW B 81, 155439 /H208492010 /H20850 155439-9ment, which are on average /H1101110% lower than theory. Sig- nificantly, an embedding calculation in which just the outer-most surface layer is explicitly included in the embeddedvolume yields the same surface state dispersion and spin-orbit splitting as a calculation in which the three outermostlayers are included. Since over 40% of the splitting origi-nates from relativistic interactions deeper that the surfacelayer, 30this demonstrates that the relativistic embedding po- tential correctly replicates the influence of the extended sub-strate on states within the embedded region. VI. SUMMARY Inglesfield’s embedding method is a powerful tool for electronic-structure studies in extended systems which maybe naturally divided into two or more distinct regions. In thepresent work we have presented extensions to the embeddingmethod that enable practical calculations to be performed forsurface and interface systems when relativistic effects areexpected to be important. First, the concept of the transfermatrix used in the nonrelativistic theory has been adapted tomatch boundary values of solutions to the Dirac equationacross an embedded region. In contrast to the second ordernature of the Schrödinger equation, which means that theconventional transfer matrix is defined in terms of an ampli-tude and normal derivative on the surface, the Dirac equationis first order and no normal derivative boundary conditionneed be specified. Consequently the new transfer matrix in-volves only the amplitude of the four component wave func-tion on the surface. We have shown that as in the nonrelativ-istic case the transfer matrix may be written in terms ofGreen’s functions for the embedded region; by choosingthese to satisfy zero-amplitude small-component boundaryconditions a relatively simple form for the transfer matrix isobtained that formally resembles the nonrelativistic versionbut which generally differs in the c→/H11009limit. Application of the transfer matrix to the problem of find- ing the complex band structure including relativistic effectshas been described and demonstrated; also the use of theeigenvectors of the transfer matrix in constructing an embed- ding potential that can replace semi-infinite substrates in sub-sequent embedding calculations. The embedding potential, a2/H110032 matrix function in the relativistic formulation, is also shown to play an important role in formulating the transportproblem across an embedded region. It enables the identifi-cation of channel functions that form an orthogonal set overthe embedding surface and which form a natural representa-tion for determining transmission across interfaces. We haveused these channel functions to derive an expression fortransmission across an interface in terms of the Green’s func-tion for the embedded region and the embedding potentialsin the leads. In deriving this, the more subtle nature of theDirac theory becomes apparent as time reversal of states isnot achieved by simple complex conjugation. In the final section we have illustrated a relativistic sur- face electronic-structure calculation within the full-potentialimplementation of the embedding method. We have chosenthe well-studied Au /H20849111/H20850surface and presented the results of calculations in which just the outermost or outer three layersof atoms at Au /H20849111/H20850are explicitly included with the semi- infinite substrate replaced by an embedding potential that hasbeen obtained from the Dirac transfer matrix and with thesemi-infinite vacuum replaced by an analytic embedding po-tential. The surface state dispersion and spin-orbit splittingthat we obtain agree with those found in previous calcula-tions and photoelectron spectroscopy measurements. Bothour one- and three-layer calculations give similar results, il-lustrating the ability of the relativistic embedding potential tocorrectly replicate the influence of an extended substrate, in-cluding relativistic effects, on electronic states within theembedded region. ACKNOWLEDGMENTS The authors wish to thank H. 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PhysRevB.93.121112.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 93, 121112(R) (2016) Helicity-protected ultrahigh mobility Weyl fermions in NbP Zhen Wang,1,2Yi Zheng,1,3,4,*Zhixuan Shen,1Yunhao Lu,2Hanyan Fang,1Feng Sheng,1Yi Zhou,1,4Xiaojun Yang,1 Yupeng Li,1Chunmu Feng,1and Zhu-An Xu1,2,3,4,† 1Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China 2State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310027, People’s Republic of China 3Zhejiang California International NanoSystems Institute, Zhejiang University, Hangzhou 310058, People’s Republic of China 4Collaborative Innovation Centre of Advanced Microstructures, Nanjing 210093, People’s Republic of China (Received 20 June 2015; revised manuscript received 21 December 2015; published 28 March 2016) Noncentrosymmetric transition-metal monopnictides, including TaAs, TaP, NbAs, and NbP, are emergent topological Weyl semimetals (WSMs) hosting exotic relativistic Weyl fermions. In this Rapid Communication,we elucidate the physical origin of the unprecedented charge carrier mobility of NbP, which can reach 1 × 10 7cm2V−1s−1at 1.5 K. Angle- and temperature-dependent quantum oscillations, supported by density function theory calculations, reveal that NbP has the coexistence of p-a n dn-type WSM pockets in the kz=1.16π/c plane (W1-WSM) and in the kz=0 plane near the high symmetry points /Sigma1(W2-WSM), respectively. Uniquely, each W2-WSM pocket forms a large dumbbell-shaped Fermi surface enclosing two neighboring Weyl nodes withthe opposite chirality. The magnetotransport in NbP is dominated by these highly anisotropic W2-WSM pockets,in which Weyl fermions are well protected from defect backscattering by real spin conservation associated tothe chiral nodes. However, with a minimal doping of ∼1% Cr, the mobility of NbP is degraded by more than two orders of magnitude, due to the invalidity of helicity protection to magnetic impurities. Helicity protectedWeyl fermion transport is also manifested in chiral anomaly induced negative magnetoresistance, controlled bythe W1-WSM states. In the quantum regime below 10 K, the intervalley scattering time by impurities becomes alarge constant, producing the sharp and nearly identical conductivity enhancement at low magnetic field. DOI: 10.1103/PhysRevB.93.121112 Topological Weyl semimetals (WSMs) are regarded as the next wonderland in condensed matter physics [ 1–4]f o r exploring fascinating quantum phenomena [ 5–10]. Unlike Dirac semimetals (DSMs) [ 11,12], band crossing points in WSMs, i.e., Weyl nodes, always appear in pairs with oppositechirality, due to the lifting of spin degeneracy by breakingeither time reversal symmetry [ 1] or inversion symmetry [ 3,4]. Fermi surfaces (FSs) enclosing the chiral Weyl nodes arecharacterized by helicity, i.e., the spin orientation is eitherparallel or antiparallel to the momentum. Such helical Weyl fermions are expected to be remarkably robust against non- magnetic disorders, and may lead to novel device concepts forspintronics and quantum computing. The recent proposed noncentrosymmetric TaAs, TaP, NbAs, and NbP, have stimulated immense interests, due to the binary,nonmagnetic crystal structure. The existence of Weyl nodeshas soon been discovered in TaAs by angle-resolved pho-toemission spectroscopy (ARPES) [ 13,14], and by quantum transport measurements of NMR and a nontrivial Berry’sphase ( /Phi1 B)o fπ[15,16]. Transport studies of NbAs [ 17] and NbP [ 18] also show ultrahigh mobility and nonsaturating magnetoresistance (MR), but no convincing evidence on theexistence of Weyl fermions in these two compounds. However,ARPES resolves tadpole-shaped Fermi arcs on the (001)surface of both NbAs [ 19] and NbP [ 20]. It also shows pronounced changes in the electronic structures of NbAsand NbP compared to TaAs [ 19], mainly due to weaker spin-orbital coupling in the former two and shifting in theband crossing energy relative to the Fermi energy ( E F). In *phyzhengyi@zju.edu.cn †zhuan@zju.edu.cnTaAs, the dominant WSM electron pockets are enclosing the eight pairs of Weyl node 1 (W1) in the kz=1.18π/c plane, while WSM pockets surrounding Weyl node 2 (W2) in thek z=0 plane near the four /Sigma1points are negligible [ 15,16]. For NbAs, W2 becomes 36 meV below the nearly neutralW1 [ 19]. It is thus expected that charge transport in NbAs and NbP would notably differ from TaAs, because the W2-Weylcones are highly anisotropic in kspace compared to the relatively isotropic W1-Weyl cones [ 3,14,19]. Intriguingly, the carrier mobility of NbP (5 ×10 6cm2V−1s−1)[18]i s one order of magnitude higher than TaAs and NbAs [ 15–17]. Such striking mobility could either be due to significantlylower concentration of lattice disorders in NbP comparedto TaAs and NbAs, or indicate a protection mechanismthat effectively suppresses the backscattering of chargecarriers [ 12,21–24]. In this Rapid Communication, we unambiguously prove the existence of chiral WSM states in NbP using angle-dependent quantum oscillations of magnetoresistance, Nernstand Seebeck, compared with the density functional theorycalculations. We show that the unprecedented mobility of1×10 7cm2V−1s−1in NbP is indeed rooted in helical Weyl fermions, associated to four unusually large WSM electronpockets near the /Sigma1points in the k z=0 plane. Each of such large WSM pockets, which are negligibly small in TaAs, ishighly anisotropic in kspace and encloses one pair of the W2 nodes. Robust chiral anomaly induced NMR, another quantumsignature of WSMs, has also been demonstrated. However, itis originating in the coexistent p-type WSM pockets in the k z=1.16π/c plane. In the quantum degerate regime, helicity protection of Weyl fermions leads to sharp and nearly identicalconductivity enhancement when the magnetic field and electricfield are applied in parallel. 2469-9950/2016/93(12)/121112(6) 121112-1 ©2016 American Physical SocietyRAPID COMMUNICATIONS ZHEN W ANG et al. PHYSICAL REVIEW B 93, 121112(R) (2016) 0 50 100 150 200 250 300051015202530RRR=70 86 101 94 85 95ρxx(μΩcm) T( K )(a)(b) 0 50 100 150 200 250 3000246810×4 1%Cr3%Znμ(×106cm2/Vs) T( K )NbP ×30002468-20-15-10-50100 3075 50ρxy(μΩm) B( T )1.5K300 FIG. 1. (a) T-dependent ρcurves at B=0, showing varying sample quality. (b) Ultrahigh carrier mobility of NbP, which is sensitive to magnetic impurities. Inset: T-dependent Hall resistivity from 300 to 1.5 K, showing a pronounced transition from p-type carriers at room temperature to the coexistence of electrons and holes below 200 K. The detailed single crystal growth and structure analysis of NbP are described in the Supplemental Material [ 25]. Figure 1(a) shows the characteristic temperature ( T)- dependent resistivity measurements at zero magnetic field(B). The results of more than 20 single crystals con- sistently show a decent residual resistance ratio [RRR = ρ xx(300 K) /ρxx(1.5 K)] in the range of 70–100, similar to the previous reports in TaAs and NbAs [ 15–17]. The linear ρabove 150 K is typical for metal with dominant electron- phonon (e-ph) scattering. However, the quadratic behavior(T n,n∼2.8) ofρbelow 150 K cannot be simply explained by e-ph scattering, but indicating limiting scattering mechanismof electron-electron (e-e) interactions. Below 30 K, ρbecomes linear again, which will be correlated to the helicity protectionmechanism. Using the single-band theory [ 26], we estimate the electron concentration at 1.5 K from Hall signals to be ∼2× 10 18cm−3[27]. Noticeably, the deduced electron Hall mobility at 1.5 K is very weakly dependent on RRR. With RRR =95, the sample in Fig. 1(b) (red squares) has a stunning mobility exceeding 1 ×107cm2V−1s−1, while a polished sample with RRR=25 showing 5 ×106cm2V−1s−1[25]. The ultrahigh mobility of NbP is comparable to DSM Cd 3As2[12], despite that RRR is nearly two orders of magnitude higher in the latter.Since RRR is a direct measure of defect concentrations, theobservation indicates that the main charge carriers in NbP areeffectively protected from defect scattering at zero field. To getinsights into the protection mechanism, we have synthesized∼1% Cr- and ∼3% Zn-doped NbP single crystals [ 25] to study the effects of chemical impurities on the mobility. As shown inFig. 1(b), with the presence of minimal magnetic impurities, the mobility of Nb 0.99Cr0.01P (blue circles) is degraded by almost three orders of magnitude, while significantly higherconcentration of nonmagnetic Zn yields comparable mobilityto pristine NbP with similar RRR. The chemical dopingexperiments strongly suggest that the dominant charge carriersin NbP are spin polarized [ 28], which is consistent with the existence of helical WSM pockets in NbP. We further studied quantum oscillations in NbP to confirm the WSM origin of the spin-polarized carriers. Like the otherTaAs family members, NbP has rather complex FSs due to thecoexistence of multiple charge carrier pockets. Using densityfunctional theory (DFT) calculations [ 25], we found that the FSs of NbP consist of p-type W1-WSM, n-type W2-WSM, and eight large trivial hole pockets along the Z-Slines. Uniquely, each pair of W2-WSM pockets, enclosing W2 nodeswith the opposite chirality, are emerged into a dumbbell -shaped continuous FS with an inner FS of trivial electrons, whileall W1-WSM pairs are much smaller and well separated inkspace. Experimentally, we have analyzed the FSs of NbP using angle-dependent Shubnikov–de Haas (SdH) oscillations.Different from the literature, we rotated Bperpendicular to the electric field ( E), and thus define the rotation angle ( θ)b yt h e orientation of Band the caxis [Fig. 2(a)]. Such configuration not only allows us to get robust SdH oscillations for all θset points, but also excludes possible signals from chiral anomaly. With θ=0 ◦(B/bardblc), pristine NbP at 2 K is char- acterized by unusually large magnetoresistance, MR = [ρxx(B)−ρxx(0)]/ρxx(0), and extremely strong SdH oscil- lations [Fig. 2(a)]. The SdH peaks become visible once B exceeds 0.5 T, and all measured crystals show nonsaturatingquasilinear MR, which can reach striking 10 000 at 15 T.Similar to the electron mobility, the amplitudes of the SdHoscillations are also weakly correlated to RRR, while thelinear MR is rather sample dependent [Fig. 2(a) and [ 25]]. This also implies that the SdH oscillations are controlled bythe spin-polarized carriers with ultrahigh mobility, while thelinear MR may require compensation mechanism [ 29]f r o mt h e large, low-mobility hole pockets. Indeed, the chemical dopingof Zn greatly suppresses RRR, but the SdH oscillations remainrobust in Nb 0.97Zn0.03P. In contrast, quantum oscillations are completely absent in Nb 0.99Cr0.01P. By performing fast Fourier transformation (FFT), we get four major oscillation frequencies of F0=6.8T ,F1=13.9T , F2=31.8 T, and F3=64.4 T, respectively [Fig. 2(b)]. Since B is applied in the direction with the fourfold rotation symmetry,the same type of carrier pockets are quantized equivalently.In this case, SdH oscillations detect the FS cross sectionsof different carrier pockets in the k x-kyplane [Fig. 2(c)]. By comparing the experimental oscillation frequencies to the DFTcalculations, we can correlate the FFT peaks to the existingpockets, as summarized in Table I. Note that the F 3trivial hole pocket is surrounded by a much larger hole FS of 133 T, thus 121112-2RAPID COMMUNICATIONS HELICITY-PROTECTED ULTRAHIGH MOBILITY WEYL . . . PHYSICAL REVIEW B 93, 121112(R) (2016) 02468 1 0 1 2 1 40246810 B(T)MR ( ×103) Frequency (T) (c)θ=0º S1 S2 20 40 60 80 100 120 1400.00.20.40.60.81.0 F1Amplitude (a.u.)F1bF1aF0 F3bF2+F3a F0 F3F0F2F0 F2θ=0º S1 S2 S7S7F1 F3F1F2F320 40 60 80 100 120 1400.00.51.0 (b)(a) θ=90 º S7 S8(d) (e) 20 40 60 80 100 120 140 Amplitude (a.u.)7590 80 5070 65 60 55 3040 20 10 Frequency (T)θ=0 FIG. 2. Angle-dependent SdH oscillations of NbP. (a) Extremely strong SdH oscillations in NbP, with amplitudes independent of RRR. Inset: Schematic of the angle rotation. (b) Four major oscillation peaks of F0=6.8T ,F1=13.9T ,F2=31.8T ,a n d F3=64.4 T with B/bardblc. (c) Schematic of FS cross sections for different pockets with B/bardblc. (d) SdH oscillation peaks with θ=90◦. (e) The evolution of F0,F 1,F 2,a n d F3as a function of θ. forming the complex inner and outer FSs similar to the case ofF1andF2[23]. The assignments of FFT peaks have also been cross-checked by complementary magnetic oscillationsof Nernst and Seebeck coefficients as well as the de Haas–vanAlphen (dHvA) effect [ 25]. As shown in Table I, all three techniques quantitatively agree with the SdH results. Using angle- and T-dependent SdH oscillations [ 30], we determined indispensable information on the anisotropy of theindividual FS and the effective mass ( m ∗) of the corresponding carriers, respectively. As shown in Figs. 2(d) and 2(e),w e have observed dramatic θ-dependent changes in F2, which monotonically shifts up to 118 T when θis increased from 0◦to 90◦[the red dashed line in Fig. 2(e)], agreeing with the highly anisotropic nature of the W2-Weyl cones. In contrast,F 0gradually reaches a maximum frequency of 35 T at 90◦. Such isotropic FS is also expected for the W1-WSM pockets, TABLE I. Quantum oscillation frequencies ( T) determined by experiments and DFT with B/bardblc. W2-WSM Hole Pockets W1-WSM ( F0) Inner (F1) Outer-WSM (F2) Inner (F3) Outer DFT 4.1 (Hole) 12.5 35.6 66.2 133.8 SdH 6.8 13.9 31.8 64.4 130.1 Nernst 6.9 14.0 31.7 63.8 127.3 Seebeck 7.1 12.2 31.6 61.7 123.8dHvA (5 T) 7.3 12.3 31.0 62.0 N.A. 121112-3RAPID COMMUNICATIONS ZHEN W ANG et al. PHYSICAL REVIEW B 93, 121112(R) (2016) 81 0 1 2 1 4-0.50.00.51.01.5 0 5 10 15 20 250.00.20.40.60.81.020 40 60 80 100 120 1400.00.20.40.60.8 02468 1 00.00.10.20.3S2S2S2Amplitude (a.u.)NF2=2 NF3=4Δρxx/ρ0(×103) B( T )13.4T 10.9TNF2=3 NF3=6 NF3=5S2 (d) (c)(b) F2+F3 T(K)13.4T (F2+F3) 10.9T (F3) S1 0.09me/0.52me 0.38me S2 0.10me/0.47me 0.47me S17 0.15me/0.50me - F3F1 F2 F3 1/B(T-1)Amplitude (a.u.) Frequency (T)1.5K 3.3 5.3 8.3 11.3 14.3 19.3 F0(a) Exp. LLs Linear fitting (N>5) Simulation LL Index Nγ~0.13 FIG. 3. T-dependent SdH oscillations and Berry’s phase. (a) FFT of T-dependent SdH oscillations of S2. (b) T-dependent SdH oscillations in the field range of 8–15 T. (c) The LK model fitting of the 10.9 and 13.4 T peaks. The latter is the superposition of F2andF3. (d) Deviation of Berry’s phase from πdue to nonideal relativistic fermions in the W2-WSM pockets. which are date-like ellipsoids [ 14,16]. Noticeably, the trivial pockets of F1andF3are both splitting into two oscillation peaks when θ> 30◦, which are typical for ellipsoid-shaped FSs of parabolic energy bands. The lower frequency part of F3 is coincident with the W2-WSM peak to form the dominant peak of 118 T at θ=90◦. -8 -6 -4 -2 0 2 4 6 8110100 Classical NMRWAL250200 150 100 75 50ρ(B)/ρ(0) B(T)10 1.5K30300 Quantum NMR FIG. 4. Robust chiral anomaly induced NMR in NbP as a function ofT, showing T-independent NMR below 10 K and strong T dependence above 30 K.For WSM pockets, the linear energy dispersion results in significantly smaller m∗for Weyl fermions, compared to trivial pockets. Taking the quantization condition of /planckover2pi1ωc/kBT/greaterorequalslant1, in which kBis the Boltzmann constant and ωc=eB/m∗is the cyclotron frequency, we can expect the SdH oscillationsof WSM pockets to be persistent at much higher Tthan the trivial ones. Indeed, F 0andF2remain robust at 20 K, while F1andF3are not discernible above 10 K [Fig. 3(a)]. We have determined the effective mass m∗forF2andF3,u s i n gt h e Lifshitz-Kosevich (LK) formula for three-dimensional (3D)systems: A(B,T)∝exp/parenleftbigg −2π 2kBTD /planckover2pi1ωc/parenrightbigg2π2kBT//planckover2pi1ωc sinh(2 π2kBT//planckover2pi1ωc),(1) where A(B,T) is the SdH amplitude, and TDis the Dingle temperature. As shown in Fig. 3(b), there are four prominent SdH peaks in the field range of 8–15 T. The 9.4 and 13.4 Tpeaks are the superimposition of the N=3 and N=2 Landau levels (LLs) of F 2and the N=6 and N=4o fF3, respectively. The fitting of 9.4 and 13.4 T peaks requires twodistinct effective mass of m ∗ F2=0.1meandm∗ F3=0.47me, respectively [Fig. 3(c)]. For the 10.9 T peak, which is solely contributed by F3(N=5), single LK fitting yields 0 .45me, agreeing well with the double LK fitting. For F0andF1,i ti s difficult to extract m∗directly from MR oscillation peaks due to the superimposition of much higher frequencies of F2and F3. Instead, we analyzed the T-dependent FFT amplitudes and gotm∗ F0=0.06meandm∗ F1=0.29me[25]. Like F3,F1also 121112-4RAPID COMMUNICATIONS HELICITY-PROTECTED ULTRAHIGH MOBILITY WEYL . . . PHYSICAL REVIEW B 93, 121112(R) (2016) becomes vanishingly small when Tis above 10 K, supporting its origin in the inner trivial electron FS of the W2-WSMpockets. Distinctively, the SdH oscillations of NbP at 20 K arecharacterized by strong second harmonic peaks of F 0andF2,a manifestation of low m∗and small TDof Weyl fermions [ 25]. Nontrivial Berry’s phase of πis the quantum signature of the linear energy bands [ 31] in DSMs and WSMs. We have determined the /Phi1Bof the W2-WSM state, using the Landau fan diagram extracted from the SdH peaks from six differentsamples. Surprisingly, a simple linear fitting with the constraintfrequency of F 2gives an odd Onsager phase of γ∼0.3[25], which is considerably exceeding the geometrical correctionfactor for a 3D FS, following the Lifshitz-Onsager relation ofγ=1/2−/Phi1 B/2π+δ[23,32]. A detailed examination of the experimental data reveals that for high LL index N> 5, linear fitting of the the LL fan diagrams produces γ∼0.13, which gradually shifts to the odd number of 0 .3a sNapproaching the ultraquantum limit. Such deviation is a manifestationof the nonideal relativistic fermions [ 33] in the W2-WSM pockets, which have non-negligible m ∗ F2=0.1meand relative low Fermi velocity of vF∼1.8×105m/s. Using the method proposed by Taskin et al. [33], we are able to simulate the systematic changes in γ(N) ,a ss h o w nb yt h er e ds o l i dl i n ei n Fig. 3(d). With the presence of chiral Weyl node pairs in NbP, NMR induced by the Adler-Bell-Jackiw anomaly [ 5] would be expected when B/bardblE. Surprisingly, we have observed very robust NMR in NbP far above the quantization temperatureregime. As shown in Fig. 4, the MR curves form a sharp negative dip below 2 T with T< 50 K. The overall MR change at 1.5 K is about −80%, in contrast to −30% reported in TaAs [ 16]. The NMR becomes weaker and broader when Tincreases, but persists up to 150 K. Considering that each W2-WSM pocket is continuous FSs enclosing one W2 pair,an extra chiral current channel between different W2 nodes isnot allowed. In contrast, each pair of the W1-WSM pocketsare well separated in kspace. Using F 0andm∗=0.06me,w e estimated the EFis∼15 meV below the W1 nodes. By taking the thermal activation energy of 13 meV (150 K) into account,the results qualitatively agree with the DFT calculations whichsuggest a dome structure of 25 meV below the W1 nodes alongthe internode direction. It is distinctive that the NMR effects are identical below 20 K. In this quantum regime, the longitudinalchiral conductivity is expressed by /Delta1σ c∝B2v3 F E2 Fτ(EF), (2) where τ(EF) is the elastic intervalley scattering time by impurities [ 34]. In this regime, Weyl fermions are strictly protected by helicity from the scattering of nonmagneticimpurities. It leads to large constant τand thus identical NMR behavior. At elevated temperatures above 20 K, however, theintravalley inelastic scattering between electrons cannot beignored, and the chiral anomaly induced conductivity enhance- ment becomes /Delta1σ c∝B2v3 F T2τT, in which the T-dependent τTis modified by electron-electron inelastic scattering τe[34]. Such T-dependent NMR behavior is consistent with the zero-field resistivity measurements in Fig. 1(a). Our discovery not only proves the existence of exotic WSM states in NbP, but also provides unambiguous evidence incorrelating the ultrahigh mobility to the spin conservation ofhelical Weyl fermions. Unlike the pseudospin in graphene,which is vulnerable to lattice defects and atomic-scale dis-orders, the helical spin textures in NbP are topologicallyprotected by the noncentrosymmetric symmetry, and thusare remarkably robust against nonmagnetic disorders. Forpristine NbP, the doping is nearly intrinsic despite that thereare small amounts of excessive P in crystals (P:Nb =50.5: 49.5±3% [ 25]). Nevertheless, a strategy to continuously tune the doping level in NbP would be highly desirable, and mayopen enormous opportunities for exploring various topologicalquantum phenomena and spin device concepts. We thank Yayu Wang, Kai Wang, and Fuchun Zhang for in- sightful discussions. This work was supported by the NationalBasic Research Program of China (Grants No. 2014CB921203and No. 2012CB927404), the National Science Foundationof China (Grants No. 11190023, No. U1332209, and No.11574264), and the Fundamental Research Funds for theCentral Universities of China. 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PhysRevB.82.165334.pdf

Electric and thermoelectric phenomena in a multilevel quantum dot attached to ferromagnetic electrodes M. Wierzbicki and R. Świrkowicz Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland /H20849Received 12 July 2010; published 28 October 2010 /H20850 Nonequilibrium Green’s function formalism is used to study electron and energy transport in the Coulomb- blockade regime through a two-level quantum dot/molecule attached to ferromagnetic leads. Interplay betweenmagnetic and thermoelectric properties is investigated in the nonlinear-response regime and strong spin effectsin the nonlinear thermopower are found. Significant spin thermopower S spincan be generated in a relatively wide region of temperature difference /H9004T. It is also shown that spin asymmetry due to presence of one half-metallic and one nonmagnetic electrodes plays an important role and strongly influences both charge andspin thermopower. S spinvaries considerably, if the roles of both leads are interchanged. Spin thermopower is enhanced in the region of higher values of /H9004T, if half-metallic electrode acts as an energy drain. On the other hand, in such a configuration, Pauli spin blockade occurs in electron transport. DOI: 10.1103/PhysRevB.82.165334 PACS number /H20849s/H20850: 73.23.Hk, 73.50.Lw, 85.80.Fi I. INTRODUCTION Recently, thermoelectric phenomena in nanostructure sys- tems have been intensively studied both experimentally andtheoretically. 1–5In particular, a new field of research, focused on spin effects in energy transport has been developed inview of future application in spintronic devices. 6–17Spin- dependent heat current was experimentally investigated inmagnetic multilayer nanowires. 7,8Magnetothermoelectric power /H20849MTEP /H20850was measured, as well as magnetothermogal- vanic voltage was determined, which describes ac voltageresponse to a small temperature oscillations in a presence ofdc current. A strong dependence on spin asymmetry wasfound. “Three-current model,” which joins heat current andcharge currents in two spin channels corresponding to spinsup and down, was developed to discuss the experimentaldata. 7Spin Seebeck effect was also observed for a ferromag- netic slab and spin voltage generated by temperature gradientwas measured. 9It was theoretically shown that in ferromag- netic tunnel junctions thermal conductance and Peltier effectvary with configuration of magnetic moments. 10Moreover, at low temperatures, a strong suppression of electrical andthermal currents was observed for antiparallel orientation ofmagnetizations. With use of finite-element theory a signifi-cant dependence of thermal coefficients on relative align-ment of magnetic moments was also obtained for magneticmultilayer nanostructures. 11Spin-transfer torque due to a flow of spin-polarized heat current through a multilayer sys-tem was found, which indicates that current-induced magne-tization switching can be affected by thermal phenomena. 12 Moreover, very recently, with use of scattering theory,domain-wall dynamics has been studied in a presence oftemperature gradient and thermally induced motion of thewalls has been observed. 13 Thermoelectric properties of nanoscale systems contain- ing quantum dots /H20849QDs /H20850are strongly influenced both by quantum confinement and Coulomb-blockade effect, whichlead to distinct phenomena like oscillations of thermal coef-ficients with gate voltage. 18–21Additional fine structure due to discrete levels was found for small dots.22It is also worthnoting that experiments performed on QDs in the Kondo regime reveal that thermopower is strongly influenced byspin correlations. 23Interplay between spin effects and ther- mal transport through a single-level QD attached to ferro-magnetic leads was also studied. 16,17The investigations re- veal a significant dependence of thermal coefficients on spin-polarization factor in electrodes as well as on the relativeorientation of magnetic moments. 17Spin Seebeck effect and spin-dependent thermal efficiency, described by the spin-dependent figure of merit was also discussed. 16,17Studies of the thermal properties of strongly correlated QD attached toferromagnetic leads were also performed in the Kondoregime. 15It was found that thermopower, calculated for par- allel configuration of magnetic moments, is strongly sup-pressed at low temperatures due to splitting of the Kondoresonance. It should be pointed out that all these calculations,performed in the Coulomb blockade and in Kondo regime,correspond to the linear-response region only. To study transport phenomena in systems subject to con- siderable voltage and temperature gradient, a more generalapproach is required, which allows to describe nonlinear ef-fects. Various methods were developed to investigate chargetransport for a variety of systems in the nonlinear regime. Inparticular, Pauli spin-blockade effects in two coupled QDs,asymmetrically attached to ferromagnetic electrodes were in-vestigated in the sequential tunneling approximation with useof density-matrix approach 24–26as well as in the cotunneling regime.27Diagrammatic technique was also used to study spin blockade in a two-level QD attached to one half-metallic ferromagnet /H20849HMF /H20850and one nonmagnetic /H20849NM /H20850 electrodes. 28It is worth noting that very recently, in a semi- conductor QD attached to one FM and one NM electrodesnegative differential conductance /H20849NDC /H20850has been experi- mentally observed for both forward- and reverse-biasvoltages. 29When electrons tunnel from NM electrode to FM one the Pauli spin-blockade effect occurs but for the reversebias a different mechanism can be expected. Thermal phenomena in the nonlinear transport through QD were mainly investigated for nonmagnetic systems, inwhich spin effects are irrelevant. 30–35Some investigationsPHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 1098-0121/2010/82 /H2084916/H20850/165334 /H208499/H20850 ©2010 The American Physical Society 165334-1were also performed for the Kondo regime.34The nonlinear thermoelectric properties of molecular junctions are of spe-cial interest. First of all, strong nonlinearity was observedexperimentally in organic molecules. 3Moreover, it was shown that optimal thermoelectric operation of the moleculardevice can be achieved in the nonlinear, nonequilibriumregime. 35Additional advantage of molecular systems is that the corresponding phonon contributions to thermal propertiesmay be small. 3,36In particular, such contributions to heat current become negligible when tunnel coupling to the leadsis symmetric and larger than phonon couplings. 35 Here, we present studies of spin-dependent phenomena in the nonlinear transport through a two-level QD/molecule inthe Coulomb-blockade regime, based on the nonequilibriumGreen’s function formalism. The analysis may also hold fortwo electrostatically coupled QDs. Similar approach wasused in our previous work to investigate thermoelectric prop-erties of nonmagnetic system in the linear-response regime. 37 In the present paper we focus on the spin-related phenomenain the nonlinear charge transport and thermopower. The ap-proach allows us to study blockade effects in the chargetransport and to show that in junctions with strong spinasymmetry, in which one of the levels is weakly coupled toHMF and NM electrodes, negative differential conductancecan be observed for both polarizations of bias voltage. Furthermore, spin effects in the nonlinear thermopower are discussed. Up to now, in the Coulomb-blockade regionspin-dependent thermoelectric phenomena were mainly in-vestigated for a single-level dot within the linear-responseregime, 16,17though some preliminary results obtained in non- linear regime were also presented.17Here, we show that in a wide region of temperature difference between electrodes asignificant spin thermopower can be generated. In particular,a considerable and practically constant spin thermopowercan be observed in the wide region of /H9004Tfor junctions with strong spin asymmetry, which show Pauli spin blockade incharge transport. The paper is organized as follows. In Sec. IIthe model is presented and the transmission expressed in terms of Green’sfunctions is given. Charge current flowing in the system dueto bias voltage applied is discussed in Sec. IIIwhereas re- sults obtained for thermopower in a presence of temperaturegradient are discussed in Sec. IV. Final conclusions and sum- mary are given in Sec. V. II. MODEL System under consideration, composed of a multilevel quantum dot/molecule attached to ferromagnetic leads, is de-scribed by the following Hamiltonian: H=H D+He+HT. The first term HDcorresponds to the dot and is taken in the form HD=/H20858 j/H9268/H9255jdj/H9268+dj/H9268+1 2/H20858 ij/H9268/H9268/H11032Uijdi/H9268+di/H9268dj/H9268/H11032+dj/H9268/H11032, /H208491/H20850 where /H9255j=/H9255j0+Vgis the energy of the level jwhich can be shifted by the gate voltage Vg.UjandUijdescribe intralevel and interlevel Coulomb correlations, respectively. dj/H9268+/H20849dj/H9268/H20850 represents creation /H20849annihilation /H20850operator of electron in thestate j/H9268. The term He=/H20858/H9252=L,R,k/H9268/H9255k/H9252/H9268ck/H9252/H9268+ck/H9252/H9268corresponds to the noninteracting electrons in the left /H20849/H9252=L/H20850and right /H20849/H9252 =R/H20850leads whereas HT=/H20858/H9252kj/H9268/H20849Vkj/H9268/H9252ck/H9252/H9268+dj/H9268+Vkj/H9268/H11569/H9252dj/H9268+ck/H9252/H9268/H20850de- scribes tunneling effects between the dot and electrodes. Vkj/H9268/H9252 are elements of the tunneling matrix corresponding to the level jandck/H9252/H9268+/H20849ck/H9252/H9268/H20850denotes creation /H20849annihilation /H20850opera- tor of electron with wave vector kand spin /H9268in/H9252electrode. In general, electrochemical potential of the lead /H9252with temperature T/H9252can be spin dependent, as temperature gradi- ent can generate spin-dependent voltage.9Thereby, we as- sume that electrochemical potentials of left and right elec- trodes are equal to: /H9262L/H9268=/H92620+1 2V/H9268,/H9262R/H9268=/H92620−1 2V/H9268, respectively. V/H9268includes the charge and spin voltages and /H92620 corresponds to the Fermi energy of the system in equilib- rium. The charge and spin currents I=I↑+I↓,Ispin=/H6036/e/H20849I↑ −I↓/H20850can be introduced and I/H9268denotes here the current flow- ing in a channel corresponding to spin /H9268. The calculate current I/H9268we apply nonequilibrium Green’s function formalism based on the equation-of-motion method.Introducing retarded G r/H20849advanced Ga/H20850and lesser G/H11021 Green’s functions one obtains the following expression:38,39 I/H9268=ie 2/H6036/H20885dE 2/H9266/H20858 j/H20853/H20851/H9003j/H9268LfL/H9268/H20849E/H20850−/H9003j/H9268RfR/H9268/H20849E/H20850/H20852/H20851Gj/H9268r/H20849E/H20850−Gj/H9268a/H20849E/H20850/H20852 +/H20849/H9003j/H9268L−/H9003j/H9268R/H20850Gj/H9268/H11021/H20849E/H20850/H20854, /H208492/H20850 where f/H9252/H9268=/H20851exp /H20849E−/H9262/H9252/H9268/H20850/kT/H9252+1/H20852−1is the Fermi-Dirac dis- tribution function in the /H9252electrode and Gr/H20849a/H11021/H20850/H20849E/H20850represents the Fourier transform of the appropriate Green’s function. /H9003j/H9268/H9252=/H9003j/H208491+/H9268ˆp/H9252/H20850determines here the spin-dependent cou- pling strength of the dot level jwith electrode /H9252,p/H9252is re- lated to the lead’s polarization and /H9268ˆ=1 for spin index /H9268 =↑or/H9268ˆ=−1 for /H9268=↓./H9003jdenotes the coupling strength of the level jwith electrodes and is treated as a parameter indepen- dent of energy. Green’s function Gj/H9268r=/H20855/H20855dj/H9268,dj/H9268+/H20856/H20856is calculated with use of procedure proposed by Chang and Kuo40,41based on the equation-of-motion method. Following the procedure, justi-fied in the Coulomb-blockade regime, one can express G rin the form /H20849for details see Ref. 40/H20850 Gj/H9268r/H20849E/H20850=/H20858 k=02 pk/H208731−Nj−/H9268 E−/H9255j−Ak−/H9018jr+Nj−/H9268 E−/H9255j−Uj−Ak−/H9018jr/H20874. /H208493/H20850 The summation is over possible configurations in which level l, different from j, is occupied by zero, one or two particles, respectively. pkdenotes here the probability factor of a particular configuration kand is expressed in terms of the average one-particle Nl/H9268=/H20855nl/H9268/H20856=/H20855dl/H9268+dl/H9268/H20856and two-particle /H20855nl−/H9268nl/H9268/H20856occupation numbers. Akin the last equation denotes the sum of all interactions seen by the electron in the level jdue to other particles occupying the level lin config- uration kand expressed in terms of Ujl.40The occu- pation numbers are expressed in terms of lesser Green’s functions: Nl/H9268=−i/H20848dE /2/H9266/H20855/H20855dl/H9268,dl/H9268+/H20856/H20856/H11021, /H20855nl/H9268nl−/H9268/H20856= −i/H20848dE /2/H9266/H20855/H20855dl/H9268dl−/H9268+dl−/H9268,dl/H9268+/H20856/H20856/H11021. In the Coulomb-blockade re- gime the lesser functions can be calculated according to theM. WIERZBICKI AND R. ŚWIRKOWICZ PHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 165334-2equation-of-motion method.40,42Then, Gl/H9268/H11021takes a form: Gl/H9268/H11021=−/H20851/H20849/H9003l/H9268LfL/H9268+/H9003l/H9268RfR/H9268/H20850//H20849/H9003l/H9268L+/H9003l/H9268R/H20850/H20852/H20849Gl/H9268r−Gl/H9268a/H20850. Similar ex- pression can be written for the two-particle lesser functionwhereas the retarded /H20849advanced /H20850ones are calculated from the appropriate equation of motion. 40Finally, one obtains a set of algebraic equations for one- and two-particle occupationnumbers N l/H9268,/H20855nl/H9268nl−/H9268/H20856, which are solved numerically for each value of bias voltage and temperature difference underconsideration. Since the lesser Green’s function is deter-mined in terms of retarded and advanced ones, the electricalcurrent can be written as: I /H9268=ie /H6036/H20885dE 2/H9266/H20858 j/H9003j/H9268L/H9003j/H9268R /H9003j/H9268L+/H9003j/H9268R/H20849Gj/H9268r−Gj/H9268a/H20850/H20851fL/H9268/H20849E/H20850−fR/H9268/H20849E/H20850/H20852 /H208494/H20850 which allows one to study transport properties.III. ELECTRON TRANSPORT. SPIN-BLOCKADE EFFECTS First, we assume that temperatures of both electrodes are equal TL=TRand study electron transport in the system sub- jected to the bias voltage Vb. The basic features of the charge transport through QD are well known so we focus our dis-cussion on the system with strong spin asymmetry, in whichPauli spin blockade can be expected. Effects of spin block-ade in similar systems were studied in the sequential tunnel-ing regime with use of real-time diagrammatic technique. 28 Here, we discuss coherent transport and apply the nonequi-librium Green’s function formalism. To describe the spin-blockade effects we consider a two-level quantum dot at-tached to external electrodes. The left electrode is a HMFwith polarization factor p L=0.95 while the right one is NM andpR=0. The coupling strengths of both levels are the same FIG. 1. /H20849a/H20850Differential charge conductance Gversus gate and bias voltages, /H20849b/H20850I-Vbcharacteristics /H20849solid line /H20850andG/H20849dashed line /H20850,/H20849c/H20850 magnetic moment induced on the dot, and /H20849d/H20850occupation numbers as a function of bias voltage for Vg=0.2 meV and polarization factors: pL=0.95, pR=0. Parameters of the junction: /H925510=−0.05 meV, /H925520=0.55 meV, U1=U2=2 meV, U12=1 meV, /H9003=0.01 meV, and kTL =kTR=0.02 meV.ELECTRIC AND THERMOELECTRIC PHENOMENA IN A … PHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 165334-3and are equal /H9003=0.01 meV. Energy levels are assumed as: /H925510=−0.05 meV, /H925520=0.55 meV. Positions of both levels can be coherently shifted with use of a gate voltage. Theintralevel and interlevel correlation parameters are taken as:U 1=U2=2 meV and U12=1 meV, respectively. Studying the charge transport we assume that temperatures of bothleads are equal with kT L=kTR=0.02 meV. Differential charge conductance G=dI/dVbas a function of gate voltage Vgand bias voltage Vbis presented in Fig. 1/H20849a/H20850. Since polarization factors of both electrodes are differ- ent,pL/HS11005pR, the coupling strengths /H9003j/H9268Land/H9003j/H9268Ralso differ strongly, and the obtained patterns are more complex thantypical Coulomb diamonds. For the assumed parameters thesystem shows negative differential conductance representedby white lines in the figure. When V gis close to zero NDC can be observed for negative bias voltage whereas for largevalues of /H20841V g/H20841it appears for positive Vb. There is also a re- gion of gate voltages for which blockade effects leading toNDC do not occur and current increases monotonically withincrease in bias voltage. This situation takes place in themiddle of the Coulomb gap with /H20841V g/H20841close to Coulomb pa- rameter U=2 meV. Moreover, the conductance shows an inversion symmetry with respect to the Coulomb gap. Spin-blockade effect is well illustrated in Figs. 1/H20849b/H20850and 1/H20849d/H20850, where I-Vbcharacteristics, Gand occupation numbers of particular states are presented for Vg=0.2 meV. For posi- tive bias voltage electrons tunnel from the left, HMF, elec-trode to the right, NM, one and the current increases mono-tonically showing characteristic Coulomb steps. As thetunneling rate to the right lead for electrons with spin up ismuch lower than to the left one, these electrons start to ac-cumulate on the dot. At first, they are located on the level 1and then also on the level 2, which results in a large positivemagnetic moment m=N ↑−N↓induced on the dot at higher values of bias voltage /H20851Fig.1/H20849c/H20850/H20852. The situation is completely different for negative voltages, when the left, half-metallic,electrode acts as a drain. Since practically there are no statesin this electrode, corresponding to spin down, electrons ac-cumulate on the dot leading to NDC. For small V bthe current shows a sharp peak associated with a rapid increase in N1↓ from zero practically to 1 /H20851Fig. 1/H20849d/H20850/H20852. For higher voltages both levels with energies /H92551and/H92552are almost fully occupied with electrons of spin down. All these features, shown by thecurrent, with negative conductance correspond to Pauli spin blockade. In general, the results are consistent with thoseobtained in sequential tunneling approximation 28but the Green’s function formalism applied here, allows us to takeinto account higher order effects, which enhance the currentin the blockade regime. Such an enhancement was observedexperimentally and discussed in two QDs in the cotunnelingregime. 27,43Electrons with spin up, the majority spin in the half-metallic electrode, practically do not accumulate on thedot for small reverse-bias voltage, which results in a strongnegative magnetic moment induced on the dot /H20851Fig. 1/H20849c/H20850/H20852. The moment is strongly asymmetric with respect to V brever- sal. Next, we consider transport through a molecule attached to external electrodes. In this case, coupling strengths /H9003jfor two molecular levels j=1,2 can be dif ferent due to different spatial distribution of the corresponding wave functions.44,45 To describe such a dependence, we express /H9003jin the form /H9003j=/H9003/H208511− /H20849−1/H20850jQ/H20852. For Q=0 both levels 1 and 2 are equally coupled to electrodes whereas for 0 /H11021Q/H110211 one of the levels becomes weakly coupled. Similar approach can be also ap-plied to two QDs which are capacitively coupled and at-tached to external electrodes. The coupling strengths to theleads can be, then, varied for each dot separately. AssumingQ=0.9 we study influence of the level-dependent coupling strength on the results obtained in the presence of Pauli spinblockade. I-V bcharacteristics and the conductance Gare pre- sented in Fig. 2forVg=0.2 meV. The current strongly in- creases in the region of small positive voltages, then, it de-creases showing a significant NDC effect. It should be notedthat NDC occurs both for negative and positive bias volt-ages. An analysis of N 1/H9268andN2/H9268shows that in the region of negative voltages the main role plays Pauli spin blockade butforV b/H110220 the NDC is of different origin. The strong suppres- sion of the current and NDC effect can be observed when thesecond level, partly decoupled from electrodes enters thebias window and becomes active in the transport. The dot isthen occupied by electrons with spin up coming from thehalf-metallic electrode. The superposition of two effectsleading to NDC gives a complex I-V bcharacteristics which may be observed in molecular junctions or in two dot sys-tems. IV. NONLINEAR THERMOPOWER To study thermoelectric phenomena we assume that tem- perature of the right electrode is kept constant and equal toT Rwith kTR=0.02 meV whereas in the left electrode it is increased by /H9004TsoTL=TR+/H9004T. In a presence of temperature gradient voltage Vis generated and the effect is described by the Seebeck coefficient S. Usually, Sis determined under the condition of vanishing charge current, I=0, and in the nonlinear-response regime, when the temperature difference/H9004Tbetween two electrodes is relatively large, one can intro- duce the differential thermopower defined as: S=dV /dT.I n systems with magnetic electrodes, when the spin relaxationtime is long enough, spin accumulation in the leads becomesimportant. 9,17Temperature gradient generates, then, spin- dependent voltage V/H9268=V+/H9268ˆVspinand the charge Vas well as-0.04-0.0200.020.04 -10 -5 0 5 10-0.0100.01G( e2/_h) current (e/_h) Vb(meV) FIG. 2. I-Vbcharacteristics /H20849solid line /H20850andG/H20849dashed line /H20850at Vg=0.2 meV for asymmetric junction with pL=0.95, pR=0, and Q=0.9. Other parameters are the same as in Fig. 1.M. WIERZBICKI AND R. ŚWIRKOWICZ PHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 165334-4spin Vspinvoltages are induced. V/H9268corresponds here to the difference in electrochemical potentials of two leads in thespin channel /H9268. Since, the two channels are independent, the voltage V/H9268and hence thermopower can be calculated under the condition of vanishing current in each spin channel, I/H9268 =0, or equivalently in the situation in which the charge cur- rentIand spin current Ispinvanish simultaneously. Therefore, for a given value of /H9004Tcorresponding to temperature differ- ence between electrodes the set of equations I↑=0, I↓=0 /H20851with I/H9268expressed by formula /H208494/H20850/H20852is solved numerically and generated voltages V↑,V↓are found. After the relation V/H9268 =V/H9268/H20849/H9004T/H20850is determined the spin-dependent differential ther- mopower S/H9268=dV/H9268/d/H20849kT/H20850is calculated with use of numerical procedures. One can consider the charge and spin ther-mopower defined by relations: 17S=1 /2/H20849S↑+S↓/H20850=dV /d/H20849kT/H20850 andSspin=1 /2/H20849S↑−S↓/H20850=dVspin /d/H20849kT/H20850. A. Symmetrical system First, we study the symmetrical system with magnetic moments parallel in both electrodes and polarization factor pL=pR=0.5. The charge and spin voltages generated in the system due to temperature gradient /H9004Tas well as the appro- priate charge and spin thermopower are depicted in Fig. 3. Since the induced voltages and thermopower strongly oscil-late as a function of V g, only absolute values are presented. Maxima /H20849minima /H20850appear in a vicinity of states, given by the poles of the Green’s function /H20851Eq. /H208493/H20850/H20852. If one of the statesapproaches the resonance, electrons tunnel due to tempera- ture gradient giving rise to the voltage drop as well as tothermopower. When the energy level reaches the resonancethe induced voltage vanishes, as currents due to electronsand holes compensate. The situation is similar for other reso-nant states and a number of peaks with different intensitiesdescribed by the probabilities of particular one and two-particle configurations can be observed. Four resonances which correspond to levels /H9255 1,/H92552+U12, and their Coulomb counterparts with energies /H92551+U12+U,/H92552+2U12+Udomi- nate the structure as the transport is mainly supported bythese levels /H20849see also Ref. 37/H20850. Significant changes in /H20841V/H20841and /H20841V spin/H20841can be observed in the region of small /H9004T, where the voltages rapidly increase. However, they saturate and remainpractically constant for large temperature differences. Withincrease in temperature the levels broaden and start to over-lap other, less significant, forming relatively wide bands,which dominate the whole structure /H20851Fig. 3/H20849a/H20850/H20852. On the other hand, the more rich structure can be observed for spin volt-age, mainly due to the fact that generated /H20841V spin/H20841is relatively low and peaks with small intensity can be distinguished. The charge and spin thermopower are presented in Figs. 3/H20849c/H20850and 3/H20849d/H20850. One can observe that a considerable ther- mopower is induced in the region of relatively small /H9004T.S sharply varies with Vgshowing maxima /H20849minima /H20850in the vi- cinity of energy levels typical for the system. With increasein temperature peaks in vicinity of four levels, which mainlysupport the transport, broaden significantly as the bands are FIG. 3. /H20849a/H20850Charge and /H20849b/H20850spin voltages as well as /H20849c/H20850charge and /H20849d/H20850spin thermopower as a function of Vgand/H9004TforpL=pR=0.5. Only absolute values are given. Parameters of the junction are: /H925510=−0.05 meV, /H925520=0.55 meV, U1=U2=2 meV, U12=1 meV, /H9003 =0.01 meV, and kTR=0.02 meV.ELECTRIC AND THERMOELECTRIC PHENOMENA IN A … PHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 165334-5formed. At higher values of /H9004Tthe thermopower decreases and regions, where Sis close to zero, considerably broaden. However, for certain values of gate voltages quite significantthermopower can be still observed. A cross section taken forV g=−1.25 meV is presented in Fig. 4/H20849a/H20850. For particular gate voltage the thermopower is negative. It is also small in thelimit of /H9004T→0. The chosen V gcorresponds to the situation, when Fermi level in equilibrium is lying between two stateswith energies /H9255 1+U12,/H92552+U12and the transport is strongly reduced at low temperatures. Absolute value of thermopowersignificantly increases with /H9004T, achieves a narrow maximum and approaches zero for large temperature difference. Ther-mopower varies with polarization factor p. The most pro- nounced changes can be observed in the vicinity of the maxi-mum, where /H20841S/H20841increases with p/H20851Fig. 4/H20849a/H20850/H20852. Moreover, the curve is shifted toward smaller values of /H9004T, which indicates that the generated voltage easily saturates in the system withhighly polarized electrodes. It should be noted that thecharge thermopower practically does not vary with pfor higher temperature differences. Thermopower Sdepends on the relative configuration of magnetic moments in the electrodes, and the magnetother-moelectrical power MTEP= /H20851S/H20849P/H20850−S/H20849AP /H20850/H20852/S/H20849AP /H20850can be in- troduced. S/H20849P/H20850and S/H20849AP /H20850correspond here to differential thermopower determined for parallel and antiparallel con-figurations, respectively. MTEP calculated for symmetricjunctions and two different polarization factors in electrodesis presented in Fig. 4/H20849b/H20850. It is positive showing that S/H20849P/H20850is greater than S/H20849AP /H20850. Moreover, MTEP strongly increases with an increase in polarization factor. Next, we discuss spin effects in thermopower. S spinin the parallel configuration calculated for different gate voltagesand/H9004Tis presented in Fig. 3/H20849d/H20850. Note that in the system under consideration it is possible to generate quite consider-able spin thermopower, especially in the region of low /H9004T. With increase in temperature difference thermopower de-creases and wide regions with S spinpractically equal to zero can be seen. The structure of spin thermopower is complex,as it changes the sign with /H9004Tincreasing. The cross section forV g=−1.25 meV and several different polarizations are presented in Fig. 5. In the region of small /H9004Tspin ther- mopower is positive so the sign of Sspinis opposite to the sign of the charge thermopower, what is consistent with re-sults obtained in the linear-response regime for a single-level dot.17Maximum of Sspinis well correlated with maximum of /H20841S/H20841. Moreover, the intensity strongly increases with leads’ polarization. For high-temperature differences spin ther-mopower becomes negative and polarization dependence isless pronounced. B. System with a strong spin asymmetry Now, the junction with one HMF and one NM electrodes is investigated. Charge thermopower, generated in such asituation takes quite significant values in a wide region of/H9004T. The appropriate curves, presented in Fig. 6for two cases with HMF electrode acting as the energy source or energydrain, are relatively broad and flat. This is in contrast tosymmetrical case, where /H20841S/H20841shows a narrow maximum for small /H9004T, and then decreases rather fast. In symmetric sys- tem, the state with vanishing current can be easily achieved,in which currents flowing due to temperature gradient anddue to generated voltage compensate. At first, generated volt-age significantly increases, but it practically saturates forsmall /H9004T. With further increase in /H9004Tonly very small volt- age changes are necessary to keep such a state. On the otherhand, in system with strong spin asymmetry, the number of-5-4.5-4-3.5-3-2.5-2-1.5-1-0.50 0 0.05 0.1 0.15 0.2 0.25S (k/e) k∆T (meV)P=0.0 P=0.9 -0.0500.050.10.150.20.250.30.350.40.45 0 0.05 0.1 0.15 0.2 0.25MTEP k∆T(meV )P=0.8 P=0.5 FIG. 4. /H20849a/H20850Charge thermopower Sand /H20849b/H20850MTEP as a function of /H9004Tfor several values of p=pL=pR.Vg=−1.25 meV. Other parameters are the same as in Fig. 3. -0.500.511.522.5 0 0.05 0.1 0.15 0.2 0.25Sspin(k/e) k∆T(meV )P=0.0 P=0.9 FIG. 5. Spin thermopower as a function of /H9004Tfor several values ofp=pL=pR/H20849polarization is varied from p=0 to p=0.9 with the step 0.1 /H20850,Vg=−1.25 meV. Other parameters are the same as in Fig. 3.M. WIERZBICKI AND R. ŚWIRKOWICZ PHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 165334-6hot electrons with spin up, coming from HMF source, sig- nificantly increases with /H9004T, but only a part of them can enter the NM drain electrode. To compensate this increasingelectron flow, the voltage, which considerably increases withtemperature, will be generated and the saturation cannot beeasily obtained. It leads to a significant thermopower in awide temperature region. Similar situation can be observedwith NM electrode acting as an energy source. Next, we discuss the behavior of spin thermopower. The appropriate curves are presented in Figs. 7/H20849a/H20850and7/H20849b/H20850. Note thatS spinessentially varies in the region of higher values of/H9004T, if the role, which HMF electrode plays in the junction, is changed. Namely, the spin thermopower is positive in thewhole temperature region for junction, in which the rightelectrode, with fixed temperature, is a half-metallic ferro-magnet. Since NM electrode acts as an energy source, thejunction is supplied with hot electrons of both spin direc-tions. Electrons with spin up can be easily transmittedthrough strongly broadened by temperature, unoccupiedlevel with energy /H9255 2+U12and take empty states in the HMF electrode. This electron flow should be compensated due togenerated voltage V ↑. However, for higher values of /H9004T, par- ticipation of holes in the transport increases. Namely, thelow-lying, broadened levels, which become partially occu-pied can also support the transport, especially the level withenergy /H9255 1+U12. Then, the tunneling of holes will suppress the induced voltage V↑. On the other hand, hot electrons with spin down cannot enter the HMF electrode and they shouldreturn to the source under an influence of the induced voltageV ↓. V oltages V↑and V↓, which lower the electrochemical potential of the left, hot electrode are presented in Fig. 7/H20849c/H20850. It can be observed that induced voltages differ considerablyin the whole temperature region. The corresponding spin thermopower calculated as S spin=1 2d/H20849V↑−V↓/H20850/d/H20849kT/H20850slowly changes with /H9004Tand remains positive. When the NM elec- trode acts as a drain, the induced voltages do not stronglydepend on spin. The generated spin thermopower is rela-tively small and becomes negative for higher values of /H9004T /H20851Fig. 7/H20849a/H20850/H20852. Consider now the situation presented in Fig. 7/H20849b/H20850,i n which the second level becomes weakly coupled to elec--5-4.5-4-3.5-3-2.5-2-1.5-1-0.50 0 0.05 0.1 0.15 0.2 0.25S (k/e) k∆T(meV )pL=0.95,pR=0.0 pL=0.0, pR=0.95 pL=pR=0.8 FIG. 6. Charge thermopower as a function of /H9004Tfor symmetric and asymmetric junctions with indicated polarizations, Q=0.Vg= −1.25 meV. Other parameters are the same as in Fig. 3. -0.500.511.52 0 0.05 0.1 0.15 0.2 0.25Sspin(k/e) k∆T(meV )Q=0.0pL=0.95,pR=0.0 pL=0.0, pR=0.95 pL=pR=0.8 -1-0.500.511.52 0 0.05 0.1 0.15 0.2 0.25Sspin(k/e) k∆T(meV )Q=0.6pL=0.95,pR=0.0 pL=0.0, pR=0.95 pL=pR=0.8 -0.6-0.5-0.4-0.3-0.2-0.10 0 0.05 0.1 0.15 0.2 0.25V(meV ) k∆T(meV )VdnVdnVupVuppL=0.0, pR=0.95 Q=0.0 Q=0.6 -0.6-0.5-0.4-0.3-0.2-0.10 0 0.05 0.1 0.15 0.2 0.25V (meV) k∆T(meV )Vdn Vdn VupVuppL=0.95, pR=0.0 Q=0.0 Q=0.6 FIG. 7. /H20851/H20849a/H20850and /H20849b/H20850/H20852Spin thermopower and /H20851/H20849c/H20850and /H20849d/H20850/H20852induced spin-dependent voltage V/H9268=/H9262L/H9268−/H9262R/H9268as a function of /H9004Tfor indicated values of polarization and Q.Vg=−1.25 meV. Other parameters are the same as in Fig. 3.ELECTRIC AND THERMOELECTRIC PHENOMENA IN A … PHYSICAL REVIEW B 82, 165334 /H208492010 /H20850 165334-7trodes, what corresponds to Q/HS110050. For symmetrical junction the spin thermopower does not considerably varies with Q. Similarly, in system with a strong spin asymmetry, changesin the spin thermopower with Qin the region of small /H9004Tare not very pronounced. However, different behavior can beobserved for high-temperature differences. When the NMelectrode acts as a source, spin thermopower starts to in-crease, achieving quite significant values. Transmission ofhot electrons with spin up through the broadened and par-tially decoupled level /H9255 2+U12is suppressed, but tunneling probability of holes increases, as the level /H92551+U12is strongly coupled to electrodes. Then, the induced voltage V↑, which blocks the current I↑, is significantly suppressed /H20851Fig. 7/H20849c/H20850/H20852. On the other hand, changes in V↓are weaker. Thereby, the generated voltage strongly depends on spin and a significant,positive spin thermopower arises. The different behavior canbe observed if HMF electrode acts as an energy source.Though, the level /H9255 2+U12is weakly coupled, transport through this broadened level for the majority spins does notchange significantly in the region of high /H9004T, as this channel is strongly supplied by the HMF hot electrode. Thereby, thegenerated voltage V ↑weakly varies with Qwhereas the changes in V↓are much more pronounced and /H20841V↓/H20841strongly diminishes. It leads to a quite significant spin thermopower,which in this case is negative. Therefore, in junctions withstrong spin asymmetry, containing one half-metallic elec-trode, spin thermopower can achieve quite significant valuesin the region of high /H9004Tand can be positive or negative in dependency on the role of this electrode. The most pro-nounced effect can be obtained for middle values of Q,a sf o r large Qthe level /H9255 2+U12, nearest to the Fermi level, be- comes practically decoupled from electrodes. V. SUMMARY AND CONCLUSIONS Studies performed in the paper for a two-level QD/ molecule attached to ferromagnetic electrodes show that insuch systems interesting and different features can be ob-served both in electron and energy transport. The analysis ofI-V bcharacteristics shows that NDC effect due to Pauli spinblockade can be obtained in junctions with one HMF and one NM electrodes, which is consistent with result found inthe sequential tunneling regime. 28NDC appears, then, for the forward or reverse bias in dependency on the gate voltage.However, if the level with higher energy is weakly coupledto electrodes, more complex characteristics can be obtainedand NDC occurs for both bias polarizations, generated by two different mechanisms. Thermoelectric phenomena in a two-level QD/molecule attached to ferromagnetic electrodes are also interesting.First of all, in systems with symmetrical junctions significantcharge and spin thermopower can be generated. Spin effectsare the most pronounced in the region of small /H9004T, where both charge and spin thermopower increase with leads’ po-larization. Moreover, magnetothermoelectric power can beobserved, indicating that, similarly to charge transport, ther-mopower is suppressed in systems with antiparallel orienta-tion of magnetic moments. Spin asymmetry of the junction due to the presence of one HMF electrode enhances thermopower in the region ofhigher values of /H9004T. Spin thermopower strongly varies in this temperature region, if the role of half-metallic electrodeis changed from the energy source to the energy drain. Themost interesting is the case when the HMF electrode acts ascharge or energy drain. In both cases electrons with spindown emitted by NM source cannot enter the HMF electrodedue to the lack of spin-down states so they must accumulateon the dot or return to the source. When the bias voltage V b is applied to the junction, electrons mainly accumulate on the dot leading to Pauli spin blockade. In a presence of tempera-ture gradient electrons with spin down will return to the NMsource giving rise to a considerable spin thermopower. Theeffect can be especially pronounced in the molecular junc-tions or two-dot systems with one of the levels /H20849dots /H20850weakly coupled to the leads. Temperature gradient applied to thejunction in such a way that HMF electrode acts as an energydrain will generate a significant spin voltage. Thereby, thesystem could be considered as an effective spin battery,which would allow to convert the heat into spin voltage inspintronic devices. 1A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Gar- nett, M. Najarian, A. Majumdar, and P. Yang, Nature /H20849London /H20850 451, 163 /H208492008 /H20850. 2K. Baheti, J. A. Malen, P. Doak, P. Reddy, S. Y . Jang, T. D. Tilley, A. Majumdar, and R. A. 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PhysRevB.94.060507.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 94, 060507(R) (2016) Generic Weyl phase in the vortex state of quasi-two-dimensional chiral superconductors Tomohiro Yoshida and Masafumi Udagawa Department of Physics, Gakushuin University, Tokyo 171-8588, Japan (Received 29 April 2016; revised manuscript received 26 July 2016; published 29 August 2016) We study the collective behavior of Majorana modes in the vortex state of chiral p-wave superconductors. Away from the isolated vortex limit, the zero-energy Majorana states communicate with each other on a vortexlattice, and form a coherent band structure with a nontrivial topological character. We reveal that the topologicalnature of Majorana bands changes sensitively via quantum phase transitions in two-dimensional (2D) systems,by sweeping magnetic field or Fermi energy. Through the idea of dimensional reduction, we show the existenceof a generic superconducting Weyl phase in a low magnetic field region of quasi-2D chiral superconductors. DOI: 10.1103/PhysRevB.94.060507 Topological superconductivity is one of the central topics in modern condensed matter physics [ 1–16]. A fascinating feature of topological superconductivity is the fully gappedbulk spectrum accompanied by topologically protected gaplessboundary states. As a prototypical example, a zero-energyMajorana state appears in an isolated vortex core of spinlesschiral p-wave superconductors [ 1,2,17]. While the existence of a stable zero-energy Majorana mode is striking in itself, its nontrivial statistical properties, such as anon-Abelian statistics, show up only in multiparticle systems.For example, the manipulation of qubits for topologicalquantum computation requires an entanglement of two or moreparticles. In this light, it is desirable to elucidate the collectivebehavior of Majorana modes. The interaction between Majorana modes, however, intro- duces a new problem. Namely, the zero-energy property maynot be protected in the presence of other particles. In fact,this problem has been examined by several groups [ 18–23] in chiral superconductors. They focused on the effect ofintervortex tunnelings, and showed that they tend to perturbthe Majorana modes off the zero energy. This fragility of thezero-energy state may be disappointing in terms of an ap-plication to, e.g., topological quantum computation, however,the relevance of the interaction implies a possibility of novelcooperative phenomena inherent in Majorana many-bodysystems. In this Rapid Communication, we focus on coherent band formation in the vortex state of chiral superconductors. In thiscontext, Majorana modes in each vortex core play the roleof atomic orbitals in the band formation in a crystal solid.From this perspective, these Majorana modes have severalfascinating properties that are absent in normal solids. First,the fundamental degrees of freedom obey Majorana, ratherthan Fermi, commutation relations. This makes a differencein the symmetry classification of the resultant band structure.Second, the (magnetic) unit cell contains two vortices, sinceeach superconducting vortex carries flux π. Accordingly, the theoretical description of Majorana bands needs a doublyenlarged magnetic unit cell with a coupling to the gaugefield, which results in a fertile possibility of metal-insulatortransitions with a topologically nontrivial character. And third,the band parameters are easily tuned by the magnetic fieldand electron density. As we discuss later, these features leadto the existence of successive quantum phase transitions intwo-dimensional (2D) chiral superconductors.Furthermore, this ubiquity of quantum phase transitions in 2D systems is connected to a topological property in higher-dimensional systems. Topological phenomena in differentdimensions are sometimes closely related to each other.Through the idea of dimensional reduction, we show theexistence of a generic superconducting Weyl phase in quasi-2Dchiral superconductors. In order to explore the topological aspect of chiral p-wave superconductors in the vortex state, we start with an attractiveextended Hubbard model on square and layered square lattices,whose Hamiltonian is given by H=/summationdisplay i/summationdisplay δti,i+δc† ici+δ−μ/summationdisplay ic† ici+1 2/summationdisplay i,jVijc† ic† jcjci. (1) For simplicity, we consider a spinless fermion, and define cias its annihilation operator at site i. For the moment, we consider only the square lattice, and set its coordinate as i=ixx+iyy. Here, we set the lattice space a=1, and define x(y)a sau n i t vector in the x(y) direction. The summation over δis taken for the vectors connecting nearest-neighbor sites, δ=± x,±y. We apply magnetic field Hin thezdirection, and adopt the Landau gauge for the vector potential A(r)=(0,Hx, 0). Here, we assume a limit of type-II superconductivity, and ignoreinternal magnetic fields. We incorporate the effect of magneticfield in the hopping term with Peierls substitution, t i,i+x=−t andti,i+y=−te−i2π φ0Hix, with φ0=hc/e a flux quantum. We analyze the model with the Bogoliubov–de Gennes (BdG) equation, /summationdisplay j/parenleftBiggHij/Delta1ij /Delta1† ij−H∗ ij/parenrightBigg/parenleftbigguν(j) vν(j)/parenrightbigg =Eν/parenleftbigguν(i) vν(i)/parenrightbigg , (2) withHij=/summationtext δti,jδi+δ,j−μδi,j. Here, we define the order parameter, /Delta1ij=Vij/angbracketleftcjci/angbracketright, with /angbracketleft ···/angbracketright , a thermal average at the temperature T=0.001t. We assume that superconducting vortices form a square lattice. Since each vortex carries a flux π, a magnetic unit cell must contain two vortices. Accordingly, we set the magneticunit cell of dimension N x×Ny=N×2N, corresponding to a magnetic field H=φ0/N2. We Fourier transform the BdG equation ( 2), with magnetic wave vectors k=(kx,ky), kx,y∈(−π/N x,y,π/N x,y], by imposing a boundary condition 2469-9950/2016/94(6)/060507(4) 060507-1 ©2016 American Physical SocietyRAPID COMMUNICATIONS TOMOHIRO YOSHIDA AND MASAFUMI UDAGAW A PHYSICAL REVIEW B 94, 060507(R) (2016) FIG. 1. Spatial profile of the px-ipy-wave order parame- ter at the magnetic field (a) H=Hl=8.68×10−4φ0and (b) H=Hh=3.47×10−3φ0, and the temperature T=0.001. Cor- responding energy spectra are shown in (c) and (d). We show the energy spectrum near the Fermi energy along the symmetric points, /Gamma1[k=(0,0)]→X[k=(π/N x,0)]→M[k= (π/N x,π/N y)]→Y[k=(0,π/N y)]→/Gamma1→M. For comparison, we show the energy spectrum for the s-wave order parameter in (e) at the low magnetic field Hl. (f) Magnetic field dependence of the energy gap Egapfors-wave and p-wave superconductors at the /Gamma1and Ypoints. on the Fourier components of uν(i) anduν(i), to fit with the nonperiodic spatial variation of A(r). As to the interaction, we assume a nearest-neighbor attraction, Vij=−Vp/summationtext δδi+δ,j, which selects the chiral p-wave superconducting state, among the five irreducible representations in the point group D4h[24], We also consider the on-site attractive interaction, Vij=−Vsδi,j,f o rt h e s-wave state as a reference. We take t=1 as a unit of energy, and we typically choose μ=− 2, which yields the electron filling n∼0.18. To facilitate the comparison, we assume slightly differentvalues for the magnitudes of attractive interactions, V s=2.5, andVp=2.4, that yield rather close values for the critical temperatures, Tc=0.17 for s-wave and Tc=0.15 for p-wave pairings. First, we show the spatial profiles of the order parameter. Of the two possible chiral pairings, the px-ipystate is favored under a magnetic field, whose order parameter /Delta1(i) is defined as/Delta1(i)=(/Delta1i,i+x−/Delta1i,i−x−i/Delta1 i,i+y+i/Delta1 i,i−y)/2. We plot /Delta1(i)f o r Hl/φ0=8.68×10−4andHh/φ0=3.47×10−3in Figs. 1(a) and 1(b), respectively. In both cases, /Delta1(i)i s suppressed around the vortex cores, and the core regionsextend to the y=±xdirections, reflecting the geometry of the square lattice. The amplitudes of /Delta1(i)h a v eas m a l ldifference between these two fields. Meanwhile, the spatial variation of /Delta1(i) differs considerably. Reflecting the smaller intervortex distance, /Delta1(i) is reduced even near the boundary of the magnetic unit cell at the higher field H h. The difference in spatial variation considerably affects the fermionic energy spectrum of the system. We plot the energyspectra at H=H landHhin Figs. 1(c) and1(d), respectively. ForH=Hl, we find low-energy states near the Fermi energy, isolated from the other high-energy bands. The low-energystates are composed of two nearly degenerate flat modes, whichare separated by a small gap ∼0.002. These states are remnants of zero-energy Majorana states in the isolated chiral vortexcores [ 1,17]. While the states are exactly degenerate at the Fermi energy at the limit of isolated vortices, however, the intervortextunneling introduced a slight separation. This tendency be-comes conspicuous when increasing the magnetic field, whereclosely spaced vortices allow larger tunnelings. At H=H h, the separation between the two low-energy bands is clearer[Fig. 1(d)], reflecting the larger spatial variation of /Delta1(i). To compare the effects of lattice formation with a conven- tional case, we show the energy spectrum for the s-wave order parameter at H=H las a reference, in Fig. 1(e). The flatness of the bands implies that there is little communication betweenthe vortices in this magnetic field. Nevertheless, there is a clearoffset from zero energy in the energy bands closest to the Fermienergy, in sharp contrast to the p x-ipycase [Fig. 1(c)]. In fact, as the magnetic field varies, the energy gaps develop in a contrastive way between the px-ipyands- wave superconductors. Figure 1(f) shows the magnetic field dependence of the energy gap Egap, defined as the separation of two band energies closest to the Fermi energy. Overall, Egap decreases (increases) with lowering magnetic field for px-ipy (s-wave) superconductors, This contrastive behavior can be ascribed to the different energy level structures in the isolatedvortex limit. In the s-wave case, there is already a finite gap from the Fermi energy at this limit. So, the lattice formationreduces this gap by a bandwidth. In contrast, in the p x-ipy case, the zero-energy states in the isolated vortices acquire finite energies through the band formation. While the overall tendency of Egapcan be understood as above, however, the field dependence of Egapshows nonmonotonicity, which cannot be captured in this picture. Inparticular, in the p x-ipycase,Egapshows oscillation [Fig. 1(f)], and sometimes approaches zero [ 25], implying the possibility of a quantum phase transition. In fact, as shown in Fig. 2(a),i f one sweeps μ, instead of the magnetic field, one will find the energy gap closes quite frequently. Moreover, these successivequantum phase transitions have a topological character: Thetotal Chern number of filled bands νjumps between 0 and −2 every time the gap closes. The series of topological quantum phase transitions (TQPTs) can be well understood in terms of an effective model.Following the procedure in Ref. [ 22], we fit the two lowest bands by an effective Majorana tight-binding model, H M=it/prime/summationdisplay n.n.λlmαlαm+it/prime/prime/summationdisplay n.n.n.λlmαlαm, (3) where αlis the Majorana operator defined at the lth vortex core,λlm=± 1i st h e Z2gauge field, and the summations 060507-2RAPID COMMUNICATIONS GENERIC WEYL PHASE IN THE VORTEX STATE OF . . . PHYSICAL REVIEW B 94, 060507(R) (2016) FIG. 2. (a) Energy gap and Chern number for H/φ 0= 1.25×10−3. Here, the order parameter /Delta1(i) obtained at μ=− 2 is used for the entire range of μ. (b)–(d) Tight-binding fitting for the lowest-energy states at (b) H/φ 0=8.68×10−4,( c )H/φ 0= 1.25×10−3,a n d( d ) H/φ 0=1.95×10−3. The transfer integrals used for the fitting are (b) t/prime=3.53×10−4,t/prime/prime=1.64×10−4,( c ) t/prime=5.75×10−4,t/prime/prime=5.11×10−5,a n d( d ) t/prime=1.46×10−3,t/prime/prime= 5.86×10−4. (e) Lattice configuration and transfer integrals for the effective tight-binding model on a general Bravais lattice. A magnetic unit cell is highlighted with a red oblique box. The arrow from site l tomindicates λlm=−λml=1. The convention of transfer integrals, t1–t4, and the two lattice vectors, a1anda2, are also depicted. are taken over pairs of sites ( l,m) for nearest- and next-nearest neighbors in the first and second terms, respectively. As shownin Figs. 2(b) and2(c), the low-energy bands are well fitted by the Hamiltonian Eq. ( 3) at low magnetic fields, by sensitively changing t /primeandt/prime/prime, as the magnetic field or μ. In particular, since the vortex cores are extended in the y=±xdirections [Figs. 1(a) and1(b)],t/prime/primetends to be larger on a square lattice. In contrast, at high magnetic fields, farther-neighbor hoppingsare also necessary [Fig. 2(d)]. These effective transfer integrals stem from the overlap integrals of localized modes in thenearby vortex cores, and rapidly oscillate as ∼sin(k FR), with Fermi wave vector kFand intervortex distance R[25,26]. Moreover, the Hamiltonian ( 3) explains the TQPT dis- cussed above. The Hamiltonian ( 3) belongs to class D[4], and its topological characters are classified with a Chern numberν M. By varying t/primeandt/prime/prime,t h eνMof the lower band changes asνM=1(−1) for t/prime/prime>0(t/prime/prime<0), while the gap vanishes between the bands at the time-reversal symmetric point t/prime/prime=0, where a Dirac point appears in the quasiparticle spectrum[27]. This change of ν Mast/prime/prime, combined with the sensitive change of transfer integrals with magnetic field or μ, explainsthe frequent occurrence of TQPTs: The topological transition of Majorana bands controls the change in the total Chernnumber ν. Indeed, the TQPT of Majorana bands is a universal feature of 2D chiral superconductors in the vortex state, which isnot related to the details of the system, e.g., the high spatialsymmetry of the square vortex lattice. In fact, for a Bravaislattice with general lattice vectors a 1anda2, the effective tight- binding model of the type ( 3) can be cast into the Hamiltonian analogous to the Qi-Hughes-Zhang (QHZ) type [ 28], H=4/summationdisplay k,ss/primeα† ksd(k)·σss/primeαks/prime, (4) with d1=t2sink2 2,d2=t3cos (k1+k2 2)+t4cos (k1−k2 2), andd3=t1sink1, where kj=k·aj. Here, we assumed only short-range hoppings, t1–t4[Fig. 2(c)]. If one sets t1=t2=t/prime andt3=t4=t/prime/prime, the result of the square lattice Hamiltonian is recovered. The QHZ Hamiltonian is the simplest model to de-scribe topological phase transitions. Indeed, the Hamiltonian(4) leads to quantum phase transitions occurring in the simple conditions, t 1=0o rt2=0o rt3+t4=0. The ubiquitous TQPT gives an important implication to the topological nature of three-dimensional (3D) chiral super-conductors. Topological phenomena in different dimensionscan often be related with each other through a dimensionalreduction. Here, in order to apply this idea to the vortex stateof quasi-2D chiral superconductors, we reconsider the startingHamiltonian ( 1) on a layered square lattice, and introduce small hopping t z,i nt h e z(/bardblH) direction. This new setting turns out to make a slight difference in the BdG equation ( 2). First, one needs to introduce the momentum in the zdirection, kz, and impose kzdependence on all the quantities appearing in Eq. ( 2). Through this change, the important observation is that the kzdependence ofHijcan be absorbed into the replacement of μwith its kz-dependent counterpart, μ/prime(kz)=μ−2tzcos(kz). This is practically the only change due to weak three dimensionality,if we assume the k zdependence of /Delta1ijis negligible for small tz. This simple correspondence between 2D and quasi-2D formulations enables us to interpret the physics in these twosystems in a unified language. In particular, the successiveTQPTs with sweeping μin 2D [Fig. 2(a)] can now be reinterpreted as successive gap closings along the k zaxis in momentum space, for the quasi-2D case. In other words, pointnodes appear at k z’s, corresponding to the quantum critical μ=μ/prime(kz). While the existence of zero points on the kzaxis was pointed out previously [ 26], the above correspondence further clarified their topological character: The Chern number defined ineachk zslice jumps at the point nodes. This means that all the point nodes are specified by topological Weyl charges.Consequently, if t zis large enough to cross the narrowly spaced quantum critical μ[Fig. 2(a)], the system shows Weyl superconductivity [ 29–33]. In Fig. 3we show the phase diagram. A superconducting Weyl phase is realizedfort z∼0.01∼0.1/lessmucht, which is reasonable for a quasi-2D system [Fig. 3(a)]. In fact, at H/φ 0=1.25×10−3, the Weyl node appears at kz=0.415 and the gap closes [Fig. 3(b)], while the gap opens in the other momenta [Fig. 3(c)]. 060507-3RAPID COMMUNICATIONS TOMOHIRO YOSHIDA AND MASAFUMI UDAGAW A PHYSICAL REVIEW B 94, 060507(R) (2016) 00.030.060.09 0.00tz x 10-30.6 1.0 1.4 1.8 H/φ( )Weyl kx kxky kyE(k) E(k)(a) (b) (c) FIG. 3. (a) Low-magnetic-field phase diagram for the quasi-2D chiral superconductor. The superconducting Weyl phase is realizedfor larger t z.T h e kz-projected dispersions are plotted for (b) kz= 0.415 and (c) kz=2.0, obtained at H/φ 0=1.25×10−3andtz=0.05. In conclusion, we have studied the topological aspect of chiral p-wave superconductors in the vortex state, with the BdG equation combined with a mapping to an effectiveMajorana tight-binding model. In the 2D case, we revealed theexistence of ubiquitous topological quantum phase transitions,which can be attributed to only a few universal properties of2D chiral superconductors, namely, the existence of Majorana modes in isolated vortices, the realization of a QHZ-typeHamiltonian from the double enlargement of the magnetic unitcell due to π-flux-carrying vortices, and a sensitive variation of the effective transfer integrals in the scale of the Fermiwavelength. Through a dimensional reduction, we furthershowed that a superconducting Weyl phase generically existsin the low-field region of quasi-2D chiral superconductors.These findings are relevant to general chiral superconductors,potentially including Sr 2RuO 4, and further explorations of their physical consequences will be awaited. In fact, thetransport phenomena associated with the low-field Weyl phase,and especially its quantum anomaly, will be a fascinatingtheme of study. The effects of lattice dislocation might also bean interesting issue, which is known to affect the topologicalnature of the system, and a vortex lattice gives a controllablestage for its study. We leave these problems for futurestudy. The authors are grateful to Y . Higashi, M. Takahashi, and Y . Yanase for fruitful discussions. This work was supportedby JSPS KAKENHI (No. JP26400339, No. JP15H05852, No.JP15K13533, and No. JP16H04026). T.Y . is supported by aJSPS Fellowship for Young Scientists. [1] N. Read and D. Green, Phys. Rev. B 61,10267 (2000 ). [ 2 ] D .A .I v a n o v , Phys. Rev. Lett. 86,268(2001 ). [3] A. Y . Kitaev, Phys. Usp. 44,131(2001 ). [4] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78,195125 (2008 ). [5] L. Fu and C. L. Kane, P h y s .R e v .L e t t . 100,096407 (2008 ). [6] Y . Tanaka, T. 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PhysRevB.90.235401.pdf

PHYSICAL REVIEW B 90 , 235401 (2014) Transforming a surface state of a topological insulator by a Bi capping layer Han Woong Yeom,1 ,2 -* Sung Hwan Kim,1 -2 Woo Jong Shin,1 ,2 Kyung-Hwan Jin,2 Joonbum Park,2 Tae-Hwan Kim,1 ,2 Jun Sung Kim,2 Hirotaka Ishikawa,3 Kazuyuki Sakamoto,3 and Seung-Hoon Jhi2 1 Center fo r Artificial Low Dimensional Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang 790-784, Korea 2 Department o f Physics, Pohang University o f Science and Technology, 77 Cheongam-Ro, Pohang 790-784, Korea 3 Department o f Nanomaterials Science, Chiba University, Chiba 263-8522, Japan (Received 15 June 2014; revised manuscript received 10 November 2014; published 1 December 2014) We introduce a distinct approach to engineer a topologically protected surface state of a topological insulator. By covering the surface of a topological insulator, Bi2Te2Se, with a Bi monolayer film, the original surface state is completely removed and three new spin helical surface states, originating from the Bi film, emerge with different dispersion and spin polarization, through a strong electron hybridization. These new states play the role of topological surface states keeping the bulk topological nature intact. This mechanism provides a way to create various different types of topologically protected electron channels on top of a single topological insulator, possibly with tailored properties for various applications. DOI: 10.1103/PhysRevB.90.235401 PACS number(s): 73.20.At, 68.37.Ef, 73.61.Ng, 79.60.Dp I. INTRODUCTION Topological insulators (TIs) are a new class of insula tor materials with unusual surface (edge) metallic electron channels [1,2]. These surface channels have massless Dirac electron character with helical spin polarization [3-12], which are robustly protected by the topological nature of the bulk [13,14]. These unique characteristics make surface states of TI, topological surface states (TSS), ideal for scattering- free carriers of spintronic information and the fault-tolerant quantum computing. The prompt application of these materials are, however, hampered by not only materials issues such as surface and bulk imperfections but also by the robustly protected nature of their surface states itself; intrinsically hard to manipulate and control [14]. So far, the most popular way of controlling a TSS is to dope them. For the cases of the most widely studied 3D TI of Bi chalcogenides, the nonmagnetic atomic and molecular dopants [4,5,7,9,12,15] were shown to shift the TSS bands. On the other hand, the magnetic impurity atoms were debatably reported to open a small band gap at Dirac points of TSS by breaking the time reversal symmetry [10,11]. However, the topological property of the materials is then destroyed and the magnetic impurities give rise to unwanted scatterings. Very recently, theoretical works suggested the changes in the effective mass of TSS upon terminating the surface with other atoms [16] and the changes in the vertical position and the Dirac point energy of TSS upon capping the surface with insulating compound films [17-19], However, none of these proposals were realized. In this work, we devise a distinct approach in engineering TSS. We show both theoretically and experimentally that a strongly interacting monolayer grown on top of a 3D TI, a simple Bi monolayer on Bi2Te 2Se (called BTS hereafter), in particular, can host new spin-helical electronic states replacing the original TSS with the bulk TI property preserved. By changing the terminating monolayer, one may be able to generate various different TSS with different dispersion and *yeom@postech.ac.krspin orientation on a single 3D TI. That is, a particular TSS can be replaced by or transformed into other helical Dirac electronic states satisfying the topological requirement of the system. This result opens a new avenue towards creating and tailoring topologically protected spin and electron channels. II. EXPERIMENTS AND CALCULATION We used cleaved single crystals of BTS, which were grown using the self-flux methods [20,21]. The BTS crystals were cleaved in ultrahigh vacuum, onto which Bi monolayers were deposited as reported before [21]. We performed scanning tun neling microscopy/spectroscopy (STM/STS) measurements using a commercial low-temperature STM. The STM topog raphy was measured in a constant-current mode, and STS spectra and maps were obtained by the lock-in technique, which minimize the effect of topographic corrugations with the current feedback turned off [21], The spin- and angle-resolved photoemission spectroscopy (SARPES) measurements were performed with a high performance hemispherical electron analyzer (VG-SCIENTA R4000) and a Mott spin detector using Xe discharge light of 8.4 eV [22]. The samples were kept at 78 K in both experiments. The ab initio calculations were carried out in the plane-wave basis within generalized gradient approximation for exchange- correlation functional [23,24]. A cutoff energy of 400 eV was used for the plane-wave expansion. The Bi/BTS structure is simulated by the supercell with Bi one monolayer (in a bilayer structure) [Fig. 1 (a)] on one surface of a slab of six quintuple layers (QLs) BTS and a vacuum layer 20 A thick between the cells [21]. During structural relaxation, the atoms of Bi monolayer and three BTS surface layers are allowed to relax until the forces are smaller than 0.01 eV/A. The van der Waals interaction is also considered [25]. III. RESULTS AND DISCUSSIONS Figure 1(b) is the band dispersion of a well-known 3D TI, BTS, whose atomic structure is depicted in Fig. 1(a). Its TSS shows the V-shape dispersion above the bulk valence band (dashed lines), exposing the characteristics of Dirac electrons. 1098-0121/2014/90(23)7235401 (5) 235401-1 ©2014 American Physical Society HAN WOONG YEOM et al. PHYSICAL REVIEW B 90, 235401 (2014) Momentum (krR , A') FIG. 1. (Color online) Experimental band dispersions along the high symmetry direction f -K. (a) Atomic structure of a Bi monolayer on Bi2Te2Se. (b) Bulk (guided by solid line) and surface state (dashed lines) band dispersions of pristine Bi2Te2Se. (c) Band dispersions of Bi-covered Bi2Te2Se along the f -K direction. The bulk bands of Bi2Te2Se are guided by solid lines and hatchings. The surface state of pristine Bi2Te2Se (dashed lines) disappears. The new surface states of B i, B2, and B3 originate from the Bi monolayer.FIG. 2. (Color online) Spin-resolved band structure calculations and their spin textures along f -K. Calculated band structure for a Bi monolayer on Bi2Te2Se along the f -K direction (a) without and [(b) and (c)] with the film-substrate interaction. In (b) the Bi monolayer is detached from the substrate further from the equilibrium position of (c) by 2 A. Gray colored region indicates the Bi2Te2Se bulk bands while the band originating from the Bi film (Bi2Te2Se) is colored in cyan (magenta), [(d) and (e)] Spin orientations of the surface state bands shown in (c). Opposite in- and out-of-plane spin components are indicated by red and blue colored dots. The size of dots represent the magnitude of the corresponding spin components, (f) Calculated band structure of the Bi monolayer on a trivial insulator In2Se3. The blue shades indicate the band gap of the substrates. The Dirac point is located at ~0.3 eV below the Fermi level as reported before [9]. On top of fresh cleaved surfaces, we epitaxially grow atomically flat Bi( 111) films [21]. The atomic structure of the monolayer film is shown in Fig. 1(a), which was confirmed by STM and ab initio calculations [21]. The electronic band dispersions of BTS are substantially changed after growing a monolayer Bi film as shown in Fig. 1(c). First, the bulk bands shift down by ~0.2 eV due to the charge transfer from the Bi film as discussed further below. This makes the edge of the bulk conduction band appear below the Fermi energy around the f point. This behavior was previously observed for various electron-doping adsorbates on Bi2Te3 and related surfaces [4,5,7,9,12]. Secondly, but most surprisingly, the TSS of BTS (green dashed lines) disappears completely and instead a A-shape band (B 3) emerges around f . In addition, two strongly dispersing states appear away from T; one (B 1) crosses the Fermi level to become metallic and the other (B2) is connected to the A-shape band (B3) at f forming a new band crossing (blue dashed lines). These data reveal greater details of the band dispersions with the improved spectroscopic resolution, which are overall consistent with the recent studies on a similar system of Bi/BioTei [26,27]. The experimental band dispersions match well with the ab initio calculation, which further unveils the origin of the newly formed bands. The two largely dispersing bands (B 1 and B2) are the pxy bands of the Bi monolayer. These bands are degenerate in the freestanding Bi( 111) monolayer [Fig. 2(a)]but split due to the inversion symmetry breaking by and the interaction with the substrate. We can artificially control the strength of such an interaction in the calculation by changing the distance between the Bi film and the TI substrate. Even at a moderate interaction [Fig. 2(b)], the TSS and the 2D electronic states of the film are strongly hybridized to make B2 and B 3 form a Rashba-type pair and Bi form a Dirac cone above E/. and disperse into the conduction band of BTS. A stronger hybridization [Fig. 2(c)] causes the TSS to totally vanish leaving only three bands within the BTS band gap (blue shades), Bj and the B 2/B3 Rashba pair. The Bi film becomes metallic due to the B 1 band dispersing into the BTS conduction band with its electrons partly transferred to the substrate. This explains the downward shift of the bands of BTS. Surprisingly and unexpectedly, the Bi/BTS complex has three surface states of only the Bi origin within the BTS band gap with the original TSS removed. The strong spin-orbit coupling of Bi itself and the Rashba- type band crossing for the newly formed surface states (B 2 + B3) suggest that they are spin polarized. This spin polarization is detailed in the band calculation (Fig. 2). In particular, along the f -K direction of the momentum space, where the band dispersion shown in Fig. 1 was measured, not only B 2 + B 3 but also the B 1 band are fully spin polarized with largely different spin orientations. While B 2 + B 3 bands have strong in-plane spin components within the surface plane, being consistent 235401-2 TRANSFORMING A SURFACE STATE OF A TOPOLOGICAL . . . PHYSICAL REVIEW B 90, 235401 (2014) FIG. 3. (Color) Spin-polarized photoelectron intensity for the B, and B2 bands for the (a) in- and (b) out-of-plane spin components measured by SARPES. The in-plane spin component measured is perpendicular to electron momenta, which is important for the spin helical system. The photoelectron energy distribution curves were taken at electron memento specified in Fig. 1(c) (the dashed box) and Figs. 2(c), 2(d), and 2(e) (vertical bars). Constant-energy contours and the spin texture of the surface state bands calculated at (c) the Fermi energy and (d) — 300 meV. The red (blue) arrows and dots denote positive (negative) spin direction and the dots are for the dominating out-of-plane spin components. The possible back scattering wave vectors connecting parallel spins, q, and q2, are indicated. with the Rashba spin splitting, Bi has its spin in the surface normal. While there is the strong out-of-plane spin component, the helical spin texture, opposite spins at opposite momenta, is obvious for all three surface states [Figs- 2(d), 2(e), 3(c), and 3(e)]. The partially out-of-plane spin polarization (B 2 in the present case) is widely found for normal TSS o f 3D TI materials, for the momentum space away from the Dirac point, and is due to the warped band structure reflecting the hexagonal crystal structure [28]. However, in the present case, the B 1 spin at the Fermi energy is unusually dominated by the out-of-plane component, except for apexes of the hexagonal Fermi surface, where the spin flips [Fig. 3(c)]. The band dispersions and their detailed spin texture obviously show that the triple bands formed within the band gap are spin helical surface states between the BTS bulk and the vacuum. We thus conclude that they play the role of TSS connecting topologically nontrivial BTS with trivial vacuum, satisfying apparently the topological requirement of an odd number of spin helical surface states crossing the band gap. The spin polarization of the major bands was directly measured by SARPES. At the electron momenta away from f , the two major surface state bands are prominent as shown in the energy distribution curves o f photoelectrons [Figs. 3(a) and 3(b)], which were taken for the B j and B 2 bands at the momenta indicated in Figs. 1(c) (the dashed box) and 2 (vertical bars). The strong spin polarization is very clear in the SARPES data. As predicted in the calculation (Fig. 2), the spin is(a ) 0.0 F 2 4 6 D istance (nm)9™ (A ’) FIG. 4. (Color online) Scatterings of surface state electrons mea sured by STM. (a) Spatially and energetically resolved STS (dl/dV ) measurement taken across a step edge of a Bi monolayer island (along the blue arrow in the inset). The crystallographic orientation and the atomic structure of the step edges were clarified in the previous work [21], which corresponds to the V-M direction in momentum space. The green arrows indicate the QPI pattern due to the electron scattering interference, (b) The topographic profile along the edge, (c) Energy-resolved Fourier transform of (a) shows electron scattering wave vectors, qt and q2 , dispersing according to the band structure given in Fig. 1(c). almost vertical, in particular for B 1, with opposite orientations between the opposite directions o f electron momenta. The strong in-plane spin polarization of the B 3 band is also clearly confirmed (data not shown), which is fully consistent with a very recent report [29]. This result clearly evidences the spin helical nature o f the three surface state bands. We measured a wide binding energy range and the agreement between the experiment and the calculation is excellent for the detailed spin orientation. The helical spin texture of the surface state is a hallmark o f a TSS and it provides the unique and important property of suppressing electron backscattering. The electron backscatter- ing can be directly confirmed by measuring the quasiparticle interference (QPI) in STM due to electron scatterings by defects. We performed STM experiments for partially covered Bi films composed of several 100 nm wide 2D islands [21], For these islands, we can observe the QPI pattern due the scattering by the edge of islands. The STS dl/dV data in Fig. 4(a) shows the spatially resolved local density o f states (LDOS) containing such electron interference patterns (indicated by arrows). This QPI depends systematically on the electron energy reflecting the band structure. The Fourier transform of the QPI patterns reveals the scattering wave vectors involved [6,30], Mainly two distinct scattering vectors are identified, q, and q2 . They match with those expected from the band structure and the spin texture discussed above; the electron backscattering is allowed only for the wave vectors connecting the spin parallel parts of the bands as shown in Figs. 3(c) and 3(d). This result, while not fully quantitative, further corroborates the calculation and the SARPES experiment. Consistent results were also obtained for the 2D QPI patterns taken at selected energies. By covering a single-atom-thick Bi film on a 3D TI BTS, we show theoretically and experimentally that the TSS of TI is totally removed and new spin helical electron states 235401-3 HAN WOONG YEOM et al. PHYSICAL REVIEW B 90, 235401 (2014) are formed in the covering film. This demonstrates that the TSS of a 3D TI can be transferred into different atomic layers covering the 3D TI. During this transfer, the band dispersion and the spin texture are substantially modified. In the present case, unique vertical spins are created. The underlying mechanism of the TSS transformation is the strong electronic hybridization between the metal monolayer and the TI including the TSS (Fig. 2). As a proof of the concept and the nontrival nature of the newly formed surface states, we performed similar calculations for a Bi film on a trivial insulator, In2Se3 [Fig. 2(f)]. This calculation yields only trivial Rashba bands of the Bi origin, four surface states paired, within the band gap. The basic concept seems similar to the change of the vertical positions of TSS for the insulating compound films covering a TI in the recent theoretical studies [17-19]. Those theoretical suggestions were not realized yet and our choice of simple and single-atom-thick film made the realization and confirmation of the TSS transformation substantially easier. We also tested various different TI substrates (data not shown), which all yield consistent results. The only difference between different substrates is the degree of hybridization of Bi and the substrates, especially for the band near the Rashba crossing. When the Rashba crossing gets closer to the conduction band minimum of the substrate, the corresponding wave functions absorb more substrate character. This different degree of the hybridization is consistent with the recent report of “hybridized Dirac cones” for Bi/Bi2Se3 and Bi/Bi2Te3, respectively [29]. Note that this crossing is not a Dirac state but a Rashba crossing and the trivial case of Bi/In2Se3 also [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). [2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). [3] D. Hsieh, D. Qian, L. Wray, Y . Xia, Y . S. Hor, R. J. Cava, and M. Z. Hasan, Nature (London) 452, 970 (2008). [4] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong, A. V . Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature (London) 460, 1101 (2009). [5] Y. L. Chen, J. G. Analytis, J. H. Chu, Z. K. Liu, S. K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, Science 325, 178 (2009). [6] J. Seo, P. Roushan, H. Beidenkopf, Y. S. Hor, R. J. Cava, and A. Yazdani, Nature (London) 466, 343 (2010). [7] Z. Alpichshev, J. G. Analytis, J.-H. Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, and A. Kapitulnik, Phys. Rev. Lett. 104, 016401 (2010). [8] S. Souma, K. Kosaka, T. Sato, M. Komatsu, A. Takayama, T. Takahashi, M. Kriener, K. Segawa, and Y . Ando, Phys. Rev. Lett. 106,216803 (2011). [9] T. Arakane, T. Sato, S. Souma, K. Kosaka, K. Nakayama, M. Komatsu, T. Takahashi, Z. Ren, K. Segawa, and Y. Ando, Nat. Commun. 3, 636 (2012). [10] Y. L. Chen, J.-H. Chu, J. G. Analytis, Z. K. Liu, K. Igarashi, H.-H. Kuo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu, M. Hashimoto, T. Sasagawa, S. C. Zhang, I. R. Fisher, Z. Hussain, and Z. X. Shen, Science 329, 659 (2010). [11] J. Zhang, C.-Z. Chang, P. Tang, Z. Zhang, X. Feng, K. Li, L.-l. Wang, X. Chen, C. Liu, W. Duan, K. He, Q.-K. Xue, X. Ma, and Y. Wang, Science 339, 1582 (2013).has such hybridized Rashba crossing near the conduction band minim [Fig. 2(f)], What is quintessential in the topological band texture of these systems is, thus, not the presence of the Rashba band crossing, but the odd number of spin helical bands within the band gap. The hybridization of the B \ band with the conduction bands of the substrate makes this odd-numbered band texture [Fig. 2(b)]. IV. SUMMARY We show that the TSS of a TI can be completely replaced by or transformed into two dimensional spin helical electronic states of a strongly interacting monolayer covering the surface through a strong electronic hybridization, in particular for a Bi monolayer film on Bi2Te2Se. In principle, with properly designed covering films, one can make various different types of TSS on a single 3D TI, providing a new degree of freedom in creating topologically protected electron channels beyond the limit of the bulk materials synthesis, with possibly tailored properties for various TI applications. ACKNOWLEDGMENTS This work was supported by Institute for Basic Science (IBS) through the Center for Artificial Low Dimensional Electronic Systems (Grant No. IBS-R014-D1) and Center for Topological Matter (Grant No. 2011-0030046), and the Basic Science Research program (Grant No. 2012-013838) of KRF. [12] S.-Y. Xu, Y. Xia, L. A. Wray, S. Jia, F. Meier, J. H. Dil, J. Osterwalder, B. Slomski, A. Bansil, H. Lin, R. J. Cava, and M. Z. Hasan, Science 332, 560 (2011). [13] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nat. Phys. 5, 438 (2009). [ 14] K. Park, J. J. Heremans, V. W. Scarola, and D. Minic, Phys. Rev. Lett. 105, 186801 (2010). [15] L. A. Wray, S.-Y. Xu, Y. Xia, D. Hsieh, A. V. Fedorov, Y. S. Hor, R. J. Cava, A. Bansil, H. Lin, and M. Z. Hasan, Nat. Phys. 7, 32(2011). [16] F. Virot, R. Hayn, M. Richter, and J. van den Brink, Phys. Rev. Lett. Ill, 146803 (2013). [17] G. Wu, H. Chen, Y. Sun, X. Li, P. Cui, C. Franchini, J. Wang, X.-Q. Chen, and Z. Zhang, Sci. Rep. 3, 1233 (2013). [18] T. V. Menshchikova, M. M. Otrokov, S. S. Tsirkin, D. A. Samorokov, V. V . Bebneva, A. Ernst, V. M. Kuznetsov, and E. V. Chulkov, Nano Lett. 13, 6064 (2013). [19] Q. Zhang, Z. Zhang, Z. Zhu, U. Schwingenschlgl, and Y. Cui, ACS Nano 6, 2345 (2012). [20] Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y . Ando, Phys. Rev. B 82, 241306 (2010). [21] S. H. Kim, K.-H. Jin, J. Park, J. S. Kim, S.-H. Jhi, T.-H. Kim, and H. W. Yeom, Phys. Rev. B 89, 155436 (2014). [22] K. Sakamoto, T.-H. Kim, T. Kuzumaki, B. Mller, Y . Yamamoto, M. Ohtaka, J. R. Osiecki, K. Miyamoto, Y . Takeichi, A. Harasawa, S. D. Stolwijk, A. B. Schmidt, J. Fujii, R. I. G. Uhrberg, M. Donath, H. W. Yeom, and T. Oda, Nat. Commun. 4, 3073 (2013). [23] G. Kresse and J. Furthmiiller, Phys. Rev. B 54, 11169 (1996). 235401-4 TRANSFORMING A SURFACE STATE OF A TOPOLOGICAL ... [24] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). [25] A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009). [26] T. Hirahara, G. Bihlmayer, Y. Sakamoto, M. Yamada, H. Miyazaki, S. I. Kimura, S. Bliigel, and S. Hasegawa, Phys. Rev. Lett. 107, 166801 (2011). [27] F. Yang, L. Miao, Z. F. Wang, M.-Y. Yao, F. Zhu, Y. R. Song, M.-X. Wang, J.-P. Xu, A. V . Fedorov, Z. Sun, G. B. Zhang,PHYSICAL REVIEW B 90, 235401 (2014) C. Liu, F. Liu, D. Qian, C. L. Gao, and J.-F. Jia, Phys. Rev. Lett. 109, 016801 (2012). [28] L. Fu, Phys. Rev. Lett. 103, 266801 (2009). [29] L. Miao, Z. F. Wang, M.-Y. Yao, F. Zhu, J. H. Dil, C. L. Gao, C. Liu, F. Liu, D. Qian, and J.-F. Jia, Phys. Rev. B 89, 155116 (2014). [30] T. Zhang, P . Cheng, X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang, X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, Phys. Rev. Lett. 103, 266803 (2009). 235401-5 Copyright of Physical Review B: Condensed Matter & Materials Physics is the property of American Physical Society and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.

PhysRevB.70.100201.pdf

Compatibility factor of segmented thermoelectric generators based on quasicrystalline alloys Enrique Maciá * Departamento de Fisica de Materiales, Facultad CC. Fisicas, Universidad Complutense de Madrid, E-28040, Madrid, Spain (Received 26 May 2004; published 13 September 2004 ) In this work we present a prospective study on the possible use of certain quasicrystalline alloys in order to improve the efficiency of segmented thermoelectric generators. To this end, we obtain a closed analyticalexpression for their compatibility factor [G. J. Snyder and T. S. Ursell, Phys. Rev. Lett. 91, 148301 (2003 )].B y comparing our analytical results with available experimental data we conclude that a promising high tempera-ture material, compatible with benchmark thermoelectric materials, can be found among AlPdMn based icosa-hedral quasicrystals. DOI: 10.1103/PhysRevB.70.100201 PACS number (s): 61.44.Br, 71.20. 2b, 72.15.Jf, 85.80.Fi The efficiency of a thermoelectric device depends on the transport properties of the constituent materials and the tem-perature difference between the hot and cold sides, whichsets its Carnot upper limit. Evaluation of new materials forthermoelectric applications is usually made in terms of thedimensionless figure of merit, ZT; sa2T ke+kph, s1d whereTis the temperature, ssTdis the electrical conductiv- ity,asTdis the Seebeck coefficient, kesTdis the charge car- rier contribution to the thermal conductivity, and kphsTdis the lattice contribution to the thermal conductivity. When considering segmented devices one should also consider thecompatibility factor defined by 1 s=˛1+ZT−1 aT, s2d since materials with dissimilar svales cannot be efficiently combined in that case. The thermoelectric compatibility ofseveral materials of current technological interest has beenrecently reviewed, concluding that a semimetallic materialwith high p-type thermopower is required for development of segmented generators. 2 In the last few years it has been progressively realized that quasicrystals (QCs)deserve some attention as potential ther- moelectric materials (TEMs ), since they naturally fulfill the Slack’s requirements for a material belonging to the ”elec-tronic crystal/phonon glass” class. 3–7Quite interestingly, relatively high, positive thermopower values s+100 −120 mVK−1dhave been reported for representatives of the icosahedral AlPdMn and AlPdRe families in the temperature range 300–600 K.8–10The main aim of this paper is to show that, by a judicious choice of sample’s stoichiometry,11suit- able candidates for a high temperature material, compatiblewith PbTe, sAgSbTe 2d0.15sGeTe d0.85(TAGS )or skutterudites, may be found among the AlPd sRe,Mn dquasicrystalline al- loys. To this end, let us start by briefly summarizing some rel- evant experimental data. In Tables I and II we list the figureof merit and compatibility factors of different QCs as re-ported in the literature.At room temperature we observe thatthe largest svalues are comparable to those observed in usualTEMs, like Bi 2Te3or SiGe ss.1V−1d.2At higher tem- peratures the most promising QC is i-AlPdMn, which exhib- its ansfactor larger than those reported for SiGe ss .1V−1dand PbTe ss.1.2 V−1d, and approaches that of TAGS ss.2.7 V−1datT=550 K.2On the other hand, the s factor corresponding to AlPdRe samples is larger at room temperature (where it exhibits a lower ZTvalue )than it is at higher temperatures (albeit it exhibits a larger ZTvalue ). This result highlights the importance of properly balancingtheZTandscontributions in designing optimized devices. In order to gain some theoretical insight into this question we obtain a closed analytical expression for the compatibilityfactor within the Kubo-Greenwood framework. 15The central information quantities are the kinetic coefficients, LijsTd=s−1di+jEssEdsE−mdi+j−2S−]f ]EDdE,s3d wherefsE,m,Tdis the Fermi-Dirac distribution function. In this formulation all the microscopic details of the system are included in the ssEdfunction. As a first approximation we will assume msTd<EF. Then, by expressing (3)in terms of the scaled variable x;sE−md/kBT, the transport coefficients can be written as6,16 ssTd=J0 4, s4d asTd=−kB ueuJ1 J0, s5d kesTd=kB2T 4e2SJ2−J12 J0D, s6d where we have introduced the reduced kinetic coefficients, JnsTd=Exnssxdsech2sx/2ddx. s7d In previous works16–18it has been shown that the experimen- talssTdandSsTdcurves of several QCs can be consistently described in terms of the two-Lorentzian spectral conductiv- ity function,19,20PHYSICAL REVIEW B 70, 100201 (R)(2004 )RAPID COMMUNICATIONS 1098-0121/2004/70 (10)/100201 (4)/$22.50 ©2004 The American Physical Society 70100201-1ssEd=B pFg1 sE−d1d2+g12+Ag2 sE−d2d2+g22G−1 ,s8d whereBis a scale factor expressed in V−1cm−1eV−1units, and the Lorentzian peaks are characterized by their height,s pgid−1, and their position, di, with reference to the Fermi level. The overall behavior of this curve agrees well with the experimental results obtained from tunneling and point con-tact spectroscopy measurements, where the presence of a dipfeature of small width (20–60 meV, narrow Lorentzian ), su- perimposed onto a broad (0.5–1 eV, broad Lorentzian ), asymmetric pseudogap has been reported. 21–25The relative importance of each spectral feature in the overall electronicstructure is tuned through the weight factor Ain Eq. (8). Making use of Eq. (8)into Eq. (7)we get 16 1J0 J1 J22=4s0110 j20 j40j0 02j102j302j10 p2 30p2 321 5j20p2 3j2002 311 bT kB bT2 b2T3 kB b2T4 b3T5 kB b3T62, s9d where s0=ssT!0dis the residual electrical conductivity, andb;e2L0, where L0=p2kB2/3e2=2.44 310−8V2K−2is the Lorenz number The phenomenological coefficients ji,a s well as the parametric functions ji=jisj3,j4d, are directly re- lated to the sample’s electronic structure. In particular, we have17 j1=1 2SdlnssEd dED EF, s10d so that, according to Mott’s expression, j1can be derived from the low temperature slope, a, of the experimental asTd curve as18j1exp.−20.5afmVK−2gseVd−1. s11d Making use of Eq. (9)into Eqs. (4)–(6)and Eqs. (1)and(2) we get s=−QF˛1 1−R−1GV−1, s12d where QsTd=P0 2bT2P1, s13d RsTd=2P1 QsP2+Cd, s14d csTd;kphsTd/s0L0T, and we have introduced the polynomi- als P0sTd;1+s2j12+Vdy+j4y2+j0y3, s15d P1sTd;j1+j3y+j1y2, s16d P2sTd;1+21 5s2j12+Vdy+j2y2, s17d where V;1 2Sd2lnssEd dE2D EF, s18d measures the curvature of ssEdat the Fermi level.17At a given temperature Eq. (12)can be regarded as a parametric function of the different phenomenological coefficients, i.e.,ss jid. We note that the thermal conductivity of QCs is mainly determined by the lattice contribution rather than the charge carriers in the considered temperature range.26Therefore, we can confidently assume kph.kin Eq. (14), where k;ke +kphis the experimentally measured total thermal conduc- tivity. In Fig. 1 we compare the room temperature compatibility factors of i-AlPdRe [k=0.7 Wm−1K−1,9s0=30 sVcmd−1 (Ref.[27])andi-AlPdMn [k=1.6 Wm−1K−1,s0 =740 sVcmd−1,(Ref. [5])as a function of j1. These curves are derived from Eqs. (12)–(17)with V=400 seVd−2(Ref. 17);j3=−2910 seVd−3,j4=17000 seVd−4,j0=105seVd−4,j1 =130000 seVd−4, andj2=−30000 seVd−4.11In order to check the feasibility of the adopted model parameters we have de- termined the value of the j1coefficient corresponding to theTABLE I. Room temperature thermopower, figure of merit, and compatibility factors for samples belonging to different quasicrys- talline families. Sample Ref. asmVK−1dZT s sV−1d AlCuFe 12 +44 0.01 0.38 AlCuRuSi 12 +50 0.02 0.66CdYb 13 +16 0.01 1.04AlPdRe 14 +95 0.07 1.21AlPdMn 5 +85 0.08 1.54TABLE II. High temperature thermopower, figure of merit, and compatibility factors for samples belonging to the AlPd sMn,Re d icosahedral family. T*denotes the temperature maximizing the fig- ure of merit. Sample Ref. T*a*smVK−1dZT*s*sV−1d AlPdMn 5 550 +105 0.25 2.04 AlPdReRu 10 700 +100 0.15 1.03AlPdRe 9 660 +90 0.11 0.90ENRIQUE MACIÁ PHYSICAL REVIEW B 70, 100201 (R)(2004 )RAPID COMMUNICATIONS 100201-2i-AlPdMn sample. Making use of the low temperature ther- mopower data reported in Ref. 5 into Eq. (11)we get j1exp= −6.49 seVd−1. By plugging this value in our analytical ex- pressions we obtain ssj1expd=1.57 V−1(the dotted line in Fig. 1)andZTsj1expd=0.080, in excellent agreement with the ex- perimental values listed in Table I. The difference between the AlPdRe and AlPdMn ssj1d curves is due to the significant difference between their re- spective residual conductivities, determining the Cvalue in Eq.(14). In the inset of Fig. 1 we plot the corresponding ZTsj1dcurves, which exhibit a deep minimum, flanked by two maxima. According to Eq. (2), bothZTandsvanish at j10=+5.76 seVd−1. Consequently, QCs can exhibit p-type sj1,j10dorn-type sj1.j10dthermopowers depending on the j1value which, according to Eq. (10), is very sensitive to the sample’s electronic structure near EF. In fact, the electronic structure of QCs is characterized by the presence of a narrowpseudogap in the density of states close to the Fermi level.Thus, when E Fis located at the left (right )of the pseudogap’s minimum j1takes negative (positive )values, and its magnitude is directly related to the ssEdslope.There-fore, the j1value can be controlled by changing the sample stoichiometry, hence shifting EFin a scale of a few meV. In this way, we can confidently expect that larger values of theroom temperature compatibility factor, close to s=2.0 V −1 may be attained in AlPdMn QCs with j1.−15 seVd−1.I n addition, a significant enhancement of the sfactor is ex- pected for AlPdMn QCs at higher temperatures, as it is shown in Fig. 2. The value ssj1expd=3.7 V−1(dotted line )is better than that reported for both TAGS and skutterudites at T=550 K. Nonetheless, we should keep in mind that our rigid band model, which does not take into account the tem-perature dependence of E F, will be hardly applicable at tem- peratures beyond the Debye one (QD.450 K for AlPdMn ).4 Further experimental and theoretical work is then appealing in order to fully exploit the unusual transport properties ofquasicrystalline alloys in thermoelectric devices. I warmly thank Professor K. Kimura, Professor T. M. Tritt, Dr. K. Kirihara, and Dr. A. L. Pope for sharing usefulmaterials. I acknowledge M.V. Hernández for a critical read-ing of the manuscript. This work was supported by the UCMthrough Project No. PR3/04-12450. *Electronic address: emaciaba@fis.ucm.es 1G. Jeffrey Snyder and T. S. Ursell, Phys. Rev. Lett. 91, 148301 (2003 ). 2G. Jeffrey Snyder, Appl. Phys. Lett. 84, 2436 (2004 ). 3G. A. Slack, CRC Handbook of Thermoelectrics , edited by D. M. Rowe (CRC Press, Boca Raton, FL, 1995 ). 4T. M. Tritt, A. L. Pope, M. A. Chernikov, M. Feuerbacher, S. Legault, R. Cagnon, and J. Strom-Olsen, in Quasicrystals , edited by J. M. Dubois, P. A. Thiel, A-P. Tsai, and K. Urban, MRSSymposia Proceedings No. 553 (Materials Research Society, Pittsburgh, 1999 ), p. 489. 5A. L. Pope, T. M. Tritt, M. A. Chernikov, and M. Feuerbacher, Appl. Phys. Lett. 75, 1854 (1999 ).6E. Maciá, Appl. Phys. Lett. 77, 3045 (2000 ). 7E. Maciá, Phys. Rev. B 64, 094206 (2001 ). 8F. Morales and R. Escudero, Bull. Am. Phys. Soc. 44,1(1999 ). 9K. Kirihara and K. Kimura, J. Appl. Phys. 92, 979 (2002 ). 10T. Nagata, K. Kirihara, and K. Kimura, J. Appl. Phys. 94, 6560 (2003 ). 11E. Maciá, Phys. Rev. B 69, 184202 (2004 ). 12F. S. Pierce, S. J. Poon, and B. D. Biggs, Phys. Rev. Lett. 70, 3919 (1993 ); F. S. Pierce, P. A. Bancel, B. D. Biggs, Q. Guo, and S. J. Poon, Phys. Rev. B 47, 5670 (1993 ). 13A. L. Pope, T. M. Tritt, R. Gagnon, and J. Strom-Olsen, Appl. Phys. Lett. 79, 2345 (2001 ). 14K. Kirihara, T. Nagata, and K. Kimura, J. Alloys Compd. 342, FIG. 1. (Color online )Room temperature dependence of the compatibility factor (main frame )and the thermo-electric figure of merit (inset )as a function of the phenomenological coefficient j1 fori-AlPdRe (dashed line ), andi-AlPdMn (solid line ). FIG. 2. (Color online )Dependence of the compatibility factor as a function of the phenomenological coefficient j1fori-AlPdRe with k=1.2 Wm−1K−1atT=660 K (dashed line );9and AlPdMn with k=2.1 Wm−1K−1atT=550 K (solid line )(Ref. 5 ).COMPATIBILITY FACTOR OF SEGMENTED PHYSICAL REVIEW B 70, 100201 (R)(2004 )RAPID COMMUNICATIONS 100201-3469(2002 ). 15D. A. Greenwood, Proc. Phys. Soc. London 71, 585 (1958 );R . Kubo, J. Phys. Soc. Jpn. 12, 570 (1957 ). 16C. V. Landauro, E. Maciá, and H. Solbrig, Phys. Rev. B 67, 184206 (2003 ). 17E. Maciá, Phys. Rev. B 66, 174203 (2002 ). 18E. Maciá, J. Appl. Phys. 93, 1014 (2003 ). 19C. V. Landauro and H. Solbrig, Mater. Sci. Eng.,A 294–296, 600 (2000 ). 20H. Solbrig and C. V. Landauro, Physica B 292,4 7 (2000 ). 21R. Escudero, J. C. Lasjaunias, Y. Calvayrac, and M. Boudard, J. Phys.: Condens. Matter 11, 383 (1999 ). 22T. Klein, O. G. Symko, D. N. Davydov, and A. G. M. Jansen, Phys. Rev. Lett. 74, 3656 (1995 ).23D. N. Davydov, D. Mayou, C. Berger, C. Gignoux, A. Neumann, A. G. M. Jansen, and P. Wyder, Phys. Rev. Lett. 77, 3173 (1996 ). 24X. P. Tang, E. A. Hill, S. K. Wonnell, S. J. Poon, and Y. Wu, Phys. Rev. Lett. 79, 1070 (1997 ). 25T. Klein, O. G. Symko, D. N. Davydov, and A. G. M. Jansen, Phys. Rev. Lett. 74, 3656 (1995 ); D. N. Davydov, D. Mayou, C. Berger, C. Gignoux, A. Neumann, A. G. M. Jansen, and P. Wy-der,ibid.77, 3173 (1996 ). 26M. A. Chernikov, A. Bianchi, and H. R. Ott, Phys. Rev. B 51, 153(1995 ). 27The mean value of the low temperature data reported by J. Dela- haye, C. Berger, and G. Fourcaudot, J. Phys.: Condens. Matter 15, 8753 (2003 ).ENRIQUE MACIÁ PHYSICAL REVIEW B 70, 100201 (R)(2004 )RAPID COMMUNICATIONS 100201-4

PhysRevB.83.144203.pdf

PHYSICAL REVIEW B 83, 144203 (2011) Multiscale study of the influence of chemical order on the properties of liquid Li-Bi alloys J.-F. Wax Laboratoire de Physique des Milieux Denses, Universit ´e Paul Verlaine - Metz, 1, Boulevard F . D. Arago, F-57078 Metz Cedex 3, France M. R. Johnson Institut Laue Langevin, 6 rue Jules Horowitz, F-38042 Grenoble Cedex 9, France L. E. Bove Physique des Milieux Denses, IMPMC, CNRS-UMR 7590, Universit ´e Pierre et Marie Curie, F-75252 Paris, France M. Mihalkovi ˇc Institute of Physics, Slovak Academy of Sciences, Dubravska cesta 9, 84511 Bratislava, Slovakia (Received 22 July 2010; revised manuscript received 22 November 2010; published 13 April 2011) The static structure and diffusion properties of Li-Bi liquid alloys at three compositions are investigated by molecular dynamics simulations. Due to the strong chemical order shown by these alloys, a multiscaleapproach is applied, fitting empirical pair potentials to data from ab initio molecular dynamics simulations to subsequently perform large-scale classical simulations. In this way, the partial structure factors as well as the self-diffusion and interdiffusion coefficients can be computed with sufficient accuracy to be discussed quantitatively.This approach is validated by comparing our predictions with experimental structure factor measurements. Amarked heterocoordination is observed, which strongly influences the diffusion properties. These observationsare consistent with an evolution toward ionic bonding at the Li 75-Bi 25composition. DOI: 10.1103/PhysRevB.83.144203 PACS number(s): 61 .25.Mv, 61 .20.Ja, 66.10.cg, 34.20.Cf I. INTRODUCTION When computing the static and dynamic structure of a liquid alloy by molecular dynamics (MD) simulation, two conditionshave to be fulfilled. The first one is to accurately describe theinteractions between the atoms inside the liquid and the secondone to consider a box large enough over a time long enough toensure good statistics and sufficient resolution into space andtime. Ab initio MD (AIMD) fully meets the first condition but often struggles with the second one, while classical MD(CMD) using pair potentials is faced with the oppositedilemma. Consequently, accurately simulating physical prop-erties of a liquid alloy is not a trivial task. The fact is that, for themost part, the alloys for which an accurate description of theinteractions in terms of pair potentials is available are rathersimple mixtures with no strong chemical order. 1,2However marked homocoordination or heterocoordination tendenciesstrongly affect the physical properties of a mixture, makingsuch systems more appealing. Therefore, it is interesting to setup a methodology giving an accurate description of the interactions in terms of empiricalpair potentials working for such complicated systems. We have applied a force-matching technique in the same spirit as the pioneering work of Ercolessi and Adams, 3but adapted by Mihalkovi ˇcet al.4to obtain efficient pair potentials in the case of metallic alloys. Starting from AIMD performed with the V ASP code,5,6three analytic partial pair potentials are fitted to reproduce the ab initio computed forces and energy differences in the liquid under given thermodynamic conditions. Thesepair potentials are then introduced in a CMD simulationto increase the size of the simulated system. Consideringfirst the static structure, it is possible to reach qvalues low enough to get clear insight about chemical order. Moreover,being able to run simulations over longer times, self-diffusion and interdiffusion properties (and more generally, dynamicproperties) can also be investigated. We have chosen to apply this multiscale approach to the Li-Bi system for the following reasons. First, there is a strong difference between the valencies of both components,which presumably will induce marked heterocoordinationtendencies. The electronegativity difference between bothspecies may induce some ionic nature to the bonding betweenLi and Bi atoms in this mixture. Indeed, the electronic transportproperties, which we will discuss in more detail later, indicate a strong departure from a simple metallic behavior. Considering the phase diagram, 7we can point out the congruently melting intermetallic compound, Li 3-Bi, which is likely to have repercussions in the liquid phase.8Second, Li 30-Bi 70has been studied recently by inelastic neutron scattering9and it has been shown that the high mass ratio ( MBi/M Li=30.1) is responsible for the peculiar behavior of the collective excitations. This is likely also to be the case for the diffusion properties. To examine the influence of the composition ofthe mixture, we have considered three compositions, namelyLi 30-Bi 70,L i 57-Bi 43, and Li 70-Bi 30; the extreme compositions have been the subject of experimental investigations thatare partially published 9while the intermediate has still to be measured. From the aforementioned measurements, an estimation of the static structure factor can be derived. This will enable us to check the validity of the description of theinteractions by comparing the simulation and experimentalresults. Thus Li-Bi alloys constitute an interesting benchmarkfor our approach for which we have computed both the staticstructure and the diffusion properties. This paper is organized as follows. After this Introduction, the next section will gather all the useful technical details. 144203-1 1098-0121/2011/83(14)/144203(11) ©2011 American Physical SocietyW AX, JOHNSON, BOVE, AND MIHALKOVI ˇC PHYSICAL REVIEW B 83, 144203 (2011) Then, in Sec. III, we will display our results which will be analyzed following two lines, the first one being the checkof the accuracy of the approach and the second one beingthe study of several properties of these nonsimple mixtures,namely their electronic and static atomic structures as well astheir diffusion properties. Finally, the main conclusions willbe summarized in Sec. IVwhich will end with an overview of possible, future applications of this approach. II. FORMALISM AND SIMULATION DETAILS A.Ab initio simulations and density determination We have considered three compositions, namely Li 30-Bi 70, Li57-Bi 43, and Li 70-Bi 30. There is one important thermody- namic datum to be considered when dealing with metallicsystems, namely the density since it is well established thatinteratomic interactions in these systems are strongly densitydependent. For alloys however, experimental values are oftenlacking in the literature and this is the case of the Li-Bi ones.Indeed, to our knowledge, density data are available only forcompositions between 4 and 26.8 at % Bi. 10 We could have estimated the density as a function of composition by linearly interpolating between the pure liquidvalues, but since we were suspecting strong heterocoordinationtendencies, this seemed to be too crude an approximation. Inaddition, a reduction of the volume by about 30% is observedat the Li 75-Bi 25composition, which was interpreted as a consequence of ionic-like bonding. Therefore, we proceeded inthe following way to estimate it more realistically at each com-position and temperature. We performed AIMD simulations atseveral densities and computed the pressure in each state allow-ing the equilibrium density to be determined by interpolation.These simulations were performed using 50 particles in thecanonical ensemble (constant N,V , andT). The time step was 3 fs (we checked that results were not different using a 1 fstime step) and the runs were typically 1000 steps long, aduration sufficient to reach equilibrium. Due to the small sizeof the simulated systems, fluctuations of pressure were ratherimportant, but this approach nevertheless allowed a reliableestimate of the density. Electronic and ionic first-principles calculations were carried out using the projector-augmented wave (PAW)formalism 11of the Kohn-Sham density functional theory (DFT)12,13at the generalized gradient approximation level (GGA), implemented in V ASP . The GGA was formulated by the Perdew-Burke-Ernzerhof (PBE)14,15density functional. A single kpoint (the gamma point) was used in the electronic structure determination. The density values obtained are gathered in Table I.T h e y are quite different from those provided by a linear interpolationbetween the values of the pure metals (9.38 and 0.45 g /cm 3for Bi and Li at 1073 K, respectively). We stress that this is only away to estimate the density in view of the lack of experimentaldata. Moreover, as shown in Fig. 1, they agree with the scarce, available experimental data. B. Determination of the interaction potentials Interactions are accurately described in AIMD simulations since the electronic structure is computed from first principles.TABLE I. Temperatures and densities at which the three systems were studied. System Density (g /cm−3) T(K) Li30-Bi 70 8.28 673 Li57-Bi 43 6.45 1073 Li70-Bi 30 4.73 1073 Therefore, such runs can be considered as reference data that one may try to mimic by simpler analytical expressions. In our approach, the first step consisted of performing AIMD simulations under the desired thermodynamic condi-tions and compositions. These runs were performed using the V ASP code. The box contained 200 particles and was sized to reproduce the desired density (Table I). The setup of the simulation was mentioned above except that they lasted 10 000time steps. These simulations were used to get the electronicdensity of states for each system, as well as the partialpair distribution functions g ij(r), which will be presented in Sec. III. Our three pair potentials uij(r) were fitted to the six- parameters form of Ref. 4, namely uij(r)=C(1) ij rη(1) ij+C(2) ij rη(2) ijcos(kijr+φij). (1) This form has a repulsive core (first term in the equation involving parameters C(1) ijandη(1) ij) and an oscillating tail (second term involving parameters C(2) ij,η(2) ij,kij, and φij). While this expression has been designed for simple-metalrich mixtures with transition metals, the interplay between therepulsive part and the oscillating tail provides for flexibility andefficiency in parametrization, which prove appropriate in manyother situations. In the Li-rich case, for instance, the oscillatingtail turned out to be redundant for Li-Li interactions and theywere fitted using a repulsive shell only (two parameters). For each of the three compositions of interest, we fitted the parameters of the potentials to reproduce both the forces and 0 2 04 06 08 0 1 0 00246810density (g/cm3) atomic percent of Bi FIG. 1. Density of liquid Li-Bi alloys as a function of composi- tion. Full squares are predictions of NPT ab initio simulations while open circles are experimental data from Ref. 10. The line is a guide for the eyes interpolating between pure elements and simulation values. 144203-2MULTISCALE STUDY OF THE INFLUENCE OF CHEMICAL ... PHYSICAL REVIEW B 83, 144203 (2011) TABLE II. Features of the fitting procedure for each composition (see text). Energy rms(F) rms(E) Samples Composition Forces diff. [eV /˚A] meV /atom at T[K] Li70-Bi 30 7812 9 0.145 2.6 300; 600; 900; 1500 Li57-Bi 43 7830 17 0.151 3.9 300; 600; 900; 1500 Li30-Bi 70 2400 3 0.165 0.4 1500 the energy differences for the 200 atom configurations from the V ASP simulations. More precisely, the parameters were fitted to reproduce the forces undergone by each particle inseveral selected snapshots. Since fits from forces alone areinsufficiently constrained, we also required energy differencesto be reproduced to improve them. The energy differencesare defined as the variation of the total energy between a lowtemperature system and a high temperature one, both withthe desired composition and density. The low temperaturereferences considered here were always T=300 K. Details about the fitting procedure are summarized in Table IIwhere columns “forces” (force components for all atoms in all samples) and “energy diff.” specify the number ofdata points considered for each quantity. The columns “rms”denote r.m.s. deviations of the respective fit, and the column“samples” lists the temperatures at which snapshots weretaken to determine energy differences. The overall efficiencyis illustrated in Fig. 2where ab initio and fitted forces are plotted for each considered configuration. The parameters are gathered in Table III, and readers are referred to Ref. 4for further details about the fitting procedure. C. Large-scale classical simulations Once the fitted pair potentials are known, CMD simulations can be performed to investigate more accurately the staticstructure as well as the diffusion properties. Among them,interdiffusion, which is a collective dynamic property -2 0 6 F [eV/Å] VASP-2-1012F [eV/ Å] FITTED Bi7Li3Bi43Li57Bi3Li7 4 2 FIG. 2. (Color online) Fitted forces versus ab initio ones for each configuration considered. Data corresponding to Li 57-Bi 43and Li30-Bi 70are shifted by 2 and 4 eV /˚A along the horizontal axis, respectively.definitely beyond the scope of AIMD. So, in the spirit of recent work on liquid Na-K and K-Cs mixtures, we launchedtwo different kinds of NVE classical simulations for eachcomposition. The first one, involving 13 500 atoms over 100 000 time steps, allowed us to get g ij(r) over a wide spatial range, leading to the low- qbehavior of the static structure factors down to q less than about 0.1 ˚A−1. The second one with 2048 particles during 5 000 000 time steps was used to compute the self-diffusion and interdiffusion properties. Due to the very lightmass of lithium atoms, the time step was chosen as low as 0.1fs to ensure the stability of the temperature during the NVEsimulations. Thus, the observed temperature drift along the500 ps simulations was less than 0.5%. As we shall see in Sec. III, the accuracy of these simulations was ascertained by comparing their predictions with availableexperimental and ab initio results. Further details concerning both the classical simulations and the computation of thephysical properties of interest can be found in Refs. 1and2. D. Experimental data The measurement of the static structure factor S(q)o fal i q - uid as a function of the wave vector qis usually performed with elastic x rays or neutrons scattering experiments. However, itcan also be determined from inelastic scattering provided alarge-enough frequency range is available. Indeed, S(q)=/integraldisplay +∞ −∞S(q,ω)dω, (2) where S(q,ω) is the dynamic structure factor. Thus, we took advantage of the existence of such experimental inelastic data,performed to investigate collective excitations in liquid Li-Bialloys to evaluate S(q). The experimental quantities reported in this article are the energy-integrated dynamic structure factor for Li 70-Bi 30, and the total scattering factor as measured directly on thedetector (i.e., without energy discrimination) for Li 30-Bi 70. In the first case, the quantity has been deduced fromthe measured dynamic structure factor. In the second case, thetotal scattering factor has been directly measured during theexperiment campaign as a test for melting, but the data werenot reported in Ref. 9. For greater convenience, we recall a few characteristics of these experiments and interested readers are requested to referto Refs. 9and18for further details. Inelastic neutron scattering measurements were carried out at the three-axis spectrometerIN1 of the Institut Laue-Langevin (ILL, Grenoble, France).The Li 30-Bi 70and Li 70-Bi 30metal alloys were prepared by melting high purity Li (99.99%) and Bi (99.99%) at 673 144203-3W AX, JOHNSON, BOVE, AND MIHALKOVI ˇC PHYSICAL REVIEW B 83, 144203 (2011) and 1073 K, respectively, until complete mixing of both components. The intensity scattered by the sample was thenmeasured at several wave vector transfer values, betweenq=0.5 and 2 ˚A −1for Li 70-Bi 30and up to 2.5 ˚A−1for Li30-Bi 70. The so-derived dynamic structure factors showed two distinct doublets peaked at different frequencies. Thepresence of two well-defined coexisting modes showing anoptic- and acoustic-like behavior, respectively, is in line withthe heterocoordinated nature of the alloy. The measured dynamic structure factors were then inte- grated over an extended energy range to obtain an estimation ofthe total static structure factor to be compared with the presentsimulations. Since these inelastic scattering experiments wereperformed separately for Li 30-Bi 709and Li 70-Bi 30, the quality of the experimental determination of S(q) was not the same. We will come back to this point when considering the results. III. RESULTS A. Electronic structure The first physical property we discuss is the electronic density of states. The ab initio computed results are displayed in Fig. 3. As can be seen, valence electrons split into two bands. This was also observed in pure liquid bismuth -15 -10 -5 0 5050100150 Li70Bi30 E-EF (eV)-15 -10 -5 0 5050100150 Li57Bi43 DOS (states/eV)-15 -10 -5 0 5050100150 Li30Bi70 FIG. 3. Electronic density of states for each alloy under consid- eration as obtained from ab initio simulations.both experimentally16and theoretically.17Considering the computed number of electronic states, the lowest energy bandis filled with the two 6 selectrons of bismuth atoms, while the highest energy one is filled with both the three 6 pelectrons of bismuth and the single 2 selectron of lithium. The energy gap between both bands is quite high (about 5 eV) and prevents any transition of electrons from the 6 s band to the Fermi level. Moreover, at the Fermi energy, apseudogap appears that deepens as the bismuth concentrationdecreases. Consequently, the metallic character of the alloybecomes impaired when going from 30 to 70 at. % of lithium.Since liquid lithium is a good electrical conductor, we canpredict a reversal of this trend between 70 and 100 at. %of Li. This evolution versus composition is confirmed by elec- trical resistivity measurements. 10While liquid Li and Bi are metallic at melting with respective electrical resistivities ofabout 25 and 130 μ/Omega1/ cm and positive temperature coeffi- cients, the observed value at the stoichiometric compositionLi 3Bi grows up to 2000 μ/Omega1/ cm and the temperature co- efficient becomes negative. This unambiguously indicates aloss of the metallic nature of the electronic structure and astrengthening of the ionic character, which is corroborated byspin relaxation 19and magnetic susceptibility20measurements. It also is consistent with the phase diagram of the alloy7ex- hibiting a congruently melting Li 3Bi intermetallic compound. We will come back to this point later when considering theother physical properties of interest. B. Pair potentials This nonsimple electronic structure, far from nearlyfree- electron-like, prevents a self-consistent screening formalismfrom being applied to obtain effective pair potentials fromthe second-order perturbation method, as is possible in simplemetals like liquid lithium, for instance. Indeed, the densityof states (DOS) is too different from a parabolic one, sothat these alloys cannot be considered as simple metals. Theuse of empirical pair potentials is justified in this way. Asmentioned above, we have fitted potentials from ab initio V ASP simulations using expression ( 1). The values of the parameters are displayed in Table IIIand the corresponding curves are shown in Fig. 4, which reveal some interesting features. The repulsion range of uBiBi(r) is longer than that of uLiLi(r), and it exceeds the ratio of the corresponding ionic core radius. As a consequence, whatever the composition, bismuthatoms will repel each other in such a way that a lithium atomwill easily intercalate in between. The interactions are notadditive in the sense that u LiBi(r) is not the mean of the other two, especially concerning the repulsion. In addition, whenincreasing the lithium concentration, it clearly appears thatlithium-bismuth attraction becomes preponderant. It is alsointeresting to point out that the lithium-lithium interaction,as fitted by our method in the case of Li 70-Bi 30, is purely repulsive. Even if the value of η(1) 12does not correspond to a Coulombic interaction, this trend has to be related to theabove-mentioned evolution toward an ionic character of theinteractions in this range of composition. 144203-4MULTISCALE STUDY OF THE INFLUENCE OF CHEMICAL ... PHYSICAL REVIEW B 83, 144203 (2011) TABLE III. Values of the parameters of the pair potentials. Units are such that distances are in ˚A and energies in eV . System i−jC(1) ij η(1) ij C(2) ij η(2) ij kij φij Li-Li 509 .05117 8 .72399 19 .65249 5.90901 3.26752 4.07121 Li30-Bi 70 Li-Bi 54 .45000 5 .88803 −149.35784 7.18473 2.47563 4.04811 Li-Li 452 .88095 6 .02955 −42.98817 5.30268 3.41151 2.01862 Li-Li 968 .07392 10 .44178 33 .66214 6.35754 3.07433 4.97542 Li57-Bi 43 Li-Bi 56 .90363 6 .26837 −85.04891 6.43670 2.52266 4.36167 Bi-Bi 173 .12510 5 .08537 −183.41762 6.16428 3.02983 3.10465 Li-Li 396 .02798 8 .73571 0 .00000 0.00000 0.00000 0.00000 Li70-Bi 30 Li-Bi 2612 .40127 12 .46890 −55.12306 5.44045 2.20358 5.53860 Bi-Bi 2080 .53828 7 .28471 −58.80280 4.67069 2.53289 5.26020 Of course, these fitted pair potentials have to be checked by comparing their predictions to experimental or ab initio data to validate the fitting procedure. C. Static structure The static structure results are used in two ways, first to check the reliability of the fitted pair potentials and second toinvestigate the atomic order in the alloys. 02468 1 0-1500150300450Li70-Bi30 r (Å)02468 1 0-20020406080100 uij(r) (meV)Li57-Bi4302468 1 0-10010203040 Li30-Bi70 FIG. 4. Pair potentials fitted from ab initio MD simulations and used to mimic the behavior of the systems in the classical simulations performed in this study (Li-Li: solid; Li-Bi: dashed; Bi-Bi: dotted).1. Check of the potentials We first consider the V ASP simulations used to fit the pair potentials as reference data. We thus discuss the agreementbetween classical and ab initio simulations. The partial pair distribution functions of each alloy obtained in both ways arepresented in Fig. 5. For each composition, the partial structure functions g ij(r) are qualitatively well reproduced. Peak positions coincide, so 02468 1 001234 r (Å)02468 1 001234 Li70-Bi30Li57-Bi43gij(r)02468 1 001234 Li30-Bi70 FIG. 5. Partial pair distribution functions obtained for the three alloys under consideration. Symbols are used for ab initio results (Li-Li: squares; Li-Bi: stars; Bi-Bi: circles) and lines correspond to classical simulations (Li-Li: solid; Li-Bi: dashed; Bi-Bi: dotted). 144203-5W AX, JOHNSON, BOVE, AND MIHALKOVI ˇC PHYSICAL REVIEW B 83, 144203 (2011) the topological order is recovered. The hierarchy between the peak heights is also obeyed and so is the chemical order. Thereis not a quantitative agreement between the intensities of thepeaks, except in the case of Li 57-Bi 43for which a remarkably good agreement is observed. Nevertheless, according to theextremely peculiar structure of these mixtures, which wewill discuss later, we consider that the way such simple pairpotentials (with no many-body effects) succeed in mimickingthe structure of the alloys is adequate. We now discuss the realism of the interactions by con- sidering the experimental results. Although the number ofatoms was quite high for an AIMD simulation, it was notsufficient to compute the partial structure functions over arange large enough to allow the Fourier transform to beperformed accurately to get the Ashcroft-Langreth partialstructure factors 21defined as Sij(q)=δij+√cicjρ/integraldisplay∞ 0[gij(r)−1]sinqr qr4πr2dr, (3) where ciis the concentration of the ith species and ρ=N/V is the number density. This was also true if one tried to computeit straight from the configurations using S ij(q)=1/radicalbigNiNj/angbracketleftBiggNi/summationdisplay b=1Nj/summationdisplay a=1exp(i/vectorq·/vectorRab)/angbracketrightBigg , (4) because the simulated time was not long enough to reasonably lower the statistical noise. So the only calculated structurefactors that we can consider stem from the CMD simulationsof larger boxes performed using the fitted potentials. We recombined the partial structure factors (which will be discussed later) to evaluate the total structure factors asmeasured in neutron diffusion experiments S(q)=c 1b2 1S11(q)+2√c1c2b1b2S12(q)+c2b2 2S22(q) c1b2 1+c2b2 2,(5) withbicorresponding to the coherent diffusion length describ- ing the diffusion of an incident neutron by an atom of type i (bLi=− 1.90 fm and bBi=8.532 fm). Interestingly, the total S(q) obtained with neutrons are rather distorted, which can be interpreted as a signature of an underlying order. As can beobserved in Fig. 6, the total structure is very sensitive to the composition: the position of the first peak shifts to higher q values as c Biincreases and its height simultaneously decreases. This is qualitatively recovered in the simulation data whichmoreover display minor second peaks or shoulders in thefirst one. While the agreement with the experimental data is excellent for Li 70-Bi 30, this is not the case for the Bi-rich composition. As we explained in Sec. II, the experimental data were not obtained from the same experiment campaign. In the earlycase of the Bi-rich alloy, inelastic data were not collectedwith the intention to deduce the total structure factor andthe data reported here have been obtained from a melting-check scan, performed removing the three-axis analyser, tocollect directly the full scattered intensity on the detector.In such a way, we used the three-axis spectrometer as atwo-axis diffractometer, with no energy analysis. Anyway,neither detector efficiency correction nor multiple scatteringcorrections have been performed on the raw data. Therefore,02468012 q (Å-1)02468012 Li70-Bi30Li57-Bi43Sneutrons(q)02468012 Li30-Bi70 FIG. 6. Total structure factors for neutrons (full line: simulation; symbols: experiment). we believe that these early results are rather of qualitative than quantitative value. Conversely, the Li-rich compositiondata have been collected and analyzed following the rigorousprocedure described in Sec. IIand are thus more reliable. Nevertheless, the overall evolution is recovered and theagreement with the reliable experimental data is good, whichvalidates our whole approach. Finally, the fitted pair potentials are well suited to re- produce, at least qualitatively, the microscopic behavior ofthis alloy. Indeed, their predictions are in agreement withboth experimental and AIMD simulation results. This state-ment allows us to investigate now in more detail the staticstructure. 2. Static structure analysis Let us first consider the V ASP results (Fig. 5). The partial structures are very different from those observed in nearlyrandom alloys like K-Cs (Ref. 1)o rN a - K( R e f . 2). At each composition under study, the alloy exhibits a strongheterocoordination as indicated by the first peak of the cross-function g LiBi(r) which is higher than the corresponding peaks of the two other functions. Among these alloys, Li 57-Bi 43 shows the usual hierarchy between the successive peaks ofeach partial function. But, in both others, the first peak of the 144203-6MULTISCALE STUDY OF THE INFLUENCE OF CHEMICAL ... PHYSICAL REVIEW B 83, 144203 (2011) Li70-Bi30Li57-Bi43Li30-Bi70 FIG. 7. Snapshot of the simulation boxes in AIMD simulations (Li: black circles; Bi: grey circles). partial function between the minority species is lower than the second one. This could be understood in the following way;the atoms of the less numerous species tend to be surroundedby those of the other kind. So the first neighbor is most oftenof the other kind, which explains the shape of the curves. Thisfeature is particularly pronounced in the case of Li 70-Bi 30and the above explanation is confirmed looking at a snapshot ofthe system (Fig. 7). Looking at the partial Ashcroft-Langreth structure factors S ij(q) (Fig. 8), their usual large- qasymptotic limits are recovered. For each composition, the Li-Li function is unusual,presenting either a prepeak or two peaks with the same height.Considering their low- qlimit, it should be noticed that the three partial functions tend to zero (except the Li-Li partialfunction in Li 30-Bi 70), a feature which is not observed in random mixtures like Na-K or K-Cs. Using these partial functions, we computed the Bhatia- Thornton structure factors22which give insight into02468 1 0-1012 q (Å-1)02468 1 0-1012 Li70-Bi30Li57-Bi43Sij(q)02468 1 0-1012 Li30-Bi70 FIG. 8. Ashcroft-Langreth partial structure factors obtained from the classical simulations (Li-Li: solid; Li-Bi: dashed; Bi-Bi: dotted). topological and chemical order (Fig. 9) Snn(q)=c1S11(q)+c2S22(q)+2√c1c2S12(q), (6) Snc(q)=c1c2/bracketleftbigg S11(q)−S22(q)+c2−c1√c1c2S12(q)/bracketrightbigg , (7) Scc(q)=c1c2[c2S11(q)+c1S22(q)−2√c1c2S12(q)].(8) The topological functions Snn(q) are very common, without any prepeak, low- qdivergence, or shoulders of any kind. So, we do not observe any particular global topological feature.The lack of prepeak indicates that, at any composition, thealloys do not contain some permanent substructures likeclusters or polyatomic ions. The coupling between topologicaland chemical order as depicted by S nc(q) is also very common, looking like the corresponding function in liquid Na-K, forinstance. But, considering S cc(q), we have a clear indication of the heterocoodinated character of these mixtures. It unambigu-ously dips under the ideal mixture limit c Li·cBiasqtends to zero. Extrapolating smoothly the curves to q=0, we estimate the limits as 0.08, 0.019, and 0.011 ( ±10%) for Li 30-Bi 70, Li57-Bi 43, and Li 70-Bi 30, respectively. This corroborates the picture we draw from the study of partial gij(r). One more feature of Scc(q) to be mentioned is the sharp first peak that indicates rather marked concentration fluctuations witha wavelength corresponding to twice the interatomic distance. 144203-7W AX, JOHNSON, BOVE, AND MIHALKOVI ˇC PHYSICAL REVIEW B 83, 144203 (2011) 02468 1 0012 q (Å-1)02468 1 0012 Li70-Bi30Li57-Bi43SBT(q)02468 1 0012 Li30-Bi70 FIG. 9. Bathia-Thornton partial structure factors obtained from the classical simulations ( Snn: solid; Scc: dotted; Snc: dashed). To better understand the atomic ordering in these mixtures, let us consider alloys of bismuth with alkali metals. Publishedstructure data exist for K 50-Bi 50(Refs. 23,24), Rb 50-Bi 50 (Ref. 24), Cs 50-Bi 50, and Cs 75-Bi 25(Ref. 25). In these three systems, experimental structure factors display prepeaks,which are attributed to the existence of Bi polyanions. Sucha feature is not observed in our neutron structure factors,neither from simulation nor from experiments. However, thisdoes not invalidate our results. Indeed, let A denote an alkalielement. First, we recall that K-Bi, Rb-Bi, and Cs-Bi all displaytwo congruent intermetallic compounds (A 3-Bi and A-Bi 2)a s against only one (namely A 3-Bi) in the case of Li-Bi and Na-Bi. Second, considering the electronic transport properties of thesefive systems, 26,27an evolution clearly occurs when going from Li- to Cs-Bi alloys. While the resistivity versus compositionshows a single peak at A 75-Bi 25in the case of Li and Na alloys, a second one shows up at A 60-Bi 40in the case of K and Rb; this doublet moves to a single broad peak at the secondcomposition in the case of Cs-Bi. This had been interpreted asfollows: While the liquid phases of Li-Bi and Na-Bi displayonly the so-called A 75-Bi 25“octet compounds” in a given range of concentration, Cs-Bi liquid mixtures contain quiteexclusively bismuth polyanions. As for K-Bi and Rb-Bi, theyexhibit both features. The prepeaks were associated with theexistence of the polyanions, which are absent in the case of Li and Na-Bi alloys and our results corroborate these assertions. However, if stable Li 3-Bi “clusters” were to exist, there should appear a prepeak in the S nn(q) structure factor. Moreover, these clusters should be of trigonal planar kind,characterized by Li-Bi-Li angles equal to 120 ◦and we would expect to see some effect on the coordination numbers. Asalready mentioned, we do not see any prepeak in the staticstructure factor, which means that there are no permanentstructures on a scale larger than the interatomic distance.We also computed the coordination numbers which do notexhibit any significant behavior. Indeed, the total coordinationnumber is above 11 whatever the composition and the kindof the central atom; such values are typical of random closepacked structures. Looking also at the bond-angle distributionfunctions (not shown), we do not see any peak at 120 ◦, especially in the partial Li-Bi-Li function. Thus, we believethat stable Li 3Bi entities do not exist in the liquid, but a strong heterocoordination is confirmed at every stage of ourinvestigations. Finally, our static structure results show that all three compositions considered in this study display a marked hete-rocoordination. This is all the more true when the compositiongets closer to that of the intermetallic compound. Althoughpermanent Li 3Bi clusters are not seen in this case, we cannot rule out the possibility that correlation between motion ofatoms of different kinds exist. Therefore, it is interesting toconsider now the diffusion properties. D. Diffusion properties Since these properties are highly sensitive to temperature, the results displayed now have been obtained at the sametemperature for each composition, namely 1073 K, to be ableto discuss the influence of the composition only. The self-diffusion coefficients have been computed from the velocityautocorrelation function (V ACF) 28,29 ψ(t)=1 Nlim τ→∞1 τ/integraldisplayτ 0N/summationdisplay i=1vi(t0)·vi(t0+t)dt0, (9) which also contains some information about the motion of the atoms. In this expression, vi(t) denotes the velocity of the ith atom at time t. The functions are displayed in Fig. 10for each component at each considered composition. In the same figure,we also display the spectral densities of these V ACFs ˜ψ(ω)=/integraldisplay +∞ −∞ψ(t) exp(−iωt)dt. (10) These curves merit several remarks. First, ψ(t=0)= 3kBT/m , explaining the ratio between the values in Li and Bi curves. The first minimum of the curves, when it exists,indicates the extent of the backscattering undergone by theparticles. As can be seen, Li atoms, which are lighter, are morestrongly backscattered than Bi ones. While a light particlewill mostly be backscattered in a collision with heavier ones, aheavy atom will tend to break through a cage of light neighbors. These arguments should also explain the evolution from the Bi-rich mixture to the Li-rich one. Indeed, in the first one, the 144203-8MULTISCALE STUDY OF THE INFLUENCE OF CHEMICAL ... PHYSICAL REVIEW B 83, 144203 (2011) . . . .. . . ........ FIG. 10. Velocity autocorrelation function (left column) and corresponding spectral densities (right column) for lithium (top row) and bismuth atoms (bottom row) in Li 30-Bi 70(solid lines), Li 57-Bi 43(dashed lines), and Li 70-Bi 30(dotted lines) mixtures. rebound should be amplified due to the presence of a larger proportion of heavy Bi atoms. This is observed with lithium,but the situation of bismuth is rather unexpected. Although onecould have conjectured that Bi would diffuse more easily in Li-rich mixtures since the surrounding medium is less resistant,the opposite is observed. This might be correlated to the strongheterocoordination existing in this mixture. In agreement withthe conclusions drawn on the structure of this melt since thediffusion of Bi atoms is hindered, we could imagine that Biatoms do not diffuse alone, but with surrounding Li ones stuckon them. As we have already discussed, the bond is not stableand permanent; therefore we shall refer to this situation as Biatoms loosely bound with some Li atoms. On one hand, thissituation impedes bismuth atoms diffusion, and on the otherhand, it improves that of Li which they transport, reducingtheir rebound. Damped oscillations are also usually present in the long time tail of the V ACF. When they exist, they are related tothe oscillatory behavior of the atoms which are more or lessconfined in the cage of their surrounding neighbors (the so-called cage effect). When it exists, this oscillatory aspect of theindividual motion of atoms is clearly recovered as a peak inthe spectral density, the position of which corresponds to thecollision frequency of the atoms. Of course, Li atoms, whichare lighter, vibrate at a higher frequency than Bi atoms. For Liatoms, such a peak is clearly present in each case. Bi atomsdisplay different behavior as a function of composition. Theoscillations are clearly indicated as a peak or as a shoulder inthe case of Li 30-Bi 70and Li 70-Bi 30alloys, while they nearly disappear in Li 57-Bi 43. Moreover, the vibrational feature is the more pronounced in the Li-rich mixture. The motion of an atom in a liquid is also diffusive and this feature is recovered considering the low-frequency limitof the spectral density which is related to the self-diffusioncoefficient as D=˜ψ(ω=0)/6. The fact that this limit is nonzero proves that the mixture is liquid and not solid. Theestimated values of the coefficients are gathered in Table IV. Unsurprisingly, Li atoms diffuse more easily than Bi ones dueto their lower mass. We also recover that the diffusion of Liatoms is easier in Li-rich environments than in Bi-rich ones, aswell as the surprising, opposite evolution of the diffusion of Bi.It is important to recall that these results were obtained at thesame temperature to avoid any temperature-related variations. Finally, we have also computed the interdiffusion coeffi- cient involved in Fick’s law that describes the ability of bothspecies to mix. This was obtained from the autocorrelation TABLE IV . Self-diffusion and interdiffusion coefficients in the liquid alloys under consideration in ˚A2/ps at 1073 K. System DLi DBi Scc(0) DLi/Bi Did DDark Li30-Bi 70 0.381 0.323 0.08 0.727 0.364 0.954 Li57-Bi 43 0.460 0.307 0.019 3.703 0.373 4.860 Li70-Bi 30 0.716 0.204 0.011 7.598 0.358 6.834 144203-9W AX, JOHNSON, BOVE, AND MIHALKOVI ˇC PHYSICAL REVIEW B 83, 144203 (2011) function of the microscopic flux2not presented here DLi/Bi=c1c2 Scc(0)/integraldisplay∞ 0VD(t)dt, (11) with VD(t)=1 3Nc 1c2lim τ→∞1 τ/integraldisplayτ 0vd(t0)·vd(t0+t)dt0,(12) where the microscopic diffusion velocity reads vd(t)=c2N1/summationdisplay i=1vi(t)−c1N2/summationdisplay j=1vj(t). (13) The results are displayed in Table IVas well as the values obtained from the ideal mixture model ( Did=c2D1+c1D2) and from Darken’s approximation [ DDark=(c2D1+c1D2)· (c1c2)/Scc(0)]. Whatever way it is computed (exact or Darken) the coefficient appears to be strongly composition dependent.The ideal mixture model is definitively ruled out, mainly dueto the departure of S cc(0) from its ideal mixture value c1·c2. Darken’s approximation neglects the correlation between thevelocities of particles of different species. Its predictions departfrom the correct interdiffusion values by up to 30%. Moreover,a change in the sign of the difference is also noticeable whenconsidering the Li-rich alloy. This means that the displacementof particles of different kinds are correlated and that a specialdiffusion regime characterizes this latter composition. Thisobservation is consistent with the hypothesis we put forwardaccording to which Li and Bi atoms could diffuse in looselybound states. Anyway, the high values of D Li/Biindicate that both species mix easily and this is consistent with the strongheterocoordination observed in the structure. IV . CONCLUSION In this study, we have been interested in the structure and diffusion properties of Li-Bi alloys. These mixtures are morecomplex than nearly random ones and their description usingclassical simulations requires multiscale approaches ratherthan usual models of potential obtained from linear screeningformalism. 1 The methodology built up by Mihalkovi ˇcet al. has been successfully tested in the case of Li-Bi for three differentcompositions. We have reproduced the partial structures asobtained by ab initio methods and in agreement with theexperiment. This double check allowed us to investigate in more detail the behavior of the mixtures, especially thechemical order and the diffusion properties. LiBi is characterized by a strong chemical order correlated to a complex electronic structure. The conduction band issplit into two subbands and a pseudogap exists at the Fermilevel. The static structure reveals strong heterocoordinationtendencies as confirmed by the very low values of S cc(0). In the case of Li 70-Bi 30, this feature is related to a change in the nature of the chemical bond between atoms of different types.Its ionic nature increases, in agreement with the deepening ofthe pseudogap of the electronic structure and the expressionof the corresponding pair potential. However, we did not findany sign of the existence of stable Li 3Bi compound, neither in the structure factors nor in the bond angle distribution. Nevertheless, the study of the diffusion properties pointed out an anomalous evolution of the self-diffusion coefficientof bismuth as the lithium concentration is increased. Bismuthatoms are found to diffuse less easily in this composition range.This could be explained considering that bismuth and lithiumatoms behave as if they were loosely bound, in agreementwith the above-mentioned ionic nature of the interaction. Theinterdiffusion also shows that the mixture is not ideal at all andthat both species have a strong propensity to mix. The present study could not have been undertaken without a multiscale approach and this work opens several newperspectives. First, we have pointed out an anomaly in thediffusion properties, but it would be interesting to consider thedynamic structure factor since recently published experimentalresults have revealed the unique behavior of the collectiveexcitations. Second, we have studied here a heterocoordinatedmixture, but homocoordinated ones should also be considered,as well as alloys involving nonsimple metals, like transitionmetals, especially in view of their industrial applications.Validating the approach in these cases would open a widescope of investigations. ACKNOWLEDGMENTS The PMMS (P ˆole Messin de Mod ´elisation et de Simulation) is greatly acknowledged for providing us with computertime. M.M. was supported by Grant No. VEGA 2/0111/11.F. Sacchetti is acknowledged for suggesting such a theoreticalstudy on LiBi alloys. 1J. F. Wax and N. Jakse, Phys. Rev. B 75, 024204 (2007). 2J. F. Wax, Physica B 403, 4241 (2008). 3F. Ercolessi and J. B. Adams, Europhys. Lett. 26, 583 (1994). 4M. Mihalkovi ˇc, C. L. Henley, M. Widom, and P. Ganesh, e-print arXiv:0802.2926v2 . 5G. Kresse and J. Furthm ¨uller, Comput. Mater. Sci. 6,1 5 (1996). 6G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 7J. Sangster and A. D. Pelton, J. 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Barocchi, P h y s .R e v .B 71, 014207 (2005). 19P. Heitjans, G. Kiese, C. van der Marel, H. Ackermann, B. Bader, P. Freil ¨a n d e r ,a n dH .J .S t ¨ockmann, Hyperfine Interact. 15–16 , 569 (1983). 20K. Hackstein, S. Sotier, and E. L ¨uscher, Journal de Physique (Colloques) 41(C8), 49 (1980). 21N. W. Ashcroft and D. C. Langreth, Phys. Rev. 155, 685 (1967). 22A. B. Bhatia and D. E. Thornton, Phys. Rev. B 2, 3004 (1970). 23K. Hochgesand, R. Kurzh ¨ofer, and R. Winter, Physica B 276–278 , 425 (2000).24K. Hochgesand and R. Winter, J. Chem. Phys. 112, 7551 (2000). 25S. A. van der Aart, V . W. Verhoeven, P. Verkerk, and W. van der Lugt, J. Chem. Phys. 112, 857 (2000). 26J. A. Meijer and W. van der Lugt, J. Phys. Condens. Matter 1, 9779 (1989). 27R. Xu, R. Kinderman, and W. van der Lugt, J. Phys. Condens. Matter 3, 127 (1991). 28M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1990). 29J. M. 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PhysRevB.101.165132.pdf

PHYSICAL REVIEW B 101, 165132 (2020) Nonequilibrium pseudogap Anderson impurity model: A master equation tensor network approach Delia M. Fugger,1Daniel Bauernfeind,1,2Max E. Sorantin,1and Enrico Arrigoni1,* 1Institute of Theoretical and Computational Physics, Graz University of Technology, Petersgasse 16 /II, 8010 Graz, Austria 2Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA (Received 18 February 2020; accepted 24 March 2020; published 23 April 2020) We study equilibrium and nonequilibrium properties of the single-impurity Anderson model with a power-law pseudogap in the density of states. In equilibrium, the model is known to display a quantum phase transition froma generalized Kondo to a local moment phase. In the present work, we focus on the extension of these phasesbeyond equilibrium, i.e., under the influence of a bias voltage. Within the auxiliary master equation approachcombined with a scheme based on matrix product states (MPS) we are able to directly address the current-carrying steady state. Starting with the equilibrium situation, we first corroborate our results by comparing to adirect numerical evaluation of ground-state spectral properties of the system by MPS. Here, a scheme to locatethe phase boundary by extrapolating the power-law exponent of the self energy produces a very good agreementwith previous results obtained by the numerical renormalization group. Our nonequilibrium study as a functionof the applied bias voltage is then carried out for two points on either side of the phase boundary. In the Kondoregime the resonance in the spectral function is split as a function of the increasing bias voltage. The localmoment regime, instead, displays a dip in the spectrum near the position of the chemical potentials. Similarfeatures are observed in the corresponding self energies. The Kondo split peaks approximately obey a power-lawbehavior as a function of frequency whose exponents depend only slightly on voltage. Finally, the differentialconductance in the Kondo regime shows a peculiar maximum at finite voltages, whose height, however, is belowthe accuracy level. DOI: 10.1103/PhysRevB.101.165132 I. INTRODUCTION The single-impurity Anderson model (SIAM) was origi- nally introduced to address the properties of metals with dilutemagnetic impurities, which displayed an unusual resistanceminimum upon decreasing the temperature [ 1,2]. This effect was termed the Kondo effect and it was traced down to theformation of a highly entangled ground state of the model,namely, a singlet state between the localized impurity electronand the conduction electrons of the host metal screening theimpurity spin. This has important consequences, such as theexistence of a regime, in which physical quantities obey a setof universal scaling laws, which are independent of the mi-croscopic details of the actual physical system. In the Kondoregime, i.e., well below the so-called Kondo temperature T K, the SIAM also behaves as a Fermi liquid. Above this energyscale, the impurity spin is no longer screened and the modeldisplays a crossover from the Kondo to a local moment (LM)regime. In the impurity spectrum, this crossover is signaled bya strong suppression and broadening of the Kondo resonance,which, however, never completely vanishes. It is important to mention that there is no true quantum phase transition (QPT) in this model [ 3,4]. In the last decades, the SIAM has drawn renewed in- terest due to its application in dynamical mean-field the-ory (DMFT), which has paved the way to understand the *arrigoni@tugraz.atproperties of a variety of correlated materials [ 5,6]. It has further drawn attention due to its capability to capture thephysics of quantum dots, which can now be faithfully fab-ricated in the laboratory [ 7,8]. These applications have in common that they usually deal with a structured density ofstates (DOS) of the host material, instead of a flat one, as in the original model. In contrast to metals, materials with a band gap cannot (fully) display the Kondo effect since a finite DOSin a small region around the Fermi energy is crucial for itsoccurrence. However, there are also materials, such as pecu-liar semiconductors and superconductors [ 9,10], that display a pseudogap (PSG), i.e., a DOS vanishing exactly at the Fermienergy with a certain power law ∝|ω| r, but remaining finite, elsewhere. For these types of materials, the interaction of band fermions with a magnetic impurity produces more intriguingeffects [ 11]. The corresponding PSG SIAM displays a rich zero-temperature phase diagram. In particular, for 0 <r< 1 2it features a second-order QPT [ 12] from a Kondo screened phase to a LM phase depending on the interplay betweenthe power-law exponent r, the interaction, and hybridization strengths. In this model, the depletion of host states at the Fermi energy prevents the impurity spin from being entirely screened by the conduction electrons. As a consequence, thePSG SIAM does not behave as an ordinary Fermi liquid inthe Kondo phase. Its behavior is captured by a natural, butnontrivial generalization of Fermi liquid theory, and the phaseis referred to as a generalized Kondo (GK) phase. Also in thiscase, a Kondo scale and a set of universal laws for the physical observables in terms of this scale is found, which is distinct from the ordinary SIAM [ 11,13–35]. 2469-9950/2020/101(16)/165132(17) 165132-1 ©2020 American Physical SocietyDELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) In this paper, we are interested in understanding the prop- erties of the PSG SIAM, when a bias voltage φis applied to drive the system out of equilibrium [ 36–43]. This model has been studied in previous works as well, with different degreesof approximation and addressing different physical questions.In [44], the PSG SIAM was studied after a local quench within a time-dependent Gutzwiller variational scheme. The authorfound that the system thermalizes within the GK phase, butwhen quenching across the phase boundary, thermalizationdoes not occur, and a highly nontrivial dynamical behavioris observed. The authors of [ 45,46] both deal with universal scaling in the nonequilibrium steady state of the PSG Kondomodel, employing variants of the renormalization group andlarge- Ntechniques, respectively. In the LM phase, close to the phase boundary, the authors of [ 45] reported universal scaling of the differential conductance, spin susceptibility, andconduction electron Tmatrix as a function of φ/T K.I n[ 46], on the other hand, it was discovered that the differentialconductance, spin susceptibility, and Kondo-singlet strengthreproduce their equilibrium behavior in the scaling regimesof the fixed points of the model, when expressed in terms ofa fixed-point-specific effective temperature T eff. The authors of [47], in contrast, focused on the steady-state impurity spectrum and differential conductance, the main quantitiesthat we are also interested in within this work. Employingsecond-order perturbation theory, the authors found a cuspor dip structure in the impurity spectrum in the GK andLM phase, respectively, when a finite bias voltage is applied.However, in [ 47], when increasing the bias voltage, these structures remain located at zero frequency and no splittingoccurs. According to the authors, this is because the systemis not in the limit of large interaction strength. The resultsof our present work, while confirming the presence of thesefeatures, present a different scenario: the structures do splitas a function of voltage. One should point out that, while ourcalculations are carried out for values of the parameters veryclose to the ones used in [ 47], there is a difference in the way the DOS pseudogap evolves as a function of voltage. Morespecifically, in [ 47] the pseudogap is fixed at zero frequency also at finite bias voltages and only the chemical potentialsare shifted by ±φ/2. In our work, on the other hand, we pin the pseudogap of each lead to the position of the respectivechemical potential. We study the PSG SIAM out of equilibrium by an ap- proach which is nonperturbative, neither in the interactionnor in the hybridization. Specifically, we employ the auxiliarymaster equation approach (AMEA) [ 8,48–50] in which the nonequilibrium bath is accurately represented by an openquantum system whose many-body dynamics is controlled bya Lindblad equation. This last is solved by an efficient matrixproduct states (MPS) formulation. We start by a benchmarkof the approach in equilibrium. Here, in particular, we exploitthe power-law exponent of the self energy to find the boundarybetween the GK and the LM phase. We then carry on with aqualitative analysis of the structure of the spectral functionand the self energy out of equilibrium in both the GK and LMregimes. In addition to these qualitative aspects, we try to fita power-law behavior to these quantities in a region aroundthe chemical potentials and investigate how the correspond-ing power-law exponents evolve upon increasing the biasvoltage. Finally, we address the behavior of the differential conductance in dependence of the bias voltage. Our method isnumerically exact, the main limitation being the fact that thepseudogap exponent in the bath DOS can be reproduced onlywith a limited resolution. Therefore, we are also limited inthe maximum bias voltage, in which our power-law analysismakes sense. This work is organized as follows. In Sec. IIthe model and the solution method are described, starting with the modelin Sec. II A, followed by a small overview about nonequilib- rium Green’s functions in Sec. II B, and a description of the auxiliary master equation approach in Sec. II C. Specifically, we present the Lindblad equation in Sec. II C 1 , discuss the mapping to the auxiliary system in Sec. II C 2 , and introduce the novel MPS scheme in Sec. II C 3 . Section II C 4 presents remarks about physical and auxiliary quantities. Section III contains the results of this work, in particular, the results ofthe fit, Sec. III A , and the ones of the many-body solution in equilibrium, Sec. III B 1 , as well as out of equilibrium, Sec. III B 2 . A discussion of the results obtained is found in Sec. IV. II. MODEL AND METHOD A. Model We study the single-impurity Anderson model (SIAM) in as well as out of equilibrium with electronic leads displayinga power-law pseudogap (PSG) in the density of states (DOS).Throughout this paper we use units of ¯ h=e=k B=1. The model is described by the following Hamiltonian: H=Himp+Hleads+Hcoup. (1) Himpis the Hamiltonian of the impurity. It is a single-site Hubbard Hamiltonian with on-site interaction U, accounting for the Coulomb repulsion between electrons, and on-siteenergy ε f=−U 2, producing particle-hole (PH) symmetry Himp=/summationdisplay σεff† σfσ+Un f↑nf↓. (2) f† σ/fσcreates /annihilates an impurity electron with spin σ∈ {↑,↓}andnfσ=f† σfσis the corresponding particle-number operator. Hleads is the Hamiltonian of the left and right lead, λ∈{L,R}, Hleads=/summationdisplay λkσελkd† λkσdλkσ. (3) It describes a continuum ( N→∞ ) of noninteracting energy levels ελk=εk+˜ελrigidly shifted symmetrically by half the bias voltage φ, so that ˜ ελ=±φ 2.d† λkσ/dλkσare the corre- sponding creation /annihilation operators. Finally, Hcoup=t/prime √ N/summationdisplay λkσ(d† λkσfσ+f† σdλkσ)( 4 ) is the Hamiltonian that describes the coupling of the impurity to the leads via hoppings t/prime. We assume that the leads are initially decoupled ( t/prime=0) and in equilibrium at the same temperature Tand chemical 165132-2NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) potentials μλwith an occupation given by the Fermi function, fλ(ε,T)=1 1+exp/parenleftbigε−μλ T/parenrightbig. (5) Requiring the (asymptotical) particle density of each lead to be independent of φamounts to setting μλ=˜ελ. The leads have a power-law PSG DOS at μλ, which we describe with the retarded hybridization functions Im/Delta1R λ(ω)=−πt/prime2 N/summationdisplay kδ(ω−/epsilon1λk) =−/Gamma1 2e−γ(ω−˜ελ)2|ω−˜ελ|r, (6) whose symmetric forms produce a PH symmetric occupation of the leads. Here /Gamma1is the hybridization strength and γ> 0 is used to fix the bandwidth.1The Keldysh hybridization functions are fixed by the fluctuation-dissipation theorem /Delta1K λ(ω)=2i[1−2fλ(ω,T)]Im/Delta1R λ(ω), (7) and the total hybridization function at the impurity, accounting for both the left and the right lead, /Delta1β(ω) with β∈{R,K},i s given by /Delta1β(ω)=/summationdisplay λ/Delta1β λ(ω). (8) Notice that /Delta1β(ω) encodes the combined effect of Hleadsand Hcoupon the impurity. Thus, the properties of the impurity are controlled by /Delta1β(ω) and by Himp, alone. B. Nonequilibrium Green’s function Out of equilibrium, there are two independent single- particle Green’s functions. We are especially interested in thesteady-state Green’s functions at the impurity. The lesser andthe greater ones are defined as G < σ(t)=i/angbracketleftf† σ(t)fσ/angbracketright∞, G> σ(t)=−i/angbracketleftfσ(t)f† σ/angbracketright∞.(9) Note that they have only one time argument since, in steady state (indicated by the subscript ∞), the system is time- translation invariant. After a Fourier transform to frequencyspace, G α σ(ω)=/integraldisplay Gα σ(t)e x p ( iωt)dt, (10) withα∈{<,>}, these Green’s functions may be combined to obtain the spectral function or local impurity DOS and theKeldysh Green’s function, which we are typically interestedin A σ(ω)=i 2π[G> σ(ω)−G< σ(ω)], (11) GK σ(ω)=G> σ(ω)+G< σ(ω). (12) 1A Heaviside step function would also fix the bandwidth without distorting the power law. We choose the exponential because AMEAperforms better for smooth hybridization functions.From the spectral function the retarded and the advanced Green’s function are obtained via the Kramer’s Kronig rela-tions. In the nonequilibrium Green’s function formalism G R σ(ω), GA σ(ω), and GK σ(ω) are typically arranged in a 2 ×2m a t r i x (Keldysh space), which we indicate by an underline: Gσ(ω)≡/parenleftbigg GR σ(ω)GK σ(ω) 0 GA σ(ω)/parenrightbigg . This has the advantage that Dyson’s equation is valid in the same form as in equilibrium, G−1 σ(ω)=G−1 0σ(ω)−/Sigma1(ω), G−1 0σ(ω)=g−1 0σ(ω)−/Delta1(ω).(13) Here g0σis the Green’s function of the decoupled and nonin- teracting impurity, the self energy /Sigma1(ω) accounts for the inter- action, and the hybridization function /Delta1(ω) for the coupling to the noninteracting leads. From the Green’s functions defined above, the current across the impurity can be obtained as jλ=1 2π/summationdisplay σ/integraldisplay Re/parenleftbig GR σ/Delta1Kλ+GK σ/Delta1Rλ/parenrightbig dω. (14) In steady state, the left and right-moving current must be identical, |jL|=| jR|, so we can also compute j=1 2(jR−jL). The differential conductance follows from the current via G=dj dφ. (15) C. Auxiliary master equation approach The auxiliary master equation approach (AMEA) is based upon a mapping of the model introduced in Sec. II A— which we call the physical system in the following — consisting of an impurity and an infinite bath, to a finite auxiliary open quantum system. This last system consists of the impuritycoupled to a small number of N B=N−1 auxiliary bath sites that are furthermore attached to Markovian environments. Thedynamics of the auxiliary system is governed by a Lindbladmaster equation [ 49] whose parameters are chosen such that its hybridization function /Delta1 auxapproximates the one of the physical system /Delta1phys [Eq. ( 8)] as accurately as possible. Upon solving the corresponding many-body Lindblad equa-tion, an approximation for the behavior of the interactingimpurity in the physical system is found. We stress that thismapping becomes exponentially exact, upon increasing thenumber of bath sites N B→∞ in the sense that the Lindblad bath provides an exponentially accurate representation of theoriginal Hamiltonian problem [ 50,51]. 1. Lindblad equation As outlined in [ 49,52], the Lindblad equation for a fermionic lattice model can be expressed in terms of anordinary Schrödinger equation in an augmented state spaceof twice as many sites 2 N, d dt|ρ(t)/angbracketright=L|ρ(t)/angbracketright. (16) 165132-3DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) In this augmented space, the density operator is represented by a quantum state |ρ(t)/angbracketrightand the Lindbladian iLplays the role of a non-Hermitian Hamiltonian. For our case,2it reads iL=/summationdisplay σc† σ/parenleftbigg E+i/Omega1 2/Gamma1(2) −2/Gamma1(1)E−i/Omega1/parenrightbigg cσ−2T r(E+i/Lambda1) +U/parenleftBigg nf↑nf↓−˜nf↑˜nf↓+/summationdisplay σ˜nfσ−1/parenrightBigg . (17) Here E,/Gamma1(1), and/Gamma1(2)areN×Nmatrices holding the param- eters of the Lindblad equation yet to be determined by a fit of/Delta1 auxto/Delta1physand /Omega1=/Gamma1(2)−/Gamma1(1), /Lambda1=/Gamma1(2)+/Gamma1(1).(18) The vector c† σ=(c† 1σ,..., c† Nσ,˜c† 1σ,..., ˜c† Nσ) (19) contains the creation operators c† iσand ˜c† iσin the auxiliary sys- tem, which is composed of original3“nontilde” and additional “tilde” sites. They obey the usual fermionic anticommutationrules. fis the position of the impurity site, which is typically in the center, f=(N+1)/2,n fσ≡c† fσcfσ, and ˜ nfσ, analo- gously. In this framework, steady-state expectation values as well as Green’s functions are obtained as4 /angbracketleftA(t)B/angbracketright=/angbracketleft I|AeLtB|ρ∞/angbracketright, (20) for local impurity operators A,Band times t/greaterorequalslant0. Here |ρ∞/angbracketright=limt→∞|ρ(t)/angbracketrightdefines the steady state and |I/angbracketrightis the so-called left vacuum5 |I/angbracketright=/summationdisplay {n}|n,˜n/angbracketright, |n,˜n/angbracketright≡(−i)/summationtext iσniσ(c† 1σ˜c1σ)n1σ···(c† Nσ˜cNσ)nNσ|0/angbracketright|˜F/angbracketright. (21) niσand|0/angbracketrightare the occupation numbers and the vacuum in the nontilde system and |˜F/angbracketrightis the completely filled Fock state in the tilde system. Equations ( 16)t o( 21) describe the so-called super-fermion (SF) representation. 2. Mapping procedure The mapping to the auxiliary system is outlined in [ 50,53] and we sketch it only briefly here. Starting from properinitial values, the parameters E ij,/Gamma1(1) ij,/Gamma1(2) ijare adjusted by minimizing a suitable [ 49,50] cost function. This cost function punishes deviations between the auxiliary and the physicalhybridization function and, in general, both the retarded andKeldysh components contribute. Its evaluation involves only 2See, e.g., Eqs. (9) to (11) in [ 49]. 3“Original” refers to Eqs. (9) to (11) in [ 49]. 4Here we used the fact that /angbracketleftI|L=0. 5This representation of |I/angbracketrightfollows from Eq. (13) in [ 52]v i a particle-hole transformation.the solution of a noninteracting problem, which is computa- tionally cheap. In this paper, the optimization of the Lindbladparameters is carried out with the ADAM [ 54] algorithm as implemented in the PYTHON library TENSORFLOW [55]. In principle, the best fit is obtained by allowing the Lindblad parameters to connect all pairs of lattice sites [ 50]. However, employing matrix product states (MPS) as a solverfor the many-body problem, as described in Sec. II C 3 ,i t is convenient to adopt a one-dimensional geometry, whichminimizes the entanglement. Specifically, here we adopt achain geometry with the impurity in the center. In this case,the optimal solution numerically turns out to be such that allsites to the left (right) of the impurity have /Gamma1 (2)=0(/Gamma1(1)=0) and, therefore, are almost completely empty (full) [ 56]. This situation is particularly convenient for the MPS many-bodysolution since it prevents the propagation of entanglement, asdiscussed in [ 56]. In addition, knowing this fact, it is then sufficient to fit the retarded component of the hybridizationfunction, only, as explained in Appendix A1. We start from the zero-bias, φ=0, i.e., equilibrium situ- ation and perform the fit as discussed above. The importantphysics obviously occurs in the region around ω=0 and is controlled by the power-law exponent r. Thus, it is particu- larly important to have an accurate fit there. To achieve this,we introduce a weight in the cost function, which is twiceas large on |ω|/lessorequalslant1 than on |ω|>1. For nonzero φ, we can construct the nonequilibrium fit from the equilibrium one, asoutlined in Appendix A2. This has the advantage that the accuracy of the fit to reproduce the power-law is independentof the bias voltage, which is crucial, to faithfully investigatethe crossover to finite voltage. 3. Matrix product states implementation We solve the many-body Lindblad equation employ- ing MPS in combination with the time-dependent densitymatrix renormalization group (tDMRG) algorithm [ 57,58]. MPS are especially suited for one-dimensional problems,where they can provide an efficient representation with asmall bond dimension. In particular, ground states of one-dimensional gapped closed systems are conveniently ex-pressed as MPS [ 59]. On the other hand, also steady states and Green’s functions of open quantum systems in a chain geom-etry are reproduced accurately using MPS and the entangle-ment remains limited [ 56]. We decided to employ tDMRG for the time evolution here since it is conveniently implementedwith the C++ tensor network library ITENSOR [60]. Within AMEA, a chain geometry naturally results from combining a nontilde and a tilde site associated with an indexi, according to Eq. ( 19), to a single effective site with a local Hilbert space dimension of d=16 [56], see Fig. 1. Since the SIAM couples opposite spins only at the impurity, it isconvenient to separate spin-up and spin-down degrees of free-dom [ 61], which reduces the local Hilbert space dimension back to d=4. Figure 1shows the effective sites we use in this work (lower panel) and sketches the steps to obtain them.Note that in this arrangement, the Hubbard interaction is onthe bond between the spin-down and spin-up impurity sites.Furthermore, it is necessary to introduce two long-range terms 165132-4NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) ↓↑ ↓↑ ↓↑ ↓↑ ↓↑ ˜↓˜↑ ˜↓˜↑ ˜↓˜↑ ˜↓˜↑ ˜↓˜↑12345 ↓˜↓↑˜↑ ↓˜↓↑˜↑ ↓˜↓↑˜↑ ↓˜↓↑˜↑ ↓˜↓↑˜↑12345 ↓˜↓ ↓˜↓ ↓˜↓ ↓˜↓ ↓˜↓ ↑˜↑ ↑˜↑ ↑˜↑ ↑˜↑ ↑˜↑1254334521nontilde tilde combined spin separatedd=4 d=1 6 d=4 FIG. 1. Construction of effective sites for the MPS time evolu- tion. The impurity sites are displayed as red circles, the full andempty bath sites as blue and white ones. As discussed in the text, by “full” and “empty” we mean sites for which /Gamma1 (1)=0o r/Gamma1(2)=0, respectively, for details, see Appendix A1. Each site is labeled with an index and its spin and tilde degrees of freedom. The upper panel of this figure shows the sites and their couplings occurring in the Lindblad equation in the augmented state space. Here the upper(lower) part of this ladder structure is formed by nontilde (tilde) sites. Lines connecting these two sets of sites represent /Gamma1terms, while lines within the same set are hoppings. The central panel shows theeffective sites used in [ 56] that result from combining nontilde and tilde sites with the same index. Finally, the lower panel shows the effective sites used in this work that result from the combined sitesby separating the spin degrees of freedom. The advantage of this representation is that the local Hilbert space has a dimension of 4, instead of 16 as in our previous work. On the other hand, it introducestwo long-range hopping terms. between the empty bath sites and the impurity, violating the linear geometry. We encode the left vacuum |I/angbracketrightas well as a proper initial state|ρ(t=0)/angbracketrightas MPS on these effective sites. We choose |ρ(0)/angbracketright∝| I/angbracketrightsince this has proved convenient in our previous work [ 8,56]. Taking |n1↓˜n1↓...nf−1↓˜nf−1↓nN↓˜nN↓...nf↓˜nf↓/angbracketright ⊗|nf↑˜nf↑...nN↑˜nN↑nf−1↑˜nf−1↑...n1↑˜n1↑/angbracketright(22) as basis states, we can express the corresponding expansion coefficients ψ({niσ,˜niσ}) of any required state as products of local matrices ψ({niσ,˜niσ})=An1↓˜n1↓...Anf−1↓˜nf−1↓AnN↓˜nN↓...Anf↓˜nf↓ ×Anf↑˜nf↑...AnN↑˜nN↑Anf−1↑˜nf−1↑...An1↑˜n1↑. (23) In the case of |I/angbracketright, only matrices with niσ=1−˜niσare nonzero. Specifically, comparing to Eq. ( 21), the correspond- ↓˜↓↓ ˜↓↓ ˜↓↓ ˜↓↓ ˜↓↑ ˜↑↑ ˜↑↑ ˜↑↑ ˜↑↑ ˜↑12543345211 time stepΔt 2 Δt 2 Δt Δt 2 Δt 2odd even swap even oddspin down spin up FIG. 2. Single step in the MPS time evolution of the (PSG) SIAM with separated spin degrees of freedom. The impurity sites are represented as red circles, the full and empty bath sites as blueand white ones. The same coloring also classifies the time evolution gates that are represented as boxes. A time evolution step /Delta1tconsists of five layers, labeled “odd,” “even,” and “swap.” In each layer, a siteiσ, with index iand spin σ, is touched only by one gate. In the swap layer, swap gates displayed as crossing time lines are employed to cope with the long-range couplings between the empty bath sites andthe impurity sites. ing expansion coefficients read ψ({niσ,˜niσ})=/productdisplay iσδniσ,1−˜niσ(−i)niσ, (24) resulting in the 1 ×1, i.e., scalar matrices A01=1 and A10= −i. Having expressed the relevant states as MPS, we can proceed with the time evolution of the auxiliary system. In tDMRG the time evolution of the system |ρ(t)/angbracketright= exp ( Lt)|ρ(0)/angbracketrightis decomposed into a Trotter sequence of small time evolutions on bonds induced by gates. After the applica- tion of a gate, the original structure of the MPS, Eq. ( 23), is restored with a singular value decomposition. As is usual atthis step, the smallest singular values are neglected, defininga truncated weight, which is the sum of all discarded squaredsingular vales. Then the next gate may be applied in the sameway [ 59]. Figure 2shows the sequence of gates we use in this work to evolve one time step /Delta1t. There are five layers, labeled “odd,” “even,” and “swap,” and the gates within them are displayedas boxes. To understand them, we identify the following termsas building blocks of the Lindbladian, Eq. ( 17): iL iσjσ=(E+i/Omega1)ijc† iσcjσ−2/Gamma1(1) ij˜c† iσcjσ +2/Gamma1(2) ijc† iσ˜cjσ+(E−i/Omega1)ij˜c† iσ˜cjσ, iLf↑f↓=U(nf↑nf↓−˜nf↑˜nf↓+˜nf↑+˜nf↓). (25) Within the odd layers, all on-site terms in Eq. ( 25)a sw e l l as the two-site terms on every second bond, according toFig. 2, including the impurity bond, are grouped, exponenti- ated, and applied as gates, see Eq. ( 26). In the even layers, the two-site gates on the remaining bonds are applied, excludingthe long-range bonds between the impurity and the empty 165132-5DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) baths, which are taken care of in the swap layer [ 59,62]. In the swap layer, the innermost sites of the empty bathsare swapped with their nearest neighbors, i.e., they changepositions until they are next to the impurity sites. Then thetime evolution gates are applied, before they are swappedback to their original positions. Swap gates are displayed ascrossing time lines. Summarizing: odd :⎧ ⎪⎪⎨ ⎪⎪⎩exp/bracketleftbig (L iσjσ+Ljσiσ+Liσiσ+Ljσjσ)/Delta1t 2/bracketrightbig , (i,j)={(1,2),(4,5),...} exp/bracketleftbig (Lf↑f↓+Lf↑f↑+Lf↓f↓)/Delta1t 2/bracketrightbig , even : exp/bracketleftbig (Liσjσ+Ljσiσ)/Delta1t 2/bracketrightbig , (i,j)={(3,4),...}, swap : exp/bracketleftbig (Lf−1σfσ+Lfσf−1σ)/Delta1t/bracketrightbig .(26) To complete the time step, also the constant in Eq. ( 17) has to be taken into account, so we multiply the MPS withexp{i/Delta1t[2Tr( E+i/Lambda1)+U]}. Notice that the described sequence of gates may be em- ployed, provided that N Bis even, as reasonable at PH symme- try, otherwise the sequence needs to be adjusted accordingly.Since this sequence is derived from a second-order Suzuki-Trotter decomposition, an error O(/Delta1t 3) is acquired in the time evolution, which is further proportional to the commutators ofthe Lindbladians, Eq. ( 26), in different layers. Additionally, there is an error from the truncation of the singular values afterthe application of each gate. In this work, we employ the tDMRG scheme as follows: We first determine the steady state |ρ ∞/angbracketright∝exp(Lt∗)|I/angbracketright6via the time evolution of the initial state with tDMRG up to a time t∗, for which expectation values of static observables, such as single and double occupancies, are converged. Afterwards, wecompute, e.g., the lesser impurity Green’s function, G < σ(t)= i/angbracketleftI|c† fσexp(Lt)cfσ|ρ∞/angbracketright, by applying cfσto the steady state, employing tDMRG again, applying cfσto|I/angbracketright, and calculating the overlap. G< σ(ω) is obtained in the frequency domain via Fourier transformation of G< σ(t) after linear prediction [ 63]. 4. Physical versus auxiliary quantities The observables obtained directly by the MPS treatment of the auxiliary system are called “auxiliary” quantities inthe following. The auxiliary Green’s functions are used asan approximation for the Green’s functions of the physicalmodel. As discussed, this approximation becomes exponen-tially exact upon increasing the number of bath sites. We canget an even better approximation by extracting the self energyfrom Dyson’s equation for the auxiliary system, assuming/Sigma1 phys(ω)≈/Sigma1aux(ω) and reentering Dyson’s equation with the (approximated) physical self energy and the (exact) physicalhybridization function. The Green’s functions extracted in thisway are refereed to as “physical” in the following. 6/angbracketleftI|ρ(t)/angbracketright=1 must be fulfilled for all tsince this corresponds to Trρ(t)=1.III. RESULTS Here we present results obtained with AMEA for the parameters r=0.25,U=6,T=0.05, and /Gamma1=1i nt h eG K phase and /Gamma1=0.25 in the LM phase. In equilibrium, we compare the results to the ones obtained with a direct MPStime evolution of the Hamiltonian Eq. ( 1), at T=0[61]. For clarity, we refer to this procedure as “Hamiltonian MPS”(HMPS) to distinguish it from AMEA, which is also treatedvia MPS. Of course, HMPS cannot be used to achieve thesteady state since the system is finite. Since HMPS is faster,we also provide equilibrium results for different values of r andUobtained with that approach. A. Fit We start by fitting the equilibrium hybridization function with the auxiliary Lindblad system, as described in Sec. II C 2 . As discussed above, we use a weight function, such that thehybridization function is reproduced better at low frequen-cies. We also concentrate on reproducing the power law asaccurately as possible, while putting less emphasis on themultiplicative factors as well as on the large- ωbehavior. The results of the fit are displayed in Fig. 3. From Fig. 3(a) we can see that the auxiliary (AMEA) retarded hybridization function accurately matches the phys-ical one for |ω|/greaterorsimilar0.2, which, on the other hand, behaves approximately as Im/Delta1 R(ω)∝|ω|r(27) for|ω|/lessorsimilar1.2. It follows that Im /Delta1R aux(ω) displays a power law on the interval /Omega1≡(0.2<|ω|<1.2), but the exponent is slightly underestimated. In fact, a fit by Eq. ( 27) on the interval /Omega1yields r/prime=0.23, whereas its value should be equal to r= 0.25. Note that the behavior of −Im/Delta1R aux(ω) is qualitatively acceptable7even down to |ω|≈0.02, which is one order of magnitude smaller than the lower edge of the power-lawinterval /Omega1. Below this value, though, it bends towards a constant, −Im/Delta1 R aux(ω=0)≈0.39/Gamma1, instead of going to zero. Figure 3(a) also shows the hybridization function used in HMPS, for comparison. Here it is plotted using a Lorentzianbroadening of η=0.1. 8It features a good representation of the power law, roughly on the same interval /Omega1as AMEA, but−Im/Delta1R HMPS (0) is larger for this value of η. Note that for HMPS many more bath sites are necessary to get such ahigh resolution. Specifically, on |ω|<10 we use N B=1301 for HMPS in comparison to NB=10 or 20 for AMEA9to achieve roughly the same accuracy. In Fig. 3(b) the auxiliary distribution function faux, obtained from /Delta1R aux(ω) and/Delta1K aux(ω) 7In the sense that it is decreasing 8In HMPS, in the Fourier transform of the Green’s function Eq. ( 10), a modified kernel exp ( iωt−η|t|) with a finite broadening ηis used instead of the mathematically exact limit η→0. Note that inω-space this is equivalent to a convolution of the exact (finite-size) result with a Lorentzian distribution of width η. Here ηis chosen such that the hybridization function, the spectral function, and theself energy are smooth. 9NB=20 in nonequilibrium, see Appendix A. 165132-6NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) (a) (b) FIG. 3. Equilibrium ( φ=0) fit results. (a) Retarded hybridiza- tion function −Im/Delta1R(ω) in units of the hybridization strength /Gamma1 and (b) distribution function fdetermined from Im /Delta1Rand Im /Delta1K via Eq. ( 7). The power-law exponent r/primeis obtained by fitting the AMEA hybridization function with Eq. ( 27) on the interval /Omega1.T h e same procedure applied to the HMPS result yields quite the same exponent (up to a deviation of ≈0.01).|ω|ris plotted for comparison, see Eq. ( 27). These curves are hardly distinguishable (black vs. red dots). via Eq. ( 7), is plotted. It compares well to the Fermi function, i.e., the distribution function in the physical system. Since /Omega1identifies the interval, where we can faithfully represent the power law in AMEA and in HMPS with anexponent r /prime≈r, it is also the interval where we should study other quantities, such as the spectral function A(ω)o rt h es e l f energy /Sigma1R(ω). With a bias voltage applied, the interval /Omega1 shrinks to /Omega1(φ)=/parenleftbigg 0.2+φ 2<|ω|<1.2−φ 2/parenrightbigg ,forφ/greaterorequalslant0,(28) since the hybridization functions /Delta1R Land/Delta1R Rare shifted by φwith respect to each other. This also limits the values of the bias voltage, in which we can reasonably estimate a power-lawbehavior to φ/lessorsimilar0.6. This estimate is obtained by assuming that we need a frequency interval of width /epsilon1=0.4, in which to fit power-law exponents.B. Many-body solution After carrying out the fit, we solve the resulting Lindblad equation (or Schrödinger equation in the case of HMPS) anddetermine the steady state (or just equilibrium for HMPS)Green’s functions, as described in Sec. II C 3 (or [ 61]). We are especially interested in the spectral function as well as theself energy, as there are predictions about their behavior inequilibrium [ 23] and in the differential conductance. Unless stated otherwise, our plots display the physical spectral func-tions and not the auxiliary ones, according to the definition inSec. II C 4 . Due to the Trotter and truncation errors, the MPS results break PH symmetry. Therefore, the curves we showare PH symmetrized and the shadings indicate an estimate ofthese errors obtained from the deviations from PH symmetry,see Appendix Bfor a more detailed discussion. 1. Equilibrium case The equilibrium case has been extensively studied in the literature [ 11,17–35]. It is well established that in a certain range of r,U, and /Gamma1the system displays a Kondo-like behavior, the so-called generalized Kondo effect. In the GKphase, the spectral function and the retarded self energy aresupposed to show a power-law behavior at small frequencies|ω|[23] A(ω)∝|ω| −s,s=r, (29) Im/Sigma1R(ω)∝|ω|κ,κ > r. (30) First, we would like to address the following question: To which extent are these properties reproduced by our calcu-lations, and, in particular, how are these affected by the factthat AMEA cannot reproduce the pseudogap exactly down toasymptotically low energies? Therefore, we study one set ofparameters in the GK phase, according to the phase diagramin Fig. 5of [24], which is reproduced in Fig. 6.o ft h e present paper. Specifically, we solve the many-body problemforr=0.25,U=6, and /Gamma1=1 (and a small temperature T=0.05) and compute the spectral function and retarded self energy. Then we fit Eqs. ( 29) and ( 30) to these quantities on the interval /Omega1and extract the corresponding power-law exponents. In the following, we denote their numerical valuesass /primeandκ/prime, respectively. The results are plotted in Fig. 3. together with the ones obtained by an HMPS treatment of themodel for T=0 andη=0.1. From the results plotted in Fig. 4. we conclude that ex- ponents extracted from the two methods, AMEA and HMPS,agree quite well. We can also see that κ /prime>ris fulfilled, but s/prime is significantly larger than the value rgiven in the literature. This is because the interval /Omega1used to determine the exponent lies at too large frequencies |ω|.10On the other hand, it is not reasonable to go to smaller |ω|values since the power-law is 10This is checked easily by calculating the U=0 spectral function for the exact physical hybridization function. The outcome showsthat we need a good representation of the power-law exponent in Im/Delta1 Rdown to |ω|values that are at least one to two orders of magnitude smaller than the lower edge of /Omega1and this is not feasible within AMEA or HMPS at the moment. 165132-7DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) (a) (b) FIG. 4. Equilibrium ( φ=0) (a) spectral function A(ω)a n d (b) retarded self energy −Im/Sigma1R(ω) in the GK phase. The power-law exponents s/primeandκ/primeare obtained by fitting the AMEA results with Eqs. ( 29)a n d( 30) on the interval /Omega1. The same procedure applied to the HMPS results yields quite the same exponents (up to a deviationof≈0.01). A power-law ∝|ω| −sis plotted for comparison, see Eq. ( 29). The error shadings, hardly to be seen in this figure, are estimates of the PH symmetry errors, see Appendix B. not well represented there in the hybridization function, see Fig. 3(a). Possibly, a more appropriate way to proceed here would be to use a logarithmic energy discretization as in NRG.However, without the possibility to integrate out high-energydegrees of freedom, this is no use here, and indeed the AMEAfit becomes quite unstable. It is also well established in the literature that upon in- creasing U, the system undergoes a QPT from the GK to an LM phase, where the Kondo-like behavior is absent. Ournext goal is to reproduce the phase boundary from Fig. 5 in [24], i.e., to numerically calculate the critical value U c, which depends on rand/Gamma1, see Fig. 6. We would like to exploit Eqs. ( 29) and ( 30) for that purpose. Since we find that it is difficult to extract the correct exponent s/primefrom the impurity spectral function, we choose to use the one of the self energyκ /primeinstead. In [ 23] it was shown that, in the GK phase, this exponent must be larger than r. In the equilibrium case, it is(a) (b) FIG. 5. Determination of the phase boundary by linear extrap- olation of the power-law exponent κ/primeof the HMPS self energy in the GK phase. First, (a) κ/primeis extrapolated to vanishing values of the broadening ηto extract κ/prime 0=κ/prime(η→0) for various values of the interaction strength U. Second, (b) the critical interaction strength is determined from a second extrapolation, Uc=U(κ/prime 0→r). convenient to use the HMPS solver rather than AMEA for the numerical calculations. In Figs. 3and4we already checked that both methods provide essentially the same values for theexponents up to a small deviation ( ≈0.01). The HMPS solver is suitable for the equilibrium case, and, since it is based ona Hamiltonian time evolution, it is easier to employ and a bitfaster, even for this large number of 1301 bath sites. Specifically, we compute the Green’s functions for dif- ferent values of the interaction strength Uand extract the corresponding self energies from Dyson’s equation ( 13)f o r various Lorentzian broadenings η. Then we fit Im /Sigma1 R(ω)o n /Omega1and determine κ/primeas a function of η. The results of this procedure are illustrated for r=0.25 in Fig. 5(a). We can see thatκ/primedisplays a significant dependence on η(in contrast to r/primeands/prime)11and that it is almost a linear function of ηfor all considered values of U. To extract the result without artificial 11This is due to the fact that the self energy is extracted from an inversion of the Green’s function. 165132-8NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) broadening, we perform a linear extrapolation κ/prime 0=κ/prime(η→ 0). In Fig. 5(b) the obtained values for κ/prime 0are plotted and we find again an almost linear dependence on the interactionstrength. According to the condition in Eq. ( 30), the system should leave the GK phase at the value of Ufor which κ /prime 0=r. Thus we perform a second linear extrapolation to extract thecritical interaction strength as U c=U(κ/prime 0→r). The phase boundary estimated in this way agrees well with the ones obtained by the numerical renormalization group, seeFig. 6. In particular, the deviations within the results obtained from different NRG calculations are of the same size as thedeviation of the HMPS results from the NRG results for theconsidered values of r. 12It is notable, though, that the HMPS scheme tends to overestimate the critical interaction strength,yielding slightly smaller values U r−1 cin Fig. 6. This could be improved by taking into account that κ/prime(η) is not strictly a linear function. By accounting for its curvature, one obtainsslightly larger values κ /prime 0[see Fig. 5(a)]. This, in turn, results in smaller critical interaction strengths [see Fig. 5(b)] and thus in larger values of Ur−1 c, closer to the corresponding NRG results. From the literature it is known that the GKphase can occur only for 0 <r<0.5, see, e.g., [ 28]. Close to the phase boundary at r→0.5, the HMPS calculations are more involved, the quantities κ /prime(η) and U(κ/prime 0) are much more difficult to obtain and the extrapolation scheme describedabove breaks down. Therefore, in Fig. 6the HMPS results are plotted only up to r=0.45. It is remarkable that our results reproduce the NRG phase boundary to this level of accuracy, even though the low-energypart of the bath hybridization function used in our calculationis not reproduced perfectly and the Kondo effect is, of course,especially dependent on the hybridization function at ω≈0. The encouraging performance of the HMPS scheme and thegood agreement between the results obtained from HMPS andfrom AMEA prompts us to use AMEA to study the system inits nonequilibrium steady state, for which HMPS cannot beused. 2. Nonequilibrium steady state We now present nonequilibrium steady-state results ob- tained by applying a finite bias voltage. Since the calculationsare more demanding than the conventional HMPS ones, wefocus on two points in the (equilibrium) phase diagram Fig. 6, one in the GK and another in the LM phase, instead of doing acomplete sweep of parameters. Specifically, we take r=0.25, T=0.05,U=6 and/Gamma1=0.25 and 1, respectively. We start by studying the behavior of the Kondo peak as a function of voltage. Therefore, we plot in Figs. 7(a) and 7(b) the spectral function and the imaginary part of the self energy. In the Kondo regime, we observe that upon increasingthe bias voltage from φ=0 the equilibrium Kondo peak is suppressed and broadened and, at some value of the volt-age, it splits in two peaks. The split peaks then move apart 12Even though the phase diagram of [ 24] was obtained under the assumption U/lessmuchD,w h e r e Dis the bandwidth, while in this work we have U/lessorsimilarD.F o r U/lessmuchD,Dis irrelevant as the energy scale and the phase boundary is solely determined by /Gamma1,U,a n d r,s e e[ 24].FIG. 6. Phase diagram adapted from [ 24] (with kind permission) displaying different NRG results.13On top of this we present our HMPS results for the phase boundary obtained via the extrapolationscheme discussed in the text. We also indicate the two points we consider in AMEA, i.e. r=0.25,U=6a n d /Gamma1=0.25 and /Gamma1=1. IfUis much smaller than the bandwidth, the phase boundary for a given ris expected to depend on /Gamma1U r−1only [ 24]. together with the chemical potentials and they are further suppressed and broadened. Qualitatively, this is very similar tothe situation observed for the nonequilibrium SIAM withouta pseudogap [ 49,56,64–72]. In our data, the splitting becomes visible for φ/greaterorequalslant0.8 in the spectral function and, even before, forφ/greaterorequalslant0.6 in the self energy. A measure for the accuracy of the mapping between Eq. ( 1) and the auxiliary open system, which is at the basis of theAMEA approach, can be read off from the deviations betweenthe physical and the auxiliary spectral functions, defined inSec. II C 4 . In the limit in which the mapping to the auxiliary system becomes exact, i.e., for large N B, these quantities become identical. Our data show that AauxandAphysdiffer only slightly for φ/greaterorequalslant0.3. Decreasing the voltage below φ=0.3 increases this deviation, especially for ωbetween the chemical potentials, and it is largest at φ=0, where the exact physical spectral function is expected to diverge at ω=0. Here we expect the accuracy of the AMEA mapping to be less reliable. It is notable that as soon as the Kondo split peaks appear, they are very broad and poorly defined, even more in Aphys, but also in Aaux. They are first located at |ω|values slightly below |μλ|=φ/2, which they reach monotonically upon increasing the bias voltage. The physical spectral functiondisplays additional features, namely two cusps at ±φ/2, not to be confused with the Kondo split peaks. We believe these to beartifacts originating from the difference between the auxiliaryand the physical system and we expect them to disappear uponimproving the accuracy. Figures 8(a) and8(b) are obtained for the same parameters as Figs. 7(a) and7(b), but a reduced hybridization strength of /Gamma1=0.25, instead of /Gamma1=1. According to the phase diagram in Fig. 6, the equilibrium system is in the LM phase, here. 13Results obtained by the local moment approach were removed here since they are not relevant to the present discussion. 165132-9DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) (a) (b) (c) FIG. 7. Nonequilibrium ( φ> 0) quantities in the Kondo regime, (a) spectral function, (b) retarded self energy, and (c) differential conductance. The solid lines are the physical quantities and the dotted lines the auxiliary ones, see Sec. II C 4 . Notice that the two curves are often indistinguishable. The error shadings and error bars are estimates based on symmetry considerations, see Appendix B. This is confirmed by our results which, indeed, do not show signatures of the Kondo effect anymore, neither in equilibriumnor at finite bias voltage. 14Specifically, at nonzero φ,w e observe dips in the spectral function located almost exactlyat the values of the chemical potentials that appear to emergeas images of the dips in the leads’ density of states. Also inthis case, the physical and auxiliary spectral functions agreevery well with each other, thus making us confident about the 14Notice that it is not clear whether a true phase transition or rather a crossover occurs between the two phases at finite voltage.accuracy of our results. Artificial cusps at |μλ|are also present inAphys, but they are much smaller than the cusps in the Kondo regime.15These essentially lie within the error shadings of Aauxand are notable only upon zooming in. Figures 7(a) and8(c) display the differential conductance G, obtained from Eqs. ( 14) and ( 15) as a function of the bias voltage at parameters corresponding to the Kondo and the LMregime. A notable difference with respect to the conventional 15Here a smaller truncated weight was chosen in the SVDs in the MPS time evolution, which could explain this improved accuracy. 165132-10NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) (a) (b) (c) FIG. 8. Nonequilibrium ( φ> 0) quantities in the LM regime. Conventions are as in Fig. 7. SIAM is that in the Kondo regime, the maximum of G(φ) appears to be shifted to a finite voltage of φ≈0.2. On the other hand, for φ/greaterorsimilar0.2,G(φ) decreases logarithmically, as usual. The unusual structure of the differential conductance inthe Kondo regime is probably due to the fact that the positionof the pseudogap is shifted along with the bias voltage. Onthe other hand, one should be aware of the fact that, dueto the relatively large error bars, 16it is not clear, whether the maximum at finite voltage is a genuine feature: strictly 16The error bars as well as the error shadings are estimated from the violation of PH symmetry of the corresponding quantities, as discussed in Appendix B. Violation of PH symmetry via protocolspeaking, also a maximum at φ=0 would be consistent with the error bars. Furthermore, we already noticed in Fig. 7(a) that the deviations between AphysandAauxare large at φ/lessorequalslant0.2 compared to the other bias voltages and this is exactly wherethe peculiar behavior of G(φ) sets in. In contrast to the Kondo 2 produces a slight difference between the left- and right-moving current |jL|and|jR|, which is clearly unphysical for the steady state. Since G(φ) is obtained by numerical differentiation of the jλ(φ) curves, its error is amplified. This explains why the error bars in theG(φ)a r es ol a r g e . 165132-11DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) FIG. 9. Nonequilibrium ( φ> 0) effective power-law exponents as a function of the bias voltage φ. The three pairs of exponents are extracted from a fit of the auxiliary retarded hybridization function(r /prime,r/prime/prime), the spectral function ( s/prime,s/prime/prime) and the retarded self energy (κ/prime,κ/prime/prime)w i t hE q s .( 31)t o( 33). The single and double primes cor- respond to different fitting intervals /Omega1(φ)a n d/Omega11(φ), see text. regime, Fig. 8(a) shows that in the LM regime the differential conductance increases with the bias voltage, as expected.17 We now attempt extracting “effective” power-law expo- nents in the Kondo regime, as we do in equilibrium, by carry-ing out a fit of the nonequilibrium curves. More specifically,in analogy to Eqs. ( 27), (29), and ( 30), we fit the behavior Im/Delta1 R(ω)∝|ω−μL|r+|ω−μR|r, (31) A(ω)∝|ω−μL|−s+|ω−μR|−s, (32) Im/Sigma1R(ω)∝|ω−μL|κ+|ω−μR|κ. (33) The finite voltage and the imperfect pseudogap set a low- frequency cutoff to this behavior, which we expect not tohold down to zero frequency. The exponents, r /prime(φ),s/prime(φ), andκ/prime(φ), obtained by a fit on the interval /Omega1(φ), defined in Eq. ( 28), are presented in Fig. 9. Since this interval shrinks upon increasing the bias voltage, we can faithfully performthe fit only for voltages φ/lessorsimilar0.6, as discussed below Eq. ( 28). Thus we can just catch the beginning of the interesting voltageregion, where the Kondo split peaks start developing at φ≈ 0.6, according to Fig. 7(a). Moreover, due to the lower cutoff 17There seems to be a flat region up to φ/lessorsimilar0.1i nG(φ). However, we do not believe this to have a particular physical meaning. This apparent behavior may be due to the fact that the numerical eval-uation of Gis quite challenging. Gis obtained from finite current differences using three-point Lagrange polynomials to approximate the derivative and cubic splines to interpolate the result. However,we only have a coarse mesh of voltage points, with φ=0.1b e i n g the first point at nonzero voltage (notice that only the points with errorbars correspond to datapoints). On the other hand, using a finermesh does not make sense due to the limited accuracy of the fit. In addition, this apparently flat behavior is enhanced by the logarithmic scale of the φaxis. In a linear plot, the region φ/lessorsimilar0.1 obviously looks much thinner and the curve displays a quadratic shape there.in energy, the extracted exponents can only provide a rough semi-quantitative estimate. In the range φ/lessorsimilar0.6 the exponents depend only slightly on the bias voltage. Nevertheless, it isnotable that r /prime(φ) and s/prime(φ) are almost parallel. This may indicate that deviations in Im /Delta1R(ω) (such as between Im /Delta1R aux and Im /Delta1R phys) mainly translate into deviations in the spectral function, affecting Im /Sigma1R(ω) in a minor way.18Indeed, if the self energy is more stable against numerical inaccuracies thanthe spectral function, one could try to exploit this to study thephase transition or crossover also out of equilibrium, with ascheme similar to the one presented in Sec. III B 1 . However, to do this, it would be necessary to resolve a larger fractionof the interesting voltage region φ/greaterorsimilar0.6, which, on the other hand, would require a larger /Omega1interval where the power law in the auxiliary hybridization function is accurately resolved. Figure 9also displays the power-law exponents r /prime/prime(φ), s/prime/prime(φ), and κ/prime/prime(φ) fitted on a larger interval /Omega11(φ)=0.2+ φ 2<|ω|<1.2+φ 2, which is obtained by a rigid shift of the equilibrium interval /Omega1byφ 2. In the region φ/lessorsimilar0.6, where both kinds of exponents ( /primeand/prime/prime) are defined, their values lie very close to each other. This confirms that the influence of theexponential factor in the hybridization function is negligibleon these frequency and voltage intervals. IV . SUMMARY AND CONCLUSION In this work we addressed the single-impurity Anderson model with leads displaying a power-law pseudogap in thedensity of states (PSG SIAM) by means of a nonperturba-tive approach to deal with nonequilibrium steady states, theauxiliary master equation approach (AMEA). We studied thegeneralized Kondo (GK) and the local moment (LM) phaseof this model in equilibrium as well as their extension out ofequilibrium. To assess the validity of our approach, we first compared the results to the ones obtained with a direct MPS timeevolution of the Hamiltonian (HMPS) [ 61]. HMPS is faster than AMEA and it can treat a larger number of bath sitesin equilibrium, but, on the other hand, it cannot deal with anonequilibrium steady state due to the lack of a dissipationmechanism. We found that the spectral function, the selfenergy, and the power-law exponents of these quantities agreevery well between AMEA and HMPS, see Fig. 4. Further- more, we implemented a scheme to find the phase boundaryupon linear extrapolation of the power-law exponent of theself energy in the GK phase. The phase boundary obtainedin this way agrees quite well with previous NRG results, seeFig. 6. Out of equilibrium, we observe a splitting of the Kondo peak in the spectral function and in the self energy as afunction of the bias voltage, see Figs. 7(a) and7(b),a si nt h e case of the conventional Kondo effect. On the other hand, thedifferential conductance appears to display a peculiar maxi-mum at finite bias voltage, Fig. 7(c), which could be caused by 18This argument is supported by the fact that /Delta1enters G0andGin the same way in Dyson’s equation. This is easily seen by comparingthe general form of Eq. ( 13) to the result for zero self energy G=G0. 165132-12NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) the shift of the hybridization functions at finite bias voltages. Due to the error bars, it is not clear whether this maximumcan be considered a genuine feature of the model. We are notaware of any other work on this model displaying this feature.For example, in [ 47], the conductance maximum occurs at zero bias. However, this work also does not show a splittingof the Kondo resonance at finite bias voltages. The authorsattributed this to the fact that the system is not in the limitof large interactions. A comparison to our results is difficultsince the position of the pseudogap as a function of voltage isconsidered differently in our paper. More specifically, whilein [47] the pseudogap is fixed at ω=0, in our case it moves with the chemical potentials of the two leads, consistent witha rigid shift of the two leads. Strictly speaking, what we observe in the Kondo regime is the result of a superposition of the (pseudogap) GK effectwith a small contribution from the ordinary one. This is, dueto the fact that the imperfect mapping produces a nonzeroresidual /Gamma1 resid=− Im/Delta1R aux(0)≈0.39, even at zero bias volt- age. However, the contribution from this residual DOS isnegligible since the resulting Kondo temperature T K,resid≈ 0.0025 is much smaller than the temperature of our data T≈20TK,resid.19Therefore, the Kondo resonances shown in Figs. 7(a) and7(b) are clearly dominated by the pseudogap GK effect. It would be clearly desirable to be able to extend an accu- rate mapping of the hybridization function down to smaller|ω|values. This would further reduce the contribution of the ordinary Kondo effect and it would allow for a moreaccurate analysis of the low-frequency behavior. In previousworks [ 50,56] we demonstrated that the accuracy of the map- ping increases exponentially upon increasing the number ofbath sites. However, this is only true if we find good enoughminima of the cost function measuring the difference between/Delta1 auxand/Delta1phys. This has, so far, turned out to be difficult for the PSG model studied here. To resolve the power law withthe cusp, bath sites on all energy scales would be required, asused in NRG. To make progress in this direction, we tried tofit the hybridization function on a logarithmic frequency gridand/or include its power-law exponent explicitly into the cost function, but without success so far. The fit seems to be quiteunstable in all of these cases. On the technical side, this work presents a development of the AMEA Lindblad many-body impurity problem withina matrix product states algorithm. Due to the reduced localHilbert space obtained by separating the degrees of free-dom, the present implementation is faster and more stablethan the one of our previous work, see [ 56]. On the other hand, the disadvantage of the structure used here is that additionallong-range couplings between the impurity and the baths areintroduced, as illustrated in Fig. 2, and the entanglement must be carried across the sites in between, which causes the bonddimension to increase. An obvious way to avoid this is a“fork” structure, in particular, a “double fork,” which has threebonds at the impurity instead of two. This structure naturally 19TKis estimated with the widely used formula from [ 4],TK=√/Gamma1U/2e x p [ −πU/(8/Gamma1)], assuming a constant lead DOS with a hybridization strength of /Gamma1=/Gamma1resid.takes into account the spin separation as well as the separation between full and empty baths and, at the same time, only hasnearest-neighbor couplings. The disadvantage of this schemeis that it cannot be represented by MPS because of the thirdbond, and it thus requires the implementation of a new tensornetwork, similar to the one described in [ 61]. Work along these lines is in progress. ACKNOWLEDGMENTS We would like to thank Franz Scherr for providing a first implementation of the AMEA mapping using the PYTHON library TENSORFLOW . This work was supported by the Aus- trian Science Fund (FWF) within the Projects No. P26508 andNo. Y746 (START Programme), as well as NaWi Graz. Thenumerical results presented here have been carried out on theD-Cluster Graz and on the VSC-3 HPC Cluster Vienna. APPENDIX A: CONSTRUCTION OF A NONEQUILIBRIUM SYSTEM FROM EQUILIBRIUM BATH PARAMETERS Here we show two results concerning the representation of a noninteracting fermionic bath in terms of Lindbladopen systems, focusing on a geometry that is suitable fora treatment with MPS. As discussed in our previous work,see [ 56], for the sake of an MPS treatment, it is convenient to connect the impurity to a bath which is full and one which isempty. Each of the two baths should have a one-dimensionalchain geometry and couple on each side of the impurity. Thisgeometry guarantees a slower propagation of entanglement.For this reason, in Appendix A1we show, how to represent an arbitrary hybridization function as originating from a fulland an empty bath. This is valid both for a nonequilibriumas well as for an equilibrium φ=0 hybridization function. In our paper, it is convenient to start with such a representationfor the fit of an equilibrium bath and then use this solution toproduce a full-empty representation for a finite voltage φ/negationslash=0. How this is done, is shown in Appendix A2. 1. Splitting into a full and empty bath The effects of an arbitrary noninteracting fermionic bath on a single-site impurity are completely described by itshybridization function /Delta1 (ω) in Keldysh space. Here we show that any (equilibrium or nonequilibrium) /Delta1can always be split as/Delta1=/Delta1F+/Delta1E, where /Delta1Fdescribes a full ( F) and/Delta1Ean empty ( E) (equilibrium) bath. As discussed above, these two baths are represented by a Lindblad equation, where /Gamma1(1)=0 or/Gamma1(2)=0, respectively. For better readability, we omit the frequency argument ωand introduce the two components of the hybridization function /Delta1Ri≡Im/Delta1R,/Delta1Ki≡/Delta1K 2i. (A1) In equilibrium, these two components are linked via the fluctuation-dissipation theorem, /Delta1Ki=/Delta1Ri[1−2f(ω−μ)], (A2) 165132-13DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) where fis the Fermi function and μthe chemical potential. For a full /empty equilibrium bath the relation /Delta1Ki F/E=∓/Delta1Ri F/E (A3) follows from Eq. ( A2)f o r f≡1(F)o r0( E), respectively. We can, therefore, decompose /Delta1Ki=/Delta1Ki F+/Delta1Ki E=−/Delta1Ri F+/Delta1Ri E, /Delta1Ri=/Delta1Ri F+/Delta1Ri E, which gives /Delta1Ri F/E=/Delta1Ri∓/Delta1Ki 2. (A4) Note that Eqs. ( A2) and ( A3) are equilibrium properties. Therefore, these are valid for any component of each one ofthe two (uncoupled) baths, EandF, and in particular for the Green’s function matrix. Moreover, a matrix inversion pre-serves these relations. However, for a matrix A β,β∈{R,K}, such as the Green’s function or self-energy matrix, one has toreplace the imaginary part ( A1) with the anti-Hermitian part, i.e., A Ri=1 2i(AR−AR†), AKi=1 4i(AK−AK†). (A5) Notice that the Keldysh component AKis anti-Hermitian anyway. In this case Eq. ( A3) becomes AKi F/E=∓ARi F/E. (A6) Applying Eq. ( A5) to the Green’s function matrix of one of the two uncoupled baths (cf. Eqs. (40) and (41) in [ 49]) (G−1)R=ωI−E+i(/Gamma1(1)+/Gamma1(2)), (G−1)K=− 2i(/Gamma1(2)−/Gamma1(1)), results in (G−1)Ri=/Gamma1(1)+/Gamma1(2), (G−1)Ki=/Gamma1(1)−/Gamma1(2).(A7) Inserting this result further into Eq. ( A6) yields that a full bath has/Gamma1(1)=0 and an empty one /Gamma1(2)=0, as expected, (G−1)Ri F=/Gamma1(2),(G−1)Ri E=/Gamma1(1). (A8) Notice that this splitting procedure does not change the properties of the impurity. Furthermore, it can be carried outalso for an equilibrium bath or for a situation in which theleads are partially full or partially empty. A crucial point isthat in MPS, it is always convenient to split the baths in thisway because the entanglement is less severe, see [ 56]. 2. From equilibrium to nonequilibrium As discussed, we start by fitting a bath in equilibrium and then we split it into a full and an empty one, see Fig. 10.I n fact, it turns out that such a geometry naturally comes out fora chain geometry fit.μ (a) (b) FIG. 10. (a) Impurity (red sphere) coupled to a partially filled bath (semicircle) at chemical potential μ. (b) The same hybridization function can be obtained by coupling the impurity to a full and emptybath with appropriate DOS. For the situation depicted in Fig. 10(b) the fit produces the following Lindblad matrices, assuming PH symmetry, E=⎛ ⎜⎜⎜⎝˜Eτ 0 t0 0tεft0 0t 0˜E⎞ ⎟⎟⎟⎠(A9) and /Gamma1(1)=⎛ ⎝˜/Gamma1τ00 00 000 0⎞ ⎠,/Gamma1 (2)=⎛ ⎝000 000 00 ˜/Gamma1⎞ ⎠. Here ˜Eand ˜/Gamma1areNB/2×NB/2 block matrices and each matrix AτisAwith the order of indices inverted and different signs, see Eq. (27) in [ 50] for the exact relations. For MPS, ˜Eand ˜/Gamma1should be tridiagonal, which corresponds to having nearest-neighbor hoppings and /Gamma1terms only. The retarded hybridization function of, for instance, the full bath is thengiven by /Delta1 R F(ω)=t2¯γR(ω) (A10) with the boundary Green’s function ¯γR(ω)=[(ωI−˜E+i˜/Gamma1)−1]11 (A11) and the Keldysh hybridization function /Delta1K F(ω)i sfi x e db y Eq. ( A3). The result for the empty bath follows from PH symmetry. Instead of the equilibrium situation in Fig. 10(a) , we would now like to represent a nonequilibrium one, as depicted inFig. 11(a) . If the total DOS is fixed, this is obtained by reduc- ing the hoppings to the impurity by 1 /√ 2 and by doubling the number of bath sites and shifting their on-site energiesby±/Delta1ε. Then Fig. 10(b) schematically becomes Fig. 11(b) , which can no longer be represented in a chain geometry (with 165132-14NONEQUILIBRIUM PSEUDOGAP ANDERSON IMPURITY … PHYSICAL REVIEW B 101, 165132 (2020) Δε (a) Δε (b) FIG. 11. (a) Impurity (red sphere) coupled to a partially filled left bath and a partially filled right bath (semicircles), whose chemical potentials differ by 2 /Delta1ε. (b) The same situation with two full (blue) and two empty (white) baths. tridiagonal matrices). The matrix in Eq. ( A9) becomes E/prime=⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝˜E τ+/Delta1εI 00 t/prime 00 0 ˜Eτ−/Delta1εI0 t/prime 00 0t/prime0t/primeεf t/prime0 t/prime0 00t/prime 0˜E−/Delta1εI 0 00t/prime 00 ˜E+/Delta1εI⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (A12) with t /prime=t/√ 2 and, correspondingly, /Gamma1(1)and/Gamma1(2).I nt h i s situation, Eq. ( A10) still holds, but instead of Eq. ( A11), we have /Delta1R F(ω)=t2 2[¯γR(ω+/Delta1ε)+¯γR(ω−/Delta1ε)]. However, the matrix ( A12) is not suitable for MPS, as it is not tridiagonal. To make progress, we observe that /Delta1R F(ω) can be obtained by considering the following matrix in block form: h/prime=⎛ ⎜⎜⎜⎜⎝0 t /prime0 t/prime0 t/prime 0˜E−i˜/Gamma1−/Delta1εI 0 t/prime 00 ˜E−i˜/Gamma1+/Delta1εI⎞ ⎟⎟⎟⎟⎠ and employing Dyson’s equation /Delta1 R F(ω)=ω−1 [(ωI−h/prime)−1]11. (A13) For MPS we need a tridiagonal form, as discussed above. This can be achieved with a Bi-Lanczos transformation. All weneed is that [( ωI−h /prime)−1]11remains invariant. The transfor- mation is produced by a matrix (here the upper block is 1 ×1and the lower is NB×NB) U=/parenleftbigg10 0 ˜U/parenrightbigg , (A14) where Uis, in general, nonunitary, yielding h/prime/prime=U−1h/primeU =⎛ ⎝0t/prime/prime0 t/prime/prime 0H/prime/prime⎞ ⎠. Here the non-Hermitian tridiagonal matrix H/prime/primeidentifies the new parameters of the full ( F) bath ˜E/prime/prime≡H/prime/prime†+H/prime/prime 2, ˜/Gamma1/prime/prime≡H/prime/prime†−H/prime/prime 2i,(A15) while the ones of the empty ( E) bath are obtained by PH symmetry, see Eq. ( 27)i n[ 50]. Note that, since ˜Uis not unitary, ˜E/prime/primeand ˜/Gamma1/prime/primeare not simply obtained by transforming ˜Eand˜/Gamma1, separately. This can, and in our case does, produce ˜/Gamma1/prime/primethat are not semi-positive–definite, as should be required for the Lindblad equation. Still, thesteady state we obtain is stable and the spectral functionsturn out to be causal. The reason is that the new parametersoriginate from semi-positive–definite matrices. APPENDIX B: SYMMETRY CONSIDERATIONS AND ERROR ESTIMATION In principle, we can calculate four Green’s functions in- dividually, Gα σwithσ∈{ ↑,↓}andα∈{<,>}. The system, though, is PH symmetric, which relates the lesser and thegreater Green’s functions to each other, and it is spin sym-metric. Therefore, the following relations must be fulfilled: G < σ(x)=−G> σ(−x), (B1) Gα ↑(x)=Gα ↓(x), (B2) forxbeing either torω. This reduces the number of actually independent Green’s functions to only one. Thus, to obtainthe spectral function, for example, it is in principle sufficientto calculate only one G α σ, then construct G¯α σwith ¯α/negationslash=αfrom Eq. ( B1) and evaluate Eq. ( 11). We refer to this as protocol 1 . However, if we calculate Gα σwith AMEA employing MPS, the symmetry relations, Eqs. ( B1)t o( B2), are not exactly fulfilled. This is due to the approximations within theMPS calculation, more specifically, due to the truncation andSuzuki-Trotter errors. Figure 12shows the consequences of these violations at the example of the spectral function. We can see that the spectral functions determined from only one G α σ, according to protocol 1, are symmetric by con- struction Aα σ(ω)=Aα σ(−ω), but they differ from each other, Aα σ(ω)/negationslash=A¯α ¯σ(ω)f o rα/negationslash=¯αandσ/negationslash=¯σ. The area enclosed by the four different solutions is color-shaded and the solid curvein the center is the average of these solutions, which we callthe symmetrized spectral function in this paper. The devia-tions of the borders of the shaded area from the symmetrized 165132-15DELIA M. FUGGER et al. PHYSICAL REVIEW B 101, 165132 (2020) FIG. 12. Auxiliary spectral functions A(ω) obtained from differ- ent raw data for symmetry considerations and error estimation, see text. spectral function can be used as a measure for the symmetry errors throughout the MPS calculation.In this figure, we can also see the spectral functions naively determined from two Green’s functions, G< σand G> σ,b y evaluating Eq. ( 11) directly, without enforcing PH symmetry. We refer to this as protocol 2 . These spectral functions are not exactly symmetric, Aσ(ω)/negationslash=Aσ(−ω), as discussed above, but they are close to the symmetrized spectral function and theylie almost entirely within the shaded area for almost all biasvoltages (except φ=0.8 andφ=1). Throughout this paper, we display also other, in princi- ple symmetric, quantities as symmetrized curves with errorsin the form of color-shaded areas, obtained by protocol 1.Specifically, the self energy and the differential conductanceare represented in this way, see Figs. 4,7, and 8.F o rt h e differential conductance, we also consider deviations arisingby protocol 2 and plot the corresponding errors separately, asbars, in addition to the shaded area, see Figs. 7(c) and8(a). For the other quantities these errors lie almost entirely withinthe shaded area, anyway, and their inclusion does not makeany difference. 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PhysRevB.85.144527.pdf

PHYSICAL REVIEW B 85, 144527 (2012) Gap nodes induced by coexistence with antiferromagnetism in iron-based superconductors S. Maiti,1R. M. Fernandes,2,3and A. V . Chubukov1 1Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA 2Department of Physics, Columbia University, New York, New York 10027, USA 3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA (Received 25 February 2012; published 30 April 2012) We investigate the pairing in iron pnictides in the coexistence phase, which displays both superconducting and antiferromagnetic orders. By solving the pairing problem on the Fermi surface reconstructed by long-rangemagnetic order, we find that the pairing interaction necessarily becomes angle dependent, even if it was isotropicin the paramagnetic phase, which results in an angular variation of the superconducting gap along the Fermisurfaces. We find that the gap has no nodes for a small antiferromagnetic order parameter M, but may develop accidental nodes for intermediate values of M, when one pair of the reconstructed Fermi surface pockets disappears. For even larger M, when the other pair of reconstructed Fermi pockets is gapped by long-range magnetic order, superconductivity still exists, but the quasiparticle spectrum becomes nodeless again. We alsoshow that the application of an external magnetic field facilitates the formation of nodes. We argue that thismechanism for a nodeless-nodal-nodeless transition explains recent thermal conductivity measurements of hole-doped (Ba 1−xKx)Fe 2As2[J-Ph. Read et al. , e-print arXiv: 1105.2232 ]. DOI: 10.1103/PhysRevB.85.144527 PACS number(s): 74 .20.Rp I. INTRODUCTION Exploring the different regions in the phase diagram of iron based superconductors (FeSCs) is an important step toward developing a unified understanding of the physics ofsuperconductivity in these systems. A typical phase diagramof a FeSC in the ( T,x) plane, where xis doping, shows a metallic antiferromagnetic order, also called spin-density wave(SDW), below T Natx=0. Upon doping, the SDW order parameter is suppressed and superconductivity (SC) emerges with maximum Tc(x) near the point where Tc(x) exceeds TN(x) (an “optimal doping”). The gap symmetry at optimal doping is most likely s+−, but the structure of the s-wave gap varies from one material to another: the gap is nodeless inBa(Fe 1−xCox)2As21–3and (Ba 1−xKx)Fe 2As24–6but has nodes in BaFe 2(As 1−xPx)2.7,8 In this article, we investigate the behavior of the s+− SC gap in the underdoped regime of the Ba(Fe 1−xCox)2As2 and (Ba 1−xKx)Fe 2As2systems, where Tc(x)<TN(x), and superconductivity emerges in a continuous fashion from apre-existing SDW order. The microscopic coexistence of SCand SDW below T c(x) has been reported by magnetization and NMR experiments.9–12 The electronic structure of FeSCs in the paramagnetic phase consists of near-circular hole pockets and elliptical electronpockets. We assume, following earlier works, 13–20that the interaction leading to SDW order is independent on the anglealong the FS. 21In this situation, the Brillouin zone (BZ) is reduced once the SDW order sets in, and the Fermi surface(FS) is reconstructed into four banana-like pockets (right panelin Fig. 1). As the SDW order parameter Mincreases, the reconstructed pockets shrink and eventually disappear. Thishappens in two stages: first, at M=M 1, the number of pockets shrinks from four to two (middle panel in Fig. 1), and then the remaining two pockets vanish at a larger value M=M2(left panel in Fig. 1). Because these reconstructed FSs are formed by mixing electron and hole bands, on which the s+−SC order parameter has different signs in the absence of SDW order,one could naively expect that the SC gap on the reconstructed FSs alternates between plus and minus signs, implying thatit must have nodes. However, calculations show 13that this is not the case because SDW order with, e.g., /angbracketleftSx/angbracketright/negationslash=0, mixes hole and electron FSs with opposite σzcomponents of electron spins. Because a spin-singlet s-wave gap changes sign underσ z→−σz, the sign change imposed by the s+−gap structure is compensated by the sign change due to the flip of σz, and, as a result, the gaps on the reconstructed FSs retain the samenodeless structure as in the absence of magnetic order. Experiments near the onset of the co-existence phase, where Mis small, are in agreement with this reasoning. For instance, the ratio κ/T , where κis the in-plane thermal conductivity, tends to zero in the T→0 limit, as it is expected for a superconductor with a nodeless gap. However, recentthermal conductivity measurements on (Ba 1−xKx)Fe 2As2deep in the coexistence phase23found that there is a range of dopings where κ/T remains finite at T=0, as it happens in a superconductor with line nodes. At even smaller dopings,κ/T again vanishes at T=0. Application of a magnetic field makes the doping dependence more smooth and extends therange of nodal behavior. In this paper, we explain the sequence of nodeless-nodal- nodeless behavior as a transition from a nodeless s +−gap in the region where all four reconstructed FS pockets arepresent to an s +−gap with accidental nodes in the region where two FS pockets are present and to the fully gappedquasiparticle spectrum in the region where the remaining tworeconstructed FS pockets are also gapped (see Fig. 1). We show that the pairing interaction in the coexistence phase acquiresan additional angular dependence as it gets dressed by angle-dependent coherence factors associated with the SDW order.The angular dependence of the interactions in turn gives rise toan angular dependence of the superconducting gap. This effectis weak in the limit of small M, considered in Ref. 13, when all four reconstructed Fermi pockets are present, but is strongand gives rise to gap nodes when Mbecomes large enough to 144527-1 1098-0121/2012/85(14)/144527(11) ©2012 American Physical SocietyS. MAITI, R. M. FERNANDES, AND A. V . CHUBUKOV PHYSICAL REVIEW B 85, 144527 (2012) SCSDW xT nodeless nodeless nodalM2M1SDW+SC FIG. 1. (Color online) Schematic phase diagram showing the nodeless-nodal-nodeless transitions inside the region where super-conductivity (SC) and spin-density wave (SDW) coexist. Each transition is roughly associated with the disappearance of one pair of magnetically reconstructed Fermi surface pockets. M 1andM2 refer to the values of the SDW order parameter Mat which the two pairs of pockets disappear respectively. A dip in Tcis observed near M2, when both pockets disappear. gap out one pair of Fermi pockets. At even larger M, deeper in the coexistence region, the remaining pair of Fermi pockets isgapped out. In this situation, superconductivity still developsover some range of dopings, 14but the quasiparticle excitation spectrum remains gapped irrespective of the structure of theSC gap, and κ/T again vanishes at T=0. Concurrently, the superconducting transition temperature T cshows a minimum when the remaining pockets vanish. The theoretical behavioracross the SDW-SC coexistence region is shown schematicallyin Fig. 1. The paper is organized as follows. In Sec. II, we describe the formalism listing out the details of the FS geometries, thenature of the reconstructed FSs, and the generic form of the SCgap in the coexistence region. In Sec. III, we discuss the solutions for the gap obtained within this formalism, showwhen and how nodes arise in the coexistence region, andprovide analytical explanation of the results. We also brieflydiscuss the role played by bands that do not participate inthe formation of SDW. In Sec. IV, we show that the doping range where the SC gap has nodes is enhanced by applying amagnetic field. We state our final conclusions in Sec. V. II. THE PAIRING PROBLEM ON THE RECONSTRUCTED FERMI SURFACE Our point of departure is a microscopic band model of interacting fermions located near hole or electron pockets.The electronic structure of FeSCs consists of two or three holepockets, centered at (0 ,0) in the folded zone (two Fe atoms per unit cell), and two hybridized electron pockets centeredat (π,π). The interactions between low-energy fermions are generally angle-dependent already in the normal state, beforeeither SDW or SC order sets in, which gives rise to angle-dependent SDW and SC gaps even outside the coexistenceregion.As we said, we follow earlier works and consider a simplified model in which we neglect the angular dependenciesof the interactions in the normal state, and approximate thenormal-state interactions as constants. 13–20We further neglect the hybridization and perform calculations in the unfoldedzone (one Fe atom per unit cell) in which electron pockets areellipses centered at (0 ,π) and ( π,0) and hole pockets are at (0,0) and ( π,π). Finally, we neglect the third hole pocket and only consider two circular hole pockets centered at (0 ,0) in the unfolded zone (in orbital notations these are d xz/dyzpockets). Earlier works have found17that SDW magnetic order emerges in this model with ordering vector Q=(0,π)o rQ=(π,0), mixing one hole and one electron pocket. For definiteness, wesetQ=(0,π) in which case one hole pocket at (0 ,0) and one electron pocket at (0 ,π) are mixed up. The other two pockets are not participating in SDW and remain intact once the SDWorder sets in. In line with our goal, we first consider only thetwo-pocket model with one circular hole and one ellipticalelectron pocket, which get reconstructed in the SDW phase.We discuss the role of the other pockets later in the next section. To quadratic order in the fermions, the Hamiltonian of the two-pocket model in the SDW phase is H 0+HSDW, where H0=/summationdisplay k,σ/parenleftbig εc kc† kσckσ+εf k+Qf† k+Qσfk+Qσ/parenrightbig , (1) HSDW=/summationdisplay kσM(σc† kσfk+Qσ+H.c.). Here,candfare fermionic operators for hole and electron states, respectively, σis±1, and Mis the SDW order parameter, which we treat below as a variable. In practice,larger Mcorrespond to smaller dopings, whereas smaller M correspond to doping close to the optimal one. The fermionicdispersions are ε c k=μc−k2 x+k2 y 2m,εf k+Q=−μf+k2 x 2mx+k2 y 2my, (2) where m,mx,myare the effective band masses of the fermions. For definiteness, we set the interatomic spacing to one anduseμ c=μf=μ,m=1/(2μ),mx=0.5/(2μ), and my= 1.5/(2μ). These parameters are chosen to give the FS geometry as in Fig. 2. We will also measure Min units of 2 μ. TheMterm in Eq. (1)couples candfoperators such that the eigenstates of Eq. (1)are coherent superpositions of electrons and holes. These are described by new fermionic operators aandb, with dispersion Ea,b kthat vanishes at the reconstructed FSs. The transformation to the new operators is /parenleftbiggσakσ bkσ/parenrightbigg =/parenleftbigg−σcosθk sinθk −sinθk−σcosθk/parenrightbigg/parenleftbiggckσ fk+Qσ/parenrightbigg ,(3) where cosθk=M/radicalbig M2+(E−)2,sinθk=E− /radicalbig M2+(E−)2, (4) and E−=εc−εf 2−/radicalBigg/parenleftbiggεc−εf 2/parenrightbigg2 +M2, (5) 144527-2GAP NODES INDUCED BY COEXISTENCE WITH ... PHYSICAL REVIEW B 85, 144527 (2012) −0.5 0 0.5−0.4−0.200.20.4 kx/πky/π aba b M=0.041 50 100150 −0.5 0 0.5−0.4−0.200.20.4 kx/πky/π aba b M=0.09 −0.5 0 0.5−0.500.5 kx/πky/πb bM=0.12 FIG. 2. (Color online) Reconstructed aandbFermi pockets for several values of the SDW order parameter M.T h eapockets are smaller and disappear first upon increasing M.A te v e nl a r g e r M,t h ebpockets also disappear (not shown). Each pocket was discretized into a finite number of points indicated in the left figure for one of the bpockets. Our convention is such that the numbering for each pocket starts at the tip of a “banana” and goes first along the “inner” side of the pocket and then along the “outer” side. such that cos 2θk=εc−εf 2/radicalBig M2+/parenleftbigεc−εf 2/parenrightbig2, (6) sin 2θk=−M/radicalBig M2+/parenleftbigεc−εf 2/parenrightbig2. In terms of the new operators, H0+HSDW=/summationdisplay k,σ/parenleftbig Ea ka† kσakσ+Eb kb† kσbkσ/parenrightbig , (7) where Ea,b k=εc+εf 2±/radicalBigg/parenleftbiggεc−εf 2/parenrightbigg2 +M2 (8) are the quasiparticle excitation energies of the SDW state (the plus sign corresponds to afermions, the minus sign corresponds to bfermions). The condition Ea,b k=0 defines the new reconstructed FSs. For small enough M, there are two pairs of banana-shaped reconstructed pockets, the aand the bpockets (see Fig. 2), with the latter larger than the former. At the critical M=M1 given by M1=1 4|m−my| √mmy, (9) theapockets disappear, while the bpockets remain (Fig. 2, right). At even larger M=M2given by M2=1 4|m−mx|√mmx, (10) thebpockets also disappear, i.e., all electronic states become gapped by SDW.21For our set of parameters, M1≈0.102 and M2≈0.177. We now turn to the issue of the pairing on the reconstructed FS. There are four residual interactions between the originalfermions located near the hole and electron FSs: 19the interaction between fermionic densities near hole and electronpockets [ U 1c† σcσf† σ/primefσ/prime], the exchange interaction betweenhole and electron pockets [ U2c† σfσf† σ/primecσ/prime], the umklapp pair- hopping interaction in which two fermions near a hole pocketare converted into two fermions near an electron pocket and vice versa [ U 3/2(c† σfσc† σ/primefσ/prime+f† σcσf† σ/primecσ/prime)], and the density- density interaction within each pocket [ U4/2(c† σcσc† σ/primecσ/prime+ f† σfσf† σ/primefσ/prime)]. In the absence of SDW order, only U3andU4 contribute to the pairing channel ( U3must be larger than U4for s+−pairing). In the SDW phase, candfoperators are mixed up, and all four interactions contribute to pairing vertices foraandbfermions. The interactions U 2andU4do not give rise to new physics and only renormalize the values of the pairingvertices obtained from the U 1andU3terms. To shorten the formulas, we neglect the U2andU4terms and approximate Hintby Hint=/summationdisplay [1234]U1c† 1σf† 2σ/primef3σ/primec4σ +U3 2(c† 1σc† 2σ/primef3σ/primef4σ+H.c.), (11) where/summationtext [1234] denotes the sum over all momenta subject to k1+k2=k3+k4, and the summation over repeated spin indices is implied. Converting to new operators via Eq. (3)and keeping only the interactions that contribute to the pairing, weobtain from Eq. (11), H pair=/summationdisplay q,k,σ,σ/primeUaa q,k(a† qσa† −qσ/primea−kσ/primeakσ+b† qσb† −qσ/primeb−kσ/primebkσ) +/summationdisplay q,k,σ,σ/primeUab q,k(a† qσa† −qσ/primeb−kσbk,σ+H.c.), (12) where Uaa q,k=Ubb q,k=U3 4(−1+cos 2θqcos 2θk+ssin 2θqsin 2θk), Uab q,k=Uba q,k=−U3 4(1+cos 2θqcos 2θk+ssin 2θqsin 2θk), and we defined s≡U1/U 3. We see from Eq. (12) that both intrapocket and inter- pocket pairing interactions acquire angular dependence via thecos 2θ q,cos 2θkand sin 2 θq,sin 2θkfactors. Constructing the set of linearized BCS gaps equations for the angle-dependent 144527-3S. MAITI, R. M. FERNANDES, AND A. V . CHUBUKOV PHYSICAL REVIEW B 85, 144527 (2012) gaps/Delta1a(q) and /Delta1b(q) by standard means, we find at T=Tc, /Delta1a(q)=−/integraldisplay adk/bardbl 4π2vFUaa q,k/Delta1a(k)L−/integraldisplay bdk/bardbl 4π2vFUab q,k/Delta1b(k)L, /Delta1b(q)=−/integraldisplay adk/bardbl 4π2vFUba q,k/Delta1a(k)L−/integraldisplay bdk/bardbl 4π2vFUbb q,k/Delta1b(k)L, where L≡ln(/Lambda1 Tc),/Lambda1is the upper cutoff of the theory, and the Fermi velocity vF=vF(k/bardbl) varies along aandbFermi surfaces. A generic solution of this gap equation is of the form /Delta1a(q)=g1+g2cos 2θq+g3sin 2θq, (13) /Delta1b(q)=g1−g2cos 2θq−g3sin 2θq. Note, however, that the three angular components of /Delta1a,b are not orthogonal, since, e.g.,/integraltext adq/bardblcos 2θqdoes not vanish. Substituting these expressions into the gap equation, we re-express it as a matrix equation for T candgi: ⎛ ⎜⎝g1 g2 g3⎞ ⎟⎠=U3L 2⎛ ⎜⎝NN c Ns −Nc−Ncc−Ncs −sNs−sNcs−sNss⎞ ⎟⎠⎛ ⎜⎝g1 g2 g3⎞ ⎟⎠,(14) where N=2/integraldisplay adk/bardbl 4π2vF+2/integraldisplay bdk/bardbl 4π2vF, Nc=2/integraldisplay adk/bardbl 4π2vFcos 2θk−2/integraldisplay bdk/bardbl 4π2vFcos 2θk, Ns=2/integraldisplay adk/bardbl 4π2vFsin 2θk−2/integraldisplay bdk/bardbl 4π2vFsin 2θk, (15) Ncc=2/integraldisplay adk/bardbl 4π2vFcos22θk+2/integraldisplay bdk/bardbl 4π2vFcos22θk, Nss=2/integraldisplay adk/bardbl 4π2vFsin22θk+2/integraldisplay bdk/bardbl 4π2vFsin22θk, Ncs=2/integraldisplay adk/bardbl 4π2vFcos 2θksin 2θk +2/integraldisplay bdk/bardbl 4π2vFcos 2θksin 2θk. Without SDW order, sin 2 θk=0, cos 2 θk=sign(/epsilon1c−/epsilon1f), andNs,Nsc, andNssall vanish. Then g3=0, while g2/g1= −Nc/(N+/radicalbig N2−N2c). As a result, the gaps on the inner and outer sides of the aandbpockets reduce to g1[1+Nc/(N+/radicalbig N2−N2c)] and g1[1−Nc/(N+/radicalbig N2−N2c)]. These two coincide with the gaps on the c- andf-Fermi surfaces at Tc in the paramagnetic phase, which we set in our model to be angle independent (we recall that the sign of the two gaps isthe same because we attributed an extra minus sign to the gapon the f-Fermi surface by flipping the spins of ffermions). The magnitudes of the gaps on c- and f-FSs are generally unequal as N c/negationslash=0 because of the different geometry of hole and electron pockets. Numerically, however, for our set ofparameters N c/N/lessmuch1, i.e., at M=0 the gaps on aandb FSs are essentially isotropic. Once SDW order sets in, allN iandNijbecome nonzero, g3becomes finite, cos 2 θkand sin 2θkbecome smooth functions along the pockets, and the0.05 0.1 0.15−2−1012 MNj/N Nc/N Ns/NNcc/N Nss/N FIG. 3. (Color online) The quantities Nj/N,w h e r e Nj= Nc,Ns,Ncc,Nss, plotted as functions of M. The dashed line indicates the value M=M1at which the apockets disappear. The ellipse indicates the area around M1where the ratios rapidly evolve. Note thatNc/Nremains small for all M, while Ns/NandNssincrease withMand become positive and of order one at M/greaterorequalslantM1. gaps/Delta1a(q) and/Delta1b(q) acquire smooth angular variations. The magnitudes of the variations depend on the two parameters:the ratios g 2/g1andg3/g1and the actual variation of cos 2 θk and sin 2 θkalonga- andb-Fermi surfaces. In the small Mlimit, considered previously in Ref. 13, sin 2θkis small except for narrow ranges near the tips of the bananas. Then Ns,Nsc, andNssare small, and Ncc/N≈1. Solving Eq. (14) in this limit, we find that g3/lessmuchg2/lessmuchg1, hence /Delta1aand/Delta1bremain almost constant along the aandb pockets, including the tips of the bananas. For arbitrary M,g2/g1andg3/g1are rather complex functions of the ratios Nj/Nand also s.W ep l o t Nj/Nin Fig. 3. We see that Nc/NandNcs/Nare small because the integrands for NcandNcscontain cos 2 θk, which changes sign between the inner and outer sides of the aandbpockets (see Fig. 4). At the same time, the magnitudes of Ns/NandNss/N increase with increasing Mand become of order one. In Fig. 4, we plot cos 2 θkand sin 2 θkalong the aandb pockets. We recall that the prefactors of the cos 2 θkterms in /Delta1aand/Delta1bare already small because g2/g1is small. Figure 4 shows that the variation of cos 2 θkalong each of the pockets decreases with increasing M, making the g2cos 2θkterm even less relevant. On the other hand, the range where |sin 2θk| is not small widens up with increasing M. Over some range ofM,u pt o M11=0.5(my−m)/(my+m)<M 1on the a pocket and up to M22=0.5(m−mx)/(m+mx)<M 2on thebpocket, sin 2 θkvaries between −1 and some negative value. At M11<M<M 1,s i n 2θkon the apocket varies in a more narrow interval, does not reach −1, and eventually, becomes sin 2θkaM→M1=− 2√mmy m+my=− 0.98. (16) The same behavior holds for sin 2 θkon the bpocket for M22<M<M 2, but with sin 2θkbM→M2=− 2√mmx m+mx=− 0.94. (17) 144527-4GAP NODES INDUCED BY COEXISTENCE WITH ... PHYSICAL REVIEW B 85, 144527 (2012) 0 50 100 150 200−1−0.500.51 length along pocketcos2 θk M=0.04a−pocket b−pocket 0 50 100 150 200−1−0.500.51 length along pocketsin2θkM=0.04a−pocket b−pocket 0 50 100 150−1−0.500.51 length along pocketcos2 θk M=0.09a−pocket b−pocket 0 50 100 150−1−0.500.51 length along pocketsin2θkM=0.09a−pocket b−pocketFIG. 4. (Color online) The behavior of cos 2θk(left column) and sin 2 θk(right column) along the aandbpockets for M=0.04 and M=0.09. The gap is given by Eq. (13).T h e “length along the pocket” refers to the numbers along each pocket specified in Fig. 2. III. GAP STRUCTURE IN THE COEXISTENCE STATE AS FUNCTION OF M A. From small to intermediate M: nodeless-nodal transition We now discuss in more detail the solution of the matrix equation (14) and the structure of the SC gap. We found in the previous section that the angular dependence of cos 2 θk and sin 2 θkalong the aandbpockets depends on M, while the solution of the matrix equation for gidepends on Mand also on s—which, we remind, is the ratio of the interactions s=U1/U 3. The solution of the matrix equation can be easily analyzed in the limits of small and large s.A ts m a l l s,g3is small compared to g2[g3/g2=O(s)], while g2/g1is given by g2 g1≈−Nc N+Ncc. (18) This ratio is always small because Nc/N is small for allM(see Fig. 3). As a result, the gaps /Delta1a(q) and /Delta1b(q) remain essentially constant regardless of how strong the SDWorder is. In the opposite limit s/greatermuch1, the behavior is different. Only in a narrow range of the smallest M/lessmuch1/s, the gaps remain almost angle independent. Outside this range, g 2is small compared to g3[g2/g3=O(1/s)], while g3/g1is given by g3 g1≈−Ns Nss. (19) BothNsandNssbecome nonzero at a finite Mand their ratio is of order one, i.e., g3∼g1. This leads to sizable angular dependencies of /Delta1a(q) and/Delta1b(q). For small M<M 1,Nsis smaller in magnitude than Nss(see Fig. 3), hence |g3/g1|<1 and the angular variations of the gaps do not give rise to nodes.However, as Mincreases and approaches M 1, the ratio Ns/Nss changes sign, becomes negative and its magnitude exceeds one (see Fig. 3). The combination of this behavior and the fact that for M∼M1sin 2θreaches −1 at the tip of the bananasimplies that the gap along the bpocket develops accidental nodes. This happens even before Mreaches M1because at M=M1(when the apocket disappears), |Ns/Nss|is already larger than one, as one can immediately see from Eq. (15), if one neglects in this equation the contributions from theapocket. The nodes in /Delta1 b(q) obviously survive up to M22 because at M<M 22the minimal value of sin 2 θqremains −1. At larger M22<M<M 2, this minimal value becomes smaller than one, but we checked numerically that the nodesstill survive and remain present up to M=M 2. To understand the gap structure in between the limits s/lessmuch1 ands/greatermuch1, we solve the 3 ×3 gap equation (14) numerically for several swithMas a running parameter, and for several Mwithsas a running parameter. In the two panels in Fig. 5, we plot the ratios g2/g1andg3/g1as functions of sfor two representative values of M,M=0.09<M 1andM1<M= 0.12<M 2, and as functions of Mfors=5. We see that |g2/g1|is always small, while |g3/g1|increases with increasing sand for M=0.12 becomes larger than 1 above a certain s. Once this happens, the gap on the bpocket develops accidental nodes. Overall, these results show that the behavior that weobtained analytically at large sextends to all s/greaterorequalslant1. For these values of s, the gap along the bpocket, which survives up to larger M, necessarily develops accidental nodes around M=M 1, where the apocket disappears. In Fig. 6, we show the variations of /Delta1a(q) and/Delta1b(q)f o rt w o different values of s,s=1 and 5, and three different values ofM:M/lessmuchM1,M/lessorequalslantM1, andM1<M<M 2.F o rs=1, the gaps /Delta1aand/Delta1bhave no nodes for all M, consistent with |g3/g1|<1i nF i g . 5. The only effect of the disappearance of the apockets is the switch between the maxima and the minima of the gap function along the remaining bpockets. This switch is a consequence of the sign change of both g2/g1 andg3/g1(see Fig. 5). The situation is different for larger s=5. Even for small M/lessmuchM1, the gap variation along both aandbpockets become substantial. Once Mbecomes large enough to (almost) gap out theapockets, the nodes appear near the tips of the bpocket 144527-5S. MAITI, R. M. FERNANDES, AND A. V . CHUBUKOV PHYSICAL REVIEW B 85, 144527 (2012) 0 2 4 6−2−1012 sg2,3/g1 M=0.09 M=0.12 g2/g1 g3/g1 0.05 0.1 0.15−2−1012 Mg2,3/g1g2/g1 g3/g1 FIG. 5. (Color online) Top panel: the ratios g2/g1(blue line) and g3/g1(red line) as function of sforM=0.09 (solid lines) and M= 0.12 (dashed lines). These values are slightly below and slightly above the critical M1=0.102 at which two out of four pockets disappear. Both ratios flip sign between M=0.09 and M=0.12. The key result here is that for s/greaterorequalslant1, the magnitude of g3/g1becomes larger than 1 forM=0.12. This gives rise to accidental nodes of the gap on the remaining FS pocket (see text). Bottom panel: the plot of the sameratios as functions of Mfors=5. The magnitude of g 2/g1remains relatively small, while the magnitude of g3/g1becomes greater than 1f o rM/greaterorequalslantM1. bananas. This behavior is in full agreement with the analysis ofg3/g1earlier in this section. We therefore conclude that the two conditions to obtain nodes in the superconducting gap in the coexistence phasewith stripe antiferromagnetism are (i) relatively large density-density interaction U 1(leading to magnetism) compared to the pair-hopping interaction U3and (ii) the disappearance of one of the two pairs of reconstructed Fermi pockets. Ouranalysis agrees with previous works that found fully gappedquasiparticle excitations in the coexistence state in the casesof small M(see Ref. 13) andU 1=0( s e eR e f . 15). The other interactions, which we listed in Sec. IIbut did not include into Hintin Eq. (11), modify the value of sand therefore affect whether or not the nodes appear upon increasing M. In particular, the exchange term [ U2c† σfσf† σ/primecσ/prime] changes sto s=(U1−U2)/U3.I fU2is negative, it makes the development of nodes more likely, while if it is positive and smaller thanU 1, it decreases sand may eliminate the nodes. The inclusion of this term also opens up the somewhat exotic possibility ofan e g a t i v e s. Although negative sis unlikely for FeSCs, we analyzed the s< 0 case for completeness and show the results in Fig. 7. We see that now the behavior is nonmonotonic with s: the nodes appear at some intermediate |s|and disappear at larger |s|, already at relatively small M, when all four pockets are present.B. From intermediate to large M: nodal-nodeless transition In the previous section, we showed that nodes appear at large enough sat intermediate values of M∼M1, where one of the two pairs of reconstructed pockets (the apockets) disappears. We now analyze what happens when Mbecomes larger than M2andbpockets also disappear. We find that the results depend on whether we restrict the pairing to the FS or include into the pairing problem also thestates which are already gapped out by SDW. If we restrictthe pairing problem to the FS, as we did before, we find thatthe nodes in /Delta1 bare present for all M<M 2.A tM=M2 the remaining FS disappears and Tcvanishes. If we do not restrict the pairing to the FS and take into consideration thestates already gapped by SDW, SC persists into the regionM>M 2(see Refs. 14,16, and 24). In this last region, all states are gapped already above Tc, and the opening of an additional SC gap only moves states further away from zeroenergy. The thermal conductivity and the penetration depthin this last regime show exponential behavior, typical for asuperconductor with a full gap, From this perspective, theevolution of the system response around M=M 2mimics the transformation from a nodal to a nodeless gap, irrespective ofwhether the actual SC gap by itself evolves around M 2. On a more careful look, we find that the gap /Delta1bdoes change its structure near M=M2and becomes nodeless. At around the same M,Tcis likely nonmonotonic and passes through a minimum. The argument goes as follows. Consider first thepairing confined to the FS and solve for T c. Neglecting Ncand Ncsin Eq. (14), which are small for all Mands, and solving for the linearized gap equation for L=ln (/Lambda1 Tc), we obtain 1+U3 2(sNss−N)L+/parenleftbiggU3 2/parenrightbigg2 s/parenleftbig N2 s−NNss/parenrightbig L2=0.(20) One can verify that NNss−N2 sis positive for all M>M 2. Fors/greaterorequalslant1, when /Delta1bhas nodes at M∼M1,w ea l s oh a v e sNss>N . In this situation, there is a single solution of Eq. (20) withL> 0: L=1 sU3/parenleftbig NNss−N2s/parenrightbig ×/bracketleftbig/radicalBig (sNss+N)2−4sN2s+(sNss−N)/bracketrightbig .(21) As long as N2 s/lessmuchNNss,L≈2/U 3N, i.e., Tcdoes not depend strongly on M. However, near M=M2,s i n 2 θ(k) on the bpockets tends to a constant value c,s e eE q . (17), (c=− 0.94 for our parameters) and N2 sandNNssboth tend to the same value c2N2. As a result, Ldiverges as (2/U 3s)(sNss−N)/(NNss−N2 s), i.e., Tcvanishes (see Fig. 8). One can straightforwardly show that Ldiverges at M=M2even if we solve the full 3 ×3 linearized matrix gap equation, without neglecting the terms NcandNcs. In the same limit NNss≈N2 s,w ea l s oh a v e g3 g1sin 2θq=−Ns Nsssin 2θq→−c−c c2=1. (22) We see that the angle-independent and the sin 2 θterms in /Delta1b become identical and cancel each other, i.e., /Delta1bvanishes [see Eq.(13)]. This is consistent with the vanishing of Tc. 144527-6GAP NODES INDUCED BY COEXISTENCE WITH ... PHYSICAL REVIEW B 85, 144527 (2012) −0.5 0 0.5−0.4−0.200.20.4 kx/πky/π aba b M=0.04 −0.5 0 0.5−0.4−0.200.20.4 kx/πky/π aba b M=0.09 −0.5 0 0.5−0.500.5 kx/πky/πb bM=0.12 0 50 100 150 200−0.500.511.5 length along pocketGap Structures=1 a−pocket b−pocket 0 50 100 150−0.500.511.5 length along pocketGap Structures=1 a−pocket b−pocket 0 50 100−0.500.511.5 length along pocketGap Structures=1 b−pocket 0 50 100 150 200−0.500.511.5 length along pocketGap Structures=5 a−pocket b−pocket 0 50 100 150−0.500.511.5 length along pocketGap Structures=5 a−pocket b−pocket 0 50 100−0.500.511.5 length along pocketGap Structures=5 b−pocket FIG. 6. (Color online) The gap structure for two values of s=U1/U 3and three different Mcorresponding to the cases when the aandb pockets are of comparable size (left column), when the apockets are about to disappear (middle column), and when only the bpockets are left (right column). Note the appearance of gap nodes for s=5a n dM=0.12, when one pair of reconstructed pockets disappear. These results, however, hold only as long as we restrict the integrals for N,Ns, andNssto the FS. Once we include the contributions from the gapped states, N2 sandNNssno longer tend to the same value. The contributions from thegapped states contain, instead of L=ln/Lambda1/T c, the factor LD= ln/Lambda1//radicalbig D2+T2c, where Dis the gap in the absence of SC. In a generic weak coupling case, these contributions would besmall in L D/Lcompared to the contributions from the FS, but in our case the FS contribution to NNss−N2 svanishes at M2, and the contribution from the gapped states become thedominant ones. Similarly, contributions from the gapped state break the cancellation between the angle-independent and thesin 2θ qterms in /Delta1b. Combining the contributions from the FS and from the gapped states to Ni, we have near M=M2, N=N0/parenleftbigg 1+γLD L/parenrightbigg ,N s=N0/parenleftbigg cs−γsLD L/parenrightbigg , (23) Nss=N0/parenleftbigg css+γssLD L/parenrightbigg , 0 50 100 150−1−0.500.511.5 length along pocketGap Structures=−1a−pocket b−pocket 0 50 100 150−1−0.500.511.5 length along pocketGap Structures=−1.5a−pocket b−pocket 0 50 100 150−1−0.500.511.5 length along pocketGap Structures=−5a−pocket b−pocket FIG. 7. (Color online) Gap structure for negative sandM=0.09. As |s|increases, the gap on the bpocket is pushed down, acquires the nodes, and eventually reverses sign and again becomes nodeless. For all three values of s, the FS contains both aandbpockets. 144527-7S. MAITI, R. M. FERNANDES, AND A. V . CHUBUKOV PHYSICAL REVIEW B 85, 144527 (2012) where cs≈−c> 0 andcss≈c2. One can introduce /epsilon1≡css− c2 sas a measure of deviation from M2, with /epsilon1=0a tM2.T h e constants γiare all positive, at least if they predominantly come from gapped apockets. Since sin 2 θqalong the apockets is quite close to −1( s e eF i g . 4), we have γ2 s=γγssto a good accuracy. Substituting these forms into the result for L, Eq.(21), and assuming for simplicity that s/greatermuch1, we obtain for small /epsilon1, /epsilon1˜L2+K1˜L−K2=0, K1=(γc2+γss+2γs|c|)˜LD−c2, (24) K2=γss˜LD>0, where ˜L=L(N0U3/2),˜LD=LD(N0U3/2). If we set ˜LD= 0, i.e., neglect the contributions from the gapped states, K2= 0,K1=−c2, and we obtain ˜L=c2//epsilon1, as before. If instead we set /epsilon1=0, we find that Lremains finite and positive and, to leading order in ˜L−1 D<1, is given by ˜L/epsilon1=0≈γss γss+γc2+2γs|c|. (25) The behavior of ˜Lat small but finite /epsilon1is rather involved, but its main features can be understood directly from Eq. (24). Far from M2,Tcis not small, implying that ˜LDis small, i.e., and K1<0 and K2can be neglected. As a result, ˜Lincreases as ˜L∼1//epsilon1, leading to a decrease in Tcand, consequently, to an increase in ˜LD. With increasing ˜LD,K1 crosses zero and changes sign. At this point, K2=O(1) and ˜L∼K2/√/epsilon1∼1/√/epsilon1, i.e., it still increases with decreasing /epsilon1, but more slowly than before. Because ˜L/epsilon1=0∼O(1) given by Eq.(25) is certainly smaller than 1 /√/epsilon1,˜Lnecessary passes through a maximum near the point where K1changes sign and then decreases toward ˜L/epsilon1=0. Accordingly, Tc=/Lambda1e−2˜L/U 3N0 passes through a minimum at some M/lessorequalslantM2,a ss h o w n schematically in Fig. 8. Contributions from the gapped states also affect the in- terplay between the constant and the sin 2 θqterms in /Delta1b.I f we include these terms, we obtain, near M=M2, instead of FIG. 8. (Color online) (a) Schematic behavior of Tcas a function ofM,i fTcis obtained by restricting to contributions only from the FSs. This Tcdrops to zero as M→M2. The arrow indicates the direction along which Mincreases. (b) The actual Tc(solid red line) is a superposition of contributions from the FS and from thegapped states (dashed lines). This T cinitially decreases when M approaches M2but reverses the trend and remains finite at M2and even in some range of M>M 2. The dip in Tcis in the region where the contributions from the FS and from the gapped states become comparable to each other. This nonmonotonic behavior of Tcaround M2is captured by Eq. (24) (see text).Eq.(22), g3 g1sin 2θq=1+γs cLD L 1+γss c2LD L, (26) where, we remind, in our case, γs,ss>0 and c=− 0.94. We see that the numerator gets smaller than one and thedenominator gets larger than one. As a result, | g3 g1sin 2θq| becomes smaller than one and the gap /Delta1brecovers its nodeless form in the vicinity of M2. Very likely /Delta1bremains gapless also at larger M, before Tcfinally vanishes. We also note that our consideration is self-consistent in the sense that the transformation from nodal to nodeless gap andnonvanishing of T catM=M2, where /epsilon1=0, are consistent with each other. Indeed, once the gap /Delta1bbecomes nodeless, the pairing problem at M/greaterorequalslantM2is qualitatively similar to the one for fermions with the dispersion εgapped (k)=/radicalbig D2+[vF·(k−kF)]2, (27) where Dis the pre-existing gap in the excitation spectrum. For such model, Tcis nonzero at D=0 (equivalent of M=M2) and extends into the region where D/negationslash=0: Tc(D)=Tc(0)F/bracketleftbiggD Tc(0)/bracketrightbigg , (28) where F(x) is a decreasing function of xsubject to F(x→ 0)=1−x2andF(x)=− 1/ln(1.76−x) near the critical x=1.76. The outcome of this analysis is that nodal SC gap exists only in relatively narrow range of M, from M/lessorequalslantM1toM/lessorequalslant M2. At larger M, the gap is again nodeless, and, furthermore, forM>M 2all low-energy states are gapped already above Tc. The SC transition temperature Tcis non-zero at M2and decreases into M>M 2region. Before that, Tchas a minimum roughly where the gap changes from nodal to nodeless. C. Role of the nonreconstructed pockets So far, we have restricted our consideration of the pairing problem to one hole and one electron pocket reconstructed bySDW. There are additional pockets that do not participate inthe SDW state. 17In this section, we analyze to what extent these additional pockets affect our results. Specifically, weadd another elliptical electron pocket centered at ( π,0), denote fermions near this pocket by d k, and solve the set of coupled linearized gap equations for /Delta1a(q),/Delta1b(q), and /Delta1d(q). The interactions involving fermions near (0 ,0) and near the electron pocket at ( π,0) are the same as the interactions between (0 ,0) and (0 ,π), i.e., the couplings are the same U1,U2,U3, andU4. As before, we consider the model with only U3andU1=sU3. We also neglect direct interaction between electron pockets. The effective pairing interactions in the a-b-dspace are obtained by dressing the interactions by coherence factorsassociated with the transformation from candftoaand boperators for the pockets at (0 ,0) and at (0 ,π). Since the expressions for the vertices are long, we refrain frompresenting them. Yet, it is straightforward to obtain and solvethe set of linearized gap equations for /Delta1 a(q),/Delta1b(q), and /Delta1d(q). It is essential that the “dressing” does not involve d fermions, hence the interaction has no dependence on the 144527-8GAP NODES INDUCED BY COEXISTENCE WITH ... PHYSICAL REVIEW B 85, 144527 (2012) 0 100 200 300 400−1−0.500.511.5 length along pocketGap StructureM=0.04a−pocket b−pocket d−pocket 0 100 200 300 400−1−0.500.511.5 length along pocketGap StructureM=0.09a−pocket b−pocket d−pocket 0 100 200 300 400−1−0.500.511.5 length along pocketGap StructureM=0.12b−pocket d−pocket FIG. 9. (Color online) The gap structure for s=5a n dt h es a m ev a l u e so f Mas used in Fig. 6for the case when we include an additional, spectator pocket at ( π,0), which does not participate in the FS reconstruction. A comparison with Fig. 6shows that the gap structure is almost unaffected by the presence of the spectator pocket. angle along the d-FS. As a consequence, /Delta1d(q) remains angle independent. This constant /Delta1d, however, affects the interplay between the angle-independent and the cos 2 θkand sin 2 θk terms for /Delta1a(q) and/Delta1b(q). In Fig. 9, we show the gap structure obtained from the solution of the pairing problem for s=5 and the same three values of Mas in Fig. 6. Comparing Figs. 6and9, we see that the nonreconstructed dpocket has only a minor effect on the a-bgap structure—the gap on the bpocket still becomes more angle dependent with increasing Mand develops accidental nodes at M/greaterorequalslantM1. In view of this result, it is very likely that the physics of gap variation with Mis fully captured already within the two-band model of one hole and one electron pocket. IV . EFFECT OF AN EXTERNAL MAGNETIC FIELD We now consider how the SC gap structure is affected by an external magnetic field H. The specific goal is to verify whether the field increases or reduces the tendencytoward the development of nodes in /Delta1 b. This issue is related to experiments on (Ba 1−xKx)Fe 2As2, particularly to recent measurements of thermal conductivity in the presence of afield. 23 We direct the field along zand include into the Hamiltonian the Zeeman coupling between the field and the zcomponent of the spin of a fermion Sz=(1/2)c† ασz αβcβ. For an isotropic system, the SDW vector Mis oriented transverse to Hand, for definiteness, we direct it along x.A tafi n i t e H, the system also develops a nonzero uniform magnetization, which rotatesa spin at a given site toward the field and creates a canted two-sublattice structure, with orthogonal antiferromagnetic SDWcomponent M=M xˆxand ferromagnetic component along z. The quadratic part of the Hamiltonian in the presence of the field is H=/summationdisplay k,σ/bracketleftbig/parenleftbig εc k+σh/parenrightbig c† kσckσ+/parenleftbig εf k+Q+σh/parenrightbig f† k+Qσfk+Qσ/bracketrightbig +/summationdisplay kM(c† k↑fk+Q↓+c† k↓fk+Q↑+H.c.), (29) where h=μBH. We will measure hin units of 2 μ, as with other observables. The transformation to the new operators is now given by /parenleftbiggσakσ bkσ/parenrightbigg =/parenleftbigg−σcosθk,σ sinθk,σ −sinθk,σ−σcosθk,σ/parenrightbigg/parenleftbiggckσ fk+Qσ/parenrightbigg ,(30)where cosθk,σ=M/radicalbig M2+(E−σ)2,sinθk,σ=E− σ/radicalbig M2+(E−σ)2, (31) and E− σ=/parenleftbiggεc−εf 2+σh/parenrightbigg −/radicalBigg/parenleftbiggεc−εf 2+σh/parenrightbigg2 +M2. (32) Substituting the transformation into Hintin Eq. (11) and taking care of the fact that cos θk,↑and cos θk,↓are now different, we obtain Eq. (12) but with new Uaa q,k,Ubb q,k, andUab q,kgiven by Uaa q,k=Ubb q,k=−U3 2(Cq↑C−q↓Sk↑S−k↓+Sq↑S−q↓Ck↑C−k↓) +U1Cq↑S−q↓Ck↑S−k↓, Uab q,k=Uba q,k=−U3 2(Cq↑C−q↓Ck↑C−k↓+Sq↑S−q↓Sk↑S−k↓) −U1Cq↑S−q↓Ck↑S−k↓, where Ck,σ≡cosθk,σandSk,σ≡sinθk,σ. The gap structure consistent with these Uij q,kis of the form /Delta1a(q)=g1Cq↑C−q↓+g2Sq↑S−q↓ +g3Cq↑S−q↓+g4Sq↑C−q↓, /Delta1b(q)=g1Cq↑C−q↓+g2Sq↑S−q↓ −g3Cq↑S−q↓−g4Sq↑C−q↓. We constructed the set of coupled linearized gap equations forgiby standard means and solved them for various M, s=U1/U 3, andh. We found that the field enhances the angle variation of the gaps /Delta1aand/Delta1b, such that nodes appear at smaller Mand smaller s. To illustrate this, in Fig. 10,w e compare the gap structure for a given M=0.04 and s=1 (same as in the upper left panel in Fig. 6) without a field and with a field. We clearly see that the gap variations along thereconstructed FSs grow with the field, and for large enoughfield nodes appear well before Mreaches the value at which theapockets disappear. The implication is that the range where the gap has nodes widens up with the application of afield and, in particular, extends to smaller M(larger dopings), where without a field the system was a superconductor with anodeless gap. 144527-9S. MAITI, R. M. FERNANDES, AND A. V . CHUBUKOV PHYSICAL REVIEW B 85, 144527 (2012) 0 50 100 150 200−1−0.500.511.5 length along pocketGap Structure s=1 h=0a−pocket b−pocket 0 50 100 150 200−1−0.500.511.5 length along pocketGap Structure s=1h=0.08 a−pocket b−pocketFIG. 10. (Color online) The gap structure for h=0 (left panel) and h=0.08 (right panel) for M=0.04 and s=1. The application of the field increases angle variations of the gaps and leadsto the appearance of nodes at a smaller M(and a smaller s) than without a field. We emphasize that the extension of the nodal region to smaller Mis a different effect than the field-induced changes inM1andM2. These latter changes are due to the modification of the quasiparticle dispersion, which in the presence of thefield becomes four-band dispersions and takes the form E a,b k=εc+εf 2±/radicalBigg/parenleftbiggεc−εf 2±h/parenrightbigg2 +M2. (33) The two apockets are split by ±hintoa+anda−pockets, and the two bpockets are split by ±hintob+andb−pockets. The presence of ±hunder the square root splits M1andM2 intoM+ 1,M− 1andM+ 2,M− 2, where M+ 1,2=M1,2(1+2h) and M− 1,2=M1,2(1−2h), with M1andM2given by Eqs. (9)and (10) (we recall that his measured in units of 2 μ). Forh=0.08 used in Fig. 10, we get M+ 1=0.118,M− 1=0.086 instead ofM1=0.102, and M+ 2=0.205,M− 2=0.149 instead of M2=0.177. Clearly, the field-induced changes in the form of reconstructed FSs are small compared to the changes ofthe SC gap structure associated with the field-induced changeof the pairing interaction. The small effect of the field onthe reconstructed FS is in agreement with previous workson the cuprates. 25For another perspective of the increase in anisotropy of the gap structure with magnetic field see Ref. 26. V . DISCUSSION AND CONCLUDING REMARKS We showed that the superconducting gap of an s+−SC in the co-existence phase with an SDW order acquires additionalangular dependence compared to the case when SDW orderis not present. In particular, an isotropic pairing interactionbecomes angle dependent inside the SDW state. At small SDWorder parameter M, or when the density-density interaction between the hole and electron pockets is small comparedto pair hopping interaction term, this extra dependence isweak, in agreement with previous studies. 13,15However, when density-density and pair hopping interactions are comparableandMis large enough to gap out one out of two pairs of reconstructed FS pockets, the angular dependence gets quitestrong and the SC gap on the remaining FS banana-like pocketdevelops accidental nodes near the tips of the bananas. At evenlarger M, the system bounces back into the fully gapped SC state which extends into the regime where large enough M gaps out the remaining pair of FS pockets. The SC transitiontemperature T chas a dip near the point where nodal SC state becomes nodeless. In the presence of a magnetic field, thewidth of the nodal region expands, and, in particular, the nodesappear at smaller M(larger doping).Our results offer a consistent explanation for the recent experimental observations 23in underdoped (Ba 1−xKx)Fe 2As2. Performing in-plane thermal conductivity measurements, Reidet al. found a small range of doping in the SC-SDW coexistence region where the ratio κ/T is finite at T=0, i.e., κis linear inTat small T. This linearity is generally viewed as a strong indication for the presence of the nodes in the SC gap. Atsmaller and larger dopings Reid et al. found that κ/T vanishes atT=0, as it is expected for a fully gapped superconductor. They also observed that the doping range where κ/T is finite atT=0 expands upon the application of an external magnetic field. In terms of our model, the nodeless-nodal-nodeless transi- tion observed by Reid et al. can be attributed to the following sequence of events: at the optimal doping, which roughlycoincides with the onset of the coexistence range, the SC gapsin (Ba 1−xKx)Fe 2As2are almost isotropic, and the excitation spectrum is fully gapped. Decreasing xand moving to the underdoped region in which SC and SDW start to coexist isequivalent to making Mnonzero in our model. In this situation, the gaps become anisotropic, but remain nodeless over somerange of M. Moving deeper inside the coexistence region is equivalent to making Mlarger. Eventually, Mbecomes large enough such that one of the reconstructed pairs of pocketsvanishes. In this situation, we found (for s/greaterorequalslant1) that the gap on the remaining FS pockets develops nodes, which leads to afiniteκ/T atT=0. As the doping level decreases even further, Mcontinues to increase and the gap structure bounces back to nodeless, since for such pairing state T cremains nonzero even when the remaining pair of reconstructed pockets disappears.The quasiparticle spectrum becomes fully gapped again, andκ/T vanishes at T=0. In this explanation, the nodeless-nodal transition is roughly associated with the vanishing of one pair of reconstructedFS pockets. An SDW-driven electronic transition has beenobserved by ARPES as well as by transport measurementsin electron-doped Ba(Fe 1−xCox)2As2.27In that case, however, the reconstructed pocket that disappears is a much smallerholelike pocket associated with additional details of the banddispersion not captured by our simplified two-band model.It would be interesting to verify experimentally whetherthe transition from four to two pockets takes place in thehole-doped (Ba 1−xKx)Fe 2As2materials, and what is their relationship to the SC gap structure. Such transitions shouldhave distinct signatures not only in the band dispersions, butalso in transport coefficients. Furthermore, as discussed inRefs. 28and 29, the onset of accidental nodes should also affect the low-temperature behavior of several thermodynamic 144527-10GAP NODES INDUCED BY COEXISTENCE WITH ... PHYSICAL REVIEW B 85, 144527 (2012) quantities, giving rise to peculiar scaling relations. Another result of our analysis is the observation of a dip in the dopingdependence of T cnear the doping where the system undergoes the nodal-nodeless transition. Our model indeed does not include all aspects of the physics of the co-existence state. In particular, doping with holes orelectrons changes the chemical potential, which was kept con-stant in our calculation. The feedback from this change affectsthe values M 1andM2at which reconstructed pockets disappear leaving a possibility that reconstructed FSs can be presenteven for zero doping. The angular dependence of the magneticinteraction may also preserve FS pockets even for large M. 22 Still, we believe that the physics described by our model is quite genetic and should hold for more realistic models. Finally, we point out that the disappearance of two out of four pockets is not the only mechanism that can give rise tonodes in the coexistence state. When the exchange interaction between unreconstructed hole and electron pockets is strongenough, it can induce nodes even in the absence of any dramaticchange of the reconstructed FSs. Conversely, there is also aregion in parameter space (small s) in which the coexistence with SDW does not generate nodes in the SC gap. ACKNOWLEDGMENTS We are thankful to L. Taillefer, J.-P. Reid, R. Prozorov, M. Tanatar, J. Schmalian, I. Eremin, J. Knolle, for usefuldiscussions and for sharing unpublished results with us.The work was supported by NSF-DMR-0906953 (S. M andA.V .C) and by the NSF Partnerships for International Researchand Education (PIRE) program (R.M.F.). A.V .C gratefullyacknowledges partial support from Humboldt foundation. 1K. Terashima, Y . Sekiba, J. H. Bowen, K. Nakayama, T. Kawahara, T .S a t o ,P .R i c h a r d ,Y . - M .X u ,L .J .L i ,G .H .C a o ,Z . - A .X u ,H .Ding, and T. Takahashi, Proc. Natl. Acad. Sci. USA 106, 7330 (2009). 2K. Gofryk, A. S. Sefat, M. A. McGuire, B. C. Sales, D. Mandrus,J. D. Thompson, E. D. Bauer, and F. Ronning, Phys. Rev. B 81, 184518 (2010). 3R. T. Gordon, N. Ni, C. Martin, M. A. Tanatar, M. D. Vannette,H. Kim, G. D. Samolyuk, J. Schmalian, S. Nandi, A. Kreyssig, A. I.Goldman, J. Q. Yan, S. L. Bud’ko, P. C. Canfield, and R. Prozorov,P h y s .R e v .L e t t . 102, 127004 (2009). 4Y-M. Xu, Y-B. Huang, X-Y . Cui, E. Razzoli, M. Radovic, M. Shi, G-F. Chen, P. Zheng, N-L. Wang, C-L. Zhang, P-C. Dai, J-P. Hu,Z. Wang, and H. Ding, Nat. Phys. 7, 198 (2011). 5X. G. Luo, M. A. Tanatar, J.-Ph. Reid, H. Shakeripour, N. Doiron-Leyraud, N. Ni, S. L. Bud’ko, P. C. Canfield, HuiqianLuo, Zhaosheng Wang, Hai-Hu Wen, R. Prozorov, and LouisTaillefer, Phys. Rev. B 80, 140503 (2009). 6C. Martin, R. T. Gordon, M. A. Tanatar, H. Kim, N. Ni, S. L. Bud’ko, P. C. Canfield, H. Luo, H. H. Wen, Z. Wang, A. B.V orontsov, V . G. Kogan, and R. Prozorov, P h y s .R e v .B 80, 020501 (2009). 7T. Shimojima, F. Sakaguchi, K. Ishizaka, Y . Ishida, T. Kiss,M. Okawa, T. Togashi, C.-T. Chen, S. Watanabe, M. Arita,K. Shimada, H. Namatame, M. Taniguchi, K. Ohgushi, S. Kasahara,T. Terashima, T. Shibauchi, Y . Matsuda, A. Chainani, and S. Shin,Science 332, 564 (2011). 8K. Hashimoto, M. Yamashita, S. Kasahara, Y . Senshu, N. Nakata, S. Tonegawa, K. Ikada, A. Serafin, A. Carrington, T. Terashima,H. Ikeda, T. Shibauchi, and Y . Matsuda, P h y s .R e v .B 81, 220501 (2010). 9D. K. Pratt, W. Tian, A. Kreyssig, J. L. Zarestky, S. Nandi, N. Ni,S .L .B u d ’ k o ,P .C .C a n fi e l d ,A .I .G o l d m a n ,a n dR .J .M c Q u e e n e y ,P h y s .R e v .L e t t . 103, 087001 (2009). 10S. Avci, O. Chmaissem, E. A. Goremychkin, S. Rosenkranz, J.-P. Castellan, D. Y . Chung, I. S. Todorov, J. A. Schlueter, H. Claus,M. G. Kanatzidis, A. Daoud-Aladine, D. Khalyavin, and R. Osborn,P h y s .R e v .B 83, 172503 (2011). 11M.-H. Julien, H. Mayaffre, M. Horvatic, C. Berthier, X. D. Zhang, W. Wu, G. F. Chen, N. L. Wang, and J. L. Luo, Europhys. Lett. 87, 37001 (2009).12E. Wiesenmayer, H. Luetkens, G. Pascua, R. Khasanov, A. Amato,H. Potts, B. Banusch, H.-H. Klauss, and D. Johrendt, Phys. Rev. Lett.107, 237001 (2011). 13D. Parker, M. G. Vavilov, A. V . Chubukov, and I. I. Mazin, Phys. Rev. B 80, 100508 (2009). 14A. B. V orontsov, M. G. Vavilov, and A. V . Chubukov, Phys. Rev. B 81, 174538 (2010). 15R. M. Fernandes and J. Schmalian, P h y s .R e v .B 82, 014521 (2010). 16R. M. Fernandes and J. Schmalian, Phys. Rev. B 82, 014520 (2010). 17I. Eremin and A. V . Chubukov, P h y s .R e v .B 81, 024511 (2010); J. Schmiedt, P. M. R. Brydon, and C. Timm, e-print arXiv:1108.5296v1 . 18R. Fernandes, A. V . Chubukov, J. Knolle, I. Eremin, and J. Schmalian, Phys. Rev. B 85, 024534 (2012); 85, 109901(E) (2012). 19A. V . Chubukov, Physica C 469, 640 (2009). 20J. Knolle, I. Eremin, J. Schmalian, and R. Moessner, Phys. Rev. B 84, 180510(R) (2011). 21The actual situation may be more complex as at least in some orbital models SDW gap turns out to be angular dependent and vanishesalong particular directions. 21Near these directions, FS survives even when the SDW order parameter is large. 22Y . Ran, F. Wang, H. Zhai, A. Vishwanath, and D.-H. Lee, Phys. Rev. B 79, 014505 (2009). 23J.-Ph. Reid, M. A. Tanatar, X. G. Luo, H. Shakeripour, S. Ren ´e de Cotret, A. Juneau-Fecteau, N. Doiron-Leyraud, J. Chang,B. Shen, H.-H. Wen, H. Kim, R. Prozorov, and Louis Taillefer,e-print arXiv: 1105.2232 v2 (2011); R. Prozorov and M. 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PhysRevB.83.205423.pdf

PHYSICAL REVIEW B 83, 205423 (2011) Monoatomic and dimer Mn adsorption on the Au(111) surface from first principles Francisco Mu ˜noz,1,2Aldo H. Romero,3Jose Mej ´ıa-L´opez,1,2and J. L. Mor ´an-L ´opez4 1Departamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut ´onoma de M ´exico, M ´exico D.F . M ´exico 2Centro para el Desarrollo de la Nanociencia y la Nanotecnolog ´ıa CEDENNA, Avda. Ecuador 3493, Santiago, Chile 3CINVESTAV , Unidad Quer ´etaro, Libramiento Norponiente 2000, Real de Juriquilla, CP 76230, Quer ´etaro, M ´exico 4Departamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut ´onoma de M ´exico (UNAM), M ´exico D. F . M ´exico (Received 9 December 2010; revised manuscript received 14 March 2011; published 23 May 2011) A theoretical study based on the density functional theory of the adsorption of Mn monomers and dimers on a Au-(111) surface is presented. As necessary preliminary steps, the bulk and clean surface electronicstructure are calculated, which agree well with previous reports. Then, the electronic structure of the Mn adatom,chemisorbed on four different surface geometries, is analyzed. It is found that the most stable geometry iswhen the Mn atom is chemisorbed on threefold coordinated sites. Using this geometry for a single adatom asecond Mn atom is chemisorbed and the most stable dimer geometrical structure is calculated. The lowest-energyconfiguration corresponds to the molecule lying parallel to the surface, adsorbed on two topological equivalentthreefold coordinated sites. It is also found that the lowest-energy magnetic configuration corresponds to theantiferromagnetic arrangement with individual magnetic moments of 4.64 μ B. Finally, it is concluded that the dimer is not stable and should fragment at the surface. DOI: 10.1103/PhysRevB.83.205423 PACS number(s): 34 .35.+a, 75.70.Rf, 31 .10.+z, 31.15.A− I. INTRODUCTION Most of the foreseen technological applications of nanos- tructures imply the deposition of well-characterized nanopar-ticles, or thin films, on particular substrates. Thus, aftera full characterization of free clusters and of the chosensubstrate, it is imperative to know the changes introduced bythe substrate-cluster interactions. In particular, the growth ofultrathin films or nanostructures based on magnetic transitionmetals and deposited on different substrates has been thesubject of intensive research, 1–6but the effect of the substrate- nanostructure interactions is not fully understood. Depending on the substrate-adatom interaction, one may expect important modifications to the magnetic properties ofthe transition-metal adatoms and perhaps the appearance ofnew magnetic phases in structures adsorbed on nonmagneticsubstrates. One of the systems that has garnered much attentionis the adsorption of Mn on metallic surfaces because of thefollowing: (i) Mn is the transition metal with the highest atomicmagnetic moment (5 μ B) and can be used to deposit overlayers of ferromagnetically coupled atoms which could lead to tech-nological applications, (e.g., spintronics 7). (ii) A fundamental motivation is to understand the interplay between the differentorbitals on the bonding process to metallic substrates. Ingeneral, it is expected that after adsorption, the bonding andhybridization would decrease the atomic magnetic moment,unless charge transfer from the surface to the majority-spinlevels compensates that reduction. (iii) Recent calculationson free Mn dimers show that the magnetic coupling dependson the distance between the two atoms. 8The ferromagnetic (FM) and antiferromagnetic (AFM) configurations are stableabove and below 3.06 ˚A, respectively. The ground state is the AFM arrangement with a bond distance of 2.89 ˚A. Although chemisorpion may produce many changes in the electronicstructure of the deposited species, it is interesting to seeif under chemisorption, on particular substrates and surfaceorientations, the dimer bond length can be modified andinduce one coupling or the other. As mentioned above, dueto the high atomic Mn magnetic moment, it is interesting to find out whether under particular circumstances ferromagneticcoupled nanostructures or thin films can be generated. It is well known that Mn in bulk samples presents a rich variety of magnetic behaviors, which depend on crystallinestructure, temperature, and pressure. 9Thanks to that, a com- plex behavior is observed in small clusters, where the average magnetic moment shows a non-monotonic size dependence.10 The theoretical results by Mej ´ıaet al.8,11on very small clusters offer as a possible explanation for such behavior. Theypropose that complex noncollinear structures are produced due to the AFM interactions between nearest neighbors and the ferromagnetic interactions among more distant pairs. Thus, itis of great relevance to study the changes on the magneticproperties, of multiatom clusters, induced by the adsorption process. Experimentally it has been observed that the growth of Mn on noble-metal substrates has the following characteristics: For low coverage the adatoms remain at the surface onparticular sites. Upon increasing the coverage, there is atendency to exchange sites between the topmost surface atomsand Mn, forming a surface alloy. Further deposition leadsto Mn islands. This growth mechanism has been observed,by scanning tunneling microscopy (STM), on Cu(100), 12 Cu(110),13Ag(100),14and Au(111)15surfaces. From a theoretical point of view, several ab initio and semiempirical calculations have found, in addition to the FMand AFM arrangements, noncollinear magnetic behavior inboth supported 16and free Mn clusters.8,17There is a general consensus that Mn clusters supported on a metal surfacedevelop noncollinear ordering due mainly to frustration in theAFM order. 18Another feature observed on small Mn clusters when deposited on some specific metal surfaces is magneticbistability, where, due to an almost degenerate ground-stateenergy, there is a coexistence of both FM and AFM ordering. 19 Finally, it is important to note that most theoretical studiesof Mn clusters supported on noble metals neglect a proper 205423-1 1098-0121/2011/83(20)/205423(9) ©2011 American Physical SocietyFRANCISCO MU ˜NOZ et al. PHYSICAL REVIEW B 83, 205423 (2011) structural optimization of the substrate, mainly due to high computational costs. Nevertheless, we are convinced that aclear interpretation of the chemisorption process of a singleadatoms or dimers of molecules, including the structureoptimization, is of vital importance in understanding thegrowth and properties of larger nanostructures. Here, we report a set of total energy calculations of the adsorption of Mn adatoms and dimers on Au(111) surfaces.We calculate the electron density distribution, the magneticmoment, and the bond lengths at various chemisorptionsites. In Sec. IIa brief description of the computational approach is presented. Then, in Sec. III, we discuss the clean Au(111) surface properties and compare our findings withresults previously reported in the literature. The energetics andstructure of a single Mn atom adsorption on the Au surface arepresented in Sec. IV. The results on the adsorption of the Mn dimer are contained in Sec. V. Finally, we discuss our results and present the conclusions in Sec. VI. II. COMPUTATIONAL METHODS We performed a density functional theory (DFT)20study as implemented within the framework of the Vienna Ab-initio Simulation Package ( V ASP ).21–24We consider only valence electrons and describe them with projector-augmented-wave(PAW) types of pseudopotentials 25,26to take into account spin-orbit interactions (SOIs). It is worth noticing that althoughSOIs in the manganses atoms are important, as compared toother noble metals, they are much smaller than the SOIs inAu atoms and influence only to a small extent our numericalresults. For the exchange correlation we use the Perdew-Burke-Erhenzof (PBE) description. 27The energy cutoff for the plane waves was set at 260 eV in all calculations. This value assuresa force convergence of less than 0.01 eV /˚A. Our study started by testing the exactness of the assump- tions and approximations made. For that purpose, we started bycalculating the ground-state electronic structure of bulk gold.As a result, we found that Au crystallizes with a fcc geometrywith a lattice parameter of 4.17 ˚A, a value that compares well with the experimental one, 4 .08˚A. 28We also noticed that the calculated electronic structure reproduces very well previousreports. With the confidence that our approximation describesthe bulk system well, next we calculated the electronicstructure of the Au(111) surface. We modeled the surfaceregion as a slab of five layers; two of them were kept fixedto the bulk parameters and the other three were allowed torelax. After the total relaxation of the atomic positions of thethree topmost layers we found that the surface layer expands2%, and the second one compresses 0.8%. This behavioralso compares well with the reported experimental values of3.3% and 1%, respectively. 29It is well documented that the Au (111) surface reconstructs with a long periodicity,29,30and its simulation requires to consider a (22 ×√ 3) cell.31Since we are interested in the chemisorption of only one and two Mnatoms, the surface reconstruction is not taken into account inthis study. To simulate the monomer and dimer Mn adsorption, we employed a 3 ×3 supercell (1 /9 and 2 /9 coverage) along the surface plane and considered the five layers for theAu surface slab. Due to the fact that the wave function isexpanded in plane waves, we have to consider a large empty space between periodic images perpendicular to the surface.In our calculation we took a distance of 12 ˚A(equivalent to five surface layers) and checked that our results did notdepend on this specific value. The surface energy changes byless than 0.2% when we increase the vacuum thickness to14˚A. Furthermore, in the calculation of the surface electronic structure and the adsorption process, the geometry was relaxeduntil the forces were smaller than 0.03 eV /˚A. Finally, due to the metallic character of the surface, we considered a K-mesh of 12×12×1, which allowed us to obtain a 0.01 eV accuracy in the total energy of the 3 ×3 supercell. III. Au CLEAN SURFACE The Au(111) surface electronic structure was obtained after a geometrical relaxation perpendicular to the surface. Wepresent in Fig. 1the electronic redistribution in the atoms close to the surface. Here, we plot the difference betweenthe converged charge density of the system ρ system , and the superposition of the noninteracting free atomic chargedensities ρ atomic , δρ=ρatomic−ρsystem. (1) This function is helpful in order to visualize the electronic rearrangement due to bonding and local symmetry. In the left-hand panel we show the atoms at the surface layer. Positivevalues (light yellow) mean a higher electron density in thecomposed system as compared to the atomic distribution, andnegative values (red) denote zones where the opposite occurs. In the three right-hand panels we show the charge redis- tribution on planes perpendicular to the surface and passingthrough the various stacking lines of atoms. The first one showsa side view of the electronic redistribution on the plane thatpasses through the surface atoms (type A), and those on thefourth layer. One can observe that the surface atoms retainmore the atomic electron configuration due to the smallercoordination and the metallic character. The next panel showsthe redistribution on a plane that passes through the atoms at thesecond stacking layer (type B). One can notice a slight effectproduced by the surface. The rightmost panel correspondsto the electronic redistribution on a plane passing through thethird layer corresponding to atoms in the fcc stacking sequenceof type C. As mentioned above, the relaxation produces an expansion of 2% of the surface layer, and a compression of 0.8% of thesecond layer. This can be explained by noticing that the lowcoordination of the outermost layer produces a high attractionto the electronic cloud by the ions and a repulsion by theelectronic density of the second layer (see Fig. 1). The surface energy, defined by E s=Esystem−NE bulk 2N, (2) where Nis the number of surface atoms involved, was calculated and is given in Table I, where our results are compared with previous theoretical studies, based on twodifferent theoretical methods. 32,33There is also a recent publication34in which a surface energy of 0.5 eV per atom 205423-2MONOATOMIC AND DIMER Mn ADSORPTION ON THE ... PHYSICAL REVIEW B 83, 205423 (2011) 02468 0.300.250.200.150.100.050.000.050.100.15 02 468 02468 0 2 4 6 8505 1015 052025 AB C FIG. 1. (Color online) Charge redistribution δρof a Au(111) relaxed surface. The left-hand image shows the topmost surface plane passing through the relaxed surface atoms. The images A, B, C show the electronic redistribution at planes perpendicular to the surface and passing through the atoms of the three different layers (ABC fcc stacking). The distances are in ˚A, and the color bar in e/˚A3. is reported. This value compares also well with our results (0.45 eV/atom). The surface reconstruction of the Au(111) surface has been well characterized by diverse experimental methods.29,30,35 The reconstruction consists of a periodic displacement of 46 surface atoms (two rows of 23), where close to 2 /3a r ei n a fcc arrangement and ∼1/3 in hcp locations. To simulate the observed experimental reconstruction it is necessary toconsider a very large (22 ×√ 3) cell,31a calculation that demands large computational resources. We believe that thislong-range atomic redistribution is not necessary to take intoaccount in this study since our cell is much smaller than the oneneeded to model the reconstruction. Furthermore, it has beenshown that the chemical activity of the surface is dominatedby the topmost surface atoms located at fcc stacking sites. 31 Although, the SOIs present in the gold atoms are not taken into account in the Mn adsorption, for the sake ofcompleteness we calculated the surface energy including theseinteractions. We found that the surface energy gets increased to0.059 eV /˚A 2, but the changes in the geometrical structure are minor; the interatomic distances differ only by 0.001 ˚A. Thus, we ignore the SOI in the rest of the study. Even still, we didspecific testing of the adsorption geometries by consideringthis correction and the adsorption energies changed by lessthan 5%. TABLE I. Calculated surface energy ( Es)i n eV/˚A2compared to published results. Es(This work) 0.044 Es(Ref. 32) 0.055 Es(Ref. 33) 0.049IV . SINGLE Mn ATOM ADSORPTION Once the Au (111) surface had been characterized, we proceeded with the adsorption of a single Mn atom. As shownin Fig. 2, the (111) fcc surface offers four different symmetric adsorption sites: on top of a surface Au atom (A), in thebridge position between two surface Au atoms (AA), or inthree coordinated sites. The three coordinated sites are oftwo types, one that follows the hcp sequence (B), and otherthat follows the fcc stacking order (C). In Fig. 2, the purple circles A, B, C, and AA denote the chemisorption sites. Thefirst, second and third layer surface atoms are denoted asdark, medium and light gray circles, respectively. To findthe equilibrium geometry, we performed an optimization ofthe position of both the adatom and the atoms at the three sur-face layers (but keeping the two further layers fixed, as before). A AA BC FIG. 2. (Color online) Adsorption sites on a fcc (111) surface. The site of type A is on the top of a surface atom, B and C, are abovethe surface and coordinated to three surface atoms, and AA is bonded to two surface atoms. The difference between B and C is that B has a Au neighbor in the second layer while C does not. 205423-3FRANCISCO MU ˜NOZ et al. PHYSICAL REVIEW B 83, 205423 (2011) T A B L EI I .A d s o r p t i o ne n e r g yi ne V ,t h eM n magnetic moment in μB, and the distance between the Mn atom and the surface plane ( dMn-surf )i n˚A. Site EA μd Mn-surf A −1.95 5.07 2.44 AA −2.73 4.83 2.52 B −2.80 4.83 2.58 C −2.81 4.82 2.58 In this process, we allowed the relaxation of the spin degrees of freedom at the Mn atoms, and calculated the local magneticmoment by using an atomic sphere with radius 1.32 ˚A. In Table IIwe present the results for the adsorption energy, E A=ETot−Esurf−NE Mn N, (3) where Nis the number of adsorbed Mn atom for each of the nonequivalent adsorption sites. We also give the results for theMn atomic magnetic moment in μ B, and the distance between the Mn atom and the surface plane in ˚A,dMn-surf . We see that the highest adsorption energy corresponds to a chemisorption site of type C, which is the threefold coordinatedsite that follows the fcc gold stacking sequence. The bondingin a B site is very similar and almost degenerate to C. Themagnetic moment is also very similar in both sites, 4.82and 4.83 μ B, respectively. Furthermore, the distance from the adatom to the Au surface is the same. The bonding energy of a Mn chemisorbed at a bridge site (AA) is smaller, as well as the distance dMn-surf ,b u tt h e magnetic moment is the same as in the C case. It is important tonotice that in this case, the Mn atom is bonded to two nearestneighbors on the surface and to two next-nearest neighborsin the same layer. The distance at which the nearest andnext-nearest neighbors are located differs only by a small amount, i.e., 2.52 and 2.95 ˚A, respectively. The adsorption in an A site produces a large electronic localization around the Mn atom, which is probably responsi-ble for the weakest bonding energy among all the consideredsites. From these results we conclude that the adsorption of Mnclusters on gold surfaces is ruled by the Mn-surface interaction,since the bonding energies are at least four times larger thanthe Mn-Mn free dimer bonding energy ( ∼0 . 5e V ) .W ea l s o note that the magnetic properties are almost independent ofthe adsorption site, but it still holds that the lower the energy,the smaller the magnetic moment. Furthermore, the bridge typeis also competitive with respect to B and C, which indicatesthat in the growing process, due to its dynamical dependence,the adsorption on B, C, and AA is competitive between them,and the adsoption type will depend mostly on the site surfacedensity. To better understand the adsorption at an electronic level, one can notice the differences at each site by plotting thedifference δρbetween the converged charge density, including the Mn atom, and the superposition of the free atomic chargedensities. From Fig. 3one can see marked differences between the adsorption over a type A site and the other sites: TheA-site adsorption affects more notoriously the Au atom belowand displaces it inside the solid. One can also notice thatsome electrons are pulled from the surface neighbors andaccumulated mainly close to the Mn and the Au atom below,and that there is a small accumulation of electronic charge onthe Au surface neighbors (Mn loses on the order of 1 /4e − after adsorption, mostly from s-like orbitals). In the B, C, and AA cases, the charge redistributions are very similar, giving rise only to small differences. In thesecases the bonding is more uniform: The Mn not only shares itselectrons with its nearest Au, but also does it with the metallicsurface electronic cloud. This fact makes E A∼1 eV larger than 02468 0.300.250.200.150.100.050.000.050.100.15 02 468 02468 024681015 052025 AB C A A FIG. 3. (Color online) Charge redistribution δρfor each adsorption site of a (111) Au surface. The red circles denote the Mn and the substrate atoms located at the same plane. The distances are in ˚A, and the color bar in e/˚A3. 205423-4MONOATOMIC AND DIMER Mn ADSORPTION ON THE ... PHYSICAL REVIEW B 83, 205423 (2011) FIG. 4. (Color online) Local electronic density of states (LDOS) on Mn and on the neighbor Au (111) atoms. Each adsorption site is indicated by a superscript in the legend. Au surfis the DOS of the nearest-neighbor surface Au atom. The LDOS in the lower panel corresponds to the spin-minority states. that for the chemisorption on an A site. One can also observe that the electronic rearrangements due to the adatom has littleeffect in layers deeper than the second Au atomic layer. Figure 4shows the local electronic density of states (LDOS) for each adsorption site on the (111) Au surface. The LDOSof the surface nearest-neighbor Au atoms reflects the presenceof the Mn atom. In the A case, the Au LDOS presents, inthe low-energy part, peaks produced by the the interactionwith the Mn atom chemisorbed at the top of it. Cases Band C show only very slight differences. A more continuousDOS is obtained in the AA case, due to the higher effectivecoordination number. In the Au atoms, we only notice a rathersmall asymmetry between the up and down states, a fact thatproduces only small magnetic moments ( ≈0.05μ Bin the nearest Mn neighbors, and one order of magnitude smalleron Au second neighbors). In contrast, the Mn LDOS presents larger differences for the various adsorption sites, although, as expected, thesedifferences are much smaller between the B and C sites.However, the atomic character of the Mn is still clearlyidentified (half-filled dorbitals below E F). In particular, the A site yields more localized Mn dstates. The LDOS around the Fermi level has few states, but a closer look (checking theorbital contributions to the electronic bands) shows that it ispopulated manly with selectrons. The magnetic moment of the Mn atom has a contribution from the selectrons and yields a value of 5.07 μ B. Furthermore, it is worth noticing that in the AA case, the interaction of the Mn atom with four atoms(two nearest-neighbor surface atoms and two next-nearest neighbors located at the same layer) produces a LDOS withfeatures similar to the C and B cases, but with smaller peaks.The magnetic moment values for the AA, B, and C are almostidentical ( ∼4.83μ B). V . Mn DIMER ADSORPTION We proceeded considering the adsorption process of a second Mn by accounting for two different cases: one wherethe dimer is directly adsorbed and another one where twodissociated but closely positioned atoms are placed on thesurface. The presence of a gold surface adds more degrees offreedom, produced by the bond dimer orientation with respectto it and the orientation of the magnetic moments of bothmanganese atoms. Let us mention that from our calculation ofthe free Mn dimer, we obtain as a result that the two atoms arecoupled antiferromagnetically, with a binding energy per atomof−0.53 eV and a bond length of 2.6 ˚A. The ferromagnetically ordered dimer gives the same bond length, but a weakerbinding energy of ≈−0.28 eV , close to previous calculations. 8 As mentioned above, it is worth noting that the competition between energetics and spin orientation is determined by thedimer bond length. 8Even still, we do recognize that this effect remains a topic of debate. Since the chemisorption energy of a single Mn atom is approximately four times the binding energy of the freeMn dimer, its electronic properties, once deposited on the 205423-5FRANCISCO MU ˜NOZ et al. PHYSICAL REVIEW B 83, 205423 (2011) FIG. 5. (Color online) Adsorption geometries considered for Mn 2 on a Au (111) surface. Site I correspond to the dimer chemisorbed, perpendicular to the surface, on a threefold coordinated site. Incases II and III the dimer is chemisorbed parallel to the surface, on nonequivalent and equivalent triangles, respectively. surface, are ruled by the Mn-surface interaction. Thus, one must optimize all the nonequivalent geometrical and magneticconfigurations. The considered adsorption geometries for Mn 2on the Au surface are shown Fig. 5. Based on our results for a single atom adsorption, which show that the B and C sites are the moststable, we locate in one of those sites the first Mn atom. Thus,we optimize all possible configurations with one Mn atom overa B or C site and the other on a B, C, or AA site, or on top of thefirst Mn atom. After the optimization, the chemisorption on anAA bridge site became unstable, i.e., when an Mn atom wasinitially over an AA bridge, it diffuses toward a B or a C site.Similar to the monomer case, there were only slight differencesbetween the dimer chemisorption on B and C sites. Therefore,we have considered B and C sites as equivalent. We must pointout that this is a conclusion reached after the correspondingcalculations and not an ap r i o r i approximation. We present the results only of the three geometries shown in Fig. 5.I n geometry I, the dimer is chemisorbed perpendicular to thesurface. Then, by fixing one of the manganese atoms on one ofthose positions, the other is placed either on a nonequivalentTABLE III. Mn 2adsorption on (111) Au: The adsorption energy is in eV , the magnetic moment is in μB, the Mn-Mn bond length (dMn-Mn ), and Mn-surface ( dMn-surf ) distances are given in ˚A. EA μd Mn-Mn dMn-surf Site\Mag FM AFM FM AFM FM AFM FM AFM I −3.67−3.93 9.45 0.25 2.71 2.48 2.61 2.59 II −5.35−5.40 9.22 0.00 2.67 2.56 2.52 2.53 III −5.45−5.51 9.28 0.00 2.84 2.74 2.60 2.59 triangular site facing the first (geometry II), or on a neighbor equivalent triangular site (geometry III). In Table IIIwe present results for the dimer chemisorption assuming FM or AFM ordering, for the three geometries. Weobserve that the weakest energy of adsorption correspondsto case I, due to the fact that one of the Mn bonds to asingle Au atom and the other is threefold coordinated. Thisgeometry has also the largest energy difference between bothmagnetic couplings (FM and AFM). The adsorption energy forthe geometries parallel to the surface differs in ≈0.1 eV for the two types of magnetic ordering. The strongest chemisorptionenergy corresponds to geometry III in the AFM ordering. Themagnetic moment of each Mn atom in this case is 4.64 μ B, and for the bond distance of the chemisorbed dimer we obtain2.74 ˚A, a value that is similar to the free Mn dimer. The electronic redistribution after adsorption, for geome- tries I and III, is shown in Fig. 6. The two left-hand panels contain the results for geometry I assuming FM and AFMcoupling. Geometry I is interesting, due the great deformationof the electron cloud in the upper Mn, which arises fromsharing its valence selectron with the lower Mn. Also, site I shows a significant difference between FM and AFM states. Inthe two left-hand panels the corresponding results for geometry 02468 0.300.250.200.150.100.050.000.050.100.15 02 468 02468 024681015 052025 I I III III FM AFM FM AFM FIG. 6. (Color online) Charge redistribution of Mn 2adsorbed in (111) Au surface. The I and III geometries were minimized under two different magnetic configurations: FM and AFM ordering. The scale denotes the distance in ˚A and the color bar is in of e/˚A3. 205423-6MONOATOMIC AND DIMER Mn ADSORPTION ON THE ... PHYSICAL REVIEW B 83, 205423 (2011) FIG. 7. (Color online) Local electronic density of states of Mn 2on Au (111). The Mn superscript (in the legends) indicates the adsorption site of Mn 2, and the subscript is the magnetic order considered. III are shown. In this case the differences between the two magnetic orientations are smaller. We report the LDOS of the chemisorbed dimer in Fig. 7. We observe that the changes in the electronic structure ofthe gold nearest neighbors are small and we only observesmall changes around the Fermi energy. On the other hand, theelectronic DOS associated with the Mn 2shows interesting features. In geometry I, where the Mn dimer is depositedperpendicular to the surface, the e-LDOS shows large peaks around the bonding and antibonding states of the free dimer.In geometries II and III, where the dimer lies parallel to thesurface, the orbital hybridization with a larger number of gold FIG. 8. (Color online) Simulated STM images for a Mn dimer chemisorbed on the most probable geometry (III) on a (111) Au surface. The applied voltage is 5.0 V and the charge density is kept constant to the values 0.1 (A) and 0.3 (B) e/˚A3, respectively. The color bar indicates the depth in ˚A. 205423-7FRANCISCO MU ˜NOZ et al. PHYSICAL REVIEW B 83, 205423 (2011) atoms is more important and therefore it leads to a more intense electronic dispersion. The bonding and antibondingpeaks observed in case I are smoothed in these cases andbecome almost imperceptible. The AFM solution shows largerpeaks for case I, giving a finite value for the sum of bothmagnetic moments. This is not the case in geometries II and IIin which the two Mn are equivalent and possess equal magneticmoments of opposite sign. Finally, we simulated STM images of the most stable dimer chemisorption geometry. Figure 8contains the simulations in which the applied voltage was set to 5.0 V; the upper andlower figures correspond to electronic densities of 0.1 and0.3e/˚A 3, respectively. One clearly observes the Mn dimer above the surface and the atoms lying on the surface plane.In the lower figure one can also distinguish the Au atomsof the second surface layer. These images may be useful toexperimentalists studying this system. It has been shown thatone can manipulate single Mn atoms adsorbed on Ag(111) tobuild clusters up to tetramers. 36 VI. SUMMARY AND CONCLUSIONS We have reported a set of calculations on the monoatomic and dimer adsorption of manganese on a (111) Au surface,within the DFT theory as implemented in the V ASP code. For completeness, we characterized first the bulk and the clean(111) Au surface. As a first approximation, we ignored theSOI in the gold atom basically because we found it plays aminor role in the Mn-Au interaction. Then, we calculated thebinding energy of a Mn monomer and found that the strongestcorresponds to the adsorption on the threefold coordinatedhollow sites. We obtained a value of 4.82 μ Bfor its magnetic moment.The dimer adsorption was modeled by fixing one of the atoms to the lowest-energy configuration in the monoatomiccase (threefold coordinated hollow site) and consideringdifferent geometry configurations. From those considered, wefound only three different low-energy geometries. We foundthat the most stable configuration corresponds to the dimerlying parallel to the surface, with its atoms occupying threefoldcoordinated hollow sites. We found that the AFM arrangementis the most stable. The interaction with the surface modifiesthe value of the atomic magnetic moment, and yields 4 .6μ B per atom. Furthermore, the dimer bond is larger than the one in the isolated dimer, but this increase is not enough to producea FM ground state. Considering that the adsorption energy forthe dimer, −5.51 eV , is less than twice the adsorption energy for the monomer adsorption, one expects that the dimer willdissociate to allow the single atoms to explore the surface andbind to sites of type B or C. Finally, we simulated STM images of the Mn dimer chemisorption. These can be useful to experimentalists work-ing on Mn-Au systems. Investigations on the chemisorption ofMn atoms on Ag and Cu are in progress. ACKNOWLEDGMENTS The authors acknowledge support from FONDECYT 1100365, the Millennium Science Nucleus Basic and AppliedMagnetism P06-022-F, Financiamiento Basal para CentrosCient ´ıficos y Tecnol ´ogicos de Excelencia, under project FB 0807, and CONACYT (Mexico) through Grants No. 61417 andNo. 83247, and TAMU-Conacyt agreement. The use of com-puter resources from the Centro Nacional de Superc ´omputo (CNS), M ´exico is also acknowledged. 1B. Belhadji, S. Lounis, M. Benakki, and C. Demangeat, Phys. Rev. B69, 064431 (2004). 2G. Bihlmayer, P. Kurz, and S. Bl ¨ugel, Phys. Rev. B 62, 4726 (2000). 3C. Biswas, R. S. Dhaka, A. K. Shukla, and S. R. Barman, Surf. Sci. 601, 609 (2007). 4M. Hortamani, H. Wu, P. Kratzer, and M. Scheffler, Phys. Rev. B 74, 205305 (2006). 5P. Kr ¨uger, M. Taguchi, and S. Meza-Aguilar, Phys. Rev. B 61, 15277 (2000). 6O. Rader, T. Mizokawa, A. Fujimori, and A. Kimura, Phys. Rev. B 64, 165414 (2001). 7C. Demangeat and J. C. Parlebas, Rep. Prog. 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Uzdin, V . Uzdin, and C. Demangeat, Europhys. Lett. 47, 556 (1999). 19V . S. Stepanyuk, W. Hergert, K. Wildberger, S. K. Nayak, andP. Jena, Surf. Sci. 384, (1997). 20R. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, Cambridge, UK, 2004). 21G. Kresse and J. Hafner, P h y s .R e v .B 47, 558 (1993). 22G. Kresse and J. Hafner, P h y s .R e v .B 49, 14251 (1994). 23G. Kresse and J. Furthm ¨uller, J. Comput. Mater. Sci. 6,1 5 (1996). 24G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54, 11169 (1996). 25P. E. Bl ¨ochl, P h y s .R e v .B 50, 17953 (1994). 26G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). 27J. P. Perdew, K. Burke, and M. Ernzerhof, P h y s .R e v .L e t t . 77, 3865 (1996). 28[http://www.webelements.com/gold/crystal_structure.htlm ]. 205423-8MONOATOMIC AND DIMER Mn ADSORPTION ON THE ... PHYSICAL REVIEW B 83, 205423 (2011) 29A. R. Sandy, S. G. J. Mochrie, D. M. Zehner, K. G. Huang, and D. Gibbs, Phys. Rev. B 43, 4667 (1991). 30U. Harten, A. M. Lahee, J. P. Toennies, and Ch. W ¨oll,Phys. Rev. Lett. 54, 2619 (1985). 31Y . Wang, N. S. Hush, and J. R. Reimers, P h y s .R e v .B 75, 233416 (2007). 32J. Wan, Y . L. Fan, D. W. Gong, S. G. Shen, and X. Q. Fan, Modell. Simul. Mater. Sci. Eng. 7, 189 (1999).33S. M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 33, 7983 (1986). 34V. Z´olyomi, L. Vitos, S. K. Kwon, and J. Koll ´ar,J. Phys. Condens. Matter 21, 095007 (2009). 35W. Chen, V . Madhavan, T. Jamneala, and M. F. Crommie, Phys. Rev. Lett. 80, 1469 (1998). 36J. Kliewer, R. Berndt, J. Min ´ar, and H. Ebert, Appl. Phys. A 82,6 3 (2006). 205423-9

PhysRevB.83.085107.pdf

PHYSICAL REVIEW B 83, 085107 (2011) Electron-lattice and strain effects in manganite heterostructures: The case of a single interface A. Iorio, C. A. Perroni, V . Marigliano Ramaglia, and V . Cataudella CNR-SPIN and Dipartimento di Scienze Fisiche, Universit `a degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy (Received 16 September 2010; revised manuscript received 6 January 2011; published 23 February 2011) A correlated inhomogeneous mean-field approach is proposed to study a tight-binding model of the manganite heterostructures (LaMnO 3)2n/(SrMnO 3)nwith average hole doping x=1/3. Phase diagrams and spectral and optical properties of large heterostructures (up to 48 sites along the growth direction) with a single interface arediscussed, and the effects of electron-lattice antiadiabatic fluctuations and strain are analyzed. The formationof a metallic ferromagnetic interface is quite robust upon varying the strength of electron-lattice coupling andstrain, though the size of the interface region is strongly dependent on these interactions. The density of statesnever vanishes at the chemical potential due to the formation of the interface, but it shows a rapid suppressionwith increasing the electron-lattice coupling. The in-plane and out-of-plane optical conductivities show sharpdifferences since the in-plane response has metallic features, while the out-of-plane one is characterized by atransfer of spectral weight to high frequency. The in-plane response mainly comes from the region between thetwo insulating blocks, so that it provides a clear signature of the formation of the metallic ferromagnetic interface.Results are discussed in connection with available experimental data. DOI: 10.1103/PhysRevB.83.085107 PACS number(s): 73 .20.−r, 73.40.−c, 73.50.−h I. INTRODUCTION Transition-metal oxides are of great current interest because of the wide variety of the ordered phases that they exhibitand the strong sensitivity to external perturbations. 1Among them, manganese oxides with the formula R1−xAxMnO 3(R stands for a rare earth such as La, Arepresents a divalent alkali element such as Sr or Ca, and xis the hole doping), known as manganites, have been studied intensively both fortheir very rich phase diagram and for the phenomenon ofcolossal magnetoresistance. 2This effect is often exhibited in the doping regime 0 .2<x< 0.5, where the ground state of the systems is ferromagnetic. The ferromagnetic phase is usuallyexplained by invoking the double-exchange mechanism inwhich hopping of an outer-shell electron from a Mn 3+ to a Mn4+site is favored by a parallel alignment of the core spins.3In addition to the double-exchange term that promotes hopping of the carriers, a strong interaction betweenelectrons and lattice distortions plays a non-negligible rolein these compounds, giving rise to the formation of polaronquasiparticles. 4 Very recently, high-quality atomic-scale “digital” het- erostructures consisting of a combination of transition-metaloxide materials have been realized. Indeed, heterostructuresrepresent the first steps to use correlated oxide systems inrealistic devices. Moreover, at the interface, the electronicproperties can be drastically changed in comparison withthose of the bulk. Recent examples include the formationof a thin metallic layer at the interface between band andMott insulators, such as, for example, between SrTiO 3(STO) and LaTiO 3oxides5or between the band insulators6LaAlO 3 and STO. Very interesting examples of heterostructure are given by the superlattices (LaMnO 3)m/(SrMnO 3)nwithn/(m+n) average hole doping.7Here LaMnO 3(LMO) (one electron per Mn egstate) and SrMnO 3(SMO) (no electrons per Mn egstate) are the two end-member compounds of the alloy La1−xSrxMnO 3and are both antiferromagnetic insulating. Inthese systems, not only the chemical composition but also the thickness of the constituent blocks specified by mandnis important for influencing the properties of superlattices. Focushas been on the case m=2ncorresponding to the average optimal hole doping x=1/3. 8,9The superlattices exhibit a metal-insulator transition as a function of temperature forn/lessorequalslant2 and behave as insulators for n/greaterorequalslant3. The superlattices undergo a rich variety of transitions among metal, the Mottvariable range hopping insulator, the interaction-inducedEfros-Shklovskii insulator, and the polaronic insulator. 10 Interfaces play a fundamental role in tuning the metal- insulator transitions since they control the effective dopingof the different layers. Even when the system is globallyinsulating ( n/greaterorequalslant3), some nonlinear optical measurements suggest that, for a single interface, ferromagnetism due tothe double-exchange mechanism can be induced between thetwo antiferromagnetic blocks. 11Moreover, it has been found that the interface density of states exhibits a pronouncedpeak at the Fermi level whose intensity correlates with theconductivity and magnetization. 12These measurements point toward the possibility of a two-dimensional half-metallic gasfor the double layer 13whose properties have been studied by using ab initio density-functional approaches.14However, up to now, this interesting two-dimensional gas has not beenexperimentally assessed in a direct way by using lateralcontacts on the region between the LMO and SMO blocks. In analogy with thin films, strain is another important quantity to tune the properties of manganite heterostructures.For example, far from interfaces, inside LMO, electron localization and local strain favor antiferromagnetism and e g(3z2−r2) orbital occupation.15The magnetic phase in LMO is compatible with the Ctype.2Moreover, by changing the substrate, the ferromagnetism in the superlattice can be stabilized.16 From a theoretical point of view, in addition to ab initio calculations, tight-binding models have been used to study manganite superlattices. The effects of magnetic 085107-1 1098-0121/2011/83(8)/085107(10) ©2011 American Physical SocietyA. IORIO et al. PHYSICAL REVIEW B 83, 085107 (2011) and electron-lattice interactions on the electronic properties have been investigated going beyond adiabatic mean-fieldapproximations. 17,18However, the double layer with large blocks of LMO and SMO has not been studied much.Moreover, the effects of strain have been analyzed only withinmean-field approaches. 19 In this paper, we have studied phase diagrams and spectral and optical properties for a very large bilayer(LMO) 2n/(SMO) n(up to 48 planes relevant for a comparison with fabricated heterostructures) starting from a tight-bindingmodel. We have developed a correlated inhomogeneous mean-field approach taking into account the effects of electron-latticeantiadiabatic fluctuations. Strain is simulated by modulatinghopping and spin-spin interaction terms. We have found thata metallic ferromagnetic interface forms for a large range ofthe electron-lattice couplings and strain strengths. For thisregime of parameters, the interactions are able to change thesize of the interface region. We find the magnetic solutionsthat are stable at low temperature in the entire superlattice.The general structure of our solutions is characterized by threephases running along the growth zdirection: an antiferromag- netic phase with localized or delocalized (depending on themodel parameters) charge carriers inside the LMO block, aferromagnetic state at the interface with itinerant carriers, anda localized polaronic G-type antiferromagnetic phase inside the SMO block. The type of antiferromagnetic order insideLMO depends on the strain induced by the substrate. We have discussed the spectral and optical properties corresponding to different parameter regimes. Due to theformation of the metallic interface, the density of states is finiteat the chemical potential. With increasing the electron-phononinteraction, it gets reduced at the chemical potential, but itnever vanishes even in the intermediate to strong electron-phonon coupling regime. Finally, we have studied both thein-plane and out-of-plane optical conductivities, pointing outthat they are characterized by marked differences: the formershows a metallic behavior, the latter a transfer of spectralweight at high frequency due to the effects of the electrostaticpotential well trapping electrons in the LMO block. Thein-plane response at low frequency is mainly due to the regionbetween the two insulating blocks, so that it can be used asa tool to assess the formation of the metallic ferromagneticinterface. The paper is organized as follows: In Sec. II, the model and variational approach are introduced; in Sec. III, the results regarding static properties and phase diagrams are discussed;in Sec. IV, the spectral properties are analyzed; in Sec. V, the optical conductivities are analyzed; and conclusions arepresented in the final section. II. THE V ARIATIONAL APPROACH A. Model Hamiltonian For manganite superlattices, the Hamiltonian of the bulk H0has to be supplemented by Coulomb terms representing the potential arising from the pattern of the La and Sr ions,20 thus H=H0+HCoul. (1)To set up an appropriate model for the double layer, it is important to take into account the effects of the strain.The epitaxial strain produces the tetragonal distortion of theMnO 6octahedron, splitting the egstates into x2-y2and 3z2-r2 states.19If the strain is tensile, x2-y2is lower in energy, while if the strain is compressive, 3 z2-r2is favored. In the case of n=8 with three interfaces,15the superlattices grown on STO are found to be coherently strained: all of them are forced tothe in-plane lattice parameter of substrate and to an averageout-of-plane parameter c/similarequal3.87˚A. 15As a consequence, one can infer that LMO blocks are subjected to compressive strain(−2.2%) and SMO blocks to tensile strain ( +2.6%). For the case of the LMO block, the resulting higher occupancy of3z 2-r2enhances the out-of-plane ferromagnetic interaction owing to the larger electron hopping out-of-plane. For the caseof the SMO block, the reverse occurs. A suitable model for thebilayer has to describe the dynamics of the e gelectrons, which in the LMO and SMO blocks preferentially occupy the moreanisotropic 3 z 2-r2orbitals and more isotropic x2-y2orbitals, respectively. For this reason, in this paper we adopt an effectivesingle-orbital approximation for the bulk manganite. The model for the bulk takes into account the double- exchange mechanism, the coupling to the lattice distortions,and the superexchange interaction between neighboring local-izedt 2gelectrons on Mn ions. The coupling to longitudinal optical phonons arises from the Jahn-Teller effect that splitsthee gdouble degeneracy. Then, the Hamiltonian H0reads H0=−/summationdisplay /vectorRi,/vectorδt|/vectorδ|/parenleftBigg S/vectorRi,/vectorRi+/vectorδ 0 +1/2 2S+1/parenrightBigg c† /vectorRic/vectorRi+/vectorδ +ω0/summationdisplay /vectorRia† /vectorRia/vectorRi+gω0/summationdisplay /vectorRic† /vectorRic/vectorRi/parenleftbig a/vectorRi+a† /vectorRi/parenrightbig +1 2/summationdisplay /vectorRi,/vectorδ/epsilon1|/vectorδ|/vectorS/vectorRi·/vectorS/vectorRi+/vectorδ−μ/summationdisplay /vectorRic† /vectorRic/vectorRi. (2) Heret|/vectorδ|is the transfer integral of electrons occupying egorbitals between nearest-neighbor (NN) sites, S/vectorRi,/vectorRi+/vectorδ 0 is the total spin of the subsystem consisting of two localized spins on NN sites and the conduction electron, /vectorS/vectorRiis the spin of the t2gcore states ( S=3/2), and c† /vectorRi(c/vectorRi) creates (destroys) an electron with spin parallel to the ionic spin at the ith site in the egorbital. The coordination vec- tor/vectorδconnects NN sites. The first term of the Hamilto- nian describes the double-exchange mechanism in the limitwhere the intra-atomic exchange integral Jis much larger than the transfer integral t |/vectorδ|. Furthermore, in Eq. ( 2),ω0de- notes the frequency of the local optical-phonon mode, a† /vectorRi(a/vectorRi) is the creation (annihilation) phonon operator at the site i, and the dimensionless parameter gindicates the strength of the electron-phonon interaction. Finally, in Eq. ( 2),/epsilon1|/vectorδ|represents the antiferromagnetic superexchange coupling between twoNNt 2gspins and μis the chemical potential. The hopping of electrons is supposed to take place between the equivalent NNsites of a simple cubic lattice (with finite size along the zaxis corresponding to the growth direction of the heterostructure)separated by the distance |n−n /prime|=a. The units are such that 085107-2ELECTRON-LATTICE AND STRAIN EFFECTS IN ... PHYSICAL REVIEW B 83, 085107 (2011) the Planck constant ¯ h=1, the Boltzmann constant kB=1, and the lattice parameter a=1. Regarding the terms due to the interfaces, one considers that La3+and Sr2+ions act as +1 charges of magnitude e and neutral points, respectively. In the heterostructure, thedistribution of those cations induces an interaction term fore gelectrons of Mn giving rise to the Hamiltonian HCoul=/summationdisplay /vectorRi/negationslash=/vectorRj1 2/epsilon1de2n/vectorRin/vectorRj |/vectorRi−/vectorRj|+/summationdisplay /vectorRLa i/negationslash=/vectorRLa j1 2/epsilon1de2 /vextendsingle/vextendsingle/vectorRLa i−/vectorRLa j/vextendsingle/vextendsingle −/summationdisplay /vectorRi,/vectorRLa j1 /epsilon1de2n/vectorRi/vextendsingle/vextendsingle/vectorRi−/vectorRLa j/vextendsingle/vextendsingle, (3) withn/vectorRi=c† /vectorRic/vectorRithe electron occupation number at the Mn sitei,/vectorRiand/vectorRLa iare the positions of Mn and La3+in the ith unit cell, respectively, and /epsilon1dis the dielectric constant of the material. In our calculation, the long-range Coulomb potentialhas been modulated by a factor ηinducing a fictitious finite screening length (see Appendix). This factor was added onlyfor computational reasons since it allows us to calculate thesummations of the Coulomb terms over the lattice indices. Wehave modeled the heterostructures as slabs whose in-plane sizeis infinite. To describe the magnitude of the Coulomb interaction, we define the dimensionless parameter α=e 2/(a/epsilon1dt|/vectorδ|), which controls the charge-density distribution. The order of mag-nitude of αcan be estimated from the hopping parameter t |/vectorδ|∼0.65 eV , lattice constant a=4˚A, and typical value of the dielectric constant /epsilon1∼10 to be around 0 .2. Strain plays an important role also by renormalizing the heterostructure parameters. Strain effects can be simulated byintroducing an anisotropy into the model between the in-planehopping amplitude t δ||=t(withδ||indicating nearest neigh- bors in the x-yplanes) and out-of-plane hopping amplitude t|δz|=tz(with δzindicating nearest neighbors along the z axis).21Moreover, the strain induced by the substrate can directly affect the patterns of core spins.22Therefore, in our model, we have also considered the anisotropy between thein-plane superexchange energy /epsilon1 |δ|||=/epsilon1and the out-of-plane one/epsilon1|δz|=/epsilon1z. We have found that the stability of magnetic phases in LMO blocks is influenced by the presence ofcompressive strain, while in SMO the sensitivity to strain ispoor. Therefore, throughout the paper, we take as referencethe model parameters of the SMO layers and we will consideranisotropy only in the LMO blocks with values of the ratiot z/tlarger than unity and of the ratio /epsilon1z//epsilon1smaller than unity. Finally, to investigate the effects of the electron-lattice coupling, we will use the dimensionless quantity λdefined as λ=g2ω0 6t. (4) Throughout the paper, we will assume ω0/t=0.5. B. Test Hamiltonian In this work, we will consider solutions of the Hamiltonian that break the translational invariance in the out-of-plane zdirection. The thickness of the slab is a parameter of the system that will be indicated by Nz. We will build up a variational procedure including these features of the heterostructures. Asimplified variational approach similar to that developed inthis work has already been proposed by some of the authorsfor manganite bulks 23and films.24,25 To treat variationally the electron-phonon interaction, the Hamiltonian ( 1) has been subjected to an inhomogeneous Lang-Firsov canonical transformation.26It is defined by parameters depending on plane indices along the zdirection: U=exp⎡ ⎣−g/summationdisplay i||,iz/parenleftbig fizc† i||,izci||,iz+/Delta1iz/parenrightbig/parenleftbig ai||,iz−a† i||,iz/parenrightbig⎤ ⎦,(5) where i||indicates the in-plane lattice sites ( ix,iy), while izare the sites along the direction z. The quantity fizrepresents the strength of the coupling between an electron and the phonondisplacement on the same site belonging to the i zplane, hence it measures the degree of the polaronic effect. On the other hand,the parameter /Delta1 izdenotes a displacement field describing static distortions that are not influenced by instantaneous position ofthe electrons. To obtain an upper limit for free energy, the Bogoliubov inequality has been adopted: F/lessorequalslantF test+/angbracketleft ˜H−Htest/angbracketrightt, (6) where FtestandHtestare the free energy and the Hamiltonian corresponding to the test model that is assumed with an ansatz. ˜Hstands for the transformed Hamiltonian ˜H=UHU†.T h e symbol /angbracketleft/angbracketrighttindicates a thermodynamic average performed by using the test Hamiltonian. The only part of Htestthat contributes to /angbracketleft˜H−Htest/angbracketrighttis given by the spin degrees of freedom and depends on the magnetic order of the t2gcore spins. For the spins, this procedure is equivalent to the standardmean-field approach. The model test Hamiltonian, H test, is such that that electron, phonon, and spin degrees of freedom are not interacting witheach other: H test=Hsp test+Hph test+Hel test. (7) The phonon part of Htestsimply reads Hph test=ω0/summationdisplay i||,iza† i||,iizai||,iiz, (8) and the spin term is given by Hsp test=−gSμB/summationdisplay i||/summationdisplay izhz i||,izSz i||,iz, (9) where gSis the dimensionless electron-spin factor ( gS/similarequal2), μBis the Bohr magneton, and hz i||,izis the effective variational magnetic field. In this work, we consider the following magnetic orders modulated plane by plane: F, hz i||,iz=/vextendsingle/vextendsinglehz iz/vextendsingle/vextendsingle; A, hz i||,iz=(−1)iz/vextendsingle/vextendsinglehz iz/vextendsingle/vextendsingle; (10) C, hz i||,iz=(−1)ix+iy/vextendsingle/vextendsinglehz iz/vextendsingle/vextendsingle; G, hz i||,iz=(−1)ix+iy+iz/vextendsingle/vextendsinglehz iz/vextendsingle/vextendsingle. 085107-3A. IORIO et al. PHYSICAL REVIEW B 83, 085107 (2011) For all these magnetic orders, the thermal averages of double- exchange operator, corresponding to neighboring sites in thesame plane i zγiz;i||,i||+δ||and in different planes ηiz,iz+δz;i||, preserve only the dependence on the z-plane index: γiz;i||,i||+δ||=/angbracketleftbiggSi||,iz;i||+δ||,iz 0 +1/2 2S+1/angbracketrightbigg t=γiz, ηiz,iz+δz;i||=/angbracketleftbiggSi||,iz;i||,iz+δz 0 +1/2 2S+1/angbracketrightbigg t=ηiz,iz+δz. (11) To get the mean-field electronic Hamiltonian, we make the Hartree approximation for the Coulomb interaction. The electronic contribution Hel testto the test Hamiltonian becomes Hel test=−t/summationdisplay i||Nz/summationdisplay iz=1/summationdisplay δ||γize−Vizc† i||,izci||+δ||,iz −tz/summationdisplay i||Nz/summationdisplay iz=1/summationdisplay δzηiz,iz+δze−Wiz,iz+δzc† i||,izci||,iz+δz +/summationdisplay i||Nz/summationdisplay iz=1[φeff(iz)−μ]c† i||,izci||,iz +NxNy(T1+T2)+NxNyg2ω0/summationdisplay iz/Delta1iz. (12) In Eq. ( 12), the quantity φeff(iz) indicates the effective potential seen by the electrons. It consists of the Hartree self-consistentpotential φ(i z) (see Appendix) and a potential due to the electron-phonon coupling: φeff(iz)=φ(iz)+g2ω0Ciz, (13) with Ciz=f2 iz−2fiz+2/Delta1iz(fiz−1). (14) The factors e−Vizande−Wiz,iz+δzrepresent the phonon thermal average of Lang-Firsov operators: e−Viz=/angbracketleftbig Xi||,izX† i||+δ||,iz/angbracketrightbig t,e−Wiz,iz+δz=/angbracketleftbig Xi||,izX† i||,iz+δz/angbracketrightbig t, (15) where the operator X/vectorRireads X/vectorRi=egfiz(a/vectorRi−a† /vectorRi). Finally, the quantity T1andT2derive from the Hartree approximation (see Appendix), and NxandNydenote the size of the system along the two in-plane directions, respectively.To calculate the variational free energy, we need to knoweigenvalues and eigenvectors of H el testthat depend on the magnetic order of core spins through the double-exchangeterms. C. Magnetic order and diagonalization of the electronic mean-field Hamiltonian To develop the calculation, we need to fix the magnetic order of core spins. The pattern of magnetic orders isdetermined by the minimization of the total free energy. Byexploiting the translational invariance along the directionsperpendicular to the growth axis of the heterostructure, the diagonalization for H el testreduces to an effective unidimensional problem for each pair of continuous wave vectors ( kx,ky)=/vectork||. For some magnetic patterns, the electronic problem is charac-terized at the interface by a staggered structure. Therefore, westudy the electron system considering a reduced first Brillouinzone of in-plane wave vectors. To this aim, we represent H el test with the 2 Nzstates |kx,ky,iz/angbracketright,|kx+π,ky+π,iz/angbracketright, (16) with the wave vectors such that −π/2<kx<π / 2,−π/2< ky<π / 2, and izgoing from 1 to Nz. The eigenstates of the electronic test Hamiltonian are indicated by E(kx,ky,n), with the eigenvalue index ngoing from 1 to 2 Nz. The eigenvector related to nis specified in the following way: biz(/vectork||,n)f o r the first Nzcomponents, piz(/vectork||,n) for the remaining Nz components. The variational procedure is self-consistently performed by imposing that the total density of the system ρis given by NLa/Nz, withNLathe number of layers of the LMO block, and the local plane density χ(iz) is equal to /angbracketleftn/vectorRi/angbracketright. Therefore, one has to solve the following Nz+1 equations: ρ=1 NxNyNz/summationdisplay /vectork||/summationdisplay nnF[E(/vectork||,n)] (17) and χ(iz)=1 NxNy/summationdisplay /vectork||/summationdisplay nnF[E(/vectork||,n)]/braceleftbig/vextendsingle/vextendsinglebiz(/vectork||,n)/vextendsingle/vextendsingle2+/vextendsingle/vextendsinglepiz(/vectork||,n)/vextendsingle/vextendsingle2 +/bracketleftbig b∗ iz(/vectork||,n)piz(/vectork||,n)+p∗ iz(/vectork||,n)biz(/vectork||,n)/bracketrightbig/bracerightbig , (18) where nF(z) is the Fermi distribution function. These equations allow us to obtain the chemical potential μand the local charge density χ(iz). As a result of the variational analysis, one is able to get the charge-density profile corresponding to magneticsolutions that minimize the free energy. III. STATIC PROPERTIES AND PHASE DIAGRAMS We have found the magnetic solutions and the corre- sponding density profiles that are stable for different sizesof the LMO and SMO blocks. The inhomogeneous variationalapproach allows us to determine the values of the electron-phonon parameters f izand/Delta1izand the magnetic order of the t2gspins through the effective magnetic fields hiz. We will study the systems in the intermediate to strong electron-phononregime characteristic of manganite materials focusing on twovalues of coupling: λ=0.5 and 0 .8. The maximum value of in-plane antiferromagnetic superexchange is /epsilon1=0.01t.T h e value of the Coulomb term αis fixed to α=0.2. We will analyze the heterostructures in the low-temperature regime:T=0.05t. The general structure of our solutions is characterized by three phases running along the zdirection. Actually, according to the parameters of the model, we find Gor Cantiferromagnetic phases corresponding to localized or delocalized charge carriers inside the LMO block, respectively.The localization is ascribed to the electron-phonon coupling,which gives rise to the formation of small polarons. For 085107-4ELECTRON-LATTICE AND STRAIN EFFECTS IN ... PHYSICAL REVIEW B 83, 085107 (2011) -30 -24 -18 -12 -6 0 6 12 Sites along z-direction00.10.20.30.40.50.60.70.80.91 0Local particle density48 sites 24 sites 12 sites FIG. 1. (Color online) Comparison among density profiles cor- responding to different sizes at λ=0.5a n d /epsilon1=0.01t. The index 0 indicates the interface Mn plane between the last La plane in theLMO block and the first Sr plane in the SMO block. the values of λconsidered in this paper, a ferromagnetic phase always stabilizes around the interface. The size of theferromagnetic region at the interface is determined by theminimization of the free energy and depends on the valuesof the system parameters. Only for larger values of λand/epsilon1is the possibility of interface ferromagnetism forbidden. Insidethe SMO block, a localized polaronic G-type antiferromagnet phase is always stable. At first, we have analyzed the scaling of the static properties as a function of the size of the system along the zgrowth direction. Therefore, a comparison of the density profileshas been done with (LMO) 8/(SMO) 4,( L M O ) 16/(SMO) 8, and (LMO) 32/(SMO) 16systems. In Fig. 1, we show the density profiles in a situation where strain-induced anisotropy hasnot been introduced. It is worth noticing that we indicate theinterface Mn plane between the last La plane in the LMO blockand the first Sr plane in the SMO block with the index 0. For asufficiently large number of planes, the charge profile along z shows a well-defined shape. Indeed, the local density is nearlyunity in the LMO block, nearly zero in the SMO block, andit decreases from 1 to 0 in the interface region. The decreaseof charge density for the first planes of LMO is due to theeffect of open boundary conditions along the zdirection. In the intermediate electron-phonon coupling regime that we con-sider in Fig. 1, the region with charge dropping involves four to five planes between the two blocks. We notice that the localcharge density for (LMO) 16/(SMO) 8and (LMO) 32/(SMO) 16 systems is very similar around the interface. Furthermore, the numerical results show close values of variational freeenergy corresponding to the above-mentioned systems. Giventhe similarity of the properties of these two systems, in thefollowing we will develop the analysis on the role of theinterface studying the system (LMO) 16/(SMO) 8. For the same set of electron-phonon and magnetic couplings, the variational parameters and the Hartreeself-consistent potential along the zaxis are shown in Fig. 2. The effective magnetic fields are plotted for the most stable magnetic solution: antiferromagnetic Gorders well inside LMO (planes 1–15) and SMO (planes 19–24), andferromagnetic planes at the interface (planes 16–18). The peakin the plot of the magnetic fields signals that ferromagnetism is2 4 6 8 10 12 14 16 18 20 22 24-6-4.5-3φ(iz) 2 4 6 8 10 12 14 16 18 20 22 2400.51fiz 24 681 0 1 2 1 4 1618 20 22 24 Sites alon g z-direction81624|hz iz|Chemical potential μ FIG. 2. (Color online) Self-consistent Hartree potential φ(iz) (upper panel, in units of t), variational parameters fiz(middle panel), and effective magnetic fields |hz iz|(lower panel) along the zaxis for λ=0.5a n d/epsilon1=0.01t. quite robust at the interface. The variational electron-phonon parameters fizare small on the LMO side and at the interface, but close to unity in the SMO block. This means that, for thesevalues of the couplings, carriers are delocalized in LMO up tothe interface region, but small polarons are present in the SMOblock. The quantities /Delta1 iz, which enter the variational treatment of the electron-phonon coupling, are determined by fizand the local density /angbracketleftniz/angbracketrightthrough the equation /Delta1iz=/angbracketleftniz/angbracketright(1−fiz). The Hartree self-consistent potential /Phi1indicates that charges are trapped into a potential well corresponding to the LMOblock. Moreover, it is important to stress the energy scalesinvolved in the well: the barrier between the LMO and SMOblocks is of the order of the electron bandwidth. Furthermore,at the interface, the energy difference between neighboringplanes is of the order of the hopping energy t. As mentioned above, for these systems, strain plays an important role. To study quantitatively its effect, we haveinvestigated the phase diagram under the variation of thehopping anisotropy t z/tfor two different values of /epsilon1z(/epsilon1z= /epsilon1=0.01t,/epsilon1z=0). Indeed, we simulate the compressive strain in the LMO block increasing the ratio tz/tand decreasing /epsilon1z//epsilon1. On the other hand, the tensile strain in the SMO block favors the more isotropic x2-y2orbital and does not yield sizable effects. Therefore, for the SMO block, in thefollowing we choose t z=tand/epsilon1z=/epsilon1. For what concerns the electron-phonon interaction, we assume the intermediatecoupling λ=0.8. As shown in the upper panel of Fig. 3, upon increasing the ratio t z/tup to 1 .7f o r/epsilon1z=/epsilon1, the magnetic order in LMO does not change since it remains Gantiferromagnetic. However, the character of charge carriers is varied. Actually,forλ=0.8, in the absence of anisotropy, small polarons are present in the LMO block. Moreover, at t z/t/similarequal1.5, in LMO, a change from small localized polarons to large delocalizedpolarons occurs. For all values of the ratio t z/t, the interface region is characterized by ferromagnetic order with largepolaron carriers and SMO by Gantiferromagnetic order with small polaron carriers. It has been shown that it is also important to consider the anisotropy in superexchange ( /epsilon1 z/negationslash=/epsilon1) parameters as a conse- quence of strain.22To simulate the effect of compressive strain in LMO, a reduction of /epsilon1zwill be considered. We discuss the 085107-5A. IORIO et al. PHYSICAL REVIEW B 83, 085107 (2011) 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8-3.15-3.12-3.09-3.06Free energy 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 tz/t-3.08-3.07-3.06Free energySmall/Large/Small G-AFM/FM/G-AFMLarge/Large/Small G-AFM/FM/G-AFM Small/Large/Small G-AFM/FM/G-AFMSmall/Large/Small C-AFM/FM/G-AFMLarge/Large/Small G-AFM/FM/G-AFMεz=ε=0.01t εz=0.0 λ=0.8 λ=0.8 FIG. 3. (Color online) Phase diagram in the hopping anisotropy- energy plane for the LMO 16SMO 8system, corresponding to λ=0.8 for/epsilon1z=0.01t(upper panel) and /epsilon1z=0 (lower panel). limiting case /epsilon1z=0. For this regime of parameters, the effect on the magnetic phases is the strongest. As shown in the lowerpanel of Fig. 3,f o r1.28/lessorequalslantt z/t/lessorequalslant1.5, in the LMO block, a C- type antiferromagnetic phase is the most stable. The transitionfrom small to large polarons again takes place at t z/t/similarequal1.5. Therefore, we have shown that there is a range of parameterswhere the LMO block has C-type antiferromagnetic order with small localized polarons. Due to the effect of strain, themagnetic solution in LMO turns out to be compatible withexperimental results in superlattices. 15The interface is still fer- romagnetic with metallic large polaron features. In the figure,A/B/Crefers to magnetic orders and the character of charge carriers inside LMO (A), at interface (B), inside SMO (C). To analyze the effects of the electron-phonon interaction, a comparison between two different electron-phonon couplingsis reported in Fig. 4. We have investigated the solutions that minimize the variational free energy at a fixed valueof the anisotropy factors t z/t=1.3 and /epsilon1z=0a tλ=0.5 and 0.8. The magnetic solution in the LMO block is C antiferromagnetic until the 15th plane. For both values ofλ, polarons are small. In the SMO block, starting from the 19th plane, the solution is G-type antiferromagnetic together with localized polarons. Three planes around the interface are 24 681 0 1 2 1 4 1618 20 22 24 Sites along z-direction00.10.20.30.40.50.60.70.80.91Local particle densityλ=0.5 λ=0.8 FIG. 4. (Color online) Comparison between local particle density corresponding to λ=0.5a n d0 .8f o rtz/t=1.3a n d/epsilon1z=0.TABLE I. Ratio between the magnetization and its saturation value for λ=0.5a n d0 .8 as a function of the anisotropy ratio tz/t for/epsilon1z=0. tz/t Magnetization ( λ=0.5) Magnetization ( λ=0.8) 1.0 0.1148 0.1090 1.1 0.1182 0.1123 1.2 0.1206 0.11461.3 0.1222 0.1161 1.4 0.1233 0.1172 1.5 0.1241 0.1179 ferromagnetically ordered. For λ=0.5, all three planes at the interface are characterized by delocalized polarons, while forλ=0.8, only the plane linking the ends of the LMO and SMO blocks is with delocalized charge carriers. As shown in Fig. 4, the quantity λhas important conse- quences on the physical properties, such as the local particledensity. Actually, for λ=0.8, the transition from occupied to empty planes is sharper at the interface. Only one plane at theinterface shows an intermediate density close to 0 .5. For λ= 0.5, the charge profile is smoother and the three ferromagnetic planes with large polarons have densities different from 0and 1. The last static quantity that we have evaluated is the magnetization. In Table I, we report the ratio between the magnetization of the heterostructure and the saturationvalue as a function of the anisotropy term t z/t. We consider the case in which /epsilon1z=0. Due to the formation of a few ferromagnetic planes at the interface, the magnetization ratiois very small (of the order of 0 .1). Moreover, for λ=0.8, its value is slightly smaller than 1 at λ=0.5. The comparison with recent experimental data (see Ref. 9) on the single interface is very interesting. The order of magnitude of thecalculated magnetization (about 0.12 times the saturationvalue per manganese at t z/t=1.3) is in good agreement with the experimental value at low temperature (0.162 timesthe saturation bulk ferromagnetic value per manganese).Therefore, not only the kind of magnetic order, but also theorder of magnitude of magnetization compare quite well withexperimental data. For the analysis of the spectral and optical quantities, we will consider the parameters used for the discussion of theresults in the last figure and the table focusing on t z/t=1.3. IV . SPECTRAL PROPERTIES In this section, we will calculate the spectral properties of the heterostructure for the same parameters used in Fig. 4. Performing the canonical transformation ( 5) and exploiting the cyclic properties of the trace, the electron MatsubaraGreen’s function becomes G(/vectorR i,/vectorRj,τ)=−/angbracketleftbig Tτc/vectorRi(τ)X/vectorRi(τ)c† /vectorRj(0)X† /vectorRj(0)/angbracketrightbig .(19) By using the test Hamiltonian ( 7), the correlation function can be disentangled into electronic and phononic terms.23,24Going to Matsubara frequencies and making the analytic continuationiω n→ω+iδ, one obtains the retarded Green’s function and 085107-6ELECTRON-LATTICE AND STRAIN EFFECTS IN ... PHYSICAL REVIEW B 83, 085107 (2011) the diagonal spectral function Aixiy iz(ω) corresponding to /vectorRi= /vectorRj, Aix,iy iz(ω) =eSiz T∞/summationdisplay l=−∞Il(Siz)eβlω0 2[1−nF(ω−lω0)]gix,iy iz(ω−lω0) +eSiz T∞/summationdisplay l=−∞Il(Siz)eβlω0 2nF(ω+lω0)gix,iy iz(ω+lω0), (20) where Siz T=g2f2 iz(2N0+1),Siz=2g2f2 iz[N0(N0+1)]1 2, Il(z) are modified Bessel functions, and gix,iy iz(ω)i s gix,iy iz(ω)=2π NxNy/summationdisplay /vectork||2Nz/summationdisplay n=1δ[ω−E(/vectork||,n)]/braceleftbig/vextendsingle/vextendsinglebiz(/vectork||,n)/vextendsingle/vextendsingle2 +/vextendsingle/vextendsinglepiz(/vectork||,n)/vextendsingle/vextendsingle2+(−1)ix+iy/bracketleftbig b∗ iz(/vectork||,n)piz(/vectork||,n) +p∗ iz(/vectork||,n)bicz(/vectork||,n)/bracketrightbig/bracerightbig . (21) The density of states D(ω) is defined as D(ω)=1 NxNyNz1 2π/summationdisplay ix,iy,izAix,iy iz(ω). (22) In Fig. 5, we report the density of state of the system (LMO) 16/(SMO) 8. It has been calculated measuring the energy from the chemical potential μ. This comparison has been made at fixed low temperature ( kBT=0.05t), therefore we can consider the chemical potential very close to the Fermienergy of the system. At λ=0.5, the spectral function exhibits a residual spectral weight at μ. The main contribution to the density of states at the chemical potential μcomes from the three ferromagnetic large polaron planes at the interface.Indeed, the contributions due to the (LMO) and (SMO) blocksis negligible. For stronger electron-phonon coupling at λ=0.8, we observe an important depression of the spectral functionatμ. Hence the formation of a clear pseudogap takes place. This result is still compatible with the solution ofour variational calculation since, for this value of λ= -12 -9 -6 -3 0 3 6 91 2 Ener gy00.050.10.150.2Density of Statesλ=0.5 λ=0.8 FIG. 5. (Color online) Comparison between density of states (in units of 1 /t) as a function of the energy (in units of t) corresponding toλ=0.5a n d0 .8.0.8, there is only one plane with delocalized charge car- riers that corresponds to the plane indicated as the in-terface ( i z=17), while the two additional ferromagnetic planes around the interface are characterized by smallpolarons. The depression of the density of states at theFermi energy is due also to the polaronic localizationwell inside the LMO and SMO block. In any case, wefind that, even for λ=0.8, the density of states never vanishes at the interface, in agreement with experimentalresults. 12 In this section, we have found strong indications that a metallic ferromagnetic interface can form at the inter-face between LMO and SMO blocks. This situation shouldbe relevant for superlattices with n/greaterorequalslant3, where resistivity measurements made with contacts on top of LMO showa globally insulating behavior. In our analysis, we havecompletely neglected any effect due to disorder even if, bothfrom experiments 8,9and theories,17it has been suggested that localization induced by disorder could be the cause of themetal-insulator transition observed for n/greaterorequalslant3. We point out that the sizable source of disorder due to the random dopingwith Sr 2+is strongly reduced since, in superlattices, La3+ and Sr2+ions are spatially separated by interfaces. Therefore, the amount of disorder present in the heterostructure isstrongly reduced in comparison with the alloy. However,considering the behavior of the LMO (SMO) block as thatof a bulk with a small amount of holes (particles), oneexpects that even a weak disorder induces localization. Onthe other hand, a weak disorder is not able to prevent theformation of the ferromagnetic metallic interface favoredby the double-exchange mechanism and the charge transferbetween the bulklike blocks: the states at the Fermi level dueto the interface formation have enough density 12so that they cannot be easily localized by weak disorder. In this section,we have shown that this can be the case in the intermediateelectron-phonon coupling regime appropriate for LMO /SMO heterostructures. In the next section, we will analyze the effects of electron- phonon coupling and strain on the optical conductivity in thesame regime of the parameters considered in this section. V . OPTICAL PROPERTIES To determine the linear response to an external field of frequency ω, we derive the conductivity tensor σα,βby means of the Kubo formula. To calculate the absorption, we need onlythe real part of the conductivity, Reσ α,α(ω)=−Im/Pi1ret α,α ω, (23) where /Pi1ret α,βis the retarded current-current correlation function. Following a well-defined scheme23,24and neglecting vertex corrections, one can get a compact expression for the real partof the conductivity σ α,α. It is possible to get the conductivity both along the plane perpendicular to the growth axis, σxx, and parallel to it, σzz. To calculate the current-current correlation function, one can use the spectral function A/vectork||;iz,jzderived in the previous section exploiting the translational invariance 085107-7A. IORIO et al. PHYSICAL REVIEW B 83, 085107 (2011) along the in-plane direction. It is possible to show that the components of the real part of the conductivity become Re[σxx](ω)=e2t2 NxNy/summationdisplay kx,ky4sen2(kx)1 Nz/summationdisplay iz,jzγizγjz ×1 ω/integraldisplay∞ −∞dω 1 4π[nF(ω1−ω)−nF(ω1)] ×Akx,ky;iz,jz(ω1−ω)Akx,ky;iz,jz(ω1) (24) and Re[σzz](ω)=e2t2 NxNy/summationdisplay kx,ky1 Nz/summationdisplay iz,jz/summationdisplay δ1z,δ2zδ1zδ2zηiz,iz+δ1zηjz,jz+δ2z1 ω ×/integraldisplay∞ −∞dω 1 4π[nF(ω1−ω)−nF(ω1)] ×Akx,ky;iz+δ1z,jz+δ2z(ω1−ω)Akx,ky;iz,jz(ω1).(25) In Fig. 6, we report the in-plane conductivity as a function of the frequency at λ=0.5 and 0 .8. We have checked that the in-plane response mainly comes from the interface planes.Both conductivities are characterized by a Drude-like responseat low frequency. Therefore, the in-plane conductivity providesa clear signature of the formation of the metallic ferromagneticinterface. However, due to the effect of the interactions, wehave found that the low-frequency in-plane response is at leastone order of magnitude smaller than that of free electrons in theheterostructures. Moreover, additional structures are present inthe absorption with increasing energy. For λ=0.5, a new band with a peak energy of the order of hopping t=2ω 0is clear in the spectra. This structure can be surely ascribed to the pres-ence of large polarons at the three interface planes. 23Actually, this band comes from the incoherent multiphonon absorptionof large polarons at the interface. This is also confirmed by thefact that this band is quite broad, therefore it can be interpretedin terms of multiple excitations. For λ=0.8, the band is even larger and shifted at higher energies. In this case, at the inter-face, large and small polarons are present with a ferromagneticspin order. Therefore, there is a mixing of excitations whose neteffect is the transfer of spectral weight at higher frequencies. FIG. 6. (Color online) The conductivity [in units of e2/(mt), with m=1/(2t)] into the plane perpendicular to the growth direction of the (LMO) 16/(SMO) 8bilayer as a function of the energy (in units of t) for different values of λ.24 6 81 0 1 2 1 4 Energy00.0040.0080.012σzzλ=0.5 λ=0.8 FIG. 7. (Color online) The conductivity [in units of e2/(mt), withm=1/(2t)] along the growth direction of the (LMO) 16(SMO) 8 bilayer as a function of the energy (in units of t)f o rλ=0.5a n d0 .8. The out-of-plane optical conductivities show significant differences in comparison with the in-plane responses. InFig. 7, we report out-of-plane conductivity as a function of the frequency at λ=0.5 and 0 .8. First, we observe the absence of the Drude term. Moreover, the band at energy about 2 ω 0 is narrower than that in the in-plane response. Therefore, the origin of this band has to be different. Actually, theout-of-plane optical conductivities are sensitive to the interfaceregion. A charge carrier at the interface has to overcome anenergy barrier to hop to the neighbor empty site. As shown inFig. 2, the typical energy for close planes at the interface is of the order of the hopping t. Therefore, when one electron hops alongz, it has to pay at least an energy of the order of t.I nt h e out-of-plane spectra, the peaks at low energy can be ascribed tothis process. Of course, by paying a larger energy, the electroncan hop to the next nearest neighbors. This explains the widthof this band due to interplane hopping. Additional structures are present at higher energies in the out-of-plane conductivities. For λ=0.5, the band at high energy is broad with small spectral weight. For λ=0.8, there is an actual transfer of spectral weight at higher energies. Aclear band is peaked around 10 t. This energy scale can be interpreted as given by 2 g 2ω0=9.6tforλ=0.8. Therefore, in the out-of-plane response, the contribution at high energycan be interpreted as due to small polarons. 23,27 Unfortunately, experimental data about optical properties of the LMO /SMO bilayers are still not available. Therefore, comparison with experiments is not possible. Predictions onthe different behaviors among σ xxandσzzcan be easily checked if one uses in-plane and out-of-plane polarizationof the electrical fields used in the experimental probes.More importantly, the formation of two-dimensional gas atthe interface is expected to be confirmed by experiments madeby using lateral contacts directly on the region between theLMO and SMO blocks. The dc conductivity of the sheet coulddirectly measure the density of carriers of the interface metaland confirm the Drude-like low-frequency behavior of in-planeresponse. Finally, we have evaluated the conductivity of the entire system at zero frequency for different values of modelparameters. In Fig. 8, we report the in-plane (upper panel) and out-of-plane (lower panel) conductivity as a function ofthe anisotropy ratio t z/tforλ=0.5 and 0 .8. As expected, the 085107-8ELECTRON-LATTICE AND STRAIN EFFECTS IN ... PHYSICAL REVIEW B 83, 085107 (2011) 1 1.1 1.2 1.3 1.4 1.5 1.600.020.040.06σxxλ=0.5 λ=0.8 1 1.1 1.2 1.3 1.4 1.5 1.6 tz/t0.0010.0020.0030.0040.005σzz FIG. 8. (Color online) The conductivity at zero frequency [in units of e2/(mt), with m=1/(2t)] of the (LMO) 16(SMO) 8bilayer into the plane perpendicular to the growth direction (upper panel)and along the growth direction (lower panel) as a function of the anisotropy ratio t z/tforλ=0.5a n d0 .8. conductivity gets larger upon increasing the ratio tz/tsince ferromagnetic solutions are favored. Moreover, we point outthat the out-of-plane conductivity is one order of magnitudeless than the in-plane conductivity. The order of magnitude ofthe resistivity has been estimated considering the out-of-planecontribution of the calculated conductivity. For t z/t=1.3, the resistivity is about 0 .2/Omega1cm, a value that is comparable with experimental results at low temperature for the single interface(of the order of 1 /Omega1cm). 9It is clear that disorder effects present in the material (not included in our analysis) should increasethe value of resistivity by enhancing the scattering rate ofthe carriers. Therefore, a value of resistivity smaller than theexperimental value is compatible with the complexity of theseheterostructures. VI. CONCLUSIONS In this paper, we have discussed phase diagrams and spectral and optical properties for a very large bilayer(LMO) 2n/(SMO) n(up to 48 sites along the growth direction). A correlated inhomogeneous mean-field approach hasbeen developed to analyze the effects of electron-latticeantiadiabatic fluctuations and strain. We have shown that ametallic ferromagnetic interface is a quite robust feature ofthese systems for a large range of the electron-lattice couplingsand strain strengths. Furthermore, we have found that the sizeof the interface region depends on the strength of electron-phonon interactions. At low temperature, the general structureof our solutions is characterized by three phases running alongthe growth zdirection: antiferromagnetic phase with localized and/or delocalized charge carriers inside the LMO block,ferromagnetic state with itinerant carriers at the interface, andlocalized polaronic G-type antiferromagnetic phase inside the SMO block. The type of antiferromagnetic order inside LMOdepends on the strain induced by the substrate. Spectral and optical properties have been discussed for different parameter regimes. Due to the formation ofthe metallic interface, even in the intermediate to strongelectron-phonon coupling regime, the density of states nevervanishes at the chemical potential. Finally, in-plane andout-of-plane optical conductivities are sharply different: theformer shows a metallic behavior, the latter a transfer of spectral weight at high frequency due to the effects of theelectrostatic potential well trapping electrons in the LMOblock. The in-plane response provides a signature of theformation of the metallic ferromagnetic interface. The approach proposed in this paper is accurate for the cal- culation of static properties. With regard to dynamical quanti-ties, the role of the electron-phonon coupling is properly takeninto account, while the effect of Coulomb interactions is con-sidered only within mean field. To this aim, it could be interest-ing to improve the treatment of electron-electron interactions,for example, by using the random-phase approximation. 27It is clear that the random-phase treatment is quite complex inheterostructures due to the lack of translational invariancealong one direction. Moreover, the dynamical screening ispoor due to the presence of large insulating antiferromagneticblocks in the system. For this reason, the self-consistentmean-field approach is reasonable, simple, and able to graspthe main features of the effects of the Coulomb interactions. In this paper, we have emphasized the role of polaron quasiparticles since they represent one of the main ingredientsfor the interpretation of the data in manganites. 4Within our approach, the main contribution to the polaron formationcomes from the local interaction. Other effects, such asthose relative to cooperative interactions between vibrationalmodes, could make a contribution to the polaron formation.Moreover, coupling between Jahn-Teller modes on differentsites could also improve the analysis of strain effects inthe system. However, it is important to point out that ourtreatment of electron-phonon interaction is based on aninhomogeneous approach. Therefore, through kinetic-energyterms, correlations between different sites are assured. Finally, we have focused on static and dynamic properties at very low temperature. The approach used in the paper is validat any temperature. Therefore, it could be very interesting toanalyze not only single interfaces but also superlattices withdifferent unit cells at finite temperature. Work in this directionis in progress. APPENDIX In this Appendix, we give some details about the effective electronic Hamiltonian derived within our approach. After theHartree approximation for the long-range Coulomb interac-tions, the mean-field electronic Hamiltonian reads H el test=−t/summationdisplay i||Nz/summationdisplay iz=1/summationdisplay δ||γize−Vizc† i||,izci||+δ||,iz −t/summationdisplay i||Nz/summationdisplay iz=1/summationdisplay δzηiz,iz+δze−Wiz,iz+δzc† i||,izci||,iz+δz +/summationdisplay i||Nz/summationdisplay iz=1[φ(iz)−μ]c† i||,izci||,iz+NxNy(T1+T2) +NxNyg2ω0/summationdisplay iz/Delta1iz+/summationdisplay i||Nz/summationdisplay iz=1Ciz(g2ω0)c† i||,izci||,iz. (A1) 085107-9A. IORIO et al. PHYSICAL REVIEW B 83, 085107 (2011) The self-consistent Hartree potential is given by φ(iz)=e2 /epsilon1/parenleftBigg/summationdisplay jz>izχ(jz)S(iz−jz)+/summationdisplay jz<izχ(jz)S(iz−jz) +S1(0)χ(iz)−S2(iz)/parenrightBigg , (A2) where the quantity T1is T1=−e2 2/epsilon1/parenleftBiggNz/summationdisplay iz=1Nz/summationdisplay jz>izχizχjzS(iz−jz) +Nz/summationdisplay jz<izχizχjzS(iz−jz)+S1(0)Nz/summationdisplay izχ2 iz/parenrightBigg , (A3) andT2is T2=e2 2/epsilon1/parenleftBiggNLa/summationdisplay Iz=1NLa/summationdisplay Jz>IzS(Iz−Jz)+NLa/summationdisplay Jz<IzS(Iz−Jz)+NLaS1/parenrightBigg , (A4) withS(nz),S1(0), and S2(nz) obtained by adding the Coulomb terms on the in-plane lattice index. The summations have beenmade modulating the Coulomb interaction with a screening factor:e2 |/vectorri−/vectorrj|→e2e−ηS|/vectorri−/vectorrj| |/vectorri−/vectorrj|, where1 ηSis a fictitious finite screening length in units of the lattice parameter a. Therefore, S(nz)i s S(nz)=/summationdisplay mx,myexp/parenleftbig −ηS/radicalBig m2x+m2y+n2z/parenrightbig /radicalBig m2x+m2y+n2z, (A5) S1(0) is given by S1(0)=/summationdisplay mx,myexp/parenleftbig −ηS/radicalBig m2x+m2y/parenrightbig /radicalBig m2x+m2y, (A6) with (mx,my)/negationslash=(0,0), and S2(iz−jz)i s S2(nz)=/summationdisplay mx,mylz/summationdisplay iz=1exp/parenleftbig −ηS/radicalBig h2x+h2y+h2z/parenrightbig /radicalBig h2x+h2y+h2z, (A7) withlzthe number of planes of the LMO block, hx=mx− 0.5,hy=my−0.5, and hz=nz−iz−0.5. 1M. Imada, A. Fujimori, and Y . Tokura, Rev. Mod. Phys. 70, 1039 (1998). 2E. Dagotto, Nanoscale Phase Separation and Colossal Magnetore- sistance (Springer-Verlag, Heidelberg, 2003). 3C. Zener, Phys. Rev. 81, 440 (1951); 82, 403 (1951); P. W. Anderson and H. Hasegawa, ibid. 100, 675 (1955); P. G. de Gennes, ibid. 118, 141 (1960). 4A. J. Millis, Nature (London) 392, 147 (1998). 5A. Ohtomo and H. Y . Hwang, Nature (London) 419, 378 (2002); S. Okamoto and A. J. Millis, ibid. 428, 630 (2004). 6A. Ohtomo and H. Y . Hwang, Nature (London) 427, 423 (2004); S. Thiel, G. Hammerl, A. Schmehl, C. W. Schneider, andJ. Mannhart, Science 313, 1942 (2006); N. Reyren, ibid. 317, 1196 (2007). 7T. Koida, M. Lippmaa, T. Fukumura, K. Itaka, Y . Matsumoto,M. Kawasaki, and H. Koinuma, Phys. Rev. B 66, 144418 (2002); H. Yamada, M. Kawasaki, T. Lottermoser, T. Arima, and Y . Tokura,Appl. Phys. Lett. 89, 052506 (2006). 8A. Bhattacharya, S. J. May, S. G. E. te Velthuis, M. Warusawithana, X. Zhai, B. Jiang, J. M. Zuo, M. R. Fitzsimmons, S. D. Bader, andJ. N. Eckstein, Phys. Rev. Lett. 100, 257203 (2008). 9C. Adamo, X. Ke, P. Schiffer, A. Soukiassian, M. Warusawithana, L. Maritato, and D. G. Schlom, Appl. Phys. Lett. 92, 112508 (2008). 10C. Adamo, C. A. Perroni, V . Cataudella, G. De Filippis, P. Orgiani, and L. Maritato, P h y s .R e v .B 79, 045125 (2009). 11N. Ogawa, T. Satoh, Y . Ogimoto, and K. Miyano, Phys. Rev. B 78, 212409 (2008). 12S. Smadici, P. Abbamonte, A. Bhattacharya, X. Zhai, B. Jiang,A. Rusydi, J. N. Eckstein, S. D. Bader, and J.-M. Zuo, Phys. Rev. Lett. 99, 196404 (2007). 13N. Ogawa, T. Satoh, Y . Ogimoto, and K. Miyano, Phys. Rev. B 80, 241104(R) (2009).14B. R. K. Nanda and S. Satpathy, Phys. Rev. Lett. 101, 127201 (2008); Phys. Rev. B 79, 054428 (2009). 15C. Aruta, C. Adamo, A. Galdi, P. Orgiani, V . Bisogni, N. B. Brookes, J. C. Cezar, P. Thakur, C. A. Perroni, G. De Filippis, V . Cataudella,D. G. Schlom, L. Maritato, and G. Ghiringhelli, Phys. Rev. B 80, 140405(R) (2009). 16H. Yamada, P. H. Xiang, and A. Sawa, Phys. Rev. B 81, 014410 (2010). 17S. Dong, R. Yu, S. Yunoki, G. Alvarez, J.-M. Liu, and E. Dagotto,Phys. Rev. B 78, 201102(R) (2008); R. Yu, S. Yunoki, S. Dong, and E. Dagotto, ibid. 80, 125115 (2009). 18C. Lin and A. J. Millis, P h y s .R e v .B 78, 184405 (2008). 19B. R. K. Nanda and S. Satpathy, P h y s .R e v .B 78, 054427 (2008); 81, 224408 (2010). 20C. Lin, S. Okamoto, and A. J. Millis, P h y s .R e v .B 73, 041104(R) (2006). 21S. Dong, S. Yunoki, X. Zhang, C. Sen, J.-M. Liu, and E. Dagotto,Phys. Rev. B 82, 035118 (2010). 22Z. Fang, I. V . Solovyev, and K. Terakura, Phys. Rev. Lett. 84, 3169 (2000). 23C. A. Perroni, G. De Filippis, V . Cataudella, and G. Iadonisi, Phys. Rev. B 64, 144302 (2001). 24C. A. Perroni, V . Cataudella, G. De Filippis, G. Iadonisi, V . Marigliano Ramaglia, and F. Ventriglia, P h y s .R e v .B 68, 224424 (2003). 25A. Iorio, C. A. Perroni, G. De Filippis, V . MariglianoRamaglia, and V . Cataudella, J. Phys. Condens. Matter 21, 456002 (2009). 26I. J. Lang and Yu. A. Firsov, Sov. Phys. JETP 16, 1301 (1963). 27G. Mahan, Many-Particle Physics , 2nd ed. (Plenum, New York, 1990). 085107-10

PhysRevB.72.125128.pdf

Suppression of the metal-insulator transition in the spinel Cu 1−xInxIr2S4system Guanghan Cao, *Xiaofeng Xu, and Zhengkuan Jiao Department of Physics, Zhejiang University, Hangzhou 310027, People’ s Republic of China Hideaki Kitazawa and Takehiko Matsumoto National Institute for Materials Science, Sengen 1-2-1, Tsukuba, Ibaraki 305-0047, Japan Chunmu Feng Test and Analysis Center, Zhejiang University, Hangzhou 310027, People’ s Republic of China /H20849Received 17 October 2004; revised manuscript received 2 May 2005; published 30 September 2005 /H20850 The thiospinel Cu 1−xInxIr2S4/H208490/H33355x/H333550.25 /H20850system was studied by the measurements of crystal structure, electrical resistivity, and magnetic susceptibility. The parent compound was known to exhibit an intriguingfirst-order metal-insulator /H20849MI/H20850transition with a simultaneous spin-dimerization and charge ordering at /H11011230 K with decreasing temperature. Upon indium doping on the copper site, the conduction holes of the metallic phase are depleted, or the doped electrons occupy the antibonding state of the insulating phase,suppressing the MI transition. Moreover, the first-order transition is changed into a higher-order one forx/H333560.2. Our experimental data suggest that the higher-order phase transition is associated with an electronic transformation from small polarons to small bipolarons. Comparing the doping effects of Zn, Cd, and In, wefound that the variations of the electrical and magnetic properties depend on the lattice size, i.e., the suppres-sion of the MI transition becomes weaker for an enlarged lattice. This lattice size effect is mainly explained interms of the electron-phonon interactions, which is enhanced by the band narrowing due to the larger ionic sizeof the dopants. DOI: 10.1103/PhysRevB.72.125128 PACS number /H20849s/H20850: 71.30. /H11001h, 71.38. /H11002k, 72.80.Ga I. INTRODUCTION The spinel-type compound CuIr 2S4undergoes an intrigu- ing first-order MI transition at TMI/H11011230 K with decreasing temperature.1,2The high-temperature metallic phase crystal- lizes in a normal cubic spinel structure with the Cu, Ir cat-ions occupying tetrahedral and octahedral sites of AB 2X4, respectively. On the other hand, the crystal structure of thelow-temperature insulating phase is distorted in such an un-usual way that charge /H20849Ir 3+/Ir4+/H20850ordering and spin /H20849S=1/2 for Ir4+/H20850dimerization take place simultaneously.3The unique spin-dimerization in a three-dimensional structure as well as the complex charge-ordering pattern that consists of Ir83+S24 and Ir84+S24octamers makes the spinel very interesting and challenging. Moreover, CuIr 2S4shows additional anomalous characteristics in the pressure effect on TMI,4the transport properties,5,6electronic structures,7–9x-ray-induced phase transition10,11and doping-induced superconductivity.12–14 Therefore, CuIr 2S4represents another model system on the topic of MI transitions15,16in the field of condensed matter physics. CuIr 2S4belongs to 5 d-transition-metal sulfides. The aver- aged-dCoulomb energy of Ir 5 delectrons is about 3.5 eV, obviously smaller than those of 3 dtransition-metal oxides.17 On the other hand, the hybridization between Ir 5 da n dS3 p orbitals is very strong, resulting in a large d/H9253-d/H9255splitting and a broad d/H9255band of 7.0 eV.18Therefore, the electron correlation would not play the dominant role in the MI tran-sition. It was shown that the Fermi level E Flies near the top of the d/H9255band, leading to the metallic state with the hole conduction for the cubic spinel CuIr 2S4.18Note that the cop- per is monovalent, as revealed by the photoemission study,7the energy-band calculations,18and the Cu NMR measurement.19The electronic configuration of Ir in CuIr 2S4 is 5d/H92555.5, or more precisely, /H20851eg4/H20852/H20851a1g1.5/H20852when considered the distortion of the IrS 6octahedra. So, the highest-occupying a1gstate should be regarded as quarter-occupying with holes, which seems to be important for the simultaneous chargeordering and spin-dimerization. 20Very recently, band- structure calculations21for the insulating phase of CuIr 2S4 reveals that the band gap opening is due to the bonding-antibonding splitting of the atomic d/H9255orbitals at the dimer- ized Ir sites. However, the driving force of the MI transitionremains unclear. An elemental substitution study may give useful informa- tion on the physical mechanism as well as a route to newmaterials. In the present thiospinel system, elemental substi-tutions on different crystallographic sites with specific ele-ments have been performed. 13,22–26Among them, the A-site substitution hardly changes the IrS 6-octahedron framework that determines the electronic structure of the valence bands,therefore, this kind of substitution brings the effects of thelattice size and the carrier filling. For example, Zn 2+substi- tution for Cu+depletes the hole carriers of the metallic CuIr 2S4phase, resulting in the suppression of the MI transi- tion and the appearance of superconductivity.13Substitution for Cu+by a larger cation Cd2+similarly decreases the hole concentration in the metallic phase, however, a bipolaronicstate appears instead of the superconducting one. 26So, the electronic property is determined not only by the hole deple-tion but also by the lattice size. In order to further understandthis finding, one needs to make the Cu-site substitution bythe nonmagnetic cation with different size and valence. In 3+ has the same electronic configuration nd10, whose energyPHYSICAL REVIEW B 72, 125128 /H208492005 /H20850 1098-0121/2005/72 /H2084912/H20850/125128 /H208497/H20850/$23.00 ©2005 The American Physical Society 125128-1level is far below EF, as those of Cu+,Z n2+, and Cd2+, but it is trivalent and it has intermediate size. Therefore, we carriedout the study on the Cu 1−xInxIr2S4system. The result forms a systematic understanding on the suppression and evolutionof the MI transition by the hole depletion and the lattice size,which suggests that the strong electron-lattice interactionplays an important role for the MI transition in the thiospinelsystem. II. EXPERIMENTS Polycrystalline samples of Cu 1−xInxIr2S4/H20849x =0,0.025,0.05,0.1,0.15,0.2,0.25,0.3,0.5 /H20850were prepared by a solid-state reaction method. First, mixture of Cu /H2084999.99% /H20850,I r /H2084999.99% /H20850,I n /H2084999.99% /H20850, and S /H2084999.999% /H20850pow- ders with the nominal stoichiometry was sealed in an evacu-ated quartz ampoule. Then, the sealed ampoule was heatedslowly to 1023 K, holding for 24 hours, followed by the cal-cination at 1373 K for a period of 4 days. In order to in-crease the indium solubility, samples were quenched. 26The resulting powder was subsequently ground and pressed intopellets with the pressure of /H110112000 kg/cm 2. Finally, the pel- lets were sintered in an evacuated quartz ampoule again at1373 K for 48 hours and then quenched. Powder x-ray diffraction /H20849XRD /H20850was carried out at room temperature with Cu K /H9251radiation by employing a RIGAKU x-ray diffractometer. The crystal structure parameters wererefined by the RIETAN Rietveld analysis program. 27The electrical resistivity /H20849/H9267/H20850was measured by the standard four- probe method. The magnetization of samples was measured by using a Quantum Design SQUID magnetometer. Themeasurement was carried out using about 100 mg samples inboth cooling and heating processes between 1.8 K and300 K under the applied field of 1000 Oe. The backgroundarising from the sample holder was measured in advance andthen subtracted. III. RESULTS AND DISCUSSION A. Structural properties Powder XRD measurements indicated that samples of Cu1−xInxIr2S4with 0 /H33355x/H333550.25 contained only spinel single phase. When x/H333560.3, however, impurities started to appear. So, our study was limited to the range of 0 /H33355x/H333550.25. Figure 1 shows the XRD pattern of the x=0.25 sample, in which the structure refinement has been made using the Rietveld analy-sis program. 27The refinement demonstrates that the Cu1−xInxIr2S4system crystallizes in normal spinel structure with the space group of Fd3¯m, in which Cu and Ir occupy A andBsites, respectively. The occupancy of indium was in- vestigated by assuming that In3+and Ir3+could occupy both AandBsites. The refinement for x=0.25 sample showed that In3+occupancy at the Asite was 0.22 /H208492/H20850. For other samples it was shown that more than 90% of In3+occupies the Asite. The weighted-pattern factor Rwpfor all the monophasic samples is in the range of 7.3% to 10.0%, and the parameterSthat reflects “the goodness of the fitting” is around 1.8, indicating the reliability of structural refinements.As we know, the crystal structure of the cubic spinel is characterized by the lattice constant aand the structural pa- rameter uwhich determines the atomic position of sulfur at /H20849u,u,u/H20850. Figure 2 shows the crystal structure parameters as a function of indium content. It can be seen that both aandu increase almost linearly with increasing indium content. Thelattice expansion is due to the hole-filling /H20849or, hole-depletion /H20850 effect 13as well as the larger size of In3+than that of Cu+. The value of parameter uwould be 0.375 in the ideal case of IrS 6 regular-octahedron coordination. In the present system, u /H110150.386, indicating the stretch of IrS 6octahedra along /H20855111/H20856 directions. This leads to a negative trigonal coordination forIr, which introduces a small splitting of the t 2g/H20849d/H9255/H20850level into eganda1glevels. Moreover, the increase of ualso results in FIG. 1. /H20849Color online /H20850Profile of the room-temperature XRD Rietveld refinement for the sample of Cu 0.75In0.25Ir2S4. FIG. 2. Crystal structure parameters as a function of indium content in the Cu 1−xInxIr2S4system. The upper panel shows the lattice constant aand the parameter u, while the lower panel shows the interatomic distances and the S uIruS bond angle. The lines are guides to the eye.CAO et al. PHYSICAL REVIEW B 72, 125128 /H208492005 /H20850 125128-2the deviation of S uIruS bond angle from 90°, which weakens the d/H9255-p/H9266hybridization. The bond distance and bond angle are thought to be the important structural parameters for the related interatomichybridization. From the lattice geometry, the Cu/In uS in- teratomic distance d AXcan be expressed as dAX=/H208813a/H20849u− 0.25 /H20850. /H208491/H20850 On the other hand, the Ir uS bond length dBXcan be calcu- lated using dBX=a/H208810.0625 − 0.5 /H20849u− 0.375 /H20850+3/H20849u− 0.375 /H208502./H208492/H20850 One of the S uIruS bond angles /H20849the small one /H20850/H9008can also be obtained by sin/H20849/H9008/2/H20850= 0.5 dXX1/dBX, /H208493/H20850 where the short S uS bond distance is obtained by dXX1=2/H208812a/H208490.5 − u/H20850. /H208494/H20850 As can be seen in the lower panel of Fig. 2, the Cu/In uS interatomic distance increases obviously, while the Ir uS bond length does not increase so much with the indium dop-ing. This is in fact due to the distortion of the IrS 6octahedra, which can be seen from the variations of the S uIruS band angle. It is here stressed that the related energy band such asthed/H9255-p /H9266bands should be narrowed due to the weakening of the hybridizations. B. Electrical resistivity The temperature dependence of resistivity in Cu1−xInxIr2S4/H208490/H33355x/H333550.25 /H20850is shown in Fig. 3. As can be seen, the parent compound CuIr 2S4undergoes an abrupt MI transition identified by a three-order jump in resistivityaround 230 K with a thermal hysteresis. Upon the indiumdoping, the MI transition temperature T MIshifts to lower temperatures, and the resistivity jump becomes smaller. Inother words, the MI transition in the parent compound issuppressed by the indium doping. It is noted that the thermalhysteresis in the transition disappears for x/H333560.2, suggesting that the first-order transition is changed into a higher-orderone. The higher-order transition temperature is here labelledT *, defined as the inflexion point in the log 10/H9267-Tcurve. Let us investigate the change of the high-temperature me- tallic state in some more details. First, the room-temperatureresistivity increases monotonically with the indium doping.This is mainly due to the hole depletion, since the conduc-tivity is proportional with the carrier concentration n h/H20849stron- ger evidence will be given in the magnetic susceptibilitymeasurement below /H20850. Note that each indium atom depletes two holes, therefore, n his 0.5 /H208491−2x/H20850per Ir atom, according to the ideal stoichiometry of Cu 1−xInxIr2S4. Second, the tem- perature coefficient of the resistivity /H20849TCR /H20850at room tempera- ture decreases with the indium doping, and it changes thesign at x/H110110.2, as shown in the magnifying plot of Fig. 3. The semiconductinglike conduction at the high temperaturerange in x/H110110.2 samples is rather striking, because the carrier concentration is still high enough /H208490.3 per Ir atom /H20850.A sacomparison in the Cu 1−xZnxIr2S4system, metallic conduction is robust at room temperature up to x/H110110.9/H20849i.e.,nh/H110110.05 per Ir atom /H20850.13Finally, the sign change of the TCR coincides with the change of the transition order. Both occur at x /H110110.2. The metal-semiconductor transition induced by the hole depletion was also observed in the Cu 1−xCdxIr2S4system, where the high-temperature semiconducting phase was ten-tatively called “polaronic semiconductor.” 26In fact, the po- laronic semiconductor can be distinguished from the conven-tional semiconductor by the electrical transport behavior. 28 Resistivity of conventional semiconductor obeys the Arrhen-ius relation /H9267=Aexp/H20873Ea kBT/H20874, /H208495/H20850 where Earepresents the activated energy and kBis the Bolt- zman’s constant. For adiabatic small-polaron hopping, how-ever, the resistivity is given by 28 /H9267=kB /H9263e2d2nSPTexp/H20873E0 kBT/H20874, /H208496/H20850 where E0is the hopping energy, /H9263is the optical phonon or attempt frequency, eis the charge of the hole, dis the dis- FIG. 3. /H20849Color online /H20850Temperature dependence of resistivity for the Cu 1−xInxIr2S4samples. Note that the upper panel uses the loga- rithmic scale for the resistivity axis. The lower panel shows thenormalized resistivity in the high temperature range.SUPPRESSION OF THE METAL-INSULATOR … PHYSICAL REVIEW B 72, 125128 /H208492005 /H20850 125128-3tance between the hopping sites /H20849the nearest Ir uIr distance, being /H208812a/4/H20850, and nSPis the density of small polarons. Fig- ure 4 plots the high-temperature resistivity according to Eq./H208495/H20850and Eq. /H208496/H20850, respectively, for the x=0.25 sample. Obvi- ously, the resistivity data agree the small-polaron hoppingmuch better, suggesting the small polaron formation. Thehopping energy is fitted to be 280 K and the attempt fre-quency is about 3.8 /H1100310 11Hz. It is shown in Fig. 3 that the resistivity increases rapidly at T*for the x=0.25 sample. We argue that this is due to the formation of small bipolarons /H20849more evidence will be given in the next section /H20850. The formation of small bipolaron results in the decrease of the carrier /H20849small polaron /H20850concentration. On the other hand, since the conduction of small bipolaronsis associated with the two carriers to hop to second neigh-bors, which is a higher-order process, one expects that theconductivity coming from the bipolaron pair breaking issmall. 29Therefore, one can see a rapid increase in resistivity atT*. C. Magnetic susceptibility Figure 5 shows the temperature dependence of magnetic susceptibility /H20849/H9273/H20850in the Cu 1−xInxIr2S4/H208490/H33355x/H333550.25 /H20850system. The MI transition in the parent compound CuIr 2S4is charac- terized by the abrupt drop in magnetic susceptibility with athermal hysteresis. This is due to the quenching of Pauliparamagnetic susceptibility in the metallic state, while in theinsulating phase Ir 4+is dimerized in spin singlet. With the indium doping, it can be seen that TMIfirst decreases rapidly, and then decreases steadily in accordance with the resistivityresult. Similarly, the thermal hysteresis in the transition dis-appears for x/H333560.2, suggesting the change of the transition order. The thermal hysteresis for the MI transition is associ-ated with the structural phase transition from cubic totetragonal, 2or strictly speaking from cubic to triclinic,3in the parent compound. In the triclinic phase, ordered Ir4+ dimers form. We thus argue that such a structural phase tran-sition should not occur for x/H333560.2 because of the disappear-ance of thermal hysteresis. As a comparison, in a similar system of Cu 1−xZnxIr2S4,13the thermal hysteresis can be ob- served as long as the structural phase transition occurs. Nev-ertheless, we expect the detailed low-temperature XRD andNMR studies which will be able to check the point for thepresent system in the future. One notes that the drop of /H9273can still be seen for x/H333560.2. This suggests that disordered Ir4+dimers /H20849small bipolarons /H20850 forms below T*. At lower temperatures, /H9273increases rapidly with decreasing temperature, obeying Curie-Weiss law. Theeffective magnetic moment is only about 0.1 /H9262Bper unit for- mula, which is probably attributed to the lattice imperfec-tions. One can also see from Fig. 5 that the high-temperature susceptibility decreases with increasing the indium content.Since the magnetic susceptibility in the metallic state isdominated by the Pauli paramagnetism with /H9273Pauli =/H9262B2N/H20849EF/H20850, and the Fermi level lies near the top of the va- lence band, the decrease in the magnetic susceptibility means that holes are filled by the extra electrons coming from theindium doping. Figure 6 shows the room-temperature sus-ceptibility as a function of hole filling x /H11032or hole concentra- tion nhby the different doping with Zn2+,C d2+, and In3+, respectively. Here, x/H11032is defined as the decrease of hole con- centration /H20849per Ir atom /H20850by the A-site doping, i.e., x/H11032=x for the In doping, while x/H11032=x/2 for Zn and Cd doping. According to our previous study, /H9273Pauli /H11015/H9273300 K +6.4/H1100310−5/H20849emu/mol /H20850. When x/H11032=0.5, nh=0.5− x/H11032 =0, thus /H9273Paulivalue is zero. So, the susceptibility of the three systems tends to converge at /H9273Pauli=0 when x/H11032=0.5. How- ever, the hole-filling dependence is quite different for thethree systems. The susceptibility of the Zn-doped system de-creases mildly at x /H11032/H110210.35, but drops rapidly at the high doping level. The Cd-doped system exhibits the oppositecase, and the In-doped system shows the steady decrease inthe susceptibility. The systematic variations on the hole-filling dependence of magnetic susceptibility suggests the band structure modi- FIG. 4. High temperature resistivity in the Cu 0.75In0.25Ir2S4, fit- ted in terms of the activated conduction /H20851Eq. /H208495/H20850/H20852and the small polaron hopping conduction /H20851Eq. /H208496/H20850/H20852. FIG. 5. /H20849Color online /H20850Temperature dependence of magnetic susceptibility for Cu 1−xInxIr2S4samples. The applied field is 1000 Oe.CAO et al. PHYSICAL REVIEW B 72, 125128 /H208492005 /H20850 125128-4fications upon the different doping, since /H9273Pauli=/H9262B2N/H20849EF/H20850. First, there is a band narrowing effect mainly due to the distortion of the IrS 6octahedra associated with the increase of the lattice constant. The Cd-doped system is expected tohave the narrowest highest-occupying band. Second, theA-site doping induces Anderson-type disorder, which results in the distorted band tails. The Cd doping is expected toinduce the strongest disorder potential, leading to the broad-est band tail. Finally, the electron-phonon interactions en-hanced by the band narrowing probably play an importantrole in the electronic localizations. Although Anderson localization effect is responsible for the formation of the band tails, we argue that the Anderson-type localization is not the main mechanism for thesemiconducting behavior at high temperatures for therelatively low hole-filling level /H20849i.e., the Fermi level E Fis argued to be below the Mott mobility edges Ec at the low hole filling. However, at high hole-filling levels with relatively low hole concentrations, EFlevel may go beyond the Ecedge, leading to the Anderson-type localization.26/H20850This is because that the high-temperature /H9267/H20849T/H20850behavior of Cu 0.75In0.25Ir2S4, for example, does not sat- isfy the expression of variable-range-hopping15of localized carriers, /H9267/H11008exp/H20851/H20849T0/T/H208501/4/H20852. The/H9267/H20849T/H20850curve in Fig. 4 actually suggests that small polaron formation is mainly responsible for the high-temperature semiconducting behavior. One may be surprised at the fact that the high temperature /H9273/H20849T/H20850still exhibits Pauli paramagnetism for the x=0.25 sample /H20849see Fig. 5 /H20850. This can be explained as follows. For a localized electronic state in the nondegenerate limit, the /H9273/H20849T/H20850 data should obey Curie’s law. However, in the degenerate limit, as in the case of small-polaron narrow band, Pauliparamagnetism is still expected. Therefore, the /H9273300 K data does not show obvious changes in the metal-semiconductor boundaries shown in Fig. 6.D. Electronic phase diagram The above results can be concluded in a tentative elec- tronic phase diagram, as shown in Fig. 7. At the low dopinglevel with x/H333550.2, the high-temperature phase is metallic, which can be regarded as a large-polaron state if consideredelectron-phonon interactions. The low-temperature phase isin a charge-ordered and spin-dimerized insulating state,which can be described as bipolaron crystals. According toour results the transition from the large polarons to bipolaroncrystals, which provides a possible explanation for the MItransition, would be of first order. When the indium doping exceeds 0.2, the system shows the semiconductinglike behavior in the whole temperaturerange. The high-temperature semiconducting state can beelucidated in terms of small polarons, while the low-temperature semiconducting state is regarded as a bipolaronliquid. Based on our results the transition from the smallpolarons to small bipolarons, which explains the evolution ofthe MI transition, would be second order. Since our results were obtained in polycrystalline samples, further experiments are needed to confirm theabove polaronic features. In addition, it should be stated herethat the bipolaron formation is only one possible scenario.Another possible explanation may come from the nanoscalephase separation that can be often seen in colossal magne-toresistence manganite system. 30If one considers the coex- istence of cubic and triclinic nanophases, both the drop ofmagnetic susceptibility at the T *and the absence of thermal hysteresis in the /H9267/H20849T/H20850and/H9273/H20849T/H20850curves can also be explained. E. Hole-depletion and lattice-size effects Figure 8 plots the lattice constants and the phase transi- tion temperatures in the Cu 1−xMxIr2S4/H20849M=Zn, In, and Cd /H20850 system. Although all the doping on the Cu site depletes theholes in the metallic phase, the variations of T MIorT*are different. At the low hole-filling level up to x/H11032=0.025, TMI drops to about 175 K in all three systems. This implies that the suppression of the MI transition is mainly controlled bythe hole depletion at the low doping level. The hole depletion FIG. 6. Room-temperature susceptibility as a function of hole filling x/H11032in the Cu 1−xMxIr2S4/H20849M=Zn, In, and Cd /H20850systems. The arrows mark the metal-semiconductor boundaries judged by thesign change of TCR at high temperatures. Note that x /H11032=x/2 for the Zn-doped and Cd-doped systems. FIG. 7. Tentative electronic phase diagram in the Cu 1−xInxIr2S4 spinel system. The solid line represents the first order transition, while the dashed line denotes the second order one. It is noted thatthe polaronic picture needs further confirmations.SUPPRESSION OF THE METAL-INSULATOR … PHYSICAL REVIEW B 72, 125128 /H208492005 /H20850 125128-5for the parent compound means that the ratio of Ir4+to Ir3+ deviates from 1:1, which is not in favor of the formation of the complex charge ordering /H20849ordered bipolarons /H20850. In other words, the chemical doping induces extra electrons in theantibonding d/H9255band for the dimerized Ir. As a result, T MIis decreased rapidly by the hole depletion, depending hardly onthe size of the dopants at the low doping level. With the further hole depletion, the charge ordering can- not be maintained anymore, thus the disordered bipolaronstate /H20849forM=Cd or In /H20850or the phase separation /H20851forM=Zn /H20849Ref. 13 /H20850/H20852appears. When x /H11032/H110220.025, TMIorT*shows differ- ent hole-filling dependence. At a fixed hole-filling level, thelarger lattice constant, the higher transition temperature. Asdiscussed in Secs. III A and III C, larger lattice constant cor-responds to narrower bandwidth for the valence bands. Theenergy band narrowing results in the enhancement ofelectron-phonon interactions, which stabilizes the polaronsand the bipolarons. 31Therefore, the lattice size effect on TMI orT*indicates that the electron-phonon coupling plays an important role in the MI transition in the present system. It is also noted that superconductivity ground state was observed in the Zn-doped system for x/H11032/H333560.125. In the case of Cd and In doping, however, bipolaron liquid state appearsat low temperatures for x /H11032/H114070.125. According to the theoret- ical works,32,33Very strong electron-phonon interactions may lead to the transition from BCS Cooper pairs into small bi-polarons. Therefore, the variations of the ground state due tothe lattice size effect further suggest the existence of strongelectron-phonon interactions in the thiospinel system. IV . CONCLUSION In conclusion, we have studied the thiospinel Cu1−xInxIr2S4/H208490/H33355x/H333550.25 /H20850system by the measurements of crystal structure, electrical resistivity, and magnetic suscep- tibility. The structural analysis shows that indium preferen-tially occupies the Cu site. The In doping leads to the devia-tion of SuIruS bond angle from 90°, which weakens the d/H9255-p /H9266hybridization and makes the top valence band nar- rowed. It was shown that the In doping depletes the holes inthe valence band of the high-temperature metallic phase, andinduces extra electrons in the antibonding d/H9255band for the insulating phase, suppressing the MI transition in the parentcompound. By further doping up to x/H333560.2, the MI transition evolves into a second-order semiconductor-to-semiconductortransition. The high-temperature semiconducting phase hasthe characteristic of small-polaron transport, while the low-temperature semiconducting phase has lower intrinsic mag-netic susceptibility. Thus the semiconductor-to-semiconductor transition is argued to be associated with anelectronic transformation from small polarons to small bipo-larons. We also conclude that the systematic suppressions ofthe MI transitions in the Cu 1−xMxIr2S4/H20849M=Zn, In or Cd /H20850 system are not only related to the hole depletions, but alsolinked with the strong electron-lattice coupling that is modu-lated by the band narrowing due to the large ionic size of thedopants. This observation implies that the MI transition inthe parent compound is primarily due to a kind of latticeinstability driven by the strong electron-phonon interactions. ACKNOWLEDGMENT This work was supported by the National Science Foun- dation of China /H20849Grant No. 10104012 /H20850. *Author to whom correspondence should be addressed. Electronic address: ghcao@zju.edu.cn 1S. Nagata, T. Hagino, Y . Seki, and T. Bitoh, Physica B 194–196 , 1077 /H208491994 /H20850. 2T. Furubayashi, T. Matsumoto, T. Hagino, and S. Nagata, J. Phys. Soc. Jpn. 63, 3333 /H208491994 /H20850. 3P. G. Radaelli, Y . Horibe, M. J. Gutmann, H. Ishibashi, C. H. Chen, R. M. Ibberson, Y . Koyama, Y . S. Hor, V . Kiryukhin, andS. W. Cheong, Nature /H20849London /H20850416, 155 /H208492002 /H20850.4G. Oomi, T. Kagayama, I. Yoshida, T. Hagina, and S. Nagata, J. Magn. Magn. Mater. 140–144 , 157 /H208491995 /H20850. 5H. Kang, K. Bärner, I. V . Medvedeva, P. Mandal, A. Poddar, and E. Gmelin, J. Alloys Compd. 267,1/H208491998 /H20850. 6A. T. Burkov, T. Nakama, M. Hedo, K. Shintani, K. Yagasaki, N. Matsumoto, and S. Nagata, Phys. Rev. B 61, 10049 /H208492000 /H20850. 7J. Matsuno, T. Mizokawa, A. Fujimori, D. A. Zatsepin, V . R. Galakhov, E. Z. Kurmaev, Y . Kato, and S. Nagata, Phys. Rev. B 55, R15979 /H208491997 /H20850. FIG. 8. 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PhysRevB.77.205434.pdf

Particle-hole excitations within a self-consistent random-phase approximation D. Gambacurta1,2,*and F. Catara2,3,† 1Consortium COMETA, Via S. Sofia 64, I-95123 Catania, Italy 2Dipartimento di Fisica ed Astronomia dell’Università di Catania, Via S. Sofia 64, I-95123 Catania, Italy 3Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Via S. Sofia 64, I-95123 Catania, Italy /H20849Received 14 January 2008; revised manuscript received 2 April 2008; published 23 May 2008 /H20850 We present a method for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation /H20849RPA /H20850. The main differences with respect to previous approaches are under- lined and discussed. In particular, no use of renormalized operators is made. By means of the method oflinearization of equations of motion, we derive a set of RPA-like equations depending only on the one-bodydensity matrix. The latter is no more assumed to be diagonal and it is expressed in terms of the Xand Y amplitudes of the particle-hole phonon operators. This set of nonlinear equations is solved via an iterativeprocedure, which allows us to calculate the energies and the wave functions of the excited states. Afterpresenting our approach in its general formulation, we test its quality by using metal clusters as a good testlaboratory for a generic many-body system. Comparison to RPA shows significant improvements. We discussalso how the present approach can be further extended beyond the particle-hole configuration space, getting anapproximation scheme in which energy weighted sum rules are exactly preserved, thus solving a problemcommon to all extension of RPA proposed until now. The implementation of such approach will be afforded ina future work. DOI: 10.1103/PhysRevB.77.205434 PACS number /H20849s/H20850: 21.60.Jz, 21.10.Pc, 21.10.Re I. INTRODUCTION The advent of modern computers and the continuous im- provement of their performances, together with the increas-ing ability to exploit them by using more and more refinedalgorithms, has made possible to calculate quite accuratelythe properties of ground state and low energy excited statesof many-body systems starting from the basic interaction 1 within several methods: no-core shell model,2–4coupled cluster,5–8and the Green’s function Monte Carlo9–12calcula- tions. However, it is not clear whether in the near future anaccurate description of higher lying states can be achievedalong such paths. Another microscopic approach, particularlywell suited for the study of collective vibrational states, isbased on the Hartree–Fock /H20849HF/H20850plus random phase approxi- mation /H20849RPA /H20850with an effective interaction derived from the density-energy functional. 13–15As it is well known, in the HF approximation, one imposes that the ground state of the sys-tem is the Slater determinant minimizing the expectationvalue of the Hamiltonian. By solving the correspondingvariational equations, one gets the “best” single particlewave functions by which that Slater determinant is built. Acomplete single particle basis is then determined by solvingthe one-body Schrödinger equation with the self-consistentHF potential, both for the states occupied in the ground state/H20849holes /H20850and the unoccupied ones /H20849particles /H20850. The excited states are then studied within the RPA, describing them assuperpositions of particle-hole /H20849p-h/H20850configurations, built on top of the unknown exact ground state including correlations.However, RPA, while having several very nice properties 14,15 and being completely microscopic, contains some approxi- mations. In fact, since the explicit structure of the correlatedground state is not known, the standard RPA is obtained byreplacing, in the evaluation of the matrix elements of theRPA equations, this state by the uncorrelated HF one. Thereplacement, also called quasiboson approximation, produces a missing of some terms in the evaluation of the commuta-tors in the equations of motion and thus a violation of Pauliprinciple. Various attempts have been done to improve overthem or by using boson expansion methods, 16–24or remain- ing entirely in the fermionic space.25–37In Ref. 32, an exten- sion of RPA, named improved RPA /H20849IRPA /H20850, was presented. The starting point was the introduction of renormalized p-h operators and, by using the method of linearization of theequations of motion, a set of nonlinear RPA-like equationswas obtained. The main drawback of IRPA is the fact thatone assumes the one-body density matrix /H20849OBDM /H20850to be diagonal in the HF basis. Such limit was eliminated in Ref.33, where a more refined procedure was used. In the present paper, we move along the line of Refs. 32and33and intro- duce a more general approach, which we will call extendedRPA /H20849ERPA /H20850for brevity, not based on the renormalized RPA /H20849RRPA /H20850. We will also show some results obtained by apply- ing this method to the study of the excitation spectrum andother properties of metallic clusters within the uniform jel-lium model 38,39with bare Coulomb interaction both for the electrons with the jellium and for the electrons amongthemselves. 40We are aware that the jellium approximation is too poor for a quantitative study; on the other hand, ourpurpose here is not to make a realistic study of metal clustersbut rather to compare different levels of approximations. Infact, we believe that such a model is a very good test labo-ratory since it contains many characteristics of a generic re-alistic many-body system and, on the other hand, it allows amore clear comparison among different approaches becauseno adjustable parameter is present in it /H20849as it is often the case with effective interactions /H20850. We stress that the present ap- proach can be used to study any many-body system, the onlydifferences being the interaction and the relevant quantumnumbers. In particular, we plan to apply it to nuclei where,for example, RRPA has been shown to improve the descrip-PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 1098-0121/2008/77 /H2084920/H20850/205434 /H208499/H20850 ©2008 The American Physical Society 205434-1tion of double beta decay with respect to RPA.41,42 As discussed in Secs. II and III, the merits of the present approach with respect to previous ones32,33can be traced back to the fact that one does not move anymore in theRRPA scheme. On the other hand, the problem remains that,as in all the extensions of RPA proposed until now, violationsof the energy weighted sum rules /H20849EWSRs /H20850are still present. In this respect, we recall that the fulfillment of EWSR guar-antees that spurious excitations corresponding to brokensymmetries as, for example, the translational invariance,separate out and are orthogonal to the physical states. Weshow that by enlarging the configuration space, along thelines of Ref. 37, within the present framework, it should be possible to implement a completely self-consistent RPA pre-serving the EWSR and to overcome the difficulties pointedout in Ref. 37where the RRPA enlarged scheme was studied in a three-level Lipkin model. This possibility will be inves-tigated in the near future. In Sec. II, the basic equations of ERPA are shown and their derivation is outlined, while Sec. III is devoted to tran-sition amplitudes and sum rules. In Sec. IV , we show theresults and compare them with the RPA ones and with theavailable experimental data. An improved agreement withthe latter is obtained. Finally, in Sec. V , we draw the mainconclusions and outline the direction for future work. II. BASIC EQUATIONS OF ERPA In Ref. 32, an extension of RPA, named IRPA, was pre- sented. The starting point was to introduce renormalized p-h creation and annihilation operators /H20849for notation simplicity, we do not indicate coupling to total quantum numbers /H20850, Bph†=/H20858 p/H11032h/H11032Nph,p/H11032h/H11032ap/H11032†ah/H11032/H208491/H20850 by which one builds the operators Q/H9263†=/H20858 ph/H20849Xph/H9263Bph†−Yph/H9263Bph/H20850, /H208492/H20850 whose action on the ground state of the system /H208410/H20856generates the collective states, /H20841/H9263/H20856=Q/H9263†/H208410/H20856, /H208493/H20850 with /H208410/H20856defined as the vacuum of the Q/H9263’s, Q/H9263/H208410/H20856=0 . /H208494/H20850 By assuming that the one-body density matrix is diagonal /H208550/H20841a/H9251†a/H9252/H208410/H20856=n/H9251/H9254/H9251/H9252, /H208495/H20850 choosing Nph,p/H11032h/H11032=/H9254pp/H11032/H9254hh/H11032/H20849nh−np/H20850−1 /2/H11013/H9254pp/H11032/H9254hh/H11032Dph−1 /2, /H208496/H20850 and following the equations of motion method,14,15one finds for the excitation energies /H9275/H9263and the Xand Yamplitudes, /H20873AB −B/H11569−A/H11569/H20874/H20873X/H20849/H9263/H20850 Y/H20849/H9263/H20850/H20874=/H9275/H9263/H20873X/H20849/H9263/H20850 Y/H20849/H9263/H20850/H20874, /H208497/H20850 withAph,p/H11032hp=/H208550/H20841/H20851Bph,H,Bp/H11032h/H11032†/H20852/H208410/H20856, /H208498/H20850 and Bph,p/H11032hp=− /H208550/H20841/H20851Bph,H,Bp/H11032h/H11032/H20852/H208410/H20856, /H208499/H20850 where His the Hamiltonian of the system and the symme- trized double commutators are defined as /H20851A,B,C/H20852=1 2/H20853/H20851A,/H20851B,C/H20852/H20852+/H20851/H20851A,B/H20852,C/H20852/H20854. /H2084910/H20850 In IRPA, the Xand Yamplitudes satisfy the same orthonor- mality conditions as in RPA, /H20858 ph/H20849Xph/H9263Xph/H9263/H11032−Yph/H9263Yph/H9263/H11032/H20850=/H9254/H9263/H9263/H11032, /H2084911/H20850 following from Eqs. /H208494/H20850–/H208496/H20850. In order to make affordable the solution of Eqs. /H208497/H20850, one has to resort to some approximation. In IRPA, the linearization of the equations of motion wasused: all the two-body terms appearing in the commutator ofthe Hamiltonian with a p-hoperator are contracted with re- spect to the reference state /H208410/H20856. In this way, one obtains a quantity linear in the p-hoperators. Substituting this in the double commutators of Eqs. /H208498/H20850and /H208499/H20850, one gets an expres- sion for the matrices A ph,p/H11032h/H11032=/H208550/H20841/H20851ah†ap,H,ap/H11032†ah/H11032/H20852/H208410/H20856and B containing only the OBDM. The standard RPA equations can be derived by the same procedure but using the uncorrelatedHF ground state instead of the correlated one /H208410/H20856.I nt h e IRPA case, this method leads to a set of RPA-like equations,but with the Aand Bmatrices depending on the occupation numbers which, in turn, can be evaluated in terms of the X and Yamplitudes by using the number operator method. 43 The equations of motion are thus nonlinear and have been solved iteratively. One limitation of IRPA is the fact that oneassumes the OBDM to be diagonal in the HF basis. Suchinconsistency was eliminated in Ref. 33, where a double it- erative procedure was used: starting from a zero order ap-proximation and assuming the OBDM to be diagonal, onesolves the IRPA equations, next the number operator methodis used to calculate the OBDM with the so determined Xand Y, then the OBDM is diagonalized and the IRPA calculation is repeated in the new basis, and so on until convergence isreached. Of course, this double iterative procedure is quitedemanding from the computational point of view. On theother hand, use is still made of ansatz /H208491/H20850and choice /H208496/H20850. This is certainly a limitation. In the following, we willpresent an improved approach /H20849ERPA /H20850, in which such a limi- tation is overcome. Thus, ERPA is completely self-consistentwithin the linearization of the equations of motion method. In the present approach, after the single particle basis is fixed by solving the HF equations, we introduce phonon op-erators Q /H9263having the same form as the RPA ones, Q/H9263†=/H20858 ph/H20849Xph/H9263ap†ah−Yph/H9263ah†ap/H20850, /H2084912/H20850 avoiding thus the use of “renormalized” operators and choice /H208496/H20850. The Xand Yamplitudes are solutions of the system of equationsD. GAMBACURTA AND F. CATARA PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-2/H20873AB B/H11569A/H11569/H20874/H20873X/H20849/H9263/H20850 Y/H20849/H9263/H20850/H20874=/H9275/H9263/H20873G 0 0−G/H11569/H20874/H20873X/H20849/H9263/H20850 Y/H20849/H9263/H20850/H20874, /H2084913/H20850 where Aph,p/H11032h/H11032=/H208550/H20841/H20851ah†ap,H,ap/H11032†ah/H11032/H20852/H208410/H20856, /H2084914/H20850 Bph,p/H11032,h/H11032=− /H208550/H20841/H20851ah†ap,H,ah/H11032†ap/H11032/H20852/H208410/H20856, /H2084915/H20850 Gph,p/H11032,h/H11032=/H208550/H20841/H20851ah†ap,ap/H11032†ah/H11032/H20852/H208410/H20856, /H2084916/H20850 and satisfy the orthonormality conditions /H20858 ph,p/H11032h/H11032/H20849XphvXp/H11032h/H11032v/H11032−YphvYp/H11032h/H11032v/H11032/H20850Gph,p/H11032h/H11032=/H9254vv/H11032. /H2084917/H20850 The standard RPA equations can be obtained by replacing, in the evaluation of matrices /H2084914/H20850–/H2084916/H20850, the state /H208410/H20856with the /H20849uncorrelated /H20850HF one. In particular, the norm matrix Gac- quires the simpler form Gph,p/H11032,h/H11032/H20849HF/H20850=/H20855HF/H20841/H20851ah†ap,ap/H11032†ah/H11032/H20852/H20841HF/H20856=/H9254hh/H11032/H9254pp/H11032, /H2084918/H20850 and thus for the RPA Xand Yamplitudes, we find again the orthonormality conditions /H2084911/H20850. This substitution introduces a visible inconsistency since, on one hand, the definition ofground state /H208494/H20850as the vacuum of the Qoperators is used to derive the Eqs. /H2084913/H20850, while, on the other hand, the HF state is used in calculating the expectation values appearing in thoseequations. Furthermore, it introduces some violations of thePauli principle. 15 The explicit form of the ERPA matrices A,B, and Gin terms of the OBDM /H9267, obtained by using the linearization of the equations of motion, is Aph,p/H11032h/H11032=1 2/H20853/H9267/H20849h,h/H11032/H20850/H9280/H20849p,p/H11032/H20850+/H9267/H20849p,p/H11032/H20850/H9280/H20849h,h/H11032/H20850 −/H9254hh/H11032/H20858 p1/H9267/H20849p,p1/H20850/H9280/H20849p1,p/H11032/H20850−/H9254pp/H11032/H20858 h1/H9267/H20849h,h1/H20850/H9280/H20849h1,h/H11032/H20850 +/H20858 h1h2/H20855ph1/H20841v/H20841h2p/H11032/H20856/H9267/H20849h1,h/H11032/H20850/H9267/H20849h2,h/H20850 +/H20858 p1p2/H20855hp1/H20841v/H20841p2h/H11032/H20856/H9267/H20849p1,p/H11032/H20850/H9267/H20849p2,p/H20850 +2/H20858 h1p1/H20855h1p1/H20841v/H20841p/H11032h/H20856/H9267/H20849h1,h/H11032/H20850/H9267/H20849p1,p/H20850 +/H20849ph↔p/H11032h/H11032/H20850/H20854, /H2084919/H20850 Bph,p/H11032h/H11032=− /H20853/H20858 h1h2/H20855h1h2/H20841v/H20841p/H11032p/H20856/H9267/H20849h1,h/H11032/H20850/H9267/H20849h2,h/H20850 +/H20858 p1p2/H20855h/H11032h/H20841v/H20841p1p2/H20856/H9267/H20849p1,p/H11032/H20850/H9267/H20849p2,p/H20850 +/H20858 h1p1/H20855h1h/H20841v/H20841p1p/H11032/H20856/H9267/H20849h1,h/H11032/H20850/H9267/H20849p1,p/H20850 +/H20858 h1p1/H20855h1h/H11032/H20841v/H20841p1p/H20856/H9267/H20849h1,h/H20850/H9267/H20849p1,p/H11032/H20850/H20854, /H2084920/H20850Gph,p/H11032h/H11032=/H9254pp/H11032/H9267/H20849h,h/H11032/H20850−/H9254hh/H11032/H9267/H20849p,p/H11032/H20850, /H2084921/H20850 where tandvare the one-body and the two-body parts of the Hamiltonian, respectively. In Eq. /H2084919/H20850, the/H9280quantities are defined as /H9280/H20849/H9251,/H9252/H20850=t/H9251,/H9252+/H20858 /H9253/H9254/H20855/H9251/H9253/H20841v/H20841/H9252/H9254/H20856/H9267/H20849/H9253,/H9254/H20850, /H2084922/H20850 in which /H9251,/H9252,/H9253, and /H9254run over all single particle states. Note that when /H9267/H20849/H9251,/H9252/H20850is assumed to be diagonal and its eigenvalues equals to 0 or 1, i.e., in the HF limit, the abovequantities become /H9280/H20849/H9251,/H9252/H20850=t/H9251,/H9252+/H20858 h/H20855/H9251h/H20841v/H20841/H9252h/H20856/H11013/H20855/H9251/H20841HHF/H20841/H9252/H20856, /H2084923/H20850 where HHFis the HF one-body Hamiltonian. Since the latter commutes with the HF one-body density, /H9280/H20849/H9251,/H9252/H20850 =/H9280/H9251/H9254/H20849/H9251,/H9252/H20850, where /H9280/H9251are the HF single particle energies. The matrix /H9280/H20849/H9251,/H9252/H20850of Eq. /H2084922/H20850can be diagonalized without going to the HF limit using instead the OBDM obtained self-consistently within ERPA. In this way, one gets a kind of“generalized single particle energies.” We will discuss that inSec. IV . In the expressions for the A,B, and Gmatrices, the OBDM appears. By using the number operator method, 43its matrix elements are expressed in terms of the Xand Yam- plitudes. Therefore, Eqs. /H2084913/H20850are nonlinear. In the present approach, we have /H9267/H20849p,h/H20850/H11013/H20855 0/H20841ap†ah/H208410/H20856=0 , /H2084924/H20850 /H9267/H20849p,p/H11032/H20850/H11013/H20855 0/H20841ap†ap/H11032/H208410/H20856=/H20858 /H9263/H9263/H11032S/H20849/H9263,/H9263/H11032/H20850/H20858 p1h1/H20858 p2h2Yp1h1/H9263Yp2h2/H9263/H11032/H11569 /H11003/H20858 hG/H20849ph,p1h1/H20850G/H11569/H20849p/H11032h,p2h2/H20850, /H2084925/H20850 /H9267/H20849h,h/H11032/H20850/H11013/H20855 0/H20841ah†ah/H11032/H208410/H20856=/H9254hh/H11032−/H20858 /H9263/H9263/H11032S/H20849/H9263,/H9263/H11032/H20850/H20858 p1h1/H20858 p2h2Yp1h1/H9263Yp2h2/H9263/H11032/H11569 /H11003/H20858 pG/H20849ph,p1h1/H20850G/H11569/H20849ph/H11032,p2h2/H20850, /H2084926/H20850 where S/H20849/H9263,/H9263/H11032/H20850=/H9254/H9263/H9263/H11032−1 2/H20858 p1h1,p2h2Xp1h1/H9263Xp2h2/H9263/H11032/H11569 /H11003/H20858 p3h3G/H20849p3h3,p1h1/H20850G/H11569/H20849p3h3,p2h2/H20850. /H2084927/H20850 The above equations are exact up to order O/H20849/H20841Y/H208414/H20850. We re- mark that in Eqs. /H2084925/H20850and /H2084926/H20850, the Xand Yamplitudes as well as the norm matrix Gappear. In order to solve the equations of motions /H2084913/H20850, we use an iterative procedure. At thenth iterative step, we compute the /H9267matrix, and thus, the A,B, and Gmatrices by using the Xand Yamplitudes and the/H9267matrix of the /H20849n−1/H20850th step. As starting point, we take the solutions of the standard RPA equations and the HF /H9267 matrix. This procedure is carried out until convergence isreached.PARTICLE-HOLE EXCITATIONS WITHIN A SELF- … PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-3In the present approach, all calculations are carried out in the HF single particle basis. The OBDM is calculated in thecorrelated ground state /H208410/H20856state defined as the vacuum of the Qoperators. Another possible choice could be to work in the basis which diagonalizes OBDM /H20849as it was made in Ref. 33 in the RRPA approach /H20850or the “generalized single particle energies” /H20851Eq. /H2084922/H20850/H20852. In both cases, at each step of the itera- tive procedure, one has to diagonalize /H9267or/H9280matrix and, after that, calculate the G,A, and Bmatrices in the new basis. These operations make the procedure more cumber-some from a numerical point of view. III. TRANSITION AMPLITUDES AND SUM RULES As it is well known, if /H208410/H20856and /H20841/H9263/H20856are a complete set of exact eigenstates of the Hamiltonian, with eigenvalues E0 and E/H9263, the following identity holds: /H20858 /H9263/H9275/H9263/H20841/H20855/H9263/H20841F/H208410/H20856/H208412=1 2/H208550/H20841/H20851F,/H20851H,F/H20852/H20852/H208410/H20856, /H2084928/H20850 where /H9275/H9263=E/H9263−E0. The above equality is in general violated to some extent when /H208410/H20856,/H20841/H9263/H20856, and /H9275/H9263are calculated with some approximation. To which extent it is satisfied is a mea-sure of the adequacy of the approximation. We note that theright-hand side is a quantity which depends only on theground state properties. Let us examine the transition amplitudes /H20855 /H9263/H20841F/H208410/H20856induced by a one-body operator, F=/H20858 /H9251,/H9252/H20855/H9251/H20841F/H20841/H9252/H20856a/H9251†a/H9252, /H2084929/H20850 between the ground state /H208410/H20856and excited states /H20841/H9263/H20856. By using definition /H208493/H20850and the vacuum property /H208494/H20850, one gets /H20855/H9263/H20841F/H208410/H20856=/H208550/H20841/H20851Q/H9263,F/H20852/H208410/H20856. /H2084930/H20850 The above expression is general and it is valid independently of the explicit form of the Qoperators. When the latter has form /H2084912/H20850, only the p-hcomponents of the transition operator Fare selected. A very important feature of RPA, known as Thouless theorem,44can be described as follows. In both sides of Eq. /H2084928/H20850, the HF state is used, instead of the corre- lated /H208410/H20856one. Then, from Eq. /H2084930/H20850, one gets /H20855/H9263/H20841F/H208410/H20856=/H20858 ph/H20853Xph/H9263/H11569/H20855p/H20841F/H20841h/H20856+Yph/H9263/H11569/H20855h/H20841F/H20841p/H20856/H20854. /H2084931/H20850 i.e., the standard RPA expression. On the other hand, the right-hand side evaluated in the HF state can be completelyexpressed in terms of the Aand Bmatrices of RPA. It is then easy to show 15,44that, if the left-hand side is calculated by using Eq. /H2084931/H20850and the RPA values for the energies /H9275/H9263and theX/H9263and Y/H9263amplitudes, equality /H2084928/H20850is exactly satisfied. This result is very important also because it guarantees thatspurious excitations corresponding to broken symmetries /H20849as, for example, the translational invariance /H20850separate out and are orthogonal to the physical states. We remark that, whenthe right-hand side is evaluated in the HF state, only the p-h components of the transition operator Fappear in it.When the correlated /H208410/H20856is maintained, it is still true that only the p-hcomponents of the transition operator Fappear on the left-hand side and one has /H20855 /H9263/H20841F/H208410/H20856=/H20858 php/H11032h/H11032/H20853Xph/H9263/H11569/H20855p/H11032/H20841F/H20841h/H11032/H20856+Yph/H9263/H11569/H20855h/H11032/H20841F/H20841p/H11032/H20856/H20854Gph,p/H11032h/H11032, /H2084932/H20850 while this is no more the case on the right-hand side, where the whole structure of Fappears. This is the reason why all extensions of RPA, with only p-hexcitations, violate Eq. /H2084928/H20850. We stress that if only the p-hpart of the Foperator is taken in the double commutator and the latter is calculatedby the linearization procedure, just the ERPA matrices A,B, and Gappear on the right-hand side. Therefore, if ERPA quantities are used also on the left-hand side, the EWSR ispreserved. Since the deviations present in ERPA can be ascribed to the fact that the Q /H9263†operators have the structure of Eq. /H2084912/H20850, a possible generalization would be to assume Q/H9263†=/H20858 /H9251/H11022/H9252/H20849X/H9251/H9252/H9263a/H9251†a/H9252−Y/H9251/H9252/H9263a/H9252†a/H9251/H20850/H20849 33/H20850 where /H9251and/H9252denote generic single particle states /H20849occupied and not /H20850. As shown in Ref. 37within such “enlarged” RPA approach, equality /H2084928/H20850holds exactly. This has been also tested numerically in Ref. 37where an enlarged RRPA ap- proach has been applied to a three-level Lipkin model. How-ever, a difficulty was pointed out, namely, the appearance ofa nonphysical state, which could not be eliminated withinthat approach. Within the RRPA, such state could not beidentified a priori but, by comparison, it was found that it did not correspond to any exact eigenstate. In a realistic calcula-tion, of course, this is not possible. In the scheme we arepresenting in this paper, this difficulty may be overcomesince nonphysical states, if any, can be singled out by look-ing for zero eigenvalues of the norm matrix G, thus getting, within the method of linearization of the equations of mo-tion, a self-consistent approach. This will be the subject offuture investigations. IV. RESULTS AND DISCUSSION As mentioned above, the described method can be used to study any many-body system. In this section, we apply it tostudy the electronic properties of metal clusters. In fact, suchsystems show many characteristics of a generic realisticmany-body system and, on the other hand, a more clear com-parison among different approaches can be done since noresort to effective interactions, depending on adjustable pa-rameters, is necessary. The ERPA results will be compared with those obtained in the standard RPA approach. The calculations are done in thejellium approximation. In this model, the ionic background isdescribed as a uniform positive charge distribution, whichinteracts with the delocalized valence electrons, also interact-ing among themselves, via the bare Coulomb interaction.The jellium properties are completely determined by one pa-rameter, the Wigner–Seitz radius r s. Within this model, theD. GAMBACURTA AND F. CATARA PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-4motion of the valence electrons is determined by the Hamil- tonian H=/H20858 ihi+/H20858 i/H11021jvij, /H2084934/H20850 with hi=−/H60362 2m/H11612i2+V/H20849ri/H20850;vij=e2 4/H92661 /H20841r/H6023i−r/H6023j/H20841, /H2084935/H20850 and V/H20849r/H20850=Ze2 4/H9266/H20877/H208491/2rc/H20850/H20849r2/rc2−3/H20850forr/H11349rc −1 /rforr/H11350rc,/H20878 /H2084936/H20850 where Neis the number of electrons and rcis the radius of the jellium sphere, i.e., rc=rsNe1/3. The procedure that we have followed consists of the fol- lowing steps. First of all, we have obtained, by solving HFequations, the single particle basis, in which all the subse-quent calculations are carried out. The single particle wavefunctions have been represented as linear superposition ofharmonic oscillator ones. After that, we have solved the stan-dard RPA equations. The so obtained Xand Yamplitudes are used as input quantities in the ERPA iterative procedure. Inthe first step, we thus use them and the HF norm matrix /H2084918/H20850 to calculate the OBDM. With the new A,B, and Gmatrices, we solve the ERPA /H20851Eq. /H2084913/H20850/H20852. The new X’s and Y’s and the Gmatrix of the previous iterative step are used to recalculate the OBDM and so on until convergence is reached, namely,until the maximum relative difference in the excitations en-ergies between two successive iterations is less than a chosenlimit. In the calculations, we are going to present that thislimit has been put equal to 10 −5. We remark that, at each step of the iterative procedure, we solve the ERPA equations for multipolarities ranging fromL=0 to L=6 for both the spin S=0 and S=1 channels, and the so obtained Xand Yamplitudes, for all Land S, are used in building the new OBDM. At RPA level, in some cases,these equations are found to have imaginary solutions. Thishappens, for example, for some states with spin S=1 and it is due to the too much attractive behavior of the bare Coulombinteraction in the S=1 channel. The states having imaginary energies are thus not included in the initial steps. However,with the use of the new OBDM, we find that the final solu-tions are real for all Land Sstates. The present approach has been applied to the study of neutral and ionized closed shell Na clusters, with a numberof valence electrons ranging from 8 to 92. However, in thefollowing, we will focus our analysis on the medium-size Na 40and Na41+clusters since the qualitative behavior is the same for the other closed shell clusters. In Fig. 1, we show the centroid energies of the strength distributions for multipolarities from L=0 to L=4 and spin S=0 and S=1 in Na 40. In each panel, the values on the left and on the right-hand sides are those obtained in RPA andERPA, respectively. Note that for S=1 and L=3, RPA col- lapses. The differences among the two approaches are quitepronounced in both the S=0 and the S=1 spin channels. In the latter case, we see that all the centroids are pushed up asa consequence of the better treatment of ground state corre- lations in ERPA with respect to RPA. In particular, the col-lapse of RPA for L=3 is not present in ERPA. In Figs. 2and3, we plot the dipole strength distributions obtained by folding the discrete lines of RPA and ERPA spec-tra with a Lorentzian function. In both cases, an artificialwidth /H9003=0.1 eV has been used. In Fig. 2, we show the S =0 dipole strength distribution for the neutral case. The solidline refers to the calculations performed within the presentapproach, whereas the dashed line to those of standard RPA.The experimental distribution 45exhibits two broad peaks at about 2.4 and 2.65 eV and the first one is higher than thesecond one. As we can observe in Fig. 2, RPA predicts the first peak at an energy of about 2.45 eV , very close to theexperimental value, and the second one much higher, atabout 2.97 eV . Furthermore, the height of the second peak isabout twice that of the first one. In ERPA, both peaks are012345Energy (eV)Na40 S=0 S=1 RPA ERPA RPA ERPA0+ 4+ 2+ 3- 1- 1-2+0+4+ 0+2+3- FIG. 1. RPA and ERPA centroid energies /H20849eV/H20850of Na 40cluster for multipolarities from L=0 to L=4 and spin S=0 /H20849left panel /H20850and S=1 /H20849right panel /H20850. 2 2.4 2.8 3.2 3.6 4 E(eV)050100150200250Strengt h(arb.u nits)RPA ERPA FIG. 2. Spin=0 dipole strength for Na 40metal cluster. The solid line refers to calculations performed within the present approach,while the dashed line refers to standard RPA calculations. The ar-rows roughly indicate the positions of the experimental peaks.PARTICLE-HOLE EXCITATIONS WITHIN A SELF- … PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-5shifted to lower energies, at about 2.39 and 2.79 eV , in agree- ment with the experimental values /H20849indicated by arrows in the figure /H20850. Also, the structure of the distribution changes and a reshuffling of the strength, qualitatively in better agreementwith the experimental results, are observed. In particular, wehave an increasing of the height of the first peak and a low-ering of the second one. However, the strength distributionremains quite different from the experimental one. In Fig. 3, we show the S=0 dipole strength distribution for the ionized cluster Na 41+. Also, in this case, the calculated distribution is shifted with respect to the RPA results and it fits well withthe experimental results, 46in which a single peak at about 2.6 eV is found. The ERPA energy of this peak is about 2.64eV , rather better than the RPA one which lies at about 2.77eV . The improvement obtained in ERPA can be seen by ana-lyzing also the centroid energy associated with the strengthdistribution. The ERPA result /H208492.70 eV /H20850is lower by /H1101110% with respect to the RPA one /H208492.97 eV /H20850, quite close to the experimental value /H208492.62 eV /H20850. 46 IfFis a multipole operator r/H9261Y/H92610and the Hamiltonian contains a kinetic energy term plus a local two-body interac-tion, one gets 15 1 2/H208550/H20841/H20851F,/H20851H,F/H20852/H20852/H208410/H20856=/H60362 2m/H9261/H208492/H9261+1/H20850 4/H9266N/H208550/H20841r2/H9261−2/H208410/H20856, /H2084937/H20850 with Nas the number of particles and /H208550/H20841r2/H9261−2/H208410/H20856=1 N/H20858 /H9251,/H9252/H9267/H20849/H9251,/H9252/H20850/H20885r2/H9261−2/H9272/H9251/H20849r/H20850/H9272/H9252/H11569/H20849r/H20850d3r,/H2084938/H20850 where /H9251and/H9252stand for any single particle states with wave functions /H9272/H9251/H20849r/H20850and/H9272/H9252/H20849r/H20850. Let us now consider some results on the sum rules. As we discussed in Sec. III, Eq. /H2084928/H20850is an identity when a complete set of exact eigenstates of the Hamiltonian is used. There-fore, analyzing the deviations from equality when differentapproximate schemes are used is a good test to verify and tocompare their adequacy. We recall again that the Thoulesstheorem states that Eq. /H2084928/H20850is exactly satisfied if the mean values in the correlated ground state /H208410/H20856, appearing in both sides are approximated by those in the /H20841HF/H20856one. When themean values are calculated in the correlated ground state rather than in the HF one, some differences are expected. InTable I, we show, for Na 40and multipolarities ranging from 0 to 3, the values of the right-hand side of Eq. /H2084928/H20850calculated by using Eqs. /H2084937/H20850and /H2084938/H20850in the HF, RPA, and ERPA ground states and, correspondingly, the left-hand side calcu-lated by using Eq. /H2084932/H20850with the HF, RPA, and ERPA one- body density. As above discussed, in standard RPA, the /H20841HF/H20856 state, and thus the HF one-body density, is used in evaluatingthe RPA matrices. On the other hand, the RPA ground state,defined as the vacuum of the Qoperators, can be explicitly obtained 15within the quasiboson approximation. It has the form /H20841RPA /H20856/H11008eZˆ/H20841HF/H20856, /H2084939/H20850 with Zˆ=1 2/H20858 p1h1p2h2Zp1h1p2h2ap1†ah1ap2†ah2. /H2084940/H20850 where the Zmatrix satisfies the following relation: Z=Y/H11569X/H11569−1. /H2084941/H20850 By using the number operator method of Ref. 43, one gets /H9267RPA/H20849/H9251,/H9252/H20850=n/H9251/H9254/H9251/H9252, /H2084942/H20850 with2 2.4 2.8 3.2 3.6 4 E(eV)050100150200250300350Strengt h(arb.u nits)RPA ERPA FIG. 3. As in Fig. 2but for Na41+metal cluster.TABLE I. For each multipolarity /H9261, we report the values of the two sides of Eq. /H2084928/H20850, in units of Å2/H9261·eV, calculated in the HF /H20849first row /H20850,R P A /H20849second row /H20850, and ERPA /H20849third row /H20850ground states /H20849GS/H20850. The right-hand side is calculated by using Eqs. /H2084937/H20850and /H2084938/H20850, while the left-hand side by means of Eq. /H2084932/H20850. Equations /H2084924/H20850–/H2084926/H20850and /H2084942/H20850–/H2084944/H20850are used in the ERPA and RPA case, respectively. The results refer to Na 40. Multipolarity Left-hand side Right-hand side /H9261=0 /H20849HF GS /H20850 0.16253 /H110031040.16199 /H11003104 /H20849RPA GS /H20850 0.55083 /H110031030.23964 /H11003104 /H20849ERPA GS /H20850 0.12819 /H110031040.16551 /H11003104 /H9261=1 /H20849HF GS /H20850 0.10960 /H110031030.10951 /H11003103 /H20849RPA GS /H20850 0.14767 /H110031020.10951 /H11003103 /H20849ERPA GS /H20850 0.76098 /H110031020.10951 /H11003103 /H9261=2 /H20849HF GS /H20850 0.20282 /H110031050.20239 /H11003105 /H20849RPA GS /H20850 0.61385 /H110031040.29955 /H11003105 /H20849ERPA GS /H20850 0.15843 /H110031050.20703 /H11003105 /H9261=3 /H20849HF GS /H20850 0.25643 /H110031070.25517 /H11003107 /H20849RPA GS /H20850 0.96030 /H110031060.83273 /H11003107 /H20849ERPA GS /H20850 0.20333 /H110031070.26591 /H11003107D. GAMBACURTA AND F. CATARA PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-6nh=1−1 2/H20858 p,/H9263/H20841Yph/H9263/H208412, /H2084943/H20850 and np=1 2/H20858 h,/H9263/H20841Yph/H9263/H208412. /H2084944/H20850 We remark that the above expressions for the occupation numbers differ from those obtained by making use of the QBA by the factor of1 2. The same result was obtained in Ref. 47. From the first rows of Table I, we see that the Thouless theorem /H20851i.e., equality of the left-hand side and right-hand side of Eq. /H2084928/H20850when both are calculated in /H20841HF/H20856/H20852is numeri- cally well satisfied, better than 1%, for the shown multipo-larities. From the second and third rows, respectively, we seethat both in RPA and ERPA, equality /H2084928/H20850is not satisfied. However, in the latter approach, the deviations are muchsmaller than those found when the RPA correlated groundstate, and the corresponding Xand Yamplitudes are used. Another important quantity is the static dipole polarizabil- ity /H9251that is related to the inverse moment, S−1=/H20858 /H9263/H9275/H9263−1/H20841/H20855/H9263/H20841F/H208410/H20856/H208412, /H2084945/H20850 by S−1=1 2/H9251. /H2084946/H20850 In the case of Na 40, our RPA value is 522.42 Å3, in very good agreement with that found in Ref. 40. The ERPA value is somewhat smaller, namely, 434.34 Å3, to be compared with the experimental value of 605.46 /H1100611.40 Å3.48In this respect, however, it has to be noted that, as stressed also inRef. 49, the jellium approximation may be too poor. Indeed, in Ref. 50, it is shown that the inclusion of ionic structure effects by means of pseudopotentials leads to an increase of30% in the polarizability. A problem encountered when RPA is applied to the study of nuclear systems is that the energies of the low lying statesare found quite higher than the experimental values. This isrelated to the fact that the equations of motion contain thesingle particle HF energies, whose spacing is too large, es-pecially the gap between the last occupied and the first un-occupied levels. As said at the end of Sec. II, in the ERPAequations of motion the quantities /H9280/H20849/H9251,/H9252/H20850defined in Eq. /H2084922/H20850 appear instead of the HF energies. In order to disentanglehow much the modifications of the energies of the collectivestates are related to this, we have diagonalized /H9280/H20849/H9251,/H9252/H20850.I n Fig. 4, the so obtained generalized single particle energies are compared with the HF ones. The main difference is thatthe states above the Fermi level are lowered while thosebelow are pushed up. This might be one of the reasons whythe energies of the collective states in ERPA are lower thanthose in RPA. It would be interesting to see how much thisreduced gap can improve the results in nuclei. In Fig. 5, we show the occupation numbers n pfor particle states and the opposite of the depletion numbers nh−1 forhole states obtained in RPA /H20851see Eqs. /H2084943/H20850and /H2084944/H20850/H20852and in ERPA in the lower and upper panels, respectively. The latteris calculated by diagonalizing, at the end of the iterativeprocedure, the OBDM. All states with L=0–6, both in S =0 and in S=1 spin channels, are used in the calculation of these quantities. However, as mentioned above, RPA breaksdown in the L=3, S=1 channel /H20849as well as in the L=5, S =1 one /H20850, while ERPA solutions are found to be real for all L and Sstates. We note that the deviations from the HF limit, i.e., n h=1 and np=0, are greater in ERPA than in RPA. A different result was found in Ref. 32where, however, several further approximations with respect to the present approachwere introduced /H20849see Sec. II /H20850. It has to be noticed that the use of quasiboson approximation in the evaluation of RPA ma- trices is justified only when the HF state does not differ verymuch from the correlated one. The big deviations found inRPA show that the quasiboson approximation, and thus thestandard RPA, is not adequate. As mentioned above, theERPA occupation numbers deviate from the HF limit morethan the RPA ones. However, this is not a difficulty in ERPAsince the correlated ground state, rather the HF one, is usedas the reference state. Other useful information about the ground state properties can be obtained by looking at the electron density, /H9267/H20849r/H20850=1 4/H9266/H20858 /H9251,/H9252/H9267/H20849/H9251,/H9252/H20850/H9272/H9251/H20849r/H20850/H9272/H9252/H11569/H20849r/H20850, /H2084947/H20850-8-6-4-20Energy (eV)(P) (H) HF ERPA FIG. 4. HF single particle energies /H20849left side /H20850and generalized single particle ones /H20849right side /H20850/H20849see the text for more details /H20850. A few lowest particle states /H20849p/H20850and the hole ones /H20849h/H20850are shown above and below the dashed line, respectively.PARTICLE-HOLE EXCITATIONS WITHIN A SELF- … PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-7where /H9251and/H9252stand for any single HF particle states with wave functions /H9272/H9251/H20849r/H20850and/H9272/H9252/H20849r/H20850and normalized so that 4/H9266/H20885 0/H11009 r2/H9267/H20849r/H20850dr=Ne, /H2084948/H20850 where Neis the number of electrons. In Fig. 6, we show the electron densities calculated in HF /H20849dotted line /H20850,R P A /H20849dashed line /H20850, and ERPA /H20849full line /H20850. Note that in HF, the OBDM is diagonal and one has /H9267HF/H20849r/H20850=1 4/H9266/H20858 h/H20841/H9272h/H20849r/H20850/H208412, /H2084949/H20850 where the sum runs only over hole states. Similarly, in RPA, it is still assumed to be diagonal with occupation numbersdifferent from the HF ones /H20851see Eqs. /H2084943/H20850and /H2084944/H20850/H20852and one gets /H9267RPA/H20849r/H20850=1 4/H9266/H20858 /H9251n/H9251/H20841/H9272/H9251/H20849r/H20850/H208412. /H2084950/H20850 We can see that when RPA correlations are included, the electron density in the interior part is smaller than in the HFcase while its tail is much longer. On the contrary, the ERPAelectron density comes out to be closer to the HF one and, inparticular, the tails are almost identical. In Fig. 6, we showalso the values /H20849dotted-dashed line, denoted by D-ERPA /H20850, which are obtained by taking only the diagonal terms of theOBDM calculated in ERPA. As it is apparent, in this case theelectron density gets closer to the RPA one. The HF electrondensity shows some oscillations in the interior part, whichare due to the shell structure and are washed out in the otherapproaches. However, we underline that HF and ERPA,while having occupation numbers so different from eachother, give similar electron densities when the correspondingcomplete OBDM is used self-consistently. V. CONCLUSIONS AND OUTLOOK In conclusion, we have presented an approach /H20849ERPA /H20850, going further toward a completely self-consistent RPA. Asignificant improvement is achieved over the renormalizedRPA framework of previous extensions of RPA. In particular,it is less demanding from the point of view of computationalresources. In order to test the merits of the new approach, wehave applied it to the study of the electronic properties ofmetal clusters within the jellium approximation. By compari-son with the available experimental data, we found that itgives better results than RPA. Sizeable violations of theEWSR are still present in ERPA as in all extensions of RPAproposed until now. As discussed at the end of Sec. III, byenlarging the space of elementary excitations along the linesof Ref. 37within the present framework, it should be pos- sible to implement a completely self-consistent RPA preserv-ing the EWSR and to overcome the difficulties encounteredin Ref. 37. This will be the object of future investigations. As it is clearly pointed out, the applicability of the approach isquite general. Some results shown in the present paper likethose on “generalized single particle energies” and on sumrules make us confident that our method will display all itspotentialities when applied to the study of atomic nuclei, aswe plan to do in the near future.0.0 4.0 8.0 12. 0 r(A)10-410-310-210-1ρ(A-3)HF RPA ERPA D-ERPA oo FIG. 6. Electron density /H9267for Na 40cluster as obtained in HF /H20849dotted line /H20850,R P A /H20849dashed line /H20850, and ERPA /H20849full line /H20850. The D-ERPA result /H20849dotted-dashed line /H20850means the ERPA result when only the diagonal terms of the OBDM are taken. The arrow indicates theradius of the jellium sphere.-0.400.4(nh-1) np -0.400.4(nh-1) npERPA RPA FIG. 5. Occupation numbers npfor particle states and the oppo- site of depletion numbers nh−1 for hole states. ERPA and RPA results are reported in the upper and lower panel, respectively. TheERPA occupation numbers are calculated by diagonalizing, at theend of the iterative procedure, the OBDM, while Eqs. /H2084943/H20850and /H2084944/H20850 are used for RPA values.D. GAMBACURTA AND F. CATARA PHYSICAL REVIEW B 77, 205434 /H208492008 /H20850 205434-8ACKNOWLEDGMENTS The authors gratefully acknowledge M. Sambataro and N. Van Giai for helpful discussions. This work makes use ofresults produced by the PI2S2 Project managed by the Con-sorzio COMETA, a project co-funded by the Italian Ministryof University and Research /H20849MIUR /H20850within the Piano Opera- tivo Nazionale Ricerca Scientifica, Sviluppo Tecnologico,Alta Formazione /H20849PON 2000–2006 /H20850. 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PhysRevB.82.094516.pdf

Skyrmion lattice and intrinsic angular momentum effect in the Aphase of superfluid3He under rotation Masanori Ichioka, Takeshi Mizushima, and Kazushige Machida Department of Physics, Okayama University, Okayama 700-8530, Japan /H20849Received 18 August 2010; published 23 September 2010 /H20850 As an example of the skyrmion lattice, the structure of Mermin-Ho vortex lattice in superfluids3He is studied by self-consistent Eilenberger theory and by Bogoliubov-de Gennes theory. We identify how theintrinsic orbital angular momentum lofp-wave Cooper pairs contributes to spatial structures of the pair potential, current flow, and quasiparticle states. There are two types of vortices depending on the ldirection relative to rotation. Only one of them has zero-energy bound states appearing even in coreless vortices due tointrinsic topological reasons. DOI: 10.1103/PhysRevB.82.094516 PACS number /H20849s/H20850: 67.30.he, 74.20.Rp, 74.25.Uv I. INTRODUCTION Skyrmion is one of the emergent topological objects, ex- emplified by monopole, meron, or hedgehog, encompassinga wide range of research fields from particle physics 1to condensed-matter physics.2–4The low-lying Fermionic exci- tations associated with the localized topological objects playa major role in governing the physical behaviors which re-flect those topological nature. In the condensed-matter con-text skyrmion usually forms a periodic lattice, as shown inFig.1/H20849a/H20850, to be detected by a macroscopic observation. Ex- cept for a few example such as recent clear observation ofremarkable skyrmion lattice in a weak itinerant ferromagnetof MnSi and Fe 1−xCoxSi under a magnetic field,5,6there is not much concrete system accessible experimentally. Superfluid3He,7,8which is a typical multicomponent order-parameter system with p-wave pairing, provides us a fertile research field to investigate the interplay between thetopology and the low-lying Fermionic excitations of quasi-particles, and gains recently renewed interest because of pos-sible existence of Majorana quasiparticle 9–13with a zero- energy Fermionic excitation. Here we focus on the Aphase where the Cooper pairs have intrinsic angular momentum/H20849IAM /H20850denoted by lvector l=/H20849l x,ly,lz/H20850whose direction is degenerate in a bulk. Note that the pairing component p+ =px+ipy/H20849p−=px−ipy/H20850gives positive /H20849negative /H20850lz. IAM plays a fundamental role in describing the spatial structure ofquasiparticles and thermodynamic behaviors in a system. 7,8 As a concrete realization of the skyrmion lattice, we con- sider the Mermin-Ho /H20849MH /H20850vortex lattice and texture.14The MH vortices were observed in superfluid3HeAphase ex- perimentally under rotation,15,16and the stability of MH vor- tex lattice was supported by theoretical studies usingGinzburg-Landau theory. 17,18In Refs. 15and 18,L V 1 /H20849locked vortex 1 /H20850corresponds to the MH vortex lattice. The MH vortex is a building block embedded in the MH texture,forming a periodic array. It has a soft core in contrast withconventional singular hard core vortex in scalar order-parameter phases. Since the l-vector direction can be spheri- cally rotatable depending on positions, outside the MH vor-tex core lvector could be directed to /H20849x,y/H20850plane. The coreless vortex structure can be formed merely by rotatingthel-vector direction by 360° around the vortex, keeping thetotal order-parameter amplitude constant. At the core of the MH vortex, lvector can be directed to + zor −zdirection. The soft core MH vortex is an interesting topological ob- ject because the background Aphase is maintained through- out the whole system without any singular point. Yet MHvortex is stable under rotation. The low-lying Fermionic ex-citations associated with this remarkable MH texture are in-triguing because they reflect faithfully and directly the un-derlying topological structure of skyrmion lattice. Thereforewe can analyze the intimate interplay between the low-lyingexcitations and topology. We uncover a generic question of FIG. 1. /H20849Color online /H20850/H20849a/H20850Skyrmion structure of lvectors around MH ↑at B and MH ↓at D. /H20849b/H20850Profiles of pair potentials; /H20841/H9257+/H20849r/H20850/H20841forp+component, /H20841/H9257−/H20849r/H20850/H20841forp−component, /H20841/H9257z/H20849r/H20850/H20841forpz component, and pair amplitude /H20881/H9267/H20849r/H20850, obtained by self-consistent Eilenberger theory. /H9024=0.004 /H90240and T=0.9 Tc./H20849c/H20850Unit cell of Mermin-Ho vortex lattice, including four vortices. MH ↑/H20849MH↓/H20850vor- tices with lz/H110220/H20849lz/H110210/H20850are located at positions A and B /H20849C and D /H20850. u1−u2andu2are unit vectors. Horizontal axis in /H20849a/H20850and /H20849b/H20850is along the dashed line in /H20849c/H20850.PHYSICAL REVIEW B 82, 094516 /H208492010 /H20850 1098-0121/2010/82 /H208499/H20850/094516 /H208496/H20850 ©2010 The American Physical Society 094516-1the presence or absence of the vortex bound state in multi- component superfluids with a higher orbital pairing p/H11006, which could realize the time reversal symmetry-breakingpairing, and give rise to IAM. We study the interplay be-tween the local IAM and vortex-winding number. This inter-play yields nontrivial effects on the above question as wewill see soon. The vortex lattice structure formed by MH vortices is un- conventional, where the unit cell of vortex lattice containstwo positive l zvortices /H20849MH↑/H20850and two negative lzvortices /H20849MH↓/H20850, as shown in Fig. 1/H20849c/H20850.7,17,18In nomenclature in Ref. 8,M H ↓corresponds to MT /H20849mixed-twist /H20850vortex. It is shown below that these two kinds of the cores exhibit completelydifferent low-energy excitation spectra. So far, most of thestudy for the MH vortex lattice was done by phenomenologi-cal Ginzburg-Landau theory. 17,18According to our micro- scopic calculations based on Eilenberger theory19–22backed up by the full quantum mechanical Bogoliubov-de Gennes/H20849BdG /H20850theory, we succeeded in uncovering the nontrivial physical mechanism of the interplay between IAM and wind-ing number. We note that MH vortex structures are also ex-pected in spinor Bose-Einstein condensate in ultracold Bosegases under rotation. 3,4,23 After Sec. I, we describe our formulation by self- consistent Eilenberger theory for MH vortex lattice in Sec.II. We study the spatial structure of order parameter and mass current in Sec. III, and low-energy quasiparticle states in Sec. IV, based on the Eilenberger theory. The quasiparticle struc- ture is examined also by BdG calculations in Sec V. The last section is devoted to summary and discussions. II. EILENBERGER THEORY FOR MERMIN-HO VORTEX LATTICE The quasiclassical Eilenberger theory is quantitatively valid when /H9264/H112711/kF/H20849kFis the Fermi wave number and /H9264is the superfluid coherence length /H20850, and has been used in the study of3He superfluidity.24–27In the clean limit, the quasi- classical Green’s functions g/H20849/H9275n,k,r/H20850,f/H20849/H9275n,k,r/H20850, and f†/H20849/H9275n,k,r/H20850are calculated by the Eilenberger equation,19–21 /H20853/H9275n+vˆ·/H20849/H11612+iA/H20850/H20854f=/H9004g, /H20853/H9275n−vˆ·/H20849/H11612−iA/H20850/H20854f†=/H9004/H11569g, /H208491/H20850 using the Riccati method,22where g=/H208491−ff†/H208501/2,R e g/H110220, andvˆ=v/vF0. We consider all three orbital components of the p-wave pairing for the pair potential, as /H9004/H20849r,k/H20850=/H9257+/H20849r/H20850/H9272+/H20849k/H20850+/H9257−/H20849r/H20850/H9272−/H20849k/H20850+/H9257z/H20849r/H20850/H9272z/H20849k/H20850, /H208492/H20850 /H9272/H11006/H20849k/H20850=/H11007/H208813 2kx/H11006iky kF=/H11007/H208813 2sin/H9258ke/H11006i/H9278k, /H208493/H20850 /H9272z/H20849k/H20850=/H208813kz kF=/H208813 cos/H9258k, /H208494/H20850 where k=/H20849kx,ky,kz/H20850=kF/H20849sin/H9258kcos/H9278k,sin/H9258ksin/H9278k,cos/H9258k/H20850 is the relative momentum of the Cooper pair on the sphericalFermi surface, and ris the center-of-mass coordinate of the pair. The lvector is given by lx=/H208812R e /H20853/H20849/H9257++/H9257−/H20850/H11569/H9257z/H20854//H9267,ly =/H208812I m /H20853/H20849/H9257+−/H9257−/H20850/H11569/H9257z/H20854//H9267, and lz=/H20849/H20841/H9257+/H208412−/H20841/H9257−/H208412/H20850//H9267with pair amplitude /H9267/H20849r/H20850=/H20841/H9257+/H20849r/H20850/H208412+/H20841/H9257−/H20849r/H20850/H208412+/H20841/H9257z/H20849r/H20850/H208412.23For simplicity, we do not consider the spin components of the pair potentialdenoted as dvector since we neglect small dipole coupling oflvector and dvector. Therefore, the dipole length is in- finity in our calculation. Since the Fermi surface is spherical in 3He, the Fermi velocity is given by v=vF0k/kF. When the rotational axis is the zdirection and angular velocity of ro- tation is /H9024,A/H20849r/H20850=−1 2/H208490,0,/H9024/H20850/H11003r. Throughout this paper, length, temperature, and /H9024are scaled by R0, superfluid tran- sition temperature Tc, and /H90240, respectively. Here, R0 =/H6036vF0/2/H9266kBTc,/H90240=/H92780/2/H9266R02with circulation quantum /H92780 =h/2m.19Matsubara frequency /H9275n=/H208492n+1/H20850/H9266T, energy E, and pair potential /H9004are in a unit /H9266kBTc. The order parameter /H9257j/H20849j=+,−, z/H20850is self-consistently calculated by /H9257j/H20849r/H20850=g0N0T/H20858 0/H11021/H9275n/H11021/H9275cut/H20855/H9272j/H11569/H20849k/H20850/H20849f+f†/H11569/H20850/H20856k /H208495/H20850 with /H20849g0N0/H20850−1=lnT+2T/H208580/H11021/H9275n/H11021/H9275cut/H20841/H9275n/H20841−1./H20855¯/H20856kindicates the Fermi surface average, and N0is the density of states /H20849DOS /H20850 at the Fermi energy in the normal state. We set energy cutoffof the pairing interaction as /H9275cut=40kBTc. In the MH vortex lattice, a unit cell including four vorti- ces is square,17,18as shown in Fig. 1/H20849c/H20850where two MH ↑are located at A and B and two MH ↓at C and D. Thus, we set a unit cell as r=W1/H20849u1−u2/H20850+W2u2/H20849/H20841Wi/H20841/H110210.5, i=1,2 /H20850with unit vectors u1=/H20849ax,0/H20850,u2=/H208491 2ax,ay/H20850, and ay=1 2ax, where axay/H9024=4/H92780. To consider the periodic boundary condition and the initial value for the pair potential, we introduce Abri-kosov solution which has a single vortex within a unit cell,given as /H9023/H20849r/H20850=e i/H9266xy/axay/H208732ay ax/H208741/4 /H11003/H20858 p=−/H11009/H11009 e−/H9266/H20851/H20849y+y0/H20850/ay+p/H208522ay/ax+2/H9266i/H20851p/H20849x0/ax+/H9256p/2/H20850+/H20849y0/ay+p/H20850x/ax/H20852 /H208496/H20850 when the vortex center is located at /H20849x0,y0/H20850−1 2/H20849u1+u2/H20850. This has translational relation /H9023/H20849r+R/H20850=/H9023/H20849r/H20850ei/H9273/H20849r,R/H20850, /H208497/H20850 /H9273/H20849r,R/H20850=2/H9266/H208771 2/H20875/H20849m+n/H9256/H20850y ay−nx ax/H20876 +mn 2+/H20849m+n/H9256/H20850y0 ay−nx0 ax/H20878 /H208498/H20850 forR=mu1+nu2/H20849m,n: integer /H20850. We set /H92730/H20849r,R/H20850/H11013/H9273/H20849r,R/H20850 when the vortex center is located at /H208490,0/H20850. We prepare four Abrikosov solutions with different positions of the vortex centers. Abrikosov solution with the vortex center at /H208491 4ax,0/H20850, /H20849−1 4ax,0/H20850,/H208490,1 2ay/H20850, and /H208490,−1 2ay/H20850are, respectively, denoted as /H9023A/H20849r/H20850,/H9023B/H20849r/H20850,/H9023C/H20849r/H20850, and/H9023D/H20849r/H20850. In the MH vortex lattice,17 MH↑vortices with lz/H110220 at A and B in Fig. 1/H20849c/H20850have theICHIOKA, MIZUSHIMA, AND MACHIDA PHYSICAL REVIEW B 82, 094516 /H208492010 /H20850 094516-2phase winding /H20849w+,wz,w−/H20850=/H208490,1,2 /H20850around each vortex center, where w+,wz,w−are, respectively, phase windings of the components /H9257+,/H9257z,/H9257−around a vortex center. Other vor- tices MH ↓with lz/H110210 at C and D have the phase winding /H20849w+,wz,w−/H20850=/H208492,1,0 /H20850around each vortex center. Therefore, as the initial states for MH vortex lattice,17we use /H9257+/H20849r/H20850=/H20853/H9023C/H20849r/H20850/H9023D/H20849r/H20850/H208542, /H9257−/H20849r/H20850=/H20853/H9023A/H20849r/H20850/H9023B/H20849r/H20850/H208542, /H9257z/H20849r/H20850=/H9023A/H20849r/H20850/H9023B/H20849r/H20850/H9023C/H20849r/H20850/H9023D/H20849r/H20850. /H208499/H20850 These states have the same translational relation /H9257j/H20849r+R/H20850=/H9257i/H20849r/H20850e4i/H92730/H20849r,R/H20850. /H20849j=+, −, z/H20850. /H2084910/H20850 Starting from initial states in Eq. /H208499/H20850, we solve Eqs. /H208491/H20850and /H208495/H20850alternately, and we obtain self-consistent solutions of the MH vortex lattice for /H9257+,/H9257−, and/H9257zunder a given unit cell of the vortex lattice.20,21The unit cell is divided to 82 /H1100382 mesh points, where we obtain the quasiclassical Green’sfunctions and /H9004/H20849r,k/H20850. When we solve Eq. /H208491/H20850by the Riccati method, 22we estimate /H9004/H20849r/H20850at arbitrary positions by the in- terpolation from their values at the mesh points and by theperiodic boundary condition in Eq. /H2084910/H20850. In figures of this paper, we presented the spatial structure of the MH vortexlattice within a unit cell including MH vortices A–D asshown in Fig. 1/H20849c/H20850, where we use coordinates XandYro- tated by 45° from the original coordinates xandy. Using the obtained self-consistent solutions, the mass cur- rent is given by j/H20849r/H20850=/H20849j x,jy,jz/H20850/H11008T/H20858 0/H11021/H9275n/H20855vˆImg/H20856k. /H2084911/H20850 When we calculate the quasiparticle states, we solve Eq. /H208491/H20850 with i/H9275n→E+i/H9254. We typically use /H9254=0.01, which is small smearing effect of energy by scatterings. The local DOS/H20849LDOS /H20850for quasiparticles is obtained asN/H20849r,E/H20850=N 0/H20855Re/H20853g/H20849/H9275n,k,r/H20850/H20841i/H9275n→E+i/H9254/H20854/H20856k. /H2084912/H20850 III. STRUCTURE OF ORDER PARAMETER AND MASS CURRENT We start to discuss the structure of the pair potential. Fig- ures 1/H20849b/H20850and 2present self-consistent results for spatial structures of MH vortex lattice at T=0.9 Tcand/H9024 =0.004 /H90240, where intervortex distance is about 40 R0.R0is in the order of coherence length. Around MH ↑vortices at A and B with phase winding /H20849w+,wz,w−/H20850=/H208490,1,2 /H20850around each vortex center, /H20841/H9257z/H20841/H11008rand /H20841/H9257−/H20841/H11008r2as a function of the radius rfrom the vortex center. Since /H20841/H9257+/H20841/HS110050 at the vortex center, these vortices are coreless vortices with positive lz/H20851see Fig. 1/H20849b/H20850/H20852. The other two MH ↓vortices at C and D with phase winding /H20849w+,wz,w−/H20850=/H208492,1,0 /H20850around each vortex center are also coreless vortices but with negative lzsince /H20841/H9257−/H20841/HS110050 at the vortex center. In our calculation, as dipole length7,8is infinity, the vortex core radius is in the order of intervortex distance, evenchanging /H9024. This is a character of coreless vortex. In con- trast, if we calculate the vortex structure in a single compo-nent /H20849for example, if we set /H9257+=/H9257−=0/H20850, the core radius of the singular vortex is small, i.e., in the order of coherencelength. The pair amplitude /H9267/H20849r/H20850in Fig. 2/H20849d/H20850is almost con- stant but it is slightly suppressed at vortex core in the self-consistent calculations. This suppression of /H9267/H20849r/H20850is stronger at MH ↑of positive lzat A and B, compared with MH ↓of negative lzat C and D, which is closely related to the exis- tence of low-lying excitations in MH ↑, not in MH ↓, as dis- cussed later. We note that the differences of MH ↑and MH ↓ FIG. 2. /H20849Color /H20850Spatial structure of order parameters /H20849a/H20850/H20841/H9257+/H20849r/H20850/H20841, /H20849b/H20850/H20841/H9257−/H20849r/H20850/H20841,/H20849c/H20850/H20841/H9257z/H20849r/H20850/H20841, and /H20849d/H20850pair amplitude /H20881/H9267/H20849r/H20850within a unit cell, obtained by self-consistent Eilenberger theory. /H9024=0.004 /H90240 andT=0.9 Tc.M H ↑vortices A and B are located at maximum of /H20841/H9257+/H20849r/H20850/H20841, and MH ↓vortices C and D are located at minimum of /H20841/H9257+/H20849r/H20850/H20841in/H20849a/H20850. FIG. 3. /H20849Color /H20850Spatial structure of /H20849a/H20850lvector, and /H20849b/H20850current j within a unit cell, obtained by self-consistent Eilenberger theory./H9024=0.004 /H9024 0and T=0.9 Tc. Arrows indicates vectors /H20849lx,ly/H20850or /H20849jx,jy/H20850, and color densities are for lzorjz. FIG. 4. /H20849Color /H20850/H20849a/H20850Local spectrum N/H20849r,E/H20850of quasiparticle states at vortex center of MH ↑/H20849B in Fig. 1/H20850, vortex center of MH ↓ /H20849D/H20850, and midpoint between vortices /H20849O/H20850./H20849b/H20850Zero-energy LDOS N/H20849r,E=0/H20850within a unit cell. /H9024=0.004 /H90240andT=0.9 Tc.SKYRMION LATTICE AND INTRINSIC ANGULAR … PHYSICAL REVIEW B 82, 094516 /H208492010 /H20850 094516-3vortices come from the relative orientation of lvector and the angular velocity of rotation. Since the angular velocitypoints to zdirection, the vortex windings at MH ↑vortex and MH↓vortex are both positive, and total vortex winding around a unit cell in Fig. 1/H20849c/H20850is four /H20849/H110220/H20850. Therefore, l vector is parallel to the angular velocity of rotation at thevortex core of MH ↑with lz/H110220, and lvector is antiparallel to the angular velocity at the vortex core of MH ↓with lz/H110210. In Fig. 3/H20849a/H20850, we present the texture of the lvector around the MH vortex. There, the direction of lvector rotates around the coreless vortex, keeping almost constant pair amplitude.Four vortices in a unit cell have different flow patters of l vector. MH ↑and MH ↓have different sense of lvector’s 360° rotation. The current flow jis presented in Fig. 3/H20849b/H20850. There, amplitude of circular current is larger around MH ↑, and small around MH ↓. In the MH vortex lattice, zcomponent jzap- pears due to the so-called bending current by /H11612/H11003l.7,8There, jzflows with fourfold symmetric pattern around MH ↓while jzis small around MH ↑. IV . QUASIPARTICLE STATES Those MH ↑and MH ↓vortices exhibit a distinctive low- lying excitation spectrum, depending on lzdirection. Figure 4/H20849a/H20850presents local spectrum N/H20849r,E/H20850at positions B /H20849MH↑/H20850,D /H20849MH↓/H20850and O in the unit cell of Fig. 1/H20849c/H20850. Outside the vortex core /H20849line O /H20850, we see typical DOS spectrum N/H20849E/H20850/H11008E2for anisotropic superconductors with point nodes.28Even in the bulk states without rotation, there are low-energy quasiparti-cle states near E=0 due to the point node. Therefore, the low-energy states shift to zero-energy state by vortex contri-butions. At the vortex center of MH ↑/H20849line B /H20850, remarkably we see sharp zero-energy peak even in a coreless vortex. Thispeak structure is similar to that seen in singular vortex 20,21 while the peak height is smaller. On the other hand, at MH ↓ core /H20849line D /H20850, there is no distinctive peak structure around E=0 in the vortex core region. These low-energy spectral differences by lzdirections are clearly seen in the zero- energy LDOS N/H20849r,E=0/H20850within a unit cell, as in Fig. 4/H20849b/H20850, where we see the distinctive peaks at A and B positions,corresponding to MH ↑vortex. The difference between MH ↑vortex and MH ↓comes from the relative orientation of lvector and rotational angular ve- locity /H9024of the rotation. Therefore, to see quantitative con- tribution of the rotational speed, in Fig. 5we plot /H9024depen- dence of N/H20849r,E=0/H20850and/H9267/H20849r/H208501/2at MH ↑vortex at B, MH ↓ vortex at D and the midpoint O. In the limit /H9024→0, we see thatN/H20849r,E=0/H20850→0 and /H9267/H20849r/H20850is uniform everywhere because of coreless vortices. With increasing /H9024, from /H9024/H110110.001/H90240 we find the difference between MH ↑vortex at B and MH ↓ vortex at D, coming from the relative lzdirection to the ro- tational angular velocity. When zero-energy states appears atthe vortex core, the pair amplitude /H9267/H20849r/H20850is suppressed at the core. The differences of vortices of B and D become eminenttoward the upper critical angular velocity /H9024 c2/H20849/H110110.3/H90240at T=0.9 Tc/H20850. As shown by line B in Fig. 5/H20849b/H20850, around MH ↑vortices, /H20841/H9257+/H20841/H20849/H11011/H20881/H9267/H20850is significantly suppressed at /H9024/H110110.04/H90240. There two vortex-antivortex phase singularity appears in /H9257+/H20849r/H20850atthe vortex core of MH ↑. At the higher /H9024/H20849/H110220.04/H90240/H20850, the vortex-antivortex phase singularities moves to around MH ↑ vortices, and /H20881/H9267at vortex center B increases again. There, basal plane component /H20849lx,ly/H20850oflvector is directed to oppo- site direction between inside and outside around each MH ↑ vortex at A and B. V . BOGOLIUBOV-DE GENNES CALCULATIONS In order to understand the fundamental difference in ex- citation spectrum between two MH vortices MH ↑and MH ↓, we have performed the full quantum mechanical calculationsbased on BdG theory, assuming a single vortex in asystem. 13,29To obtain quasiparticle eigenstates labeled by /H20849/H9263,kz/H20850and eigenenergy E/H9263,kz, we solve the BdG equation30 /H20885dr2/H20875H0/H20849r1,r2/H20850/H9004/H20849r1,r2/H20850 /H9004/H11569/H20849r1,r2/H20850−H0/H20849r1,r2/H20850/H20876/H20875u/H9263,kz/H20849r2/H20850 v/H9263,kz/H20849r2/H20850/H20876 =E/H9263,kz/H20875u/H9263,kz/H20849r1/H20850 v/H9263,kz/H20849r1/H20850/H20876, /H2084913/H20850 where H0/H20849r1,r2/H20850is the kinetic energy term H0/H20849r1,r2/H20850= −/H9254/H20849r1−r2/H20850/H20853/H1161212/2m−EF/H20854with the Fermi energy EF=kF2/2m. The pair potential /H9004/H20849r1,r2/H20850is expanded to the Fourier series with respect to the relative coordinate r1−r2as /H9004/H20849r1,r2/H20850=/H20885dk /H208492/H9266/H208503/H9004/H20849r,k/H20850eik·/H20849r1−r2/H20850, /H2084914/H20850 where r=/H20849r1+r2/H20850/2 is the center-of-mass coordinate. Here, we assume the coefficient /H9004/H20849r,k/H20850to be expanded in terms of thep-wave channel as /H9004/H20849r,k/H20850=1 /H208813/H20858 m=0,/H110061/H9257m/H20849r/H20850/H9272m/H20849k/H20850e−/H20849k2−kF2/H20850/H9264p2 /H2084915/H20850 with the factor /H9272m/H20849k/H20850defined in Eqs. /H208493/H20850and /H208494/H20850. This is same as the expression in Eq. /H208492/H20850, except for the additional factor e−/H20849k2−kF2/H20850/H9264p2with the pairing size /H9264p=kF−1. This factor is necessary for the BdG Eq. /H2084913/H20850to be Hermitian.30 Since zdependence of the pair potential is uniform, we can set the wave function as FIG. 5. /H20849Color online /H20850/H9024dependence of /H20849a/H20850zero-energy LDOS N/H20849r,E=0/H20850and /H20849b/H20850pair amplitude /H92671/2/H20849r/H20850at positions B /H20849vortex center of MH ↑/H20850,D /H20849vortex center of MH ↓/H20850, and O /H20849midpoint be- tween vortices /H20850. Horizontal axis for /H9024is log scale.ICHIOKA, MIZUSHIMA, AND MACHIDA PHYSICAL REVIEW B 82, 094516 /H208492010 /H20850 094516-4/H20875u/H9263,kz/H20849r/H20850 v/H9263,kz/H20849r/H20850/H20876=/H20875u˜/H9263,kz/H20849r/H20850 v˜/H9263,kz/H20849r/H20850/H20876eikzz. /H2084916/H20850 To reproduce the pair potential of MH vortex discussed in previous sections, the order-parameter profiles are given by /H20849/H9257+,/H92570,/H9257−/H20850=/H90040ei/H9278„e/H11007i/H9278/H208491/H11006cos/H9252/H20850, /H208812 sin/H9252,e/H11006i/H9278/H208491/H11007cos/H9252/H20850… with/H9252/H20849r/H20850=/H9266r/2Rfor MH ↑and MH ↓, respectively.17From wave functions and eigenenergies obtained by the BdG equa-tion, we calculate the LDOS N/H20849r,E/H20850as N/H20849r,E/H20850=/H20858 kzN/H20849r,kz,E/H20850=/H20858 /H9263,kz/H20841u/H9263,kz/H20849r/H20850/H208412/H9254/H20849E−E/H9263,kz/H20850./H2084917/H20850 Here, N/H20849r,kz,E/H20850iskz-resolved LDOS and the energy is pre- sented in a unit of the gap amplitude’s constant /H90040. At fist sight we expect no difference between MH ↑and MH↓because here we assume that the pair amplitude /H9267/H20849r/H20850is identical. The MH vortex is almost the Aphase like every- where and coreless. However, when we see the LDOSN/H20849r,E/H20850by BdG theory shown in Fig. 6/H20849a/H20850, zero-energy peak appears at the vortex core of MH ↑, and it does not exist at MH↓vortex. This quasiparticle structure is consistent to the result of Fig. 4by Eilenberger theory. The small suppression of/H9267/H20849r/H20850obtained by self-consistent calculation in Sec. IIIis a result from the low-energy quasiparticle states N/H20849r,E/H20850. To discuss the origin of zero-energy LDOS, N/H20849r,E/H20850is decomposed to contributions from each kzon the Fermi sur- face. The kz-resolved LDOS N/H20849r,kz,E/H20850at the vortex core forMH↑and MH ↓are displayed in Fig. 6/H20849b/H20850. There, distinctive zero-energy peak only for MH ↑grows as /H20841kz/H20841increases, indi- cating that the low-energy excitations come from near thepoles of the Fermi sphere. In contrast, there is no peak insidethe gap for MH ↓. The physical reason is due to the interplay between IAM and vortex-winding number: /H20849w+,wz,w−/H20850 =/H208490,1,2 /H20850for MH ↑and /H208492,1,0 /H20850for MH ↓. The IAM has the phase winding /H20849u+,uz,u−/H20850=/H208491,0,−1 /H20850around the Fermi sur- face for each pairing component /H20849/H9272+,/H9272z,/H9272−/H20850. To discuss the possibility of low-energy bound states at the vortex core, we consider effective pair potential for qua-siparticles around vortex cores. The quasiparticles propagat-ing to the angular direction around a vortex feel effectivepair potential /H9004/H20849r,k/H20850with /H9278k→/H9278+/H9266/2. For MH ↑, /H20841/H9004/H20849r,k/H20850/H9278k→/H9278+/H9266/2/H208412=6/H20841/H90040/H20841/H20849sin2/H9258k+sin2/H9252cos2/H9258k/H20850is an in- creasing function of r, and quasiparticles feel larger confine- ment potential at vortex when /H20841kz/H20841/H20849/H11008/H20841cos/H9258k/H20841/H20850is larger. This is the origin of zero-energy peak in the LDOS coming fromnear poles of the Fermi sphere. On the other hand, for MH ↓, /H20841/H9004/H20849r,k/H20850/H9278k→/H9278+/H9266/2/H208412=6 /H20841/H90040/H20841/H20853sin2/H9258k−1 2sin 2/H9278sin 2/H9252sin 2/H9258k +sin2/H9252/H20849cos2/H9258k−sin22/H9278sin2/H9258k/H20850/H20854. This effective pair poten- tial breaks circular symmetry and does not have minimum atvortex center. This implies no bound states inside the gap forMH ↓, as shown in right side of Fig. 6/H20849b/H20850. In essence this interplay between IAM and vortex winding yields the fol-lowing algebra symbolically for the phase factors:/H20849w +,wz,w−/H20850+/H20849u+,uz,u−/H20850=/H208490,1,2 /H20850+/H208491,0,−1 /H20850→ /H208491,1,1 /H20850 +/H208490,0,0 /H20850for MH ↑while /H208492,1,0 /H20850+/H208491,0,−1 /H20850→/H208491,1,1 /H20850 +/H208492,0,−2 /H20850for MH ↓. The former /H208490,0,0 /H20850gives rise to a vortex bound state similar to the singular hard core vortex form byCaroli-de Gennes-Matricon /H20849CdGM /H20850/H20849Ref. 31/H20850while the lat- ter /H208492,0,−2 /H20850yields the angle-dependent escaping form e i2/H9278 +e−i2/H9278/H11008cos 4/H9278. As for the problem whether the zero-energy quasiparticles at MH ↑is Majorana state or not, the energy level of this state is slightly lifted to positive energy in the order of /H90042/EFin the BdG theory /H20849EFis Fermi energy /H20850, as shown in Fig. 6/H20849a/H20850. This is a character of CdGM states31,32and also confirmed by the energy distribution of discretized eigenenergy, roughly given as /H20849n+1 2/H20850/H90042/EFwith integer n, obtained from the BdG equation. That is, this CdGM state is not Majorana statewhich should exist exactly at E=0. 13,29This is understand- able from the vortex winding of each pairing component. Forthe Majorana state to appear, the chiral components of p /H11006 should have odd winding number.29However, for MH ↑, winding /H20849w+,wz,w−/H20850=/H208490,1,2 /H20850does not satisfy this criterion. VI. SUMMARY AND DISCUSSIONS In summary, we have studied the detailed spatial structure of Mermin-Ho vortex lattice state in Aphase of superfluid 3He, as a representative and concrete example of the skyr- mion lattice. In Mermin-Ho vortex lattice, there are twotypes of vortices; MH ↑with positive lzand MH ↓with nega- tivelz. The differences between MH ↑vortex and MH ↓vortex among the Mermin-Ho vortices come from the orientation ofthe local intrinsic angular momentum l=/H20849l x,ly,lz/H20850at the vor- tex core, i.e., parallel or antiparallel to the angular velocity ofrotation. Due to the orientation of lvector relative to the FIG. 6. /H20849Color /H20850LDOS N/H20849r,E/H20850/H20849a/H20850and kz-resolved LDOS N/H20849r,kz,E/H20850/H20849b/H20850at the vortex center r=0 of MH ↑and MH ↓in the BdG theory. T=0,R=40kF−1, and kF/H9264=10. MH ↑has a low-energy peak /H20849assigned by arrow /H20850inside the gap. In /H20849b/H20850, gapedge narrows toward kz=kFbecause point nodes situate at north and south poles at kz =/H11006kFon the Fermi sphere.SKYRMION LATTICE AND INTRINSIC ANGULAR … PHYSICAL REVIEW B 82, 094516 /H208492010 /H20850 094516-5rotation, Mermin-Ho vortices have different structures. By self-consistent Eilenberger theory we clarified how the in-trinsic angular momentum effect of local lorientation ap- pears in the structure of order parameters, current flow andquasiparticle states around coreless Mermin-Ho vortexstates. These effects depending on the local l zorientation become eminent with increasing angular velocity of rotation.The different current flows /H20849j x,jy,jz/H20850between MH ↑and MH ↓ can be used to distinguish two types of vortices in the MH vortex lattice. It is noted that zero-energy states appear at thecoreless vortex of positive l zonly. These properties of qua- siparticle states were also confirmed by our calculation ofBogoliubov-de Gennes theory. The differences of low-energystates between MH ↑and MH ↓are interesting also in the re- lation to the dissipation mechanism at the vortex core. It maybe detected by magnetic resonance imaging /H20849MRI /H20850 technique, 33which is to probe Fermionic excitations locally. In the MRI technique, the local position of the signal can beidentified by the analysis of resonance fields under the gra-dient of the applied magnetic field. After the position of the Mermin-Ho vortex is identified, the local low-energy quasi-particle states are measurable by the relaxation experiment ofNMR at the resonance field. If the vortex has zero energyquasiparticle states, we expect rapid relaxation at the vortexcore. We expect that among two types of Mermin-Ho vorti-ces, one of them /H20849MH ↑/H20850has rapid relaxation of NMR at the vortex core, and the other MH ↓has only slow relaxation. The present study prompts us to explore other topological objectsin condensed-matter systems, such as MnSi and Fe 1−xCoxSi where a similar skyrmion lattice is realized.5,6Their elec- tronic structure may be quite interesting. The local structureof skyrmion can be different depending on the relative ori-entation of applied fields. 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PhysRevB.93.045134.pdf

PHYSICAL REVIEW B 93, 045134 (2016) Electron spin resonance in a two-dimensional Fermi liquid with spin-orbit coupling Saurabh Maiti,1,2Muhammad Imran,1and Dmitrii L. Maslov1 1Department of Physics, University of Florida, Gainesville, Florida 32611, USA 2National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA (Received 9 October 2015; revised manuscript received 8 December 2015; published 26 January 2016) Electron spin resonance (ESR) is usually viewed as a single-particle phenomenon protected from the effect of many-body correlations. We show that this is not the case in a two-dimensional Fermi liquid (FL) with spin-orbitcoupling (SOC). Depending on whether the in-plane magnetic field is below or above some critical value, ESRin such a system probes up to three chiral-spin collective modes, augmented by the spin mode in the presence ofthe field, or the Silin-Leggett mode. All the modes are affected by both SOC and FL renormalizations. We arguethat ESR can be used as a probe not only for SOC but also for many-body physics. DOI: 10.1103/PhysRevB.93.045134 I. INTRODUCTION Electron spin resonance (ESR) spectroscopy is an invalu- able tool for studying dynamics of electron spins [ 1–3]. In a single-particle picture, ESR can be understood eitherclassically, as resonant absorption of electromagnetic (EM)energy by a classical magnetic moment precessing about themagnetic field, or quantum mechanically, as absorption ofphotons with frequency equal to the Zeeman splitting. Theabsorption rate wof an incident EM wave (with frequency /Omega1 and amplitudes of the electric and magnetic fields /vectorE emand /vectorBem, correspondingly) is given by the Kubo formula [ 4–6] w=2/summationdisplay ij/bracketleftbig σ/prime ij(/Omega1)Eem iEem j+/Omega1χ/prime/prime ij(/Omega1)Bem iBem j/bracketrightbig , (1) where σ/prime ij(/Omega1) is the real part of the conductivity and χ/prime/prime ij(/Omega1)i s the imaginary part of the spin susceptibility. If the static magnetic field ( /vectorB) is in the plane of a two- dimensional electron gas (2DEG) and there is no spin-orbitcoupling (SOC), the only resonant feature is due to a polein the second term of Eq. ( 1) at the Larmor frequency. This is a conventional (or direct) ESR. However, because the spinsusceptibility is proportional to 1 /c 2, where cis the speed of light, the direct ESR signal is very weak. SOC of Rashba[7,8] and/or Dresselhaus [ 9,10] types changes the situation drastically by producing an effective magnetic field, whichacts on the spin of an electron with given momentum /vectorpand is proportional to |/vectorp|. The driving electric field (either from a dc current or EM wave) gives rise to a flow of electronswith a nonzero drift momentum; hence the electron system asa whole experiences an effective magnetic field due to SOC.The magnitude of bare SOC is strongly enhanced by virtualinterband transitions [ 11]; as a result, the electric component of an EM field couples to electron spins much stronger than themagnetic one. This is an electric dipole spin resonance (EDSR)[12–15], which gives rise to a range of spectacular phenomena, e.g., a strong enhancement of microwave absorption in a geometry when /vectorE emis in the plane of a 2DEG [ 16] and a shift of the resonance frequency by a dc current [ 3,5]. In this article, we discuss the effect of the electron- electron interaction on the ESR signal. In the Fermi-liquid(FL) language, ESR in the absence of SOC is an excita-tion of the Silin-Leggett (spin-flip) collective mode [ 17,18];cf. Fig. 1(a). Although the very existence of this dispersive mode is due to many-body correlations, its end point atq=0—the Larmor frequency—is protected from renormal- izations by these correlations and given by the bare Zeemanenergy [ 19]. In addition, there is a continuum of spin-flip single-particle excitations [shaded region in Fig. 1(a)], whose end point corresponds to the renormalized Zeeman energy.Although the absorption rate should, in principal, contain thecontributions from both the collective mode and continuum,the latter does not contribute to ESR because its spectralweight vanishes at q=0. These two main features of the ESR signal—no many-body renormalization of the resonancefrequency and no contribution from the continuum—are due to conservation of the total spin ( /vectorS) projection onto /vectorB. The situation changes drastically in the presence of SOC, which breaks conservation of /vectorS·/vectorBand thus gives rise to fundamentally new features in the excitation spectrumdiscussed in this article. (Modification of the ESR spectrumdue to both SOC and electron-electron interaction in thequantum Hall regime was considered in Ref. [ 20] within the Hartree-Fock theory.) Depending on whether the ratioof the Zeeman energy to spin-orbit splitting is larger than,comparable with, or smaller than unity, one can define theregimes of “high,” “moderate,” and “weak” magnetic fields.We show that the ESR frequency in the high-field regimeis affected both by SOC and many-body correlations andscales nonlinearly with B[see Fig. 1(b)]. The deviation from linearity can be used to extract the amplitudes of both SOCand electron-electron correlations. In addition to the resonancepeak, the ESR signal now also shows a broad feature dueto the continuum of spin-flip excitations. In the presence ofSOC, the resonance itself is entirely a many-body effect; inthe absence of interactions, the signal comes entirely from thecontinuum [ 21]. The conventional ESR regime is reached in the limit of B→∞ . As the field gets weaker, the ESR frequency scales down and finally vanishes at a critical field B c, where the spin-split energy levels become degenerate [see insets inFig. 1(b)] and the gap in the continuum closes. The region around B cdefines the moderate-field range. For B<B c,t h e resonance appears again and two more modes split off thecontinuum as the field passes through the critical values, B c2 andBc1.A tB→0, the three modes evolve into chiral-spin resonances—collective oscillations of magnetization in theabsence of the magnetic field [ 4,22,23]. In the most general 2469-9950/2016/93(4)/045134(11) 045134-1 ©2016 American Physical SocietySAURABH MAITI, MUHAMMAD IMRAN, AND DMITRII L. MASLOV PHYSICAL REVIEW B 93, 045134 (2016) qZ~ Z(a) Z~ Z~ Z(a)++ + … i j i j i j B=0 B>BC 0<B<BC B>>BC(1)(2)(3) (1)(2)(3) (1)(2)(3) (1)(2)(3)(c) (b)(d) FIG. 1. (a) The Silin-Leggett mode (red) and continuum of spin-flip excitations (shaded, blue) for a Fermi liquid in the magnetic field. /Delta1Zand ˜/Delta1Zare the bare and renormalized Zeeman energies, correspondingly. (b) Schematically: the frequencies of the collective modes and continuum boundaries as a function of Bfor a Fermi liquid with Rashba spin-orbit coupling in the magnetic field. The gap in the continuum closes at the critical field Bc, where the spin-split bands become degenerate. For B<B c, there are three chiral-spin modes, /Omega11...3.F o rB>B c, there is one mode with a renormalized Larmor frequency, /Omega1∗ L. Insets: spin-split Fermi surfaces. (c) RPA diagrams for the spin susceptibility. (d) Evolution of polarizations of the collective modes with B. case of both Rashba and Dresselhaus SOC present, all three chiral-spin modes are ESR-active. In the prior literature, the discussion of the effect of SOC on ESR was largely limited to two aspects: D’yakonov-Perel’damping [ 24] of the signal [ 25,26] and coupling of electron spins to the electric field via the EDSR mechanism. We showin this article that the effect of SOC is much richer than thetwo aspects mentioned above. To the best of our knowledge,all the experiments thus far have been performed in the high-field limit, where the effect of SOC is quantitative rather thanqualitative. We propose to study ESR in moderate and weak-field regimes, where the SOC-induced changes are qualitative. II. MODEL AND FORMALISM We study a two-dimensional (2D) electron system with both Rashba and Dresselhaus types of SOC (RSOC and DSOC,correspondingly) and in the presence of an in-plane magneticfield. We adopt the form of Dresselhaus SOC appropriate fora GaAs [001] quantum well and choose the x 1andx2axes to be along the [1 ¯10] and [110] directions, correspondingly. The single-particle part of the Hamiltonian then reads [ 27] ˆH0=/vectork2 2mˆσ0+α(ˆσ1k2−ˆσ2k1) +β(ˆσ1k2+ˆσ2k1)−gμB 2ˆσ1B, (2) where mis the band mass, μBis the Bohr magneton, ˆ σ1,2,3 are the Pauli matrices, ˆ σ0is the 2 ×2 identity matrix, and α(β) is the Rashba (Dresselhaus) coupling constant. For simplicity, we chose the magnetic field to be along one of the two high-symmetry directions, i.e., /vectorB||ˆx1. This restriction will be relaxed in Sec. III C . The many-body part of the Hamiltonian, ˆHint, depends only on the electron positions, ˆ/vectorx. Consequently, [ ˆHint,ˆ/vectorx]=0 and the velocity operator ˆ/vectorv=i[ˆH0+ˆHint,ˆ/vectorx]=i[ˆH0,ˆ/vectorx] retains its free-electron form: ˆ/vectorv=/parenleftbiggk1 mˆσ0−(α−β)ˆσ2,k2 mˆσ0+(α+β)ˆσ1/parenrightbigg . (3)The gradient ( k1/mandk2/m)t e r m si n ˆ/vectorvgive rise to the Drude part of the conductivity, while the spin-dependent termsgive rise to its B-dependent part, σ B, which determines the EDSR signal. In the V oigt geometry ( /vectorEem||/vectorB⊥/vectorBem), the first (EDSR) term in the absorption rate [Eq. ( 1)] contains the com- ponent ( σ/prime B)11, which is related to the spin susceptibility via (σ/prime B)11=e2 (gμB)2/Omega1(α−β)2χ/prime/prime 22, (4) while the second (ESR) term contains χ/prime/prime 22/33for/vectorBem||ˆx2/3. Equation ( 4) also holds in the presence of the electron-electron interaction. The ratio of the EDSR amplitude to the ESR one isgiven by e 2(α−β)2/μ2 B/Omega12 res, where /Omega1resis the resonance fre- quency. For the chiral-spin modes, /Omega1res∼|α−β|kFand the ratio of the amplitudes is of the order of ( λF/λC)2∼108–109, where λFis the Fermi wavelength and λC=/planckover2pi1/mecis the Compton length ( meis the free electron mass) [ 4]. For the Silin- Leggett mode, /Omega1res=gμBB≡/Omega1Land the EDSR/ESR ratio is (λF/λC)2×(/Delta1SOC//Omega1L)2, where /Delta1SOCis the characteristic spin-orbit splitting and /Omega1Lis the Larmor frequency. In this work, we assume that the EDSR part of the signal dominates the ESR one, so that the absorption rate in Eq. ( 1)i s determined by ( σ/prime B)11to very high accuracy. We also assume that both the spin-orbit splitting and Zeeman energy are muchsmaller than the Fermi energy. In this case, the correspondingterms in the Hamiltonian can be treated as corrections tothe conventional, SU(2)-invariant FL, and the complicationsencountered in generalizing the FL theory for arbitrarily largespin-dependent terms [ 28,29] do not arise. The ESR signal is completely characterized by the spin susceptibility. Atq=0, the spin and charge sectors of the theory decouple because of charge conservation [ 23], and χ ij(/Omega1) can be found within the usual random-phase approximation (RPA), in whichthe Green’s functions include the B-dependent shifts of the chemical potential [see Fig. 1(c)]. For an s-wave interaction (U=const), the Matsubara form of χ ijis given by the matrix product [ 23] χij(/Omega1m)=−(gμB)2 4/Pi10 ij/prime(/Omega1m)/bracketleftbigg 1+U 2ˆ/Pi10(/Omega1m)/bracketrightbigg−1 j/primej, (5) 045134-2ELECTRON SPIN RESONANCE IN A TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 93, 045134 (2016) where /Pi10 ij(/Omega1m)=/integraltext KTr[ ˆσiˆGKˆσjˆGK+Q] with i,j∈{1,2,3}, Q=(i/Omega1m,/vector0);K=(iωm,/vectork); and/integraltext K≡T/summationtext ωm/integraltextd2k (2π)2. Fur- thermore, ˆG−1 K=(iω+μ)ˆσ0−ˆH/prime 0, where ˆH/prime 0differs from ˆH0in that the Zeeman energy is replaced by its renormalized value (see Appendix A):gμBB→gμBB/(1−u), where u≡ ν2DUis the dimensionless coupling constant and ν2D=m/2π is the density of states in 2D. For weak SOC, i.e., for |α|, |β|/lessmuchvFwithvFbeing the Fermi velocity in the absence of SOC, the system is characterized by four energy scales: /Delta1R≡2αkF;/Delta1D≡2βkF;/Delta1Z≡gμBB;˜/Delta1Z=/Delta1Z 1−u,(6) where kF=mvF. We choose the Zeeman energies to be positive, while the signs of /Delta1Rand/Delta1Dare arbitrary. We note in passing that RPA for the case of an s-wave interaction gives the same results as the kinetic equation fora FL with a Landau function that contains only the zerothangular harmonic in the spin channel (see Appendix D). III. THE ESR SPECTRUM: MANY-BODY DESCRIPTION A. ESR without spin-orbit coupling We start by revisiting the well-known case of a FL without SOC in the magnetic field [ α=β=0i nE q .( 2)]. In this case, /Pi10 1j(/Omega1m)=0(j∈{1,2,3}) because the projection of spin on the direction of /vectorBis conserved. For the rest of the components we obtain, upon analytic continuation ( i/Omega1m→/Omega1+i0+), /Pi10 22(/Omega1)=/Pi10 33(/Omega1)=2ν2D˜/Delta12 Z/(/Omega12−˜/Delta12 Z) and /Pi10 23(/Omega1)= −/Pi10 32(/Omega1)=− 2iν2D/Omega1˜/Delta1Z/(/Omega12−˜/Delta12 Z). The collective mode corresponds to a pole of Eq. ( 5), when det[1 +U 2/Pi10 ij(/Omega1)]=0 or 1+U 2/Pi10 22=±U 2i/Pi1023. The only solution of this equation outside the spin-flip continuum is the Larmor frequency:/Omega1 L=˜/Delta1Z(1−u)=/Delta1Z. On the other hand, χ/prime/prime ij(/Omega1) vanishes at the continuum ( /Omega1=˜/Delta1Z), and thus the continuum does not contribute to ESR. B. ESR with Rashba spin-orbit coupling This case is realized by setting β=0i nE q .( 2). After including the self-energy correction to the Zeeman term [ 30] (see Appendix A), the dispersions of the spin-split bands be- come ε± /vectork=k2/2m±1 2/radicalbig (2αk)2+(˜/Delta1Z)2−2˜/Delta1Z(2αk)s i nθk, where θkis the angle between /vectorkand the x1axis. Although the spin projection onto /vectorBis not conserved anymore, some off-diagonal components of /Pi10 ijstill vanish. Indeed, since /vectorBem×/vectorB=0f o r /vectorBem/bardbl/vectorB/bardblˆx1, the only two pseudovectors in the system are /vectorBemand/vectorBthemselves. The magnetization induced by /vectorBemis also a pseudovector and thus can only be parallel to /vectorB, which implies that /Pi10 1j=0f o rj=2,3. The nonzero components of ˆ/Pi10are given by (see Appendix B) /Pi10 11(/Omega1)=− 2ν2DW2(1−f) 4˜/Delta12 Z, /Pi10 22(/Omega1)=− 2ν2D/bracketleftbigg˜/Delta12 Z fW2+/parenleftbigg 1−1 f/parenrightbigg/parenleftbigg 1−W2 4˜/Delta12 Z/parenrightbigg/bracketrightbigg ,/Pi10 33(/Omega1)=− 2ν2D/bracketleftbigg 1+/Omega12 fW2/bracketrightbigg , /Pi10 23(/Omega1)=2ν2Di/Omega1 ˜/Delta1Z/bracketleftbigg1 2/parenleftbigg 1−1 f/parenrightbigg +˜/Delta12 Z fW2/bracketrightbigg =−/Pi10 32(/Omega1),(7) where f≡/radicalBig 1−4/Delta12 R˜/Delta12 Z/W4andW2≡/Delta12 R+˜/Delta12 Z−/Omega12− i0+sgn/Omega1. The formulas above reduce to the known limits [ 23] when/Delta1R→0 and/Delta1Z→0, respectively. The subband energies vary around the Fermi surface, reaching the maximum and minimum values of |˜/Delta1Z±|/Delta1R||, correspondingly. As a result, the continuum of spin-flipexcitations occupies a finite interval of frequencies |˜/Delta1 Z− |/Delta1R||</Omega1< ˜/Delta1Z+|/Delta1R|, where all /Pi10’s in Eq. ( 7)h a v e nonzero imaginary parts. This is in contrast to the case ofα=0, where the continuum has zero spectral weight [see Fig. 1(a)]. The gap in the continuum closes at a special field B csuch that ˜/Delta1Z(Bc)=|/Delta1R|and the spin-split bands become degenerate [Fig. 1(b)]. The collective modes correspond to the poles of Eq. ( 5) outside the continuum. The eigenmode equation splits intotwo: 1+U 2/Pi10 11(/Omega1)=0, (8a) /bracketleftbigg 1+U 2/Pi10 22(/Omega1)/bracketrightbigg/bracketleftbigg 1+U 2/Pi10 33(/Omega1)/bracketrightbigg =−U2 4/bracketleftbig /Pi10 23(/Omega1)/bracketrightbig2.(8b) ForB>B c,E q .( 8a) has no solutions while Eq. ( 8b) has a unique solution (see Appendix C), which is the Larmor frequency, /Omega1∗ L, renormalized both by SOC and electron- electron interaction [cf. inset in Fig. 2(a)]. At the highest fields (˜/Delta1Z/greatermuch|/Delta1R|/u), /Omega1∗ L≈/Delta1Z/bracketleftBigg 1−(2−3u)(1−u) 4u/parenleftbigg/Delta1R /Delta1Z/parenrightbigg2/bracketrightBigg . (9) When Bis just slightly above Bc, i.e., ˜/Delta1Z≈|/Delta1R|but still ˜/Delta1Z>|/Delta1R|, we get /Omega1∗ L≈(˜/Delta1Z−|/Delta1R|)/bracketleftBigg 1−u2/parenleftbig 1−3u 4/parenrightbig2 2/parenleftbig 1−u 2/parenrightbig2(1−u)2(˜/Delta1Z−|/Delta1R|)2 ˜/Delta12 Z/bracketrightBigg . (10) In the limit of u/lessmuch1, we have an additional regime defined by /Delta1R/lessmuch˜/Delta1Z/lessmuch/Delta1R/u, where /Omega1∗ L≈|˜/Delta1Z|/bracketleftbigg 1−u2˜/Delta1Z 2|/Delta1R|/bracketrightbigg . (11) ForB<B c,E q .( 8a) has one solution, /Omega1=/Omega11, which corresponds to oscillations of the x1component of the magnetization /vectorM,w h i l eE q .( 8b) has two solutions, /Omega1=/Omega12 and/Omega1=/Omega13, which correspond to coupled oscillations of the components M2andM3.T h e/Omega11and/Omega12modes run into the continuum at fields Bc1andBc2, correspondingly [cf. Fig. 1(b)]. The three modes are plotted in Fig. 2(a) for a range of fields below Bc. As the field is lowered further, these three solutions evolve into the spin-chiral resonances [ 4,22]. AtB=0,/Pi10 23 in Eq. ( 8b) vanishes by time-reversal symmetry, while /Pi10 11 045134-3SAURABH MAITI, MUHAMMAD IMRAN, AND DMITRII L. MASLOV PHYSICAL REVIEW B 93, 045134 (2016) FIG. 2. (a) Chiral-spin modes as a function of the Zeeman energy, /Delta1Z, in units of the Rashba spin splitting, /Delta1R(on a semilogarithmic scale). Inset: renormalized Larmor mode ( /Omega1∗ L) at higher fields. (b) Imaginary part of the susceptibility in the weak-field limit ( /Delta1R//Delta1Z=20). (c) Same as in the high-field limit. The dashed line marks the bare Larmor frequency ( /Omega1L). The continuum is seen as a broad hump to the right of the resonance. In panels (a)–(c), the dimensionless interaction is u=0.3. (d) Evolution of the ESR signal with u. Here /Delta1R//Delta1Z=0.5. Damping of /Gamma1=0.01/Delta1Rwas added to the Green’s functions to mimic the effect of disorder in all plots. and/Pi10 22become equal by the C∞vsymmetry. In this limit, /Omega11=/Omega12=|/Delta1R|√1−u/2 and/Omega13=|/Delta1R|√1−u[23]. In the absence of DSOC, absorption is determined entirely byχ/prime/prime 22[cf. Eq. ( 4)]. Since the /Omega11mode is decoupled from the/Omega12and/Omega13modes, it is ESR-silent. The magnetic field couples the /Omega12and/Omega13modes, both of which show up in ESR. ForB>B c, there is only one ESR-active mode, whereas for B<B cthere can be one or two active modes, depending of whether Bis smaller or larger than Bc2. In addition to a sharp peak(s), there is also a broad feature corresponding toabsorption by the continuum of spin-flip excitations.Figure 2(d) depicts the evolution of the ESR signal with increasing u. In the presence of SOC, a sharp mode occurs only due to many-body interaction, as it pushes the mode away fromthe continuum. This is in contrast to the case without SOC,where the mode exists even without interaction. Both the peakand broad hump due to the continuum have been observed inRef. [ 38], although the detailed shape of the hump is yet to be explained. As the magnetic field increases from zero to values exceeding B c, polarization of the collective modes changes qualitatively [cf. Fig. 1(d)]. AtB=0, the susceptibility is 0.09 0.30.50.75 5 15 100515 D R R1 2 3 FIG. 3. Left: Collective modes in the presence of the magnetic field, and both Rashba and Dresselhaus spin-orbit coupling. β/α=− 0.25. Below the field at which the gap in the continuum closes, there are two chiral-spin modes; above this field there is only one precessing mode.All the modes in this case are elliptically polarized. Inset: zoom of the high-field region. Right: Collective modes in the presence of Rashba and Dresselhaus spin-orbit coupling but in the absence of the magnetic field. There are three modes on either side of the gap closing point. The entire structure of the collective mode is symmetric under α→β. All three modes are linearly polarized. Inset: zoom of the region |/Delta1 D|/lessmuch|/Delta1R|. u=0.3 in both plots. 045134-4ELECTRON SPIN RESONANCE IN A TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 93, 045134 (2016) TABLE I. SOC parameters for some common quantum wells. Mn doping in Cd 1−xMnxTe provides an additional field to the 2DEG due to the localized moments on Mn. The range in gfor this material is controlled by Mn doping. Material n(1011cm−2)|gfactor|α(meV ˚A) /Delta1R=2αkF gμB(T) β(meV ˚A) /Delta1D=2βkF gμB(T) References SiGe/Si/SiGe 1–7 2 0.055 7.5–19.9 ×10−3[36] MgxZn1−xO/ZnO 2.1 1.94 0.7 0.15 [ 37] Cd 1−xMnxTe 3.5 1.6–5 3.3 0.34–1.1 4.6 0.47–1.5 [ 38] GaAs/AlGaAs 2.3 0.445 3.1 2.89 0.55 0.51 [ 39] GaAs/AlGaAs 5.8 0.27 1.5 3.7 1.4 3.4 [ 40] GaAs/AlGaAs 1.4–7 0.4 5 4.0–9.1 4 3.2–7.2 [ 41] InAs 21 7.8–8.7 67 9.7–10.8 3.5 0.5–0.6 [ 42,43] InAs 11–20 8 60 6.8–9.2 [ 44] In1−xGaxAs/In1−yAlyAs 17–24 4 65–92 21.6–25.9 [ 45] diagonal, which means that the different components of the magnetization oscillate independently and are thus linearlypolarized. For 0 <B<B c,t h eM1component is still linearly polarized, while coupled oscillations of the M2andM3 components can be decomposed into two elliptically polarized modes. For B>B c, there is only one elliptically polarized mode which evolves into a circularly polarized Silin-Leggettmode for B/greatermuchB c. C. ESR with both Rashba and Dresselhaus spin-orbit coupling Adding DSOC to RSOC lowers the symmetry from C∞vto C2v. As a result, the doubly degenerate spin-chiral resonance splits into two already at B=0. Other than that, DSOC does not change the situation qualitatively, as long as /vectorBis along the high-symmetry axis [as in Eq. ( 2)]: one of the three modes is still ESR-silent, so the signal consists of up to two lines. If/vectorBis along a generic in-plane direction [which means that ˆσ1B1in Eq. ( 2) changes to ˆ σ1B1+ˆσ2B2], all modes become ESR-active, and the signal consists of up to three lines. Thiscase can only be tackled by a numerical treatment of the generalequations presented in Appendix B. Figure 3(left) shows collective modes of a system with both RSOC and DSOCand with the field oriented at 45 ◦to the x1axis, such that B1=B2=B. The RSOC and DSOC couplings are chosen in such a way that there are only two collective modes at B=0. D. ESR with Rashba and Dresselhaus spin-orbit couplings in zero field For completeness, we also discuss the chiral-spin reso- nances in the zero-field limit but in the presence of both Rashbaand Dresselhaus couplings. (The case when only one type ofSOC is present has been thoroughly analyzed in the priorliterature; see Refs. [ 4,22,23].) In the absence of field, the time-reversal symmetry is intact and thus collective modescan only be linearly polarized. Figure 3(right) shows the collective modes as a function of the Dresselhaus coupling β (parametrized as /Delta1 D≡2βkF). Although the evolution of the spectrum with the ratio /Delta1D//Delta1Ris qualitatively similar to the evolution with /Delta1Z//Delta1Rin the B/negationslash=0 case (in terms of closing and reopening the gap in the continuum), there are crucialdifferences between these two cases, namely, (1) the B=0 spectrum is symmetric about the zero-gap point, whereas itis asymmetric in the B/negationslash=0 case; (2) the modes are linearlypolarized in the B=0, whereas they are elliptically polarized in the B/negationslash=0. These results for the B=0 case can be derived analytically; see Appendix B2. IV . CONCLUSIONS We presented a many-body theory of the ESR/EDSR effects in the presence of SOC of both Rashba and Dresselhaus types.The combined effect of the electron-electron and spin-orbitinteractions leads to a splitting of the resonance into up tothree lines, which should be observable in an experiment.These multiple resonances are the optical (massive) collectivemodes of a Fermi liquid subject to both external and spin-orbit magnetic fields. We have also shown that the Silin-Leggett mode is affected by the electron-electron interactionin the presence of SOC; this effect must be accounted forwhen extracting the gfactors and SOC parameters from the precession measurements [ 33–35]. The best platform for observing the effects predicted in this paper are the semiconductor heterostructures in theregime when the SOC energy splitting is comparable to theZeeman splitting due to an in-plane magnetic field. In Table I, we provide a summary of relevant material parameters forsome of the conventional heterostructures. The Rashba andDresselhaus SOC energy scales are presented in units of Teslasto give an idea of the strength of the field required to probe themultiple-resonance regime of the spectrum. Recent advancesin microwave technology [ 46] have greatly broadened the range of frequencies thus making ESR a promising tool forthe detection of the chiral-spin modes. ACKNOWLEDGMENTS We would like to thank C. R. Bowers, I. Paul, F. Perez, E. I. Rashba, and C. A. Ullrich for useful discussions. S.M.acknowledges a Dirac Fellowship award from the NHMFL,which is supported by the NSF via Cooperative AgreementNo. DMR-1157490, the State of Florida, and the US DOE.D.L.M. acknowledges support from the NSF via Grant No.NSF DMR-1308972 and a Stanislaw Ulam Scholarship atthe Center for Nonlinear Studies, Los Alamos NationalLaboratory. We acknowledge the hospitality of the KavliInstitute for Theoretical Physics, which is supported by theNSF via Grant No. NSF PHY11-25915. 045134-5SAURABH MAITI, MUHAMMAD IMRAN, AND DMITRII L. MASLOV PHYSICAL REVIEW B 93, 045134 (2016) APPENDIX A: SINGLE-PARTICLE HAMILTONIAN: EIGENSTATES AND SELF-ENERGY CORRECTION In this Appendix, we derive the form of the self-energy that enters the Green’s functions in the calculation of the polariza-tion tensor. It is convenient to start with the Hamiltonian inEq. ( 2) which can be rewritten as ˆH 0=/vectork2 2mˆσ0+λ/vectorkk(sinφkˆσ1−cosφkˆσ2). (A1) The parameters λ/vectorkandφkare defined by the following relations: 2λ/vectorkk=/radicalbig (2αk)2+(2βk)2−8αβk2cos 2θk+(gμBB)2−4(gμBB)(α+β)ksinθk, sinφk=α+β λ/vectorksinθk−gμBB 2λ/vectorkk, (A2) cosφk=α−β λ/vectorkcosθk, where θkis the azimuthal angle /vectorkwith respect to the x1axis. The eigenvalues and eigenvectors are given by ε± /vectork=k2 2m±λ/vectork, (A3) |/vectork,±/angbracketright =1√ 2/parenleftbigg1 ∓ieiφk/parenrightbigg . (A4) To account for renormalization of the Zeeman energy and spin-orbit parameters entering the Green’s function, one needsto find the momentum- and frequency-independent part of theself-energy, ˆ/Sigma1. For an s-wave interaction ( U=const), ˆ/Sigma1can be found in the self-consistent Born approximation as ˆ/Sigma1=−U/integraldisplay KˆGK,ˆGK=/parenleftbig/bracketleftbigˆG0 K/bracketrightbig−1−ˆ/Sigma1/parenrightbig−1,(A5) where [ ˆG0 K]−1=(iωm+μ)ˆσ0−ˆH0. By construction, ˆ/Sigma1does not depend on Kand thus can be written as ˆ/Sigma1=/summationdisplay i=1...3aiˆσi, (A6) where the coefficients aiare to be determined. Note that we dropped the coefficient a0as it would only result in a shift of the chemical potential. Solving the algebraicmatrix equation, we get a 1=u 1−ugμBB 2(where u≡mU 2π), and a2=a3=0. This amounts to changing gμBB→gμBB 1−uor /Delta1Z→˜/Delta1Z. Since a2=a3=0, the spin-orbit parameters are not renormalized. This is a special feature of the s-wave interaction approximation. The Green’s function (with the self-energy correction) is then explicitly written as ˆGK=/summationdisplay r±gr Kˆ/Omega1r,ˆ/Omega1r=1 2[ˆσ0+r(ˆσ1sinφk−ˆσ2cosφk)], (A7) where gr K=1/(iωm−˜εr /vectork) and ˜ εr /vectorkis the electron dispersion which contains the renormalized Zeeman energy: /Delta1Z→˜/Delta1Z.APPENDIX B: COLLECTIVE MODES WITHIN THE RANDOM-PHASE APPROXIMATION 1. General case In this Appendix, we provide some details of the calcu- lation of the spin-charge polarization tensor /Pi10 ij, which is needed to find the collective modes within the random-phaseapproximation (RPA). This is a challenging task in the mostgeneral case, when the magnetic field and both Rashba andDresselhaus types of spin-orbit coupling (RSOC and DSOC,correspondingly) are present. However, in the limit when boththe magnetic field and SOC are weak, i.e., when the Zeemanenergy and spin-orbit splitting of the energy bands are smallcompared to the Fermi energy, one can confine the momentumintegration to the vicinity of the Fermi surface and carry outsome of the steps analytically. We choose the magnetic field to be along an arbitrary in- plane direction. Consequently, the Zeeman term in Eq. ( 2) is replaced by −(gμ B/2)( ˆσ1B1+ˆσ2B2). The corresponding changes in the eigenvalues and eigenvectors can readily betraced down; we will refrain from giving explicit forms here.Linearizing the dispersion near the Fermi energy as ε ± /vectork−μ= ξ±/Lambda1θk, where /Lambda1θk≡2λk=kF,θkkFis the SOC splitting at the pointθkon the Fermi surface, we arrive at two types of integrals [here, kFis the Fermi momentum in the absence of both the magnetic field and SOC and Qstands for the 2 +1 bosonic momentum with the zero spatial part: Q=(i/Omega1m,0)]: 1 2/integraldisplay dξ(g+ Kg− K+Q+g− Kg+ K+Q) =1 2/integraldisplay dξ/braceleftBigg nF(ε+ /vectork)−nF(ε− /vectork) i/Omega1m+ε+ /vectork−ε− /vectork+(+→− )/bracerightBigg =−/Lambda12 θk /Omega12m+/Lambda12 θk(B1) and 1 2/integraldisplay dξ(g+ Kg− K+Q−g− Kg+ K+Q)=i/Omega1m/Lambda1θk /Omega12m+/Lambda12 θk. (B2) A general form of λ/vectorkis obtained from Eq. ( A1) by adding the second component of the magnetic field, which amountsto replacing sin φ kbyα+β λ/vectorksinθk−gμBB1 2λ/vectorkkand cos φkby 045134-6ELECTRON SPIN RESONANCE IN A TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 93, 045134 (2016) (α−β) cosθk/λ/vectork+gμBB2/(2λ/vectorkk)i nE q .( A2). Using these relations, we get /Pi10 11(/Omega1m)=− 2ν2D/integraldisplaydθk 2π/Lambda12 θk /Omega12m+/Lambda12 θkcos2φk, /Pi10 12(/Omega1m)=−ν2D/integraldisplaydθk 2π/Lambda12 θk /Omega12m+/Lambda12 θksin 2φk, /Pi10 21(/Omega1m)=/Pi10 12(/Omega1m), /Pi10 13(/Omega1m)=2ν2D/integraldisplaydθk 2π/Omega1m/Lambda1θk /Omega12m+/Lambda12 θkcosφk, /Pi10 31(/Omega1m)=−/Pi10 13(/Omega1m), (B3) /Pi10 22(/Omega1m)=− 2ν2D/integraldisplaydθk 2π/Lambda12 θk /Omega12m+/Lambda12 θksin2φk, /Pi10 23(/Omega1m)=2ν2D/integraldisplaydθk 2π/Omega1m/Lambda1θk /Omega12m+/Lambda12 θksinφk, /Pi10 32(/Omega1m)=−/Pi10 23(/Omega1), /Pi10 33(/Omega1m)=− 2ν2D/integraldisplaydθk 2π/Lambda12 θk /Omega12m+/Lambda12 θk.For Fig. 3, we considered the magnetic field to be at 45◦to thex1axis, i.e., B1=B2≡B. Solutions of the eigenmode equation det[1 +(U/2)ˆ/Pi10]=0 are shown in the left panel of Fig. 3. In general, there are no qualitative differences compared to the case of only RSOC and the magnetic field: for B<B c there are two or three modes depending on the ratio α/β, whereas for B>B cthere is only one mode. For α/β=− 0.25, as chosen in the left panel of Fig. 3, there are only two modes. 2. Collective modes in zero field Here, we present details of the derivation of collective modes in the absence of Bbut in the presence of both RSOC and DSOC. In this case, the spin-flip continuumoccupies the energy interval |/Delta1 R−/Delta1D|</Omega1</Delta1 R+/Delta1D, where /Delta1D≡2βkF(for definiteness we choose /Delta1R,D>0). The collective modes are well defined as they occur belowthe lower boundary of the continuum. The susceptibility isstill a diagonal matrix so that its 11, 22, and 33 sectorsare all decoupled. The nonzero elements of ˆ/Pi1 0are (in real frequencies) /Pi10 11(/Omega1)=F1(/Omega1)+F2(/Omega1), /Pi10 22(/Omega1)=F1(/Omega1)−F2(/Omega1), /Pi10 33(/Omega1)=2F1(/Omega1), where F1(/Omega1)=−ν2D/bracketleftbigg 1+/Omega12 W2 D/bracketrightbigg , F2(/Omega1)=−ν2D/Delta12 R+/Delta12 D 2/Delta1R/Delta1D/bracketleftBigg 1+1 W2 D/parenleftBigg −/Omega12−/parenleftbig /Delta12 R−/Delta12 D/parenrightbig2 /Delta12R+/Delta12 D/parenrightBigg/bracketrightBigg , (B4) andW2 D=/radicalbig [/Omega12−(/Delta1R+/Delta1D)2][/Omega12−(/Delta1R−/Delta1D)2]. The diagonal form of ˆ/Pi10suggests that all the modes are linearly polarized. The eigenmode equation, det(1 +Uˆ/Pi10/2)=0, leads to the following three equations: 1+UF 1(/Omega1)=0, (B5) 1±u(/Delta1R∓/Delta1D)2 4/Delta1R/Delta1D⎛ ⎝1−/radicalBigg (/Delta1R±/Delta1D)2−/Omega12 (/Delta1R∓/Delta1D)2−/Omega12⎞ ⎠=0. (B6) Solving those, we get the frequencies of the collective modes: /Omega12 i=(/Delta1R−/Delta1D)2/bracketleftbigg 1−ufi 2zi/bracketrightbigg ,i∈(1,2); where z1=1+2/Delta1R/Delta1D u(/Delta1R−/Delta1D)2,z2=1−2/Delta1R/Delta1D u(/Delta1R+/Delta1D)2,f1=1,f2=(/Delta1R+/Delta1D)2 (/Delta1R−/Delta1D)2/parenleftbigg 1−4/Delta1R/Delta1D u(/Delta1R+/Delta1D)2/parenrightbigg2 , (B7) /Omega12 3=(/Delta1R−/Delta1D)2⎡ ⎣1−u2 1−2u/radicalBig 1+z3+z3/Delta12 R+/Delta12 D 2/Delta1R/Delta1D−(1+z3) z3⎤ ⎦,z3=/parenleftbiggu 1−u/parenrightbigg2(/Delta1R−/Delta1D)2 2/Delta1R/Delta1D. These solutions are plotted in Fig. 3(right) as a function of increasing DSOC. It follows from Eq. ( B7) that /Omega11 and/Omega13graze the continuum up to the gap-closing point, where /Delta1R=/Delta1D, whereas /Omega12hits the continuum at a pointwhere f2=0, which is below the gap-closing point. The solution is symmetric under /Delta1R↔/Delta1Dand, as a result, there are three collective modes on each side of the gap-closingpoint. 045134-7SAURABH MAITI, MUHAMMAD IMRAN, AND DMITRII L. MASLOV PHYSICAL REVIEW B 93, 045134 (2016) APPENDIX C: EIGENMODE EQUATIONS FOR THE CASE WHEN BOTH RASHBA SPIN-ORBIT COUPLING AND MAGNETIC FIELD ARE PRESENT In this Appendix, we analyze some properties of the eignemode equations for the case when RSOC and magneticfield are present. 1. Proving the absence of the collective mode in the 11 sector for B>Bc The frequency of the collective mode in the 11 sector (corresponding to oscillations of magnetization along the x1 axis, i.e., along the static magnetic field) is determined from Eq. ( 8a): 1+U 2/Pi10 11(/Omega1)=0. Here, we prove that this equation has no solutions for B>B c. Explicitly, this equation reads 1 u=(1−f)W2 4˜/Delta12 Z, (C1) where f≡/radicalBig 1−4/Delta12 R˜/Delta12 Z/W4 (C2) and W2≡/Delta12 R+˜/Delta12 Z−/Omega12−i0+sgn/Omega1. (C3) Using the standard inequality of arithmetic and geometric means, we find that fis always real and <1, if we restrict ourselves to the region below the continuum boundaries,i.e., for /Omega1<|˜/Delta1 Z−/Delta1R|. This implies that the right-hand s i d e( R H S )o fE q .( C1)i ss m a l l e rt h a nW2 4˜/Delta12 Z, which on its turn can be immediately seen to be less than1 2for/Delta1R ˜/Delta1Z<1, i.e, for B>B c. Therefore, we have 0 <RHS<1/2, while the left-hand side is larger than 1 within the paramag-netic phase ( u< 1). Thus there is no solution of Eq. ( C1) forB>B c. 2. Collective modes in the 22 and 33 sectors We now analyze Eq. ( 8b), which gives the collective modes in the 22 and 33 sectors, in the various limits and derivethe results presented in Eqs. ( 9)–(11). Equation ( 8b) can be rewritten as 1 u=X+1 1−uY, (C4) where X≡˜/Delta12 Z fW2−1 4/parenleftbigg1 f−1/parenrightbigg/parenleftbigg 3−/Delta12 R ˜/Delta12 Z/parenrightbigg , (C5) Y≡/Omega12 ˜/Delta12 Z/bracketleftbigg˜/Delta12 Z fW2−1 4/parenleftbigg1 f−1/parenrightbigg/bracketrightbigg . (C6) To proceed further, we introduce the dimensionless quantities w≡/Omega1/˜/Delta1Zandr≡|/Delta1R|/˜/Delta1Z. We look for a solution of the formw2=(1−r)2−δ, where δis a new variable confined to 0<δ< (1−r)2. In these notations, W2/˜/Delta12 Z=2r+δ andfW2=√ δ√ 4r+δ. The quantities XandYcan berewritten as X=1 ab−3−r2 8(a−b)2 ab, (C7) Y=/bracketleftbigg1 ab−1 8(a−b)2 ab/bracketrightbigg [(1−r)2−δ], where a=√ δandb=√4r+δ. Its easy to see that in the limitr→0,a→bandδ=u(2−u). This makes /Omega12/˜/Delta12 Z= (1−u)2or/Omega1=/Delta1Z, which is the bare Larmor frequency. In the opposite limit of r→∞ , we find two roots: δ=r2u andδ=r2u/2. These give /Omega12=/Delta12 R(1−u)o r/Delta12 R(1−u/2), which are the frequencies of the two spin-chiral modes in the absence of the magnetic field. Equation ( 9) corresponds to the strong-field limit and is derived assuming that r/lessmuchu< 1. We skip this derivation as it is a brute force expansion in r2, which is lengthy but completely straightforward. In the moderate-field limit, where r≈1, we relabel r= 1−εwith 0 <ε/lessmuch1 and look for a solution in the region δ/lessmuchε2. In this limit, the quantities XandYreduce to X=1 2+3ε2 4√ δ, (C8) Y=ε2 4√ δ. This yields δ=ε4 4u2(1−3u/4)2 (1−u)2(1−u/2)2, (C9) which reproduces Eq. ( 10). In the weak-coupling case ( u/lessmuch1), one can identify one more interval: u/lessmuchr< 1. There, we find that δ= u2(1−r)4/r. This makes the frequency /Omega12≈˜/Delta1Z(1−r)[1− u2 2(1−r)2 r], which reproduces Eq. ( 11). APPENDIX D: COLLECTIVE MODES FROM THE QUANTUM KINETIC EQUATION The quantum kinetic equation for a Fermi liquid (FL) subject to a spatially uniform external field and in thecollisionless regime reads i∂δˆn /vectork ∂t=[δˆε/vectork,ˆn/vectork], (D1) where δˆε/vectork=/vectors/vectork·/vectorσ 2+/integraldisplay /vectork/primeTr/prime[ˆF/vectork/vectork/primeδˆn/prime /vectork]( D 2 ) is a variation of the quasiparticle energy,/integraltext /vectork/prime≡/integraltextd2k/prime (2π)2= ν2D/integraltext dε/integraltextdθ 2π,F/vectork/vectork/prime=Fa(θ−θ/prime)/vectorσ·/vectorσ/primeis the antisymmetric part of an SU(2)-invariant Landau interaction function, θand θ/primeare the angle subtended by /vectorkand/vectork/prime, correspondingly, and/vectors/vectorkparametrizes the spin-orbit and Zeeman terms of the Hamiltonian. For RSOC, /vectors/vectork=/Delta1R(sinθ,−cosθ,0); for purely Zeeman coupling, /vectors/vectork=(−˜/Delta1Z,0,0); etc. The electron distribution function can be written as ˆn/vectork=/vectors/vectork·/vectorσ 2∂n0 ∂ε+δˆn/vectork, (D3) 045134-8ELECTRON SPIN RESONANCE IN A TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 93, 045134 (2016) where n0is the equilibrium distribution function in the absence of both SOC and magnetic field, and δˆn/vectorkis the nonequilibrium part. The nonequilibrium part of the magnetization is given by /vectorM=−gμB 2/integraldisplay /vectorkTr[/vectorσδˆn/vectork]. (D4) The nonequilibrium part of the distribution function can be ex- panded either over standard or rotated Pauli matrices [ 4]. In the first way, δˆn/vectork=/vectorN(θ)·/vectorσ∂n0 ∂εsuch that Mi=gμBν2D/integraltext θNi(θ), withi∈(1,2,3) and/integraltext θ≡/integraltext2π 0dθk/(2π). The kinetic equation reads ˙/vectorN(θ)=−/vectorN(θ)×/vectorsθ−/integraldisplay θ/primeFa(θ−θ/prime)/vectorN(θ/prime)×/vectorsθ, (D5) where /vectorsθ≡/vectors/vectorkatk=kF. The time dependence of /vectorNis not explicitly specified. Equation ( D5) can be solved by decomposing /vectorNandFainto angular harmonics. Note that Miis given by the zeroth harmonic of Ni. As a demonstration, we solve Eq. ( D5) for the case of RSOC in the s-wave approximation for Fa(θ−θ/prime)=Fa 0. Equation ( D5) is then simplified to ˙/vectorN(θ)=−/vectorN(θ)×/vectorsθ−Fa 0/vectorM×/vectorsθ. (D6) Note that /vectorsθ·˙/vectorN(θ)=0 suggesting that /vectorsθ·/vectorN(θ)=constant, which can be set to zero. Integrating Eq. ( D6) over θand noticing that/integraltext θ/vectorsθ=0 for RSOC, we get ˙/vectorM=−/integraldisplay θ/vectorN(θ)×/vectorsθ. (D7) Differentiating Eq. ( D7) over time again and using Eq. ( D6) for˙/vectorN(θ) with /vectorsθ·/vectorN(θ)=0, we obtain ¨/vectorM=−/parenleftbig 1+Fa 0/parenrightbig /Delta12 R/vectorM+Fa 0/integraldisplay θ/vectorsθ(/vectorsθ·/vectorM). (D8) This yields ¨M1,2=−/parenleftbigg 1+Fa 0 2/parenrightbigg /Delta12 RM1,2,¨M3=−/parenleftbig 1+Fa 0/parenrightbig /Delta12 RM3, (D9) which coincides with the q=0 limit of the hydrodynamic equations derived in Ref. [ 22].For the field-only case, when /vectorsθ≡(−˜/Delta1Z,0,0) is isotropic in the momentum space, we obtain the familiar Bloch equation by integrating Eq. ( D5) over the angle [ 47] ˙/vectorM=/parenleftbig 1+Fa 0/parenrightbig˜/Delta1Z/vectorM׈x1=gμB/vectorM×/vectorB. (D10) Equivalence of the RPA and FL approaches in the s-wave approximation The results discussed thus far were presented in with a different choice of basis [ 4,22] and reproduced in Ref. [ 23] within the RPA approximation. We now wish to reproducethe RPA result for the case of RSOC in the presence ofthe magnetic field using the quantum kinetic equation. It isconvenient to work in the basis introduced by Ref. [ 4]. In this basis we write δˆn=N i(θ)ˆτi∂n0 ∂ε, ˆτ1=ˆσ3,ˆτ2=cosθˆσ1+sinθˆσ2,ˆτ3=sinθˆσ1−cosθˆσ2. (D11) Expanding Ni(θ) into angular harmonics as Ni(θ)=/summationdisplay mN(m) icosmθ+¯N(m) isinmθ, (D12) and using Eq. ( D4), we obtain for the magnetization compo- nents M1=gμB/parenleftbig N(1) 2+¯N(1) 3/parenrightbig , M2=gμB/parenleftbig¯N(1) 2−N(1) 3/parenrightbig , (D13) M3=gμBN(0) 1. The case of RSOC only can be solved exactly for an arbitrary form of the Landau interaction function in the spin channel,F a(θ−θ/prime), because equations for harmonics of /vectorNdecouple in this case [ 4]. However, harmonics do not decouple in the presence of the field for an arbitrary Landau function, and thusan exact solution is not possible. To proceed further, we adoptthes-wave approximation, F a 0(θ−θ/prime)=Fa 0. In this case, the kinetic equation [Eq. ( D1)] can be written as ˙N1(θ)+˜/Delta1z[N2(θ)s i nθ−N3(θ) cosθ]−/Delta1RN2(θ)=˜F[/Delta1R(M2sinθ+M1cosθ)−˜/Delta1zM2], ˙N2(θ)−˜/Delta1zN1(θ)s i nθ+/Delta1RN1(θ)=˜F(˜/Delta1zsinθ−/Delta1R)M3, (D14) ˙N3+˜/Delta1zN1(θ) cosθ=− ˜F˜/Delta1zcosθM 3, where ˜F≡Fa 0/gμBν2D. After a Fourier transform in time, we obtain for those harmonics of Nthat are relevant for magnetization N0 1=˜F 2W2˜/Delta1zf/bracketleftbig i/Omega1/braceleftbig W2(1−f)+2˜/Delta12 z/bracerightbig M2−2˜/Delta1z{(/Omega12+W2f)M3}/bracketrightbig , N1 2=˜F 8˜/Delta12z/Delta12 R/bracketleftbig 2W2/Delta12 R(f−1)−/braceleftbig W4(f−1)+2˜/Delta12 z/Delta12R/bracerightbig/bracketrightbig M1, ¯N1 2=˜F 8˜/Delta12z/Delta12 Rf/bracketleftbig/braceleftbig 2(1−f)/bracketleftbig 2˜/Delta12 z/Delta12R−W2/parenleftbig˜/Delta12 z+/Delta12 R/parenrightbig/bracketrightbig −/bracketleftbig 2˜/Delta12 z/Delta12Rf+W4(f−1)/bracketrightbig/bracerightbig M2 +2i/Omega1(1−f)/braceleftbig W2˜/Delta1z−2˜/Delta1z/Delta12 R/bracerightbig M3/bracketrightbig , 045134-9SAURABH MAITI, MUHAMMAD IMRAN, AND DMITRII L. MASLOV PHYSICAL REVIEW B 93, 045134 (2016) N1 3=˜F 8˜/Delta12z/Delta12 R/bracketleftbig/braceleftbig 2W2˜/Delta12 z(1−f)+/bracketleftbig 2˜/Delta12 z/Delta12R+W4(f−1)/bracketrightbig/bracerightbig M2+2iW2/Omega1˜/Delta1z(f−1)M3/bracketrightbig , ¯N1 3=˜F 8˜/Delta12z/Delta12 R/bracketleftbig 2˜/Delta12 z/Delta12R+W4(f−1)/bracketrightbig M1, (D15) where fandWare given by Eqs. ( C2) and ( C3), correspondingly. Combining the left-hand sides of the equations above into components of /vectorM, we obtain the eigenmode equation ⎛ ⎜⎜⎝1−Fa 0 2ν2D/Pi10 11(/Omega1)0 0 01 −Fa 0 2ν2D/Pi10 22(/Omega1)−Fa 0 2ν2D/Pi10 23(/Omega1) 0 −Fa 0 2ν2D/Pi10 32(/Omega1)1 −Fa 0 2ν2D/Pi10 33(/Omega1)⎞ ⎟⎟⎠⎛ ⎝M1 M2 M3⎞ ⎠=0, (D16) where /Pi10 ij(/Omega1)a r et h es a m ea si nE q .( 7). These are the same eigenmode equations as given by RPA, det[1 +U 2/Pi1]=0, upon replacing Fa 0→−ν2DU. [1] H. L. Stormer, Z. Schlesinger, A. Chang, D. C. Tsui, A. C. Gossard, and W. 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PhysRevB.84.224513.pdf

PHYSICAL REVIEW B 84, 224513 (2011) Pressure dependence of the low-temperature crystal structure and phase transition behavior of CaFeAsF and SrFeAsF: A synchrotron x-ray diffraction study S. K. Mishra,1R. Mittal,1,*S. L. Chaplot,1S. V . Ovsyannikov,2D. M. Trots,2L. Dubrovinsky,2Y . Su,3Th. Brueckel,3,4 S. Matsuishi,5H. Hosono,5and G. Garbarino6 1Solid State Physics Division, Bhabha Atomic Research Center, Trombay, Mumbai 400 085, India 2Bayerisches Geoinstitut, Universitat Bayreuth, Universitatsstrasse 30, D-95440 Bayreuth, Germany 3Juelich Centre for Neutron Science, IFF , Forschung. Juelich, FRM II, Lichtenbergstr. 1, D-85747 Garching, Germany 4Institut fuer Festkoerperforschung, Forschungszentrum Juelich, D-52425 Juelich, Germany 5Frontier Research Center, Tokyo Institute of Technology, 4259 Nagatsuta-cho, Midori-ku,Yokohama 226-8503, Japan 6European Synchrotron Radiation Facility, BP 220, 38043 Grenoble, France (Received 9 May 2011; revised manuscript received 5 August 2011; published 21 December 2011) We report systematic investigation of high pressure crystal structures and structural phase transition up to 46 GPa in CaFeAsF and 40 GPa in SrFeAsF at 40 K using powder synchrotron x-ray diffraction experimentsand Rietveld analysis of the diffraction data. We find that CaFeAsF undergoes orthorhombic to monoclinicphase transition at P c=13.7 GPa while increasing pressure. SrFeAsF exhibits coexistence of orthorhombic and monoclinic phases over a large pressure range from 9 to 39 GPa. The coexistence of the two phases indicatesthat the transition is of first order in nature. Unlike in the 122 compounds (BaFe 2As2and CaFe 2As2), we do not find any collapse tetragonal transition. The transition to a lower symmetry phase (orthorhombic to monoclinic)in 1111 compounds under pressure is in contrast with the transition to a high symmetry phase (orthorhombicto tetragonal) in 122-type compounds. On heating from 40 K at high pressure, CaFeAsF undergoes monoclinicto tetragonal phase transition around 25 GPa and 200 K. Further, it does not show any post-tetragonal phasetransition and remains in the tetragonal phase up to 25 GPa at 300 K. The dP c/dT is found to be positive for the CaFeAsF and CaFe 2As2, however the same was not found in case of BaFe 2As2. DOI: 10.1103/PhysRevB.84.224513 PACS number(s): 74 .70.−b, 61.50.Ks, 72 .80.Ga I. INTRODUCTION Crystal structures and phase transitions are vital to super- conductivity in iron-based compounds.1–20These compounds occur in five different structural classes2(namely, FeSe, LiFeAs, BaFe 2As2, LaFeAsO/SrFeAsF, and Sr 3Sc2O5Fe2As2) but share a common layered structure based on a planar layer of iron atoms joined by tetrahedrally coordinated pnictogen (Pn=P, As) or chalcogen (S, Se, Te) anions arranged in a stacked sequence separated by alkali, alkaline-earth, or rare-earth (Ba, Ca, Sr, Eu) and oxygen/fluorine blocking layers. It is now widely thought that the interaction that leads to the highTcsuperconductivity originates within these common iron layers, similar in nature to the common copper oxygen building block found in the copper oxide (cuprate). However, structurally three key differences are found among the FeAs and cuprate compounds. First, in the FeAs compounds the pnictogen/chalcogen anions are placed above and below the planar iron layer as opposed to the planar copper-oxygen structure of the cuprates. Second, ability to substitute or dope directly into the active pairing layers of FeAs compoundsand finally the parent compounds of the new Fe-based superconductors share a similar electronic structure with all five d orbitals of the Fe contributing to a low density of states at the Fermi level, which is in contrast to the cuprates where parent compounds are Mott insulators with well-defined local magnetic moments. These similarities have inspired a flurry of theoretical and experimental works 1–30in Fe pnictides-based materials. At ambient condition, these compounds crystallize in tetragonal symmetry with no magnetic order (i.e., paramag-netic in nature). The parent compounds of iron-pnictides un-dergo a first- or second-order structural transition below room temperature (typically in the range of 100–210 K), from tetrag-onal to orthorhombic structure, and magnetic transition fromnonmagnetic to stripe antiferro-magnetic structure. 2–7,22,24,29,30 The structural transition and magnetic ordering can happen simultaneously or successively depending on the compound.It has been confirmed both experimentally and theoreticallythat the magnetic order of Fe at low temperature is stripelikeantiferromagnetism often referred to as spin density wave(SDW). 2–7Upon changing the carrier concentration, applying external pressure or by charge neutral doping, the magnetic order suppresses, and the materials become superconducting. In the 1111-family (both RFeAsO and MFeAsF, R =rare earth and M =Ca and Sr), magnetic transition temperature ( TN)i s lower than structural transition temperature ( TS). This seems to suggest that the magnetic transition is induced by the structuraltransition. In case of M /primeFe2As2(M/prime=Ba, Sr, Ca, and Eu) compounds, (at ambient pressure with decreasing temperature) the structural and magnetic transitions are found29,30to happen simultaneously (a first-order transition). However, recenthigh pressure powder x-ray synchrotron diffraction studieson BaFe 2As2show that at low temperature (33 K), these transitions occur at different pressures.19It is not clear whether the magnetic transition is induced by the structural transitionand what is the driving force of the structural transition. These two important questions are crucial to understand the formation of the stripe antiferromagnetic order in the parentcompounds. The magnetism is intimately related to the crystal structure both in terms of the Fe-As bond length and As-Fe-As bondangle. The antiferromagnetic ordering can be suppressed by 224513-1 1098-0121/2011/84(22)/224513(12) ©2011 American Physical SocietyS. K. MISHRA et al. PHYSICAL REVIEW B 84, 224513 (2011) various ways such as by changing the carrier concentration, applying external pressure, or by charge neutral doping. Thecommon FeAs building block is considered as a criticalcomponent to stabilize superconductivity. The combinationof strong bonding between Fe-Fe and Fe-As sites (and eveninterlayer As-As in the 122-type systems), and the geometryof the FeAs 4tetrahedra plays a crucial role in determining the electronic and magnetic properties of these compounds. Forinstance, the two As-Fe-As tetrahedral bond angles seem toplay a crucial role in optimizing the superconducting transitiontemperature. The highest T cvalues are found only when an As-Fe-As/Fe-As-Fe tetrahedral bond angle is close to the idealvalue of 109.47 ◦. The detailed interplay between the crystal structure, magnetic ordering, and superconductivity is hardlyunderstood. High pressure experiments play an important role in the field of superconductivity and also provide information aboutthe understanding of its mechanism. The superconductingtransition temperature ( T c) in FeAs compounds is found to increase14–21over 50 K by application of pressure. Despite the importance of the evolution of superconducting transitiontemperature ( T c) with pressure to understand the mechanism of superconducting properties of Fe-based materials, there is lackof information on the detailed pressure dependence of theirstructural properties. Recently, we reported the pressure effectson CaFe 2As2and BaFe 2As2(122-type compounds) using powder synchrotron x-ray diffraction technique. Rietveldanalysis of the high pressure powder x-ray diffraction datashowed 19that at 300 K in the Ba-compound the collapsed tetragonal transition occurs at 27 GPa. The transition pressurevalue is found to be much higher as compared to the Ca-compound where the transition occurs at 1.7 GPa. However,at low temperature (33 K), structural phase transition fromthe orthorhombic to tetragonal phase in the Ba-compoundoccurred at about 29 GPa (while increasing pressure), whichis much higher than the transition pressure of 0.3 GPa at 40 K,as known in case of the Ca-compound. 19We have not found any evidence of a post collapsed tetragonal phase transition inCaFe 2As2up to 51 GPa (at 300 K) and 37.8 GPa (at 40 K). It is important to note that transition to a collapsed phaseoccurs in the two compounds at nearly the same values of unit cell volume and ct/atratio. Although five different types2of Fe-based superconductors have been reported in literature, the122 FeAs-based superconductors seem to be the most studied(of the five types). On the other hand 1111-type compoundshave not gotten considerable attention from the scientificcommunity. In this paper we present systematic investigationof pressure effect on crystal structure, and structural phasetransition behavior of CaFeAsF and SrFeAsF at 40 K usingpowder-diffraction technique. Detailed Rietveld analysis ofthe diffraction data shows that both the compounds undergoa structural phase transition from orthorhombic to monoclinicphase on compression. While CaFeAsF undergoes a fairlysharp transition at around 13.7 GPa, SrFeAsF exhibits alarge phase coexistence region (from 9 to 40 GPa) with theorthorhombic phase fraction continuously decreasing withincrease of pressure. Possible correlation between structureevolutions with pressure and superconductivity in Fe-basedsuperconductor is also discussed. II. EXPERIMENTS The pressure-dependent powder x-ray diffraction measure- ments were carried out using the ID-27 beam line at theEuropean Synchrotron Radiation Facility (ESRF, Grenoble,France). An applied pressure was generated by membranediamond anvil cells (DACs). We employed a stainless steelgasket preindented to the thickness of ∼40–50 μm, with a central hole of 150 μm in diameter and filled with helium as pressure transmitting media. The pressure was determinedfrom the shift of the fluorescence line of the ruby. A powderysample of ∼30–40 μm in diameter and 10 μm in thickness was situated in the center of one of the diamond anvil’s tips.The wavelength of the x-ray (0.3738 ˚A) was selected and determined using a Si(111) monochromator and the iodine K-edge. The sample-to-image plate (MAR345) detector distancewas refined using the standard diffraction data of Si. To cool theDAC, a continuous helium flow CF1200 DEG Oxford cryostatwas used. Precaution was taken to obtain stable temperatureand pressure conditions prior to each acquisition. The precision 0 1 02 03 04 05 0050100150200250300 0 1 02 03 04 05 0050100150200250300 Ortho- rhombicCaFeAsF Tetragonal Monoclinic MonoclinicTemperature (K) Pressure (GPa)Tetragona l Orthorhombic +MonoclinicSrFeAsF Ortho- rhombicPhase coexistance Pressure (GPa)Tetragonal FIG. 1. (Color online) The pressure-temperature path (indicated by blue, dashed line and arrows) as followed in our measurements on CaFeASF and SrFeAsF. Symbols correspond to the pressure-temperature conditions where measurements were made. The solid lines through the symbols are a guide to the eye. The phases, namely tetragonal, orthorhombic, and monoclinic, as identified by Rietveld refinement of the diffraction data are indicated over ranges shown by red lines and arrows. 224513-2PRESSURE DEPENDENCE OF THE LOW-TEMPERATURE ... PHYSICAL REVIEW B 84, 224513 (2011) and accuracy of the temperature measurement is better than 0.1 K and 0.2 K, respectively. During the measurements, theCaFeAsF and SrFeAsF samples were first cooled to 40 K,and then pressure was increased along a path indicated inFig. 1. Typical exposure times of 20 seconds were employed for the measurements. Preferred orientation of crystal grainsis observed along different axes in different loadings, whichis common in high-pressure experiments. The diffractionpatterns indicate preferred orientation of the samples along[211] and [132] for measurements on CaFeAsF and SrFeAsF,respectively. The two-dimensional (2D) powder images were integrated using the program FIT2D 31to yield intensity vs 2 θplot. The structural refinements were performed using the Rietveldrefinement program FULLPROF. 32In all the refinements the background was defined by a sixth-order polynomial in 2 θ.A Thompson-Cox-Hastings pseudo-V oigt with axial divergenceasymmetry function was chosen to define the profile shapefor the powder synchrotron x-ray diffraction peaks. Thescale factor, background, and half-width parameters, alongwith mixing parameters, lattice parameters, and positionalcoordinates, were refined. III. RESULT AND DISCUSSION The powder synchrotron x-ray diffraction measurements for MFeAsF (M =Ca, Sr) at ambient conditions confirmed single- 5 1 01 52 02 515.7 16.115.6 15.8 40 K 2θ (degree)Intensity (arb. units)5.8 GPa(400)O (040)O(220)T (b)(a) CaFeAsF 300 K 0.6 GPa FIG. 2. (Color online) Observed (solid black circles), calculated (continuous red/dark gray line), and difference (bottom blue/medium gray line) profiles obtained after the Rietveld refinement of CaFeAsFat (a) 0.6 GPa and 300 K in tetragonal phase (space group P4/nmm ) and (b) 5.8 GPa and 40 K in orthorhombic phase (space group Cmma ). Inset in (a) shows the (220) reflection of the tetragonal phase and in (b) shows the splitting/broadening of the (220) reflection of the tetragonal phase at 40 K and provides unambiguous signature for orthorhombic structure.phase samples consistent with the published reports.22,24–26 The effects of pressure inhomogeneity on the phase transition behavior of FeAs-based compounds have been reported inliterature. 14–19In the present measurements we have used helium as a pressure transmitting medium, which is believed33 to give the best hydrostatic conditions at present. However,effect of inhomogeneity could not be completely ruled out andcould have some influence on the results obtained on thesecompounds. A. Phase transition of CaFeAsF at 40 K At ambient condition, CaFeAsF crystallizes in tetragonal structure with space group P4/nmm without magnetic or- dering. On cooling it undergoes a tetragonal to orthorhombicphase transition at 134 K, while magnetic ordering in the formof a SDW sets in at around 114 K, respectively. Figures 2(a)and 2(b) show a Rietveld refinement of powder synchrotron data at 300 K (0.6 GPa) and at 40 K (5.8 GPa), respectively. Splittingof (220) Tpeak of tetragonal phase at 300 K unambiguously confirms the orthorhombic structure at 40 K (see inset). The pressure-dependent powder x-ray diffraction measure- ments were carried out at 40 K and at pressures between 5.8to 46.2 GPa. The data show significant changes with pressureespecially in terms of dissimilar broadening of various peaks.The most prominent changes have been observed in peaksaround 2 θ=11 ◦, which become broader with increasing pres- sure above 12 GPa. Figure 3depicts a portion of the powder 11.5 12.0 12.5 7.5 8.0 8.55.8 (GPa)11.6 (GPa) 23.0 (GPa) 34.9 (GPa) 41.5 (GPa) 15.8 (GPa) 20.3 (GPa) 29.3 (GPa) 2θ (degree)Intensity (arb. units) 40 KCaFeAsF FIG. 3. Evolution of the powder-synchrotron x-ray diffraction patterns of CaFeAsF at 40 K and selected pressure. 224513-3S. K. MISHRA et al. PHYSICAL REVIEW B 84, 224513 (2011) 2468 1 0 1 2 11.0 11.5 12.011.0 11.5 12.0 11.0 11.5 12.0 11.5 12.0 11.0 11.5 12.02468 1 0 1 2 (b) TCmma P4/nmmP4/nmm (T) + Cmma (O)Intensity (arb. units) (a) CaFeAsF Cmma 2θ (degree) (g) (c) OTP4/nmm+ Cmma (e) Pbcm 2θ (degree) 2θ (degree) (d)40 K and 20.3 GPa Pmmn(f) P2/n P2/n (M) FIG. 4. (Color online) Observed (solid black circles), calculated (continuous red/dark gray line), and difference (bottom blue/medium gray line) profiles obtained after the Rietveld refinement of CaFeAsF at 40 K and 20.3 GPa using different models, namely (a) an or-thorhombic ( Cmma ); (b), (c) a combination of tetragonal ( P4/nmm ) and an orthorhombic ( Cmma ); (d), (e) an orthorhombic ( Pmmn , Pbcm ); and (f), (g) monoclinic ( P2/n) phases, respectively. x-ray diffraction patterns of CaFeAsF at selected pressure. Detailed Rietveld refinement of the powder-diffraction datashows that diffraction patterns at 40 K could be indexed using the orthorhombic structure (space group Cmma )u p to 12 GPa. The Rietveld refinements proceeded smoothly,revealing a monotonic decrease in lattice constant and cellvolume with increasing pressure. The response of structuralparameters to pressure is strongly anisotropic (as will beshown below in Fig. 9). Further increase of pressure above 12 GPa leads to even higher compression of the interlayerspacing (lattice parameter c). However, attempts to employ the same orthorhombic structural model in the refinements[Fig. 4(a)] proved unsatisfactory, and a progressive worsening of the quality of the Rietveld fits with increasing pressure wasfound. The most apparent signature of the subtle structuraltransformation that occurs above 12 GPa is the inabilityof orthorhombic structure (space group Cmma ) to account satisfactory for the peaks around 11 ◦. For more clarity it is shown in Fig. 4(a) that the diffraction data at 20.3 GPa cannot be indexed with the orthorhombic phase. Extra broadening(splitting) of peaks suggests either lowering of the symmetryor coexistence of another high symmetry phase. Recently, we reported in BaFe 2As2compounds that low temperature orthorhombic (space group Fmmm ) and room temperature tetragonal (space group I4/mmm ) phases coexist over a large pressure range above 29 GPa.19In view of this, to account for the peak broadening in the pressure- dependence diffraction data on CaFeAsF, we have explored the possibility of coexistence of low temperature orthorhombic (space group Cmma ) and room temperature tetragonal (space group P4/nmm ) phases. The powder-diffraction data above 12 GPa are refined using these coexisting phases, and the results are shown in Figs. 4(b) and4(c). The green color line (online only and marked with T) represents the additional tetragonal phase. While the profile around 11.5◦could be 5 1 01 52 02 5 11.0 11.5 12.0 7.5 8.0 8.540 K 11.6 GPaCmma 2θ ((degree )CmmaP2/n P2/n 2θ ((degree ) 2θ ((degree )CmmaP2/n 40 K 20.3 GPa (a) 40 K41.5 GPa P2/n (c) CaFeAsF 40 K and 20.3 GPa (b) CaFeAsFIntensity (arb. units)40 K and 20.3 GPa FIG. 5. (Color online) (a) Observed (solid black circles), calculated (continuous red/dark gray line), and difference (bottom blue/medium gray line) profiles obtained after the Rietveld refinement of CaFeAsF at selected pressures and 40 K. The diffraction profiles at 11.6 GPa are refined using an orthorhombic ( Cmma ) phase, while the profiles at 20.3 and 41.5 GPa are refined monoclinic ( P2/n) phase. (b), (c) The refinement of the diffraction pattern at 20.3 GPa and 40 K with an orthorhombic phase (space group Cmma ) and monoclinic ( P2/n) phase. 224513-4PRESSURE DEPENDENCE OF THE LOW-TEMPERATURE ... PHYSICAL REVIEW B 84, 224513 (2011) TABLE I. Results of Rietveld refinement of the crystal structure for CaFeAsF at selected pressure and 40 K. The measurements were carried out using a focused x-ray monochromatic beam of wavelength =0.3738 ˚A. The data collected up to 25◦have been used to determine the reported parameters. The atomic positions for space group Cmma (No. 67): Ca (4g)(0, 1 /4,z), Fe (4b) (1 /4, 0, 1 /2), As (4g) (0, 1 /4,z) and F (4a) (1 /4, 0, 0). The atomic positions for space group P12/n1 (No. 13, unique axis band cell choice 2): Ca (2f)(3 /4,y,1/4), Fe (2e) (3/4,y,3/4), As (4g) (3 /4,y,1/4), and F (4a) (3 /4,y,3/4). At 5.72 GPa and 40 K 29.3 GPa and 40 K Orthorhombic phase (space group Cmma ) Monoclinic phase (space group P12/n1) Atoms Positional coordinates Positional coordinates XY Z X Y Z Ca 0 1 /4 0.1640(1) 3 /4 0.1750(2) 1 /4 Fe 1 /4 0 0.5000 3 /4 0.4892(1) 3 /4 As 0 1 /4 0.6618(6) 3 /4 0.6639(3) 1 /4 F1 /4 0 0.0000 3 /4 −0.0493(4) 3 /4 Lattice parameters a(˚A) 5.4012(1) 3.6733(2) b(˚A) 5.3733(2) 7.5810(2) c(˚A) 8.2897(3) 3.7156(2) β=90.83 (3)◦ Rp=14.5;Rwp=25.7; R exp=13.34; χ2=3.71 Rp=17.8;Rwp=23.4; R exp=13.39; χ2=3.05 fitted (forcefully), the calculated peaks around 2.5◦,7 . 9◦, and 9.8◦arising due to the tetragonal phase are not observed in the experiments. This unsatisfactory quality of the Rietveldfit of the diffraction data suggests that the possibility of the phase coexistence of orthorhombic and tetragonal phases is not favored. The dissimilar broadening of peaks and inability to Rietveld fit the powder-diffraction patterns using a phase- coexistence model suggests a reduction of symmetry from the orthorhombic symmetry (space group Cmma ). We have further explored various possibilities, namely, orthorhombic symmetry with space group Pbcm andPmmn , monoclinic structure with space group P12/n1, etc. [see Figs. 4(d)–4(g)], to identify the correct space group. We found that orthorhombic space groups ( Pbcm ,Pmmn ) also could not fit structural data very well, and the monoclinic structure withspace group P12/n1 (No. 13, unique axis band cell choice 2) could successfully index all the peaks [see Figs. 4(f) and 4(g)]. It is well documented in literature that many isostructural Fe-based materials undergo a structural phase transition to themonoclinic phase with temperature and pressure. 34,35Rietveld refinements employing this structural model are satisfactory for all the diffraction patterns up to the highest pressure mea-sured by us. A careful inspection and analysis of diffractiondata reveal that CaFeAsF transforms to the monoclinic struc-ture at P c=13.7 GPa even though the monoclinic distortion is quite small. The monoclinic angle as a function of pressureshows a sharp discontinuity at the transition pressure at 13.7 GPa. The monoclinic angle ( β) shows a very small change from 90.5 ◦to 91.2◦on increase in pressure from 13.7 GPa to 46.2 GPa. However, the small monoclinicity has significantlyimproved the fit quality between the observed and calculatedprofiles, as shown in Fig. 5. The detailed structural parameters and goodness of fit for CaFeAsF at selected pressure and40 K, as obtained from powder-synchrotron diffraction data, are given in Table I. Schematic diagrams of the crystal structure in the orthorhombic and monoclinic phases are shownin Fig. 6.B. Phase transition of SrFeAsF at 40 K SrFeAsF also crystallizes in tetragonal structure with space group P4/nmm at ambient condition. On cooling it undergoes a structural phase transition to orthorhombic phasearound 180 K, followed by paramagnetic to antiferromagnetic cm=co am ao bo bm Ca F Fe As FIG. 6. (Color online) The relation between the unit-cell param- eter of the low pressure orthorhombic phase and that of the high pressure monoclinic phase. ao,bo,co,a n dam,bm,a n dcmare cell edges in the orthorhombic and monoclinic phases, respectively. The relationship between the unit-cell parameter in the orthorhombic and monoclinic phases can be approximately described as ao=am+bm, bo=am−bm,co=cm. The colored solid circles represent the atoms: Ca, white; Fe, red/dark gray; As, green/gray; and F, blue/medium gray in the unit cell. 224513-5S. K. MISHRA et al. PHYSICAL REVIEW B 84, 224513 (2011) 5 1 01 52 02 515.5015.2 40 K 2θ (degree)Intensity (arb. units)5.8 GPa (b)(a) SrFeAsF 300 K 0.6 GPa FIG. 7. (Color online) Observed (solid black circles), calculated (continuous red/dark gray line), and difference (bottom blue/medium gray line) profiles obtained after the Rietveld refinement of SrFeAsF at (a) 0.6 GPa and 300 K in tetragonal phase (space group P4/nmm ) and (b) 5.8 GPa and 40 K in orthorhombic phase (space group Cmma ). Inset in (a) shows the (220) reflection of the tetragonal phase and in (b) shows the splitting/broadening of the (220) reflection ofthe tetragonal phase at 40 K and provides unambiguous signature for orthorhombic structure. transition at 133 K. Figures 7(a) and 7(b) depict results of Rietveld refinement of powder synchrotron data at 0.6 GPa and300 K and 5.8 GPa and 40 K, respectively. At low temperature,we found that refinement for SrFeAsF can be carried outin the orthorhombic structure consistent with literature. Theobservation is unambiguously confirmed by comparing the(220) Tpeak of tetragonal phase at ambient temperature (see inset). As in the case of CaFeAsF, the powder-diffraction patterns of SrFeAsF also show significant changes with pressure. Toobtain the pressure dependence of the structural parameters,detailed Rietveld refinement of the powder-synchrotron x-raydiffraction data are carried out. Similar to CaFeAsF, we noticebroadening/splitting of some of the peaks around 9 GPa.At this pressure, the diffraction data could not be refinedusing either the orthorhombic or the monoclinic phase, asindicated in Fig. 8. However, a two-phase refinement with both the orthorhombic and monoclinic space groups is foundsuccessful, and all the observed diffraction peaks could beindexed (see Fig. 8). We find that orthorhombic and monoclinic phases coexist over a large pressure range from 9 GPa to the highest pressureof 40 GPa attained in our experiment. The percentage of themonoclinic phase continuously increases with pressure andreaches 98% at 40 GPa. The coexistence of both the phasesindicates that the structural phase transition from orthorhombicto monoclinic phase is of first order. The fit between theobserved and calculated profiles is quite satisfactory, and some5 1 01 52 02 5 11.0 11.540 K & 9.0 GPaIntensity (arb. units) 2θ (degree)40 K 2θ (degree)9.0 GPa40 K 22.3 GPa40 K 34.0 GPa(b)(a) SrFeAsF 40 K 39.8 GPa Cmma + P2/nTwo Phase Two Phase Two Phase Two PhaseP2/n CmmaTwo Phase FIG. 8. (Color online) (a) Observed (solid black circles), cal- culated (continuous red/dark gray line), and difference (bottom blue/medium gray line) profiles obtained after the Rietveld refinement of SrFeAsF at selected pressures and 40 K. The diffraction profilesat 9.0, 22.3, 34.0, and 39.8 GPa are refined using a combination of orthorhombic ( Cmma ) and monoclinic ( P2/n) phases. Upper and lower vertical tick marks above the difference profiles indicatepeak positions of orthorhombic ( Cmma ) and monoclinic ( P2/n) phases, respectively. (b) The refinement of the diffraction pattern at 9.0 GPa and 40 K with an orthorhombic phase (space group Cmma ), a monoclinic ( P2/n) phase, and a combination of orthorhombic (space group Cmma ) and monoclinic ( P2/n) phases. of them are shown in Fig. 8. The detailed structural parameters and goodness of fit for SrFeAsF at selected pressure and40 K, as obtained from powder-synchrotron diffraction data,are given in Table II. In our earlier measurements for BaFe 2As2we found that the coexisting region for orthorhombic and tetragonal phasewas very large. 19In the case of the fluorine-based 1111-type Ca/Sr compounds the transition pressures are found to bevery similar. These observations are in contrast to the case ofBaFe 2As2and CaFe 2As2, where the compound with smaller ionic radii (CaFe 2As2) show phase transition at much lower pressure (0.3 GPa) at 40 K as compared to the BaFe 2As2, where transition at 40 K occurs at 29 GPa.19 C. Pressure evolution of structural parameters at 40 K As mentioned previously, superconductivity in iron- arsenide materials is associated with lattice distortion andsuppression of magnetic ordering. 19,36,37The detailed inter- play between the crystal structure, magnetic ordering, andsuperconductivity is hardly understood up to now, which isto some extent due to the lack of precise structural data. 224513-6PRESSURE DEPENDENCE OF THE LOW-TEMPERATURE ... PHYSICAL REVIEW B 84, 224513 (2011) TABLE II. Results of Rietveld refinement of the crystal structure for SrFeAsF at selected pressure and 40 K. The measurements were carried out using a focused x-ray monochromatic beam of wavelength =0.3738 ˚A. The data collected up to 25◦have been used to determine the reported parameters. The atomic positions (Wycoff) for both the space groups are the same as given in Table I. At 5.8 GPa and 40 K 22.1 GPa & 40 K Orthorhombic phase space group CmmaTwo phase: orthorhombic +monoclinic space group ( Cmma + P12/n1) Atoms Positional coordinates Positional coordinates XY Z X Y Z Sr 0 1 /4 0.1570(1) 0 /0.75 0.25 /0.1455(3) 0.1707(2) /0.25 Fe 1 /4 0 0.5000 0.25 /0.75 0 /0.5131(1) 0.5 /0.75 As 0 1 /4 0.6602(6) 0 /0.75 0.25 /0.6703(2) 0.6606(5) /0.25 F1 /4 0 0.0000 0.25 /0.75 0 /0.001(1) 0 /0.75 Lattice parameters a(˚A) 5.5380(4) 5.3716(2) /3.7651(3) b(˚A) 5.5125(3) 5.34022(2)/8.1790(2) c(˚A) 8.6680(2) 8.2119(6) /3.7993(4) β=90.43 (5)◦phase fraction (%): 65(ortho.) /35 (mono.) Rp=13.3;Rwp=21.4; R exp=12.43; χ2=2.96 Rp=10.0;Rwp=20.5; R exp=12.01; χ2=2.91 At ambient conditions, MFeAsF (M =Ca/Sr) crystallizes in the ZrCuSiAs-type structure (space group P4/nmm ,Z =2). The crystal structures of all the iron-pnictides share a common 2D FeAs layer, where Fe atoms form a 2D-square sublattice with As atoms sitting at the center of thesesquares but off the Fe plane (above and below the planealternately). These are made of alternating Ca/SrF and FeAslayers. The Fe and F atoms lie in planes, while the As andCa/Sr atoms are distributed on each side of these planesfollowing a chessboard pattern. They undergo a tetragonalto orthorhombic phase transition at 134 and 180 K followedby magnetic transitions at 114 and 133 K for CaFeAsF andSrFeAsF, respectively. 22,24Applications of internal pressure (chemical pressure) suppress both orthorhombic structure andantiferromagnetic state, leading to emergence of supercon-ductivity. For example, the critical superconducting transitiontemperature T c∼4 K in the Co-substituted Fe site in SrFeAsF; Tc∼22 K in the Co-substituted Fe site in CaFeAsF; and Tc∼31 K, 52 K, and 56 K, respectively, in the La-, Nd-, and Sm-substituted Sr site in SrFeAsF was observed.23–28In addition to this, it was also found that systematically replacingR from La, Ce, Pr, Nd, and Sm in RFeAsO 1−δresulted in a gradual decreases in the a-axis lattice parameters and an increase in superconducting transition temperature.23–28,36,37 Superconducting transition temperature ( Tc) of different Fe- based superconductors is indeed correlated to their structuralproperties. 36–40A systematic trend between Tcand the Fe-As- Fe angles may be expected because the exchange couplings aredirectly related to the Fe-As-Fe bond angle and Fe-Fe/Fe-Asbond distances. It is found that the highest T cis obtained when the Fe-As-Fe reaches the ideal value of 109.47◦for the perfect FeAs tetrahedron with the least lattice distortion.36–40This suggests that the most effective way to increase Tcin Fe-based superconductors is to decrease the deviation of the Fe-As-Febond angle from the ideal FeAs tetrahedron, as the geometryof the FeAs tetrahedron might be correlated with the densityof states near the Fermi energy. In view of this, we have carried out detailed structural analysis as a function of pressure. Figures 9and10show thepressure dependence of the structural parameters of CaFeAsF and SrFeAsF at 40 K. For easy comparison in Figs. 9–11,w e have used converted lattice parameters for the orthorhombicphase ( a=a o/√2,b=bo/√2, and c=co) and for the monoclinic phase ( a=am,b=bm, andc=cm). It is clear from Figs. 9and 10that at 40 K on increasing pressure, the lattice parameters monotonically decrease in the entirerange of our measurements for CaFeAsF, while the a-lattice parameter of SrFeAsF exhibits anomalous increase beyond 0 1 02 03 04 03.63.73.83.9 0 1 02 03 04 05 07.27.68.08.4 0 1 02 03 04 095105115125 0 1 02 03 04 05 090.090.591.091.5ba Ortho.Ortho.CaFeAsFCaFeAsF CaFeAsFMono. Mono. Ortho. Mono.Lattice parameters a,b (Å)CaFeAsF Lattice parameter c (Å)Volume (Å3) Pressure (GPa) Monoclinic angle (deg) FIG. 9. (Color online) Pressure dependence of the structural parameters (lattice parameters, volume, and monoclinic angle) ofCaFeAsF at 40 K as obtained after Rietveld analyses of data with increasing pressure. For the sake of comparison, lattice parameters along [100], [010] and volume of the orthorhombic phase are dividedby√ 2 and 2, respectively, and lattice parameters of monoclinic phases are plotted in a standard (cab: P12/n1) setting. Errors are within symbol size. 224513-7S. K. MISHRA et al. PHYSICAL REVIEW B 84, 224513 (2011) 01 0 2 0 3 0 4 03.653.703.753.803.853.903.95 01 0 2 0 3 0 4 07.88.08.28.48.68.8 01 0 2 0 3 0 4 0105115125135 01 0 2 0 3 0 4 0020406080100 01 0 2 0 3 0 4 03.653.703.753.803.853.903.95 01 0 2 0 3 0 4 090.290.490.690.891.0boaoSrFeAsF ao boOrthorhombic phase cmc0 co cmLattice parameter c ( Å) VmV0 Vorth ./2 VmonoVolume ( Å3) Pressure (GPa) Ortho. phase fraction Pressure (GPa)bmam am bmMonoclinic phaseLattice parameters a,b ( Å) Monoclinic angle (deg) Pressure (GPa) FIG. 10. (Color online) Pressure dependence of the structural parameters (lattice parameters, volume, and monoclinic angle) and orthorhombic phase fraction in SrFeAsF at 40 K as obtained after Rietveld analyses of data with increasing pressure. For the sake of comparison, lattice parameters along [100], [010] and volume of the orthorhombic phase are divided by√ 2 and 2, respectively. Errors are within symbol size. 30 GPa. Figure 11depicts the pressure evolutions of the normalized lattice parameters, volume, and the ratio 2 c/(a+ b) of CaFeAsF and SrFeAsF compounds at 40 K. It is evident from this figure that the response of the lattice parametersto pressure is strongly anisotropic with the interlayer spacing(along /angbracketleft001/angbracketright) showing a significantly larger contraction thanthe basal plane dimensions (along /angbracketleft100/angbracketrightand/angbracketleft010/angbracketright). The structural phase transition around 13.7 GPa in the CaFeAsFcompound is accompanied by the discontinuity in the pressureevolution of the monoclinic angle (Fig. 9), which clearly suggests a first-order phase transition in CaFeAsF at 13.7 GPa(at 40 K). 0 1 02 03 04 05 00.850.900.951.00 01 0 2 0 3 0 4 0 5 00.850.900.951.00 0 1 02 03 04 05 00.850.900.951.00 0 1 02 03 04 05 00.750.800.850.900.951.00 01 0 2 0 3 0 4 0 5 01.952.002.052.102.152.202.25(a) a b cNormalized lattice parametersCaFeAsF (b) a b cSrFeAsF (ortho.) (c) SrFeAsF (mono) a b c Pressure (GPa) (d) Ca Sr (Ortho.) Sr (mono)Ca Sr (Ortho.) Sr (mono)Normalized volume Pressure (GPa)(e)2c/(a+b) Pressure (GPa) FIG. 11. (Color online) Pressure evolutions of the normalized lattice parameters of (a) CaFeAsF, (b) and (c) SrFeAsF, respectively, at 40 K. Pressure evolutions of the normalized volume and the 2 c/(a+b) ratio of CaFeAsF and SrFeAsF compounds are shown in (d) and (e). The normalization is done with respect to the ambient pressure values as obtained from a fit to the data of the orthorhombic phase. Errors are within symbol size. 224513-8PRESSURE DEPENDENCE OF THE LOW-TEMPERATURE ... PHYSICAL REVIEW B 84, 224513 (2011) 0 1 02 03 04 01517192123 01 0 2 0 3 0 4 02.552.602.652.702.75 0 1 02 03 04 06080100120 01 0 2 0 3 0 4 02.12.22.32.42.5 Mono.Mono.Mono. Ortho. Ortho.Ortho. Mono. Ortho.Polyhedral volume (Å3) CaFeAsF Fe-Fe (Å)Fe-As-Fe angles (deg) Pressure (GPa) Fe-As (Å) FIG. 12. (Color online) Pressure dependence of the polyhedral volume, Fe-As-Fe bond angle, and Fe-Fe/Fe-As bond length of CaFeAsF at 40 K as obtained after Rietveld analyses of data withincreasing pressure. Errors are within symbol size. The geometry of the FeAs 4tetrahedral units has been identified as a possible control parameter of Tcin the iron pnictides, and it has been argued that it sensitively controlsthe width of electric conduction band. 5,36,38–40The pressure variation of FeAs 4polyhedral volume, As-Fe-As bond angle, and Fe-As and Fe-Fe bond lengths obtained from the Rietveldanalysis of diffraction data are shown in Figures 12and13.I ti s clear from these figures that the polyhedral volume decreaseswith increasing pressure. In the case of the CaFeAsF, the Fe-Fe and Fe-As bond lengths reveal anomaly around 11.6 GPa.However, for SrFeAsF, Fe-Fe bond lengths show anomalousincrease beyond 30 GPa. A similar anomalous feature in thestructural parameters was also observed in BaFe 2As2,19which was associated with loss of magnetism. In the present studythe angle between As-Fe-As is quite different from the idealtetrahedral angle of 109.47 ◦and shows anomalous behavior with pressure. It is also evident from Figs. 9–13that the nature of phase transition seems to be different in CaFeAsF andSrFeAsF. As stated earlier, the transition is sharp in CaFeAsF,whereas SrFeAsF exhibits coexistence of the two phases overa large range of pressure. Temperature dependence of the electrical resistivity mea- surements for CaFeAsF at different pressures was carried outby Okada et al. 20,21using the piston-cylinder-type cell and the cubic anvil press. They found that the magnetic transition issuppressed by pressure above 5 GPa, the resistivity anomalydisappears, and superconductivity is observed. Further, theresistivity loss becomes larger, and the superconductingtransition shifts to lower temperature with increasing pressureup to 20 GPa. In our experiment, at a fixed temperatureof 40 K with increasing pressure, the Fe-As-Fe bond angledecreases over 5 to 10 GPa and then increases with furtherincreasing pressure. Thus, a clear correlation of the bondangle and superconductivity is difficult to establish in view ofthe different pressure-temperature paths followed in the twoexperiments. To the best of our knowledge, the high pressureresistivity measurements are still missing for SrFeAsF. In order to determine the bulk modulus Bat zero pressure and its pressure derivative B /prime, the pressure-volume data are fit- ted by a third-order Birch-Murnaghan equation, and the resultsare given in Table III. In this table we have also compared the values for several other compounds from literature. The bulk 0 1 02 03 04 01517192123 0 1 02 03 04 02.552.602.652.702.752.80 0 1 02 03 04 06080100120 0 1 02 03 04 02.12.22.32.42.5 0 1 02 03 04 02.552.602.652.702.752.80 0 1 02 03 04 06080100120Polyhedral Volume (Å3)SrFeAsFFe-Fe (Å) Fe-As-Fe angles (degree) Fe-As-Fe angles (degree)Fe-As (Å) Pressure (GPa) Fe-Fe (Å) Pressure (GPa) Pressure (GPa) FIG. 13. (Color online) Pressure dependence of the polyhedral volume, As-Fe-As bond angle, and Fe-Fe/Fe-As bond length of SrFeAsF at 40 K as obtained after Rietveld analyses of data with increasing pressure. Solid circle and triangle symbols correspond to the orthorhombic and monoclinic phases, respectively. Errors are within symbol size. 224513-9S. K. MISHRA et al. PHYSICAL REVIEW B 84, 224513 (2011) TABLE III. Comparison of the bulk modulus ( B) and its pressure derivative ( B/prime) of various Fe-based compounds. The standard errors of the fit are given in brackets. In case of the orthorhombic phase of CaFeAsF, the fit resulted in large error bar in pressure derivative ( B/prime), and so it has been fixed at 4.0. The bulk modulus of monoclinic phase of CaFeAsF has been fitted with a fixed value of B/primeat 4.0 and also with varying B/prime, with the χ2=0.6532 and 0.3569, respectively, which is significantly lower in the latter case. Temperature Bulk modulus Pressure derivative Pressure Compound (K) at P=0 (GPa) of bulk modulus range (GPa) Phase Reference CaFeAsF 40 89.4(1.1) 4.0(fixed) 5.8–11.7 Orthorhombic Present study 40 135.2(3.4) 3.2(0.2) 13.7–46.0 Monoclinic Present study 124.1(0.9) 4.0(fixed) 300 107.7(3.5) 2.5(0.3) 13–25 Tetragonal Present study SrFeAsF 40 111.1(5.3) 4.0(0.5) 5.8–31.2 Orthorhombic Present study 40 114.2(7.3) 4.0(0.6) 11.3-36.3 Monoclinic Present study SmFeAsO 0.81F0.19 300 88.9(0.8) 4.2(0.1) 0–20 Tetragonal 38 LaFeAsO 0.9F0.1 300 78.0(2.0) 7.4(0.2) 0–32.26 Tetragonal 39 Fe1.03Se0.57Te0.43 14 36.6(0.6) 7.8(0.3) 0–9.8 Monoclinic 35 FeSe 16 30.7(1.1) 6.7(0.6) 0–7.5 Orthorhombic 35 BaFe 2As2 300 65.7(0.8) 3.9(0.1) 0–20 Tetragonal 19 300 153.1(3.0) 1.8(0.1) 32–56 Collapsed tetragonal 19 40 82.9(1.4) 3.4(0.1) 1–35 Orthorhombic 19 CaFe 2As2 300 74.8(1.2) 4.8(0.1) 4.5–56 Collapsed tetragonal 19 40 80.2(3.4) 5.4(0.2) 4–34 Collapsed tetragonal 19 modulus in the high-pressure monoclinic phases of CaFeAsF appears larger than that in the orthorhombic phase. However,the values of the bulk moduli for the two phases of SrFeAsF arequite similar. The Bvalues of tetragonal SmFeAsO 0.81F0.1938 and LaFeAsO 0.9F0.139are somewhat lower than those of CaFeAsF and SrFeAsF. The bulk modulus values of othermaterials such as FeSe and Fe 1.03Se0.57Te0.43are much smaller than those of FeAs compounds. D. Phase transition of CaFeAsF on heating at high pressure Due to experimental limitations, we could not increase pressure on CaFeAsF beyond 46 GPa at 40 K. However, to seethe effect of temperature on the newly stabilized monoclinicphase in CaFeAsF (at 40 K and around 40 GPa), we carried outmeasurements at different conditions (temperature-pressure),a ss h o w ni nF i g . 1. Careful analysis of the diffraction data show that there is an abrupt change in the diffraction pattern at25 GPa and 200 K when the monoclinic splitting/broadeningin Bragg reflections around 11.5 ◦and 16.5◦disappears (insets of Fig. 14). Detailed Rietveld analyses of the diffraction data reveal that the sample undergoes a structural phase transitionfrom monoclinic to tetragonal phase around 25 GPa and 200 K.It remains in the tetragonal phase at 25 GPa and 300 K. We notethat while a phase transition is observed with pressure at 40 K,no transition is found at 300 K. In the case of SmFeAsO 0.81F0.19 and LaFeAsO 0.9F0.138,39no phase transition is also found at 300 K up to 20 and 32 GPa, respectively. The fit betweenthe observed and calculated profiles is shown in Fig. 14. For SrFeAsF we have not measured any data point whiledecreasing the pressure and increasing of temperature. Using athird-order Birch-Murnaghan equation, the fitted values of thebulk modulus Band its pressure derivative B /primefor CaFeAsF at room temperature are 107.7 ±3.5 GPa and B/prime=2.5±0.3, respectively, as given in Table III. We have summarized our observations of various phases in CaFeAsF and SrFeAsF with pressure and temperaturein Fig. 1. It is interesting to note that at 40 K, CaFeAsF shows orthorhombic to monoclinic phase transition at Pc =13.7 GPa, whereas at room temperature tetragonal phase is stable up to 25 GPa. The observation of dPc/dT >0 is similar to that observed in CaFe 2As2where collapsed 5 1 01 52 02 511 12 16.511 12 16.511 12 16.5 160 K 2θ (degree)Intensity (arb. units) 31.6 GPa25.0 GPa 200 K25.2 GPa300 K 2θ (degree)2θ (degree)2θ (degree) FIG. 14. (Color online) Observed (solid black circles), calculated (continuous red/dark gray line), and difference (bottom blue/mediumgray line) profiles obtained after the Rietveld refinement of CaFeAsF at selected pressures and temperature. The diffraction profiles at 31.6 GPa and 160 K are refined using the monoclinic structure (spacegroup P2/n); other refinements at 25.2 GPa and 200 K and 25.0 GPa and 300 K are used in tetragonal structure (space group P4/nmm ). Insets show accountability of certain Bragg’s reflections. 224513-10PRESSURE DEPENDENCE OF THE LOW-TEMPERATURE ... PHYSICAL REVIEW B 84, 224513 (2011) tetragonal phase transition occurs at a lower pressure (0.3 GPa) at a low temperature (50 K) in comparison to 1.7 GPaat 300 K. However, this behavior of dP c/dT may not hold for Ba/Sr compounds.19In addition to this, the transition to a lower symmetry phase (orthorhombic to monoclinic) in 1111(CaFeAsF/SrFeAsF) compound under pressure is in contrastwith the high symmetry phase (orthorhombic to tetragonal) in122 (BaFe 2As2/CaFe 2As2)-type compounds. IV . CONCLUSION We have investigated the effect of pressure on the crystal structure and structural phase transition behavior at 40 Kin CaFeAsF and SrFeAsF using powder-synchrotron x-raydiffraction and Rietveld analysis technique. We found thatboth the compounds undergo structural phase transition fromorthorhombic to monoclinic phase with increasing pressure.CaFeAsF undergoes a fairly sharp orthorhombic to monoclinicphase transition at 13.7 GPa with increasing pressure. On theother hand, SrFeAsF exhibits coexistence of orthorhombicand monoclinic phases over a large pressure range from 9to 39 GPa. The coexistence of the two phases indicates that the transition is of first order. On heating from 40 K at highpressure, CaFeAsF undergoes monoclinic to tetragonal phasetransition around 25 GPa and 200 K. Further, it does not showany post-tetragonal phase transition up to 25 GPa at 300 K.We note that the 1111-compound (CaFeAsF) undergoes phasetransition to a lower symmetry phase (i.e., orthorhombic tomonoclinic) under pressure, in contrast with the transition toa higher symmetry phase (i.e., orthorhombic to tetragonal)observed in 122-type compounds (BaFe 2As2and CaFe 2As2). We have also determined the bulk modulii in these compoundsthat confirm their soft nature analogous to other compounds inthe FeAs family. ACKNOWLEDGMENTS R. Mittal and S. K. Mishra thank the Department of Science and Technology (DST), India for providing financial supportto carry out synchrotron x-ray diffraction at the EuropeanSynchrotron Radiation Facility, Grenoble, France. *rmittal@barc.gov.in 1Y . Izyumov and E. Kurmaev, High-Tc Superconductors Based on FeAs Compounds, (Springer, Verlag, Hamburg, 2010); J. S. Schilling, in Handbook of High Tempera- ture Superconductivity: Theory and Experiments ,e d i t e db y J. R. Schrieffer and J. S. Brooks (Spinger Verlag, Hamburg,2007); H. Takashi and N. Mori, in Studies of High Temperature Superconductors , edited by A. Narlikar (Nova Science, New York, 1996). 2J. Paglion and R. L. Greene, Nat. Phys. 6, 645 (2010). 3I. I. Mazin and M. D. Johannes, Nat. 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PhysRevB.99.014506.pdf

PHYSICAL REVIEW B 99, 014506 (2019) Signatures of surface Majorana modes in the magnetic response of topological superconductors Luca Chirolli1and Francisco Guinea1,2 1IMDEA-Nanoscience, Calle de Faraday 9, E-28049 Madrid, Spain 2School of Physics and Astronomy, University of Manchester, Manchester M13 9PY, United Kingdom (Received 18 September 2018; published 8 January 2019) We study the magnetic susceptibility of a two-dimensional cone of Majorana modes localized at the surface of a three-dimensional time-reversal invariant topological superconductor belonging to class DIII. A field parallelto the surface tilts the surface Majorana cone along the supercurrent direction. For fields larger than a criticalthreshold field H ∗,H>H∗, a transition from a type I to a type II Dirac cone occurs and a finite current carried by the Majorana modes starts to flow, leading to an additional diamagnetic contribution to the surface magnetization.On a curved surface, interband transitions are promoted by the Majorana spin connection that couples to theexternal field, giving rise to a finite-frequency magnetic susceptibility. DOI: 10.1103/PhysRevB.99.014506 I. INTRODUCTION Topological superconductors (TSCs) are a very promising state of matter in which a topological superconducting gapopens in the bulk of a system and leads to the confinementat the surface of unconventional Andreev states named Ma-jorana states [ 1–3]. Majorana modes constitute a class of topologically protected surface excitation appearing at theboundary of topological states of matter [ 4–11], and they represent one of the basic resources in topological quantumcomputation [ 12–14]. Recently, doped Bi 2Se3topological insulators (TIs) [ 15–18] have been suggested as candidates that may realize odd-parity, time-reversal invariant topolog-ical superconductivity [ 19–26] belonging to class DIII. In three dimensions these systems are expected to host Majoranamodes forming a Dirac cone in the basis of Majorana Kramerspartners [ 27–31]. The localized and charge neutral charac- ter of Majorana modes has induced most of the theoreticaldetection proposals and the experimental efforts to focus onlocal spectroscopy, the Josephson effect, and interferometryto find a proof of their existence [ 32–34]. As far as class DIII topological superconductors are concerned, the presence ofsurface Majorana modes is expected to produce a stronglyanisotropic spin susceptibility [ 35]. Recently, the authors have shown that for class DIII topo- logical superconductors a coupling to a vector potential canarise at finite momentum and finite energy [ 36]. In a planar geometry a magnetic field laying on the surface of the systemproduces a supercurrent that Doppler shifts the Majoranamodes and results in a tilting of the cone. The tilting directionin momentum space is orthogonal to the applied field andparallel to the supercurrent. When Majorana modes are con-fined on a curved surface an additional coupling of geometricorigin arises that involves the Majorana spin connection [ 36]. The latter generates finite matrix elements between empty andoccupied states of the surface Majorana cone and a response isexpected at finite frequency. Majorana modes can be detectedthrough the application of time-dependent orbital magneticfields.In this work we study the magnetic susceptibility and the frequency-dependent response of a Majorana cone localizedat the surface of a class DIII three-dimensional topologicalsuperconductor. The Majorana velocity determines a thresh-old field, H ∗, for which a structural change of the Majorana cone takes place, which is characterized by a transition from a type I to a type II cone. For H>H∗, in the overtilted regime, a finite Andreev current flows carried by Majorana modes, asschematically depicted in Fig. 1. The Andreev current adds to the supercurrent and participates in Meissner screening of theexternal field by generating an additional surface diamagneticmagnetization. For weakly doped Dirac insulators with a small band gap characterized by odd-parity superconductivity, such as the A 1uphase predicted in Bi 2Se3[19], the Majorana velocity affecting H∗can be tuned by changing the chemical potential, in a way that the value of H∗falls into the Meissner phase for strong doping. Signatures of Majorana modes arethen expected to appear in the magnetic susceptibility, witha signal amplitude scaling as the ratio λ/L zbetween the penetration depth λand the sample thickness Lz. In type II superconductors characterized by an appreciable ratio λ/Lz the additional signal becomes detectable. Additionally, in systems characterized by a finite surface curvature, we findthat the emergence of the Majorana spin connection in the tilt-ing term, together with a curvature-induced nonzero Zeemanterm, gives rise to a finite-frequency magnetic susceptibility in response to a time-dependent magnetic field. Our findings acquire a universal character in finite geometries and openthe way to detection of Majorana states via thermodynamicbulk measurements, in contrast with the widely used localspectroscopy probes. The work is structured as follows. In Sec. IIwe introduce the system under study, in Sec. IIIwe describe the coupling to an external magnetic field, in Sec. IVwe calculate the Andreev current for a planar geometry, and in Sec. Vwe cal- culate the associated additional orbital magnetic susceptibilitydue to the surface Majorana modes. In Sec. VIwe consider a spherical system and give the surface Hamiltonian, and in 2469-9950/2019/99(1)/014506(7) 014506-1 ©2019 American Physical SocietyLUCA CHIROLLI AND FRANCISCO GUINEA PHYSICAL REVIEW B 99, 014506 (2019) Sec. VIIwe study the finite-frequency response. In Sec. VIII we conclude the work with final remarks. II. THE SYSTEM We start our analysis by considering a specific example of an odd-parity superconductor realized in doped TIs, suchas the one proposed for doped Bi 2Se3[19]. The mean-field Hamiltonian in the Nambu basis ψk=(ck,isyc† −k)T, where (ck)s,iis a fermionic state of momentum k,s p i n s, and apz-like orbital i=T,B for the top and bottom layers, respectively, in the k·papproximation, reads (¯ h=1) H=τz(mσx+vσz(kxsy−kysx)+vzkzσy−μ)+/Delta1τxσysz, (1) where σiare Pauli matrices spanning the twofold orbital space, mis the insulating band gap, vis the Dirac velocity, and/Delta1>0 is the mean-field value of the superconducting order parameter. The Hamiltonian Eq. ( 1) is time-reversal invariant, with T=isyˆK(ˆKbeing complex conjugation), and centrosymmetric, with P=σxbeing the parity operator, and it has a full bulk gap of size /Delta1on the Fermi surface. The system realizes a TSC belonging to class DIII and hostsa surface Majorana cone localized at the boundary of thesystem. We consider a semi-infinite system of doped Bi 2Se3oc- cupying the z> 0 region. The realistic boundary condition compatible with the quintuple-layer structure of the crystal isσ zψ(z=0)=−ψ(z=0) (for the z< 0 region the realistic boundary condition is σzψ(z=0)=ψ(z=0) [37]). The system realizes a TI if the condition sign( mvz)<0 is satis- fied. Additionally, for finite /Delta1>0 a zero-energy Majorana Kramers pair is found in the region z> 0a tkx=ky=0 with the wave function ψα(z)=|α/angbracketrightφ(z)[37], with φ(z)=e−z/ξz/bracketleftbigg sin(kFz) sign(m)s i n (kFz−θ)/bracketrightbigg σ, (2) where kF=/radicalbig μ2−m2/|vz|,eiθ=(|m|+i|vz|kF)/μ, and ξz=|vz|//Delta1is the superconducting (SC) coherence length along the zdirection. The states |α/angbracketright=[(1,−α)s, −isign(vz)(1,α)s]τ, with α=± 1, are simultaneous eigenstates of the mirror helicity ˜M=−isxτz[37], with eigenvalues iα, and of the operator τysz,τysz|α/angbracketright= −sign(vz)|α/angbracketright. For definiteness we choose m< 0 and vz>0. Projecting the Hamiltonian Eq. ( 1) onto the subspace spanned by the states |ψα/angbracketrightwe find the surface Hamiltonian describing a Majorana cone: Hk=v/Delta1(kxαy+kyαz), (3) where αiare Pauli matrices in the basis |ψα/angbracketrightand the velocity of the Majorana modes is v/Delta1/similarequalv|m|/Delta1/μ2(for a TI the sign of v/Delta1is opposite to the sign of the TI surface states’ velocity, which for m< 0 is negative, −v)[37]. It follows that the strength and the sign of the velocity v/Delta1depend on the topological character of the doped insulator. III. COUPLING TO THE MAGNETIC FIELD The presence of an external magnetic field is accounted for by a minimal coupling substitution in the Nambu basis FIG. 1. Schematics: An external magnetic field His forced to lie on the surface of the system by Meissner screening. The associated supercurrent jλDoppler shifts the Majorana cone, resulting in a cone tilting along the jλdirection. For H>H∗a transition to a type II overtilted cone occurs and the supercurrent acquires the component jAcarried by the Majorana modes. k→k+eA(r)τz. For a weak field we neglect the spatial dependence of the gap and the resulting coupling Hamiltonianreads H B=HA+1 2gμBs·B, (4) HA=evσz(Axsy−Aysx)+evzσyAz, (5) with B=∇×Abeing the full induction field and gthe material gfactor. If we take the matrix elements of the orbital coupling on the states in Eq. ( 2) we find /angbracketleftψα|HA|ψα/prime/angbracketright=0. A nonzero coupling occurs at finite momentum. We then splitthe Hamiltonian Eq. ( 1) into three terms, H=H 0+Hk+ H/Delta1, with H0=τz(mσx−μ) and H/Delta1=/Delta1szσyτx. Defin- ing the in-plane effective mass ˆM−1=2v2(μ−mσx)/(μ2− m2), the effective orbital coupling at small momentum can be written as Hk·A=−HAH−1 0Hk+H.c., which amounts to Hp·A=e 2ˆM−1v2 i v2{Ai,ki}+e 2ˆM−1˜giSiBi, (6) where {Ai,pi}=Aiki+kiAi, the anisotropic gfactor is ˜gx=˜gy=vz/vand ˜gz=1, and we used a gauge for which ∇·A=0. The spin operator Sappearing in Eq. ( 6)i sS= (σxsx,σxsy,sz) and corresponds to the correct spin operator that behaves as a pseudovector under the point group D3d operations that characterize the Hamiltonian Eq. ( 1)[38]. When an external field is applied to the system, Meissner screening forces the external field to lay in the plane definedby the surface of the system. We assume the external fieldto be applied along the xdirection, H=He x, as schemat- ically depicted in Fig. 1. In the Landau gauge the vector potential corresponding to the bulk screened field reads A= (0,Hλe−z/λ,0). For an in-plane field the matrix elements of the Zeeman coupling are 0 on the Majorana Kramers pair inEq. ( 2). We are then only left with the orbital term, which has no structure in spin nor in Nambu space and can onlyamount to a diagonal term in the Majorana subspace. The fullHamiltonian projected on the Majorana cone reads H k=v/Delta1(kxαy+kyαz)+vHky, (7) where the magnetic-field-dependent velocity vH, in the limit λ/greatermuchξz, is given by vH≡/angbracketleftψα|eˆM−1Ay(z)|ψα/angbracketright=λeH 2v2/μ. (8) 014506-2SIGNATURES OF SURFACE MAJORANA MODES IN THE … PHYSICAL REVIEW B 99, 014506 (2019) Defining m∗=μ/2v2, we can write the field-dependent ve- locity as vH=eHλ/m∗, in analogy with the results of Ref. [ 36]. Introducing the coherence length ξ=v//Delta1and equating the two velocities vH=v/Delta1, we find the threshold tilting field (restoring ¯ h) H∗=/Phi10 2πξλ|m| μ=ηHc, (9) withη=|m|/μand/Phi10=h/2e. In the second equality we ex- press the threshold field in terms of the thermodynamic criticalfieldH c=/Phi10/(2πξλ )[39]. If we tune η, the threshold field can be shifted in the Meissner phase. IV . ANDREEV CURRENT The spectrum of the surface Majorana Hamiltonian has two branches, /epsilon1±,k=±v/Delta1|k|+vHky.A tH=H∗one of the two branches, /epsilon1k,−, becomes flat along the line kx=0 and a structural change of the dispersion takes place that ischaracterized by a formally diverging number of states at zeroenergy [ 36]. The current operator is obtained by the usual relation ˆj i=∂HBdG/∂Ai. At zero temperature the Andreev current density carried by the Majorana modes is given by jA y(z)=e m∗|φ(z)|2/summationdisplay α/integraldisplayd2k (2π)2ky/Theta1(−/epsilon1k,α), (10) where the /Theta1function restricts the integral to the occupied states. For H<H∗there are as many occupied states at positive kyas at negative ky, so that the current is 0. On the other hand, an Andreev current will flow for H>H∗. The tilting takes below the Fermi level previously occupiedstates which lie within an angular sector −φ ∗/lessorequalslantφ/lessorequalslantφ∗, withφ∗=cos−1(H∗/H). Defining the surface quasiparticle density ρ2D=k2 F/4π, we find jA y(z)=−evF2ρ2D ξe−2z/ξ/radicalbig 1−(H∗/H)2, (11) which predicts an abrupt diamagnetic signal as H∗is ap- proached. Formally, the derivative with respect to HofjA y(z) diverges at H=H∗, as a result of the structural change of the band structure, in which the zero-energy level becomesmacroscopically occupied. V . ORBITAL MAGNETIC SUSCEPTIBILITY In order to understand what is the effect of the Andreev current carried by the Majorana modes for H>H∗,w e have to self-consistently solve for the vector potential. Weassume Meissner screening acting locally in the bulk fol-lowing the dependence j λ y(z)=−Ay(z)/(4πλ2), where λ=/radicalbig m∗/(4πe2Ns) is the penetration depth, with Nsbeing the number of bulk superconducting electrons. Majorana modesproduce an additional contribution j A y(z)≡jA ye−2z/ξ, which is restricted to the surface on a scale of ξ/2 and adds to the bulk screening current jλ y(z), so that the total current is jy(z)=jλ y(z)+jA y(z), as schematically depicted in Fig. 1. The total current is related to the vector potential by theMaxwell equation 4 πj y(z)=−∂2 zAy(z). It follows that the self-consistent vector potential Ay(z) necessarily acquiresan additional component, Ay(z)=A0e−z/λ+A1e−2z/ξ.T h e coefficients A0andA1are obtained by imposing hx(z= 0)=H, with the induction field hx(z)=−∂zAy(z), which produces the constraint A0=λH−2λA 1/ξ. (12) The amplitude of jA yin Eq. ( 11) depends nonlinearly on the vector potential at the surface Ayvia the velocity vH= e/angbracketleftAy(z)/angbracketright/m∗=e(A0+A1/2)/m∗, and the problem amounts to solving the following nonlinear equation: A1=4π×1 2evFρ2Dξ/radicalBigg 1−/parenleftbiggH∗λ A0+A1/2/parenrightbigg2 . (13) The magnetization of the system is calculated through the Gibbs free energy of the system [ 39],G=/integraltext dr(h2−4πj· A−2H·h)/(8πSL z)=−HAy(0)/(8πLz), where Sis the surface area and Lzthe thickness of the system. The magne- tization is found as M=(B−H)/4π, where the induction field is B=− 4π∂G/∂H .F o rH<H∗we find M=M0≡ −H(1−λ/Lz)/4π, which agrees with the thermodynamic limitM=−H/4πforLz→∞ .F o r H>H∗the mag- netization acquires an additional contribution, M=M0+ δMθ (H−H∗), with δM=−λ 4πξL z/parenleftbigg A1+H∂A 1 ∂H/parenrightbigg . (14) From Eq. ( 12) we have that the dependence on A0drops from Eq. ( 13). Defining a1=A1/(ξH∗) andh=H/H∗,i n the limit λ/greatermuchξwe can cast the equation for A1in the form 2a1=/epsilon1/radicalbig 1−1/(h−2a1)2, with /epsilon1=4πevFρ2D H∗=4πevFρ2D ηH c, (15) ForH>H∗we choose the solution of A1that goes to 0 at H=H∗. In a perturbative scenario, /epsilon1/lessmuch1,A1is positive and its derivative is finite at H∗and decreases with increasing H. It follows that the system develops an additional diamagneticsignal in the form of a negative jump at H=H ∗dominated byH∂A 1/∂H . More information can be obtained by studying the susceptibility χ=∂M/∂H that, to lowest order in ξ/λ, reads χ=χ0+δM(H∗)δ(H−H∗)+δχ, (16) δχ=−λ 4πLzθ(h−1)∂h(a1+h∂ha1), (17) withχ0=∂HM0being the bare susceptibility. Formally, the susceptibility negatively diverges at H=H∗as a result of the diamagnetic jump in δM.I nF i g . 2(a) we plot δχfor different values of η. In the perturbative case /epsilon1=0.4(η=0.5) we see thatδχshows a positive peak at H∗, which is due to the strong increase of the current for H>H∗. Upon increasing /epsilon1,t h e nonlinear character of δMfully manifests and δχturns into a negative broad dip at H=H∗. We then conclude that in the overtilted regime H>H∗ the Majorana modes carry an Andreev current that adds to the bulk London current and participates in the Meissnerscreening. The resulting additional signal in the susceptibilityshows a nonlinear behavior as a function of the applied field 014506-3LUCA CHIROLLI AND FRANCISCO GUINEA PHYSICAL REVIEW B 99, 014506 (2019) (a) (b) FIG. 2. (a) Orbital susceptibility δχa sg i v e nb yE q .( 17)f o r different values of η. In the plot we chose /epsilon1=/epsilon10/η, with /epsilon10=0.2, λ/ξ=10, and λ/L z=0.02, and we have expressed the external field in units of Hc. (b) Real δχ/primepart and imaginary δχ/prime/primepart of the finite-frequency orbital surface susceptibility δχ(ω) in a spherical geometry versus the applied field for kFR=100. Fast oscillations reflect the discrete nature of the spectrum with level spacing /Delta1/kFR. White lines show the kFR→∞ asympthotics as given by Eq. ( 27). and an amplitude that scales as λ/Lz. In a system of linear size on the order of micrometers a type II superconductor mayshow a penetration depth on the order of tens of nanometers,thus making the amplitude of the additional signal accessible.For thin slabs with L z/similarequalλthe field penetrates the entire system and the critical threshold field shifts to ηHc2, which may well fall in the vortex phase, and the entire picture ceasesto be valid. VI. CURVATURE EFFECTS In a previous work [ 36] we showed that the effective Hamiltonian Eq. ( 7) acquires an additional term on the surface of a sphere, which involves the spin connection of the Ma-jorana modes. In spherical geometry it is possible to choosethe self-consistent vector potential to have only the nonzeroazimuthal component A=ˆφA φ(r), with ∇·A=0. Intro- ducing Dirac matrices γi≡(γ0,γ1,γ2)=(iαx,αz,−αy), the Hamiltonian Eq. ( 7) gives rise to a Dirac equation on a flat Minkowski space, ( γi+γ0ai)∂iψ=0, for the two- component Majorana spinor ψ, with ai=(e/m∗v/Delta1)/angbracketleftAy/angbracketrightδi y, and the only nonzero component of Aiis fixed to lie in the plane, orthogonal to the magnetic field. The Majoranaequation on the surface of the sphere [ 40,41] is then obtained by introducing the covariant derivative D μ=∂μ+/Gamma1μ, with /Gamma1μbeing the Majorana spin connection: (γμ+γ0aμ)(∂μ+/Gamma1μ)ψ=0, (18) where γμ=γieμ i,aμ=aieμ i, andei μ≡∂xi/∂xμ. In addi- tion, the Zeeman term acquires a finite component orthogonalto the surface in proximity of the poles of the sphere with thesame matrix elements of the spin connection and it can beabsorbed in the coupling a μ(we refer to Ref. [ 36] for details of the surface Majorana equation in the presence of a magneticfield and curvature).Surface magnetic susceptibility We now calculate the surface magnetic susceptibility asso- ciated with the onset of a curvature-induced coupling betweenthe external field and the Majorana cone via the Majorana spinconnection. The surface Hamiltonian is H=H 0+H1≡/Delta1 kFR(ˆH0+hˆH1), (19) withh=H/H∗andH0describing the unperturbed Majorana cone with eigenvalues /epsilon1α l=α(l+1/2)/Delta1/(kFR) associated with the eigenstates |ϒα lm/angbracketright, with α=± 1,l=1/2,3/2,..., and|m|/lessorequalslantl[36]. The perturbation ˆH1is specified by the only nonzero matrix elements: /angbracketleftϒ+ lm|ˆH1|ϒ− l±1,m/angbracketright=±i/radicalbig (l+1/2±1/2)2−m2 2(l+1/2±1/2). (20) For a weak field we can calculate the correction to the total energy at second order in perturbation theory: δE=/summationdisplay l,l/prime,m,α,βf/parenleftbig /epsilon1α l/parenrightbig −f/parenleftbig /epsilon1β l/prime/parenrightbig /epsilon1α l−/epsilon1β l/prime/vextendsingle/vextendsingle/angbracketleftbig ϒα lm/vextendsingle/vextendsingleH1/vextendsingle/vextendsingleϒβ l/primem/angbracketrightbig/vextendsingle/vextendsingle2, (21) withf(/epsilon1)=[1+exp(/epsilon1/T )]−1being the Fermi-Dirac distri- bution function. At zero temperature, summing over occu-pied states with |m|/lessorsimilarlandl/lessorsimilarl max/similarequalkFR, we find δE= −1 3/Delta1H2/(H∗)2. Thus, the bare surface response, as given by δχ0=−∂2δE/∂H2, is paramagnetic and amounts to δχ0=2 3/Delta1 (H∗)2. (22) The susceptibility beyond perturbation theory can be cal- culated by diagonalization of the full-surface Hamiltonian.Introducing the eigenvalues /epsilon1 n,mofˆH0+hˆH1, we define the dimensionless susceptibility as δ¯χ=−1 kFR∂2 ∂h2/summationdisplay n,mθ(−/epsilon1n,m)/epsilon1n,m, (23) from which it follows the zero-field susceptibility δχ0= δ¯χ(h=0)/Delta1/(H∗)2,E q .( 22). Once again, the total current is the sum of the bulk contribution, which depends on the fullself-consistent vector potential, and the Andreev contribution.The latter is given by the expectation value on the occupiedstates of ˆH 1and, through the Hellmann-Feynman theorem, it can be written as jA(r)=−evFρ2D|φ(r)|2R2/integraldisplayh 0dh/primeδ¯χ(h/prime), (24) where the Andreev bound-state wave function can be approximated as φ(r)=exp[(r−R)/ξ]//radicalbig ξR2 forξ/R/lessmuch1. Maxwell equations allow us to set A1=/epsilon1H∗ξ/integraltexth 0dh/primeδ¯χ(h/prime)/4, with /epsilon1given in Eq. ( 15). Maxwell equations also fix a constraint between the externalfieldHand the vector potential on the surface. For simplicity we can choose the same constraint equation, Eq. ( 12), that applies to the planar boundary (see the Appendix for adiscussion). This way, Eq. ( 14) applies and it predicts an additional diamagnetic contribution δMto the magnetization that arises due to the Andreev current. 014506-4SIGNATURES OF SURFACE MAJORANA MODES IN THE … PHYSICAL REVIEW B 99, 014506 (2019) The diamagnetic behavior is understood by the fact that the orbital current generated in response to a field tends toscreen the applied field, thus producing a diamagnetic signal.In particular the susceptibility for ξ/lessmuchλreads 4πχ=− 1+λ R−/epsilon1λ 2Rδ¯χ(H/H∗), (25) up to a correction on the order of hδ¯χ/primethat is negligible at small field. We find that δ¯χhas a very weak dependence on h, so that susceptibility does not depend on the field and can be approximated with its zero-field value δ¯χ(0)=2/3. We clearly see that Meissner screening results in an additionaldiamagnetic contribution that is due to the surface. The am-plitude of the additional signal is controlled by /epsilon1, which in turn is inversely proportional to η=|m|/μ, and can be tuned through doping. VII. FINITE-FREQUENCY RESPONSE Finally, we address the response of a time-dependent curvature-induced surface coupling in the spherical geometry.We assume that a small external field probes the system at fre-quency ω,δH(t)=δHcos(ωt), and couples to the Majorana cone via the term H 1(t)=δH(t)/Xi11, with/Xi11≡∂H1/∂H= ˆH1/Delta1/(H∗kFR). The finite-frequency susceptibility in linear response theory is given by δχ(ω)=−/summationdisplay l,l/prime,m,α,βf/parenleftbig /epsilon1α l/parenrightbig −f/parenleftbig /epsilon1β l/prime/parenrightbig ω+/epsilon1α l−/epsilon1β l/prime+i0+/vextendsingle/vextendsingle/angbracketleftbig ϒα lm/vextendsingle/vextendsingle/Xi11/vextendsingle/vextendsingleϒβ l/primem/angbracketrightbig/vextendsingle/vextendsingle2. (26) The relation between Eq. ( 26) and Eqs. ( 21) and ( 22) is man- ifest, in that δχ0≡δχ(ω=0)=−∂2δE/∂H2. Summing over states with |m|/lessorequalslantlandl<l max/similarequalkFR,f o rl a r g e kFR we find that the suscpetibility is approximated by δχ(ω)=δχ0/parenleftbigg 1−ω 2/Delta1ln/vextendsingle/vextendsingle/vextendsingle/vextendsingleω+2/Delta1 ω−2/Delta1/vextendsingle/vextendsingle/vextendsingle/vextendsingle+iπω 4/Delta1/parenrightbigg . (27) In Fig. 2(b) we show the real and imaginary parts of the susceptibility as a function of the frequency, calculated bynumerically evaluating Eq. ( 26), together with the envelopes provided by Eq. ( 27). We see that the susceptibility follows the behavior predicted by Eq. ( 27). In addition, the susceptibility shows fast oscillations that reflect the discrete nature of thespectrum, with level spacing /Delta1/k FR. We then conclude that by irradiation with a time-dependent field the Majorana coneresponds with a finite signal and that a detection of Majoranamodes becomes possible on a curved geometry. VIII. CONCLUSIONS In this work we presented an analysis of the magnetic response of a two-dimensional cone of Majorana modes inclass DIII topological superconductors. An in-plane appliedfield gives rise to a tilting of the Majorana cone along thedirection of the supercurrent, which results in an excess An-dreev current beyond the threshold field H ∗and an associated additional diamagnetic magnetization. The extra signal hasa nonlinear dependence on the applied field and scales asthe ratio between the penetration depth λand the samplethickness L z,λ/Lz. For type II superconductors characterized by the appreciable ratio λ/Lz, the signal becomes detectable in magnetic susceptibility measurements. On a curved surfacethe tilting term acquires a geometric contribution involvingthe Majorana spin connection, which couples positive- andnegative-energy states and adds to a curvature-induced finiteZeeman term, allowing interband transitions. A finite sus-ceptibility arises at finite frequency in response to a time-dependent magnetic field. Our findings open the way to de-tection of Majorana modes by magnetization measurementsand via the application of a time-dependent magnetic field. ACKNOWLEDGMENTS The authors acknowledge very useful discussions with F. de Juan, I. Grigorieva, and A. K. Geim. L.C. and F.G.acknowledge funding from the European Union’s SeventhFramework Program (Grant No. FP7/2007-2013) through theERC Advanced Grant NOVGRAPHENE (Grant No. 290846).L.C. acknowledges funding from the Comunidad de Madridthrough Grants No. MAD2D-CM and No. S2013/MIT-3007.F.G. acknowledges funding from the European Commissionunder the Graphene Falgship (Contract No. CNECTICT-604391). APPENDIX: SCREENED SUSCEPTIBILITY Analogously to what we have done for the planar geometry we now estimate the screened susceptibility, taking into ac-count the Meissner screening. First of all, we need to calculatethe current. This is done by taking the matrix element ofthe current operator between the occupied eigenstates of thesystem. Neglecting the tilting of the bands, we write theHamiltonian as ˆH(y)=/Delta1 kFR(ˆH0+yˆH1), (A1) where /angbracketleft/Psi1α l,m|ˆH0|/Psi1β l/prime,m/prime/angbracketright=α(l+1/2)δl,l/primeδm,m/primeδα,β(for posi- tive half-integer l=1/2,3/2,...,|m|/lessorequalslantl, andα=± 1), /angbracketleft/Psi1+ l,m|ˆH1|/Psi1− l/prime,m/prime/angbracketright=δm,m/prime/bracketleftBigg i/radicalbig (l+1)2−m2 2(l+1)δl/prime,l+1 −i√ l2−m2 2lδl/prime,l−1/bracketrightBigg , (A2) andy=A(R)/(λH∗), with A(R) being the value of the azimuthal fully self-consistent vector potential of the spheresurface. The Andreev current is obtained by expressing the action of the current operator on the surface-state wave function. Itsgeneric expression, j A(r)=ˆφe m∗/summationdisplay m,nψ† m,n(r)ˆpφψm,n(r)θ(−/epsilon1m,n), (A3) can be assumed to acquire the form jA(r)=jA(r)s i nθˆφand jA(r) can be expressed through the matrix ˆH1as jA(r)=|φ(r)|2¯h2 2m∗2π /Phi101 R/summationdisplay n,m/angbracketleft/Theta1m,n|ˆH1|/Theta1m,n/angbracketrightθ(−/epsilon1m,n), (A4) 014506-5LUCA CHIROLLI AND FRANCISCO GUINEA PHYSICAL REVIEW B 99, 014506 (2019) where |/Theta1m,n/angbracketrightare the full eigenstates of ˆH(y) with eigenvalues /epsilon1m,n,θ(−/epsilon1) is the Fermi function at zero energy, and φ(r) is the Andreev-state wave function. We then introduce adimensionless susceptibility: δ¯χ(y)≡−1 kFR∂2 ∂y2/summationdisplay n,m/epsilon1m,nθ(−/epsilon1m,n). (A5) Through application of the Hellmann-Feynman theorem we have that the Andreev current can be written as jA(r)=−evF|φ(r)|2(kFR)2/integraldisplayy 0dy/primeδ¯χ(y/prime). (A6) The total current then reads jtot(r)=−A(r) 4πλ2+jA(r)( A 7 ) and has to satisfy the Maxwell equation ∂2A ∂r2+2 r∂A ∂r−2A(r) r2=A(r) λ2−4πjA(r). (A8) Clearly, the self-consistent vector potential assumes the form A(r)=A0(r)+A1(r), with A0(r)=A0f(r), where f(r)= −Im[y−2(ir/λ )]e−R/λ>0 and yν(z) is a spherical Bessel function of the second kind. To simplify the problem we assume the Andreev wave function to have the form |φ(r)|2=2u(r)/(ξ2R), (A9) withu(r)=− Im[y−2(2ir/ξ )]e−2R/ξ. This way, A1(r)= A1u(r) and we find A1=4πevFk2 FR 2[1−ξ2/(4λ2)]/integraldisplayy 0dy/primeδ¯χ(y/prime). (A10) By matching the solution of the vector potential and its derivative for r<R with the solution for r>R at the sphere boundary r=R, we find A0=3HR−2A1[2u(R)+Ru/prime(R)] 2[2f(R)+Rf/prime(R)]. (A11) ForR/greatermuchλ,ξ we can approximate −Im[y−2(z)]∼ez/(2z), which yields the relation A0=3HR−A1. (A12)This expression provides a constraint that binds A0andA1on the surface of the sphere. As for the case of the planar geom-etry, we find that the Andreev current reduces the amplitudeof the screening term A 0. Finally we need to calculate the Gibbs free energy. By integration by parts it can be written as G=/integraldisplay surfacedS 8πV·/bracketleftbigg/parenleftbiggh(r) 2−H/parenrightbigg ×A(r)/bracketrightbigg , (A13) and we can approximate it as G=−HA(R)/(8πR), with the vector potential at the boundary given by A(R)=(A0λ+ 2A1/ξ)/(2R). It then follows that the magnetization is 4πM=−H+λ 4R2∂ ∂H[H(3HR−A1+2A1ξ/λ)], (A14) which for ξ/λ/lessmuch1 can be simplified to 4πM=−H+3λ 2RH−λ 4R2∂ ∂H(HA 1) (A15) =−H/parenleftbigg 1−3λ 2R/parenrightbigg +4πδM. (A16) Writing A1=RH∗a1, with a1=/epsilon1 2/integraltexty 0dy/primeδ¯χ(y/prime), we obtain the result 4πδM=−λH∗ 4R∂ ∂h(ha1). (A17) At this point we define y=3h/2−a1/2 and obtain the im- plicit equation y=3h/2−(/epsilon1/4)/integraltexty 0dy/primeδ¯χ(y/prime), which can be approximately solved to give a1=(/epsilon1/2)/integraltexth 0dh/primeδ¯χ(h/prime). The susceptibility is then given by 4πχ=− 1+3λ 2R−/epsilon1λ 4R[δ¯χ(h)+hδ¯χ/prime/2], (A18) which shows a diamagnetic correction to the susceptibility and agrees well with the result provided by Eq. ( 20), obtained with a simplified model. [1] E. Majorana, Il Nuovo Cimento (1924-1942) 14,171(2008 ). [2] A. Y . Kitaev, Phys.-Usp. 44,131(2001 ). [3] J. Alicea, Rep. Prog. Phys. 75,076501 (2012 ). [4] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83,1057 (2011 ). [5] Y . Ando and L. Fu, Annu. Rev. Condens. Matter Phys. 6,361 (2015 ). [6] B. A. Bernevig, Topological Insulators and Topological Super- conductors (Princeton University, Princeton, NJ, 2013). [7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82,3045 (2010 ). 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PhysRevB.83.075125.pdf

PHYSICAL REVIEW B 83, 075125 (2011) Optical study of strained ultrathin films of strongly correlated LaNiO 3 M. K. Stewart,1,*C.-H. Yee,2Jian Liu,3M. Kareev,3R. K. Smith,1B. C. Chapler,1M. Varela,4P. J. Ryan,5 K. Haule,2J. Chakhalian,3and D. N. Basov1 1Department of Physics, University of California-San Diego, La Jolla, California 92093, USA 2Department of Physics & Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA 3Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA 4Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 5X-Ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA (Received 16 September 2010; revised manuscript received 9 January 2011; published 28 February 2011) An optical study of fully strained ultrathin LaNiO 3films is presented and compared with LDA +DMFT calculations. LaNiO 3films were grown by pulsed laser deposition on LaAlO 3and SrTiO 3substrates which provide compressive and tensile strain, respectively. Optical conductivity data show a Drude peak with a spectralweight that is significantly reduced compared to that obtained from LDA calculations. The extended Drudeanalysis reveals the presence of a pseudogap around 80 meV for the film on SrTiO 3and near 40 meV , at low temperature only, for the film on LAO. An unusual temperature dependence of the optical conductivity isobserved, with the Drude plasma frequency increasing by up to 20% at low temperature due to spectral weighttransfer from bands lying 2–4 eV below the Fermi energy. Such a strong temperature dependence of the Drudespectral weight has previously been reported for correlated electron systems in which a phase transition is present.In LaNiO 3, however, no phase transition is observed. DOI: 10.1103/PhysRevB.83.075125 PACS number(s): 78 .20.−e, 71.27.+a, 72.80.Ga, 78 .30.−j I. INTRODUCTION Optical studies have been invaluable in the understanding of correlated electron systems, in part due to the opticalsum rules which allow one to monitor redistributions ofthe electronic spectral weight associated with mobile andlocalized electrons. 1–3One optical signature of correlations is especially notorious. The emergence of the conducting statein a correlated metal is usually associated with a dramatictransfer of the electronic spectral weight (SW) over manyeV . 4–7A canonical manifestation of this trend is seen in V2O3as this correlated oxide undergoes an insulator-to-metal transition (IMT). Specifically, in V 2O3one registers a SW transfer from the energy range dominated by the response ofthe Hubbard bands to the quasiparticle peak responsible for theDrude response in the metallic state, confirming the validityof the half-filled Hubbard model. 4,8,9Because the IMT in this and in other correlated systems is accompanied by structuralchanges and/or electronic/magnetic phase separation, the roleof these auxiliary factors in the SW transfer is difficult todisentangle from the direct impact of correlations. In this work, we present an optical study of LaNiO 3(LNO), a correlated oxide10–13without additional complications due to phase transitions, spin or charge order, etc. This materialis the only rare earth nickelate to remain metallic at alltemperatures 14and therefore offers the opportunity to study the dynamics of correlated electrons in its pure form, unobscuredby structural or phase separation effects. The optical responseof bulk LNO is unusual for a metal, showing no well definedDrude peak. 13,15While the ultrathin strained films discussed here do exhibit a Drude resonance, electronic correlationsdominate every aspect of charge dynamics. We show that theseemingly conventional metallic temperature dependence ofthe DC resistivity in LNO is directly linked to the unexpectedenhancement of the Drude SW at the expense of featuresattributable to Mott-Hubbard bands.II. METHODS Epitaxial LNO films were grown by reflection high-energy electron diffraction (RHEED) controlled pulsed laser depo- sition on LAO and STO substrates with −1.2% and +1.7% lattice mismatch, respectively. The films are 30 unit cells thick with c-axis parameters of 3.86◦A on LAO and 3.78◦Ao n STO (see Table I). Representative time-dependent RHEED specular intensity (RSI) is shown in Fig. 1for the LNO film on STO. A full recovery of RSI is observed after each unit-cell layer, characteristic of perfect layer-by-layer growth. The insetin Fig. 1shows the corresponding RHEED image with well defined spots and streaks at the (00) specular and the (01) and(0¯1) off-specular reflections, indicative of a smooth surface morphology. Additionally, the films have been characterized by x-ray diffraction, x-ray absorption, AFM, and TEM asshown in the Appendix and in Ref. 16. Optical studies of both the films and the bare substrates between 20 and 298 K were carried out using reflectance in the range from 6 to 85 meV and variable angle spectroscopicellipsometry (V ASE) in the range from 80 meV to 5.5 eV . Near-normal incidence reflectance measurements were performed in a Michelson interferometer (Bruker 66vs). Reflectance of the sample was first measured relative to a gold reference mirrorand then normalized by the reflectance of the gold coated sample. 17Ellipsometry measurements were performed with two commercial Woollam ellipsometers. The range from 80to 550 meV was investigated with an IR-V ASE model based on a Bruker 66vs. For the range between 0.6 and 6 eV we used a V ASE model based on a grating monochromator. Bothellipsometers are equipped with home-built UHV chambers to allow low temperature measurements. 18Ellipsometry mea- surements were performed at incidence angles of 60◦and 75◦. At each angle, the polarization state of the reflected light was measured in the form of two parameters /Psi1and/Delta1, which 075125-1 1098-0121/2011/83(7)/075125(8) ©2011 American Physical SocietyM. K. STEW ART et al. PHYSICAL REVIEW B 83, 075125 (2011) TABLE I. Comparison of relevant theoretical and experimental parameters for LNO films on LAO and STO: lattice mismatch, c-axis parameters obtained from x-ray diffraction, Drude plasma frequency and scattering rate obtained from optics data at 298 K and 20 K, and ratio of the electron kinetic energy obtained from optics data to that obtained from LDA calculations (see text for details). Lattice c-axis (◦A) ωp298 K ωp20 K 1 /τ(ω→0) 1 /τ(ω→0) mismatch (%) (XRD) (eV) (eV) 298 K (cm−1)2 0 K ( c m−1) Kexp/K LDA LaNiO 3/LaAlO 3 −1.2 3.86 1.11 1.29 468 265 0.11 LaNiO 3/SrTiO 3 +1.7 3.78 0.93 1.04 505 403 0.08 are related to the Fresnel reflection coefficients for p- and s-polarized light ( ˜Rppand ˜Rss) through the equation ˜R=˜Rpp ˜Rss=tan(/Psi1)ei/Delta1. (1) In order to obtain the optical constants from the raw reflectance and ellipsometry data, a model was created usingmultiple Kramers-Kronig consistent oscillators to describe thecomplex dielectric function of the sample. 19The parameters in the model were then fitted to the experimental data usingregression analysis with the WV ASE32 software package fromWoollam Co., Inc. In the case of the LNO films, the modelconsisted of two layers: a substrate characterized by the opticalconstants previously determined for either LAO or STO, anda thin film layer from which the optical constants of the filmalone were obtained. 19Since LAO and STO have several far-IR phonons with strong temperature dependence, the substrateswere measured and modeled at all the same temperatures asthe films. This allows us to use the appropriate temperaturefor the substrate layer in order to ensure that the temperaturedependence observed in the extracted optical conductivityis indeed caused by the LNO films. Representative rawreflectance and ellipsometry data for the film on STO areplotted in Fig. 2along with the model fit. We note that the crystal structure of the films is slightly anisotropic, with the c-axis lattice parameter about 1.5% larger (smaller) than the in-plane lattice parameter for the LNO FIG. 1. (Color online) Representative RSI evolution during the growth of the LNO film on STO. Inset: Corresponding RHEED image.film on LAO (STO). In order to assess the effects of this anisotropy in our optical data, we performed room temperatureellipsometric measurements for the LNO film on LAO in whichthe position of the polarizer was varied and tracked. In thisway it was possible to obtain the ratios of the diagonal andoff-diagonal components of the Jones matrix J=/bracketleftBigg˜R pp˜Rsp ˜Rsp˜Rss/bracketrightBigg , (2) which is a diagonal matrix when the sample is completely isotropic. We found that the magnitudes of both the real andimaginary parts of ˜R ps/˜Rppand ˜Rsp/˜Rsswere between 0.003 and 0.012 in the the range from 0.6 to 6 eV . This means that theoff-diagonal components of the Jones matrix are only about1% of the diagonal components, approximately the same asthe noise level in our measurements. Additionally, modelingusing c-axis and ab-plane optical constants obtained from LDA+DMFT calculations for strained films indicates that our measurements are dominated by the in-plane responseand the effect of this anisotropy on the measured ellipsometricparameters is only evident above ∼1.5 eV and is very small. We therefore conclude that the anisotropy of the crystal structureonly minimally affects our measurements and the use of anisotropic model for our analysis is appropriate. FIG. 2. (Color online) Raw reflectance and ellipsometry data for the LNO film on STO plotted with the model fit. 075125-2OPTICAL STUDY OF STRAINED ULTRATHIN FILMS OF ... PHYSICAL REVIEW B 83, 075125 (2011) Charge self-consistent LDA +DMFT20calculations were performed using the implementation described in Ref. 21. We used U=7.3 eV and J=1 eV for the strength of the Coulomb repulsion on Ni dorbitals, and EDC=U∗(nd− 1/2)−J(nd−1)/2 as the standard double counting energy, where nd=7.3 is the average dvalence. A range for J and Uwas determined based on previous studies of this class of compounds and then scanned to obtain the bestfit to our optics data as well as to ARPES 22and thermal measurements.23In order to compute the optical conductivity, we analytically continued the self-energy using modifiedGaussians 21and cross-checked the result with maximum entropy. The conductivity was then computed using the DFTmomentum matrix elements and convolving the correlatedGreen’s function for all values between −6 eV and 6 eV , relative to the Fermi level. III. RESULTS AND DISCUSSION A. Optical conductivity: General trends Figure 3shows the real part of the optical conductivity, given by σ(ω)=iω[1−/epsilon1(ω)] 4π, for the LNO films on LAO and STO at various temperatures. In both films, a Drude peak typical of metals is evident at low frequencies, along withthree phonon modes. Additionally, four peaks (A–D in Fig. 3) can be clearly identified at higher energies. These can beassigned to interband transitions by comparing to LDA theory.The inset in Fig. 3shows a sketch of the LNO density of states (DOS) based on LDA calculations, 13,24–26indicating a t6 2ge1 gelectronic configuration, with the antibonding e∗ gstates crossing the Fermi level. Figure 4shows σ1(ω)o ft h efi l mo n LAO at 100 K along with those obtained from LDA and fromLDA+DMFT. The latter was calculated for strained LNO on LAO at 116 K. Using transition decomposition analysis of FIG. 3. (Color online) Real part of the optical conductivity of fully strained LNO films on LAO (top panel) and STO (botton panel) substrates plotted on a log-log scale. Inset: Sketch of the LNO density of states and interband transitions based on LDA calculations.FIG. 4. (Color online) Top panel: Real part of the optical conductivity of a 30 unit cell thick LNO film on LAO substrate obtained at 100 K, plotted with the optical conductivity obtained fromLDA and LDA +DMFT calculations for strained LNO on LAO at 116 K. Inset: Ratio of the experimental electron kinetic energy at room temperature for LNO thin films of different thickness grown on LAOand STO substrates to the LDA kinetic energy. This ratio approaches zero for Mott insulators. Bottom panel: Ratio of the spectral weight obtained from experimental data to that obtained from two differenttheoretical calculations: LDA and LDA +DMFT. the LDA optical conductivity we arrive at the assignment of interband transitions reported in Table II. According to Fig. 4,L D A +DMFT provides a more accurate description of our experimental data than LDAdoes. In particular, feature A is not evident in the LDAσ 1(ω) but is present in the LDA +DMFT results. The two peaks seen at 1 eV and 1.5 eV (A and B, respectively) inLDA shift to lower energy when correlations are included inLDA+DMFT, resulting in better agreement with experiment. In this picture, feature A is due to interband transitionsfrom the t ∗ 2gande∗ gorbitals. A redshift of feature C is also evident in LDA +DMFT, consistent with the scenario in which electronic correlations suppress the energy of interbandtransitions due to the quasiparticle renormalization. We notethat even though the LDA +DMFT results reproduce the key experimental trends, the agreement is less than perfect.This is not surprising given that optics is one of the mostchallenging probes to match well theoretically. This is becausethe description of the optics data relies on the convolution of TABLE II. Assignment of interband transitions based on LDA transition decomposition analysis. Aa n dB C D e∗ g→e∗ g t∗2g→e∗ g Ni 3d→Ni 3d∗ t∗ 2g→e∗ g O2p→e∗ g Ni 3d∗→La 4f Ni 3d∗→La 5d 075125-3M. K. STEW ART et al. PHYSICAL REVIEW B 83, 075125 (2011) two Green’s functions, which in turn is very sensitive to any small errors in the individual Green’s functions. B. Extended Drude analysis In considering the low frequency part of the spectra, it is instructive to first review the usual behavior of the opticalconductivity in conventional metals. The free electron (Drude)contribution to the optical conductivity can be described interms of the electron scattering rate 1 /τusing the equation σ Drude=ne2τ m1 1−iωτ. (3) In conventional conductors, 1 /τand the carrier mass mare frequency independent and 1 /τdecreases with decreasing temperature. This results in an increase in the amplitude andnarrowing of the Drude peak. The low and high temperatureoptical conductivity curves tend to cross before the onset ofinterband transitions and the Drude plasma frequency remainsconstant with temperature. The data shown in Fig. 3, which are also plotted on a linear scale in the top panels of Fig. 6,d i f f e r from this description. While the amplitude of the Drude peakincreases at low temperature for both films, the narrowing ofthe Drude peak is not very pronounced, as discussed below. To understand how our data deviate from the conventional Drude theory, we make use of the extended Drude analysis.Within this framework, the scattering rate 1 /τ(ω) and the mass renormalization factor m ∗(ω)/mbare understood to be frequency dependent and are given by6 1 τ(ω)=−ω2 p ωIm/bracketleftbigg1 ˜ε(ω)−ε∞/bracketrightbigg , (4) m∗(ω) mb=−ω2 p ω2Re/bracketleftbigg1 ˜ε(ω)−ε∞/bracketrightbigg . (5) Herembis the carrier band mass and ωpis the Drude plasma frequency given by ω2 p 8=/integraldisplay/Omega1 0σ1dω=4πne2 m, (6) with/Omega1=125 meV . This cutoff was selected to lie between the Drude peak and the onset of interband transitions and providesan upper limit for ω p.F i g . 5(b) shows 1 /τ(ω→0) obtained from Eq. ( 4). While 1 /τdecreases at low temperature, by 20% for the film on STO and by 40% for the film on LAO,the change is not sufficient to account for the increase in DCconductivity. This deficiency is compensated by an increase intheω pof up to 20% [Fig. 5(a)], an unusual effect that can be understood as a transfer of spectral weight to the Drude peakand is discussed further in Sec. III C . Figure 6shows the results of the extended Drude analysis, along with σ 1(ω) in the same energy range after subtraction of the phonon contribution. As shown in Figs. 6(e) and6(f), the mass enhancement is strongly temperature and frequencydependent. m ∗(ω)/mbincreases with decreasing temperature and peaks around 50 meV . We note that m∗(ω)/mbis larger for the film on LAO than for the film on STO, especially atlow temperature. As seen in Figs. 6(c) and6(d), the low frequency scattering rate of the carriers is larger than their energy ω.T h i si sFIG. 5. (Color online) (a) Ratio of the Drude plasma frequency at low temperature to that at 298 K squared, obtained by integrating the real part of the optical conductivity up to 125 meV . (b) Scatteringrate in the zero frequency limit obtained from the extended Drude analysis. (c) Drude plasma frequency squared plotted as a function of the temperature squared. in contrast to the relation 1 /τ(ω)<ω characteristic of well defined quasiparticles in a Fermi liquid,6and is consistent with the argument for strong electronic correlations in LNO(see Sec. III D ). For the film on STO [Fig. 6(d)], 1/τ(ω) exhibits a peak around 70 meV which increases in magnitudeas the temperature is lowered. This type of behavior in 1 /τ(ω) has been attributed to the presence of a pseudogap in othercorrelated oxides such as underdoped cuprates 27–29and the metallic puddles in phase separated VO 2.30The redshift seen in the peak at low temperature suggests that the magnitudeof the gap decreases from 80 meV at 298 K to 65 meV at20 K. The presence of a pseudogap is also supported by theminimum in the optical conductivity seen in Fig. 6(b).F o rt h e film on LAO, a peak in 1 /τ(ω) is evident around 40 meV at 20 K and 100 K only [Fig. 6(c)], suggesting that a pseudogap develops as the temperature is lowered. The magnitude of thepseudogap is smaller than for the film on STO and a clearsign of gapping is not apparent in the optical conductivity inFig. 6(a). C. Temperature dependence To understand the origin of the anomalous enhancement of the Drude SW, it is useful to look at the ratio of σ1(ω)a tl o w temperatures to that at 298 K, shown in Fig. 7. It is apparent from Fig. 7that the temperature dependence of features A and B can be explained as a transfer of SW from one to the other 075125-4OPTICAL STUDY OF STRAINED ULTRATHIN FILMS OF ... PHYSICAL REVIEW B 83, 075125 (2011) FIG. 6. (Color online) Results of the ex- tended Drude analysis for LNO films on LAO (left panels) and on STO (right panels). (a) and (b): Real part of the optical conductivity aftersubtraction of the phonon contribution. (c) and (d): Frequency dependent scattering rate. (e) and (f): Frequency dependent mass renormalizationfactor. and is opposite in the two films. For the film on LAO (STO), the SW of feature A increases (decreases) and that of featureB decreases (increases) at low temperature. This differencebetween the two samples could be due to strain, as tensile andcompressive strains have been shown to have different effectson the e gorbitals and DOS in LNO.16The reduction in σ1(ω) between 1.5 and 3.5 eV is more significant and suggests thatthe enhanced low temperature Drude SW originates from theelectronic states responsible for feature C (see Table II). In order to quantify this, we consider the change in SW of theDrude peak /Delta1SW Drude and of feature C /Delta1SWCbetween 20 K and 298 K, as shown in Table III.T h e/Delta1SW Drude and/Delta1SWC values are within a few percent of each other for both films, providing strong evidence of SW transfer between feature Cand the Drude peak. We note that most of the changes in the FIG. 7. (Color online) Ratio of the optical conductivity at low temperatures to the optical conductivity at 298 K for the LNO films on LAO (top panel) and on STO (bottom panel).Drude part of σ1(ω) are off the scale in Fig. 7and may be more clearly seen in Fig. 5(a).G i v e n U=7.3 eV for LNO, the SW transfer seen in our data appears to be consistent withthe canonical half-filled band Hubbard model, in which a SWtransfer from the Hubbard band at U/2 to the quasiparticle peak is expected. 4Thus, a similar phenomenology of spectral weight transfer is maintained in LNO despite the fact thatthe simplest version of the Hubbard band picture needs to berevised for this multiband system with quarter filled e gbands. Temperature driven changes in ωp, along with high energy effects in σ1(ω), have been observed in several correlated electron systems such as high- Tccuprates,31,32manganites,5,7 and V 2O3.4As seen Fig. 5(c), the Drude SW plotted as a function of the temperature squared is consistent with alinear behavior. While a T 2/W(where Wis the bandwidth) dependence of the SW can be attributed to thermal smearingof the Fermi-Dirac distribution function as given by theSommerfeld model, these effects are expected to be quite mildin magnitude, unlike the 35% increase in ω 2 pseen in our data. AT2dependence has also been observed in the normal state of high- Tccuprates,33–35in which even changes on the order of 2% are considered to be large based on the bandwidthW∼2 eV . This deviation from the Sommerfeld model in the cuprates has been attributed to electronic correlations whichextend the temperature dependence of the carrier response toenergies on the order of U. 35A direct comparison between the 2D, single band model in Ref. 35and our data is not possible since LNO is a multiband, 3D system. However, given thatthee gbandwidth in LNO is ∼4 eV , the observed temperature TABLE III. SW of the Drude peak and of feature C at 20 K minus the SW at 298 K for the LNO films on LAO and STO. /Delta1SW Drude (105cm−2) /Delta1SWC(105cm−2) LNO/LAO 7.23 −7.15 LNO/STO 3.61 −3.39 075125-5M. K. STEW ART et al. PHYSICAL REVIEW B 83, 075125 (2011) dependence of ωpis quite substantial and, to our knowledge, unprecedented in the absence of a phase transition. We note that, since the rest of the lanthanide rare earth nickelates do exhibit an IMT, it is likely that LNO isalso close to localization even if it remains metallic at alltemperatures. The spectral weight transfer from the low energyDrude peak into the incoherent contribution to the opticalconductivity at higher energy is in qualitative agreement withthe basic prediction of the DMFT for systems near the Motttransition. 4Our system specific calculations indeed show that with increasing temperature, the Drude spectral weight getsredistributed to higher energy. However, due to the difficultyassociated with the analytic continuation of the Monte Carlodata from the imaginary to the real axis, the precision ofthe conductivity beyond 1 eV is limited; hence we cannotdetermine the theoretical temperature dependence of feature C. D. Electronic kinetic energy In earlier work, we reported on the optical properties of LNO films of thickness 100–200 nm, which are expected tobe relatively strain free and bulk-like. 13As in the case of LNO ceramics,15no well defined Drude peak was observed in the optical conductivity of these films, presumably due toa combination of enhanced scattering and strong electroniccorrelations in this material. The fully strained films discussedin this paper and in Ref. 12, on the other hand, do exhibit a Drude peak in the optical conductivity (Fig. 3). However, our data show that electronic correlations are still very strong inthese films, albeit possibly not as strong as in bulk LNO. Inan attempt to quantify and compare the strength of electroniccorrelations of various LNO films, it is useful to consider the ra-tio of the electron kinetic energy obtained experimentally andthat obtained from band structure calculations K exp/K LDA.36,37 This ratio is close to unity for conventional metals and becomes suppressed when strong electronic correlations come into play.K expcan be obtained from optical measurements by integrating the Drude contribution of the optical conductivity,38 Kexp=¯hc0 e2/integraldisplay/Omega1 02¯h πσ1(ω)dω. (7) Here c0is the c-axis lattice parameter and /Omega1is the frequency cutoff chosen such that interband transitions are excluded fromthe integral. The inset in Fig. 4shows K exp/K LDA for various LNO thin film samples at room temperature obtained by integratingσ 1(ω) up to 125 meV , the same cutoff used to obtain ωpin Fig. 5.T h es a m e KLDA was used for all the samples. It is clear that Kexpis very strongly suppressed with respect to the LDA prediction, regardless of the substrate and thickness ofthe film. The highest value was obtained for the 30 uc film onLAO,K exp/K LDA=0.11, which is very low even compared to other correlated oxides.37The data in the inset in Fig. 4suggest that tensile and compressive strain both reduce the strength ofcorrelations in LNO. While it may seem unexpected that bothtypes of strain have a similar effect on the LNO films, we notethat the symmetry of the crystal structure is reduced in strainedfilms relative to the bulk, regardless of whether the latticemismatch is positive or negative. This is due to the epitaxialconstraint that removes the threefold rotation about the mainR¯3/caxis 39and could be the cause for the increased metallicity of the strained films. Furthermore, it has been shown that thetendency to charge and bond disproportionation is suppressednot only when compressive strain is applied, but also undertensile strain due to the dynamic breathing mode adoptedby the oxygen octahedra. 16Since charge order is the leading cause for the insulating state in nickelates, it can be expectedthat factors that suppress charge disproportionation will alsoincrease the metallicity of the films. Our data show that thisis indeed the case and that the increase in the metallicity isassociated with a stronger coherent contribution to the opticalconductivity. TheK exp/K LDA values we have obtained are somewhat lower than those reported in Ref. 12. Specifically, for a film on LAO they report a mass enhancement of 3 equivalenttoK exp/K LDA=0.33, three times higher than our result. Although measurements for a film on STO are not reported, wecan estimate from Fig. 3 in Ref. 12that an LNO film subject to 1.7% of tensile strain could have a mass enhancement ofroughly 4, or K exp/K LDA=0.25. This difference is, at least in part, likely due to the choice of integration cutoff /Omega1.T h e authors of Ref. 12have chosen /Omega1=0.2 eV while in this work we use /Omega1=0.125 eV in order to exclude the spectral weight of feature A. Because of the scale on which the opticalconductivity is shown in Ref. 12it is hard to assess whether such a feature is present in their data. While K exp/K LDAdepends on the selection of /Omega1[Eq. ( 7)], it is important to note that the argument for strong electroniccorrelations in LNO holds regardless of what cutoff is chosen.The bottom panel in Fig. 4shows the ratio of the spectral weight SW( ω)=/integraltext ω 0σ(ω/prime)dω/primeobtained from experimental data to that obtained from LDA calculations. SW( ω) represents the effective number of carriers contributing to absorption ata given frequency and SW( /Omega1) is proportional to K exp. While SW(ω)exp/SW(ω)LDA increases with increasing frequency, it remains less than unity over the entire energy range. Thisshows that independent of the cutoff frequency used to obtainK exp, LNO is very strongly correlated. The other curve in 2.4 nm 2 µm2 µm STOLNO FIG. 8. (Color online) Top: AFM surface image of a 30 uc film of LNO on STO. Bottom: TEM image of a 10 uc film of LNO on STO. 075125-6OPTICAL STUDY OF STRAINED ULTRATHIN FILMS OF ... PHYSICAL REVIEW B 83, 075125 (2011) the bottom panel of Fig. 4shows the spectral weight ratio using LDA +DMFT instead of LDA. Including electronic correlations in the theoretical calculations provides a morerealistic description of LNO, resulting in a spectral weightratio that is closer to unity. E. Tensile vs compressive strain We now compare the effects of tensile and compressive strain on the optical and electronic properties of the LNOfilms and discuss the findings (see also Table I). Even though the symmetry of the crystal structure in both LNO strainedfilms is reduced relative to the bulk, the films on LAO andSTO adopt different lower symmetry space groups ( C2/cand P2 1/c, respectively).16,39For this reason, some differences in the optical response of the two films can be expected. Asshown in Figs. 3and6the film on LAO has higher σ 1(ω→0) than the film on STO. According to Eqs. ( 3) and ( 6), the DC conductivity is inversely proportional to 1 /τand proportional toω2 p. As seen in Fig. 5(b),1/τis higher for the film on STO than for the film on LAO. However, this difference isnot sufficient to account for the decreased DC conductivity.At 298 K, τ LAO/τSTO=1.07 while ρSTO/ρLAO=1.3. This means that an increased Drude spectral weight for the filmon LAO must be included to account for the difference FIG. 9. (Color online) Reciprocal lattice maps for 10 uc films of LNO on LAO (top) and on STO (bottom).in the conductivity [Fig. 5(a)]. While 1 /τdecreases more rapidly for the film on LAO with decreasing temperature,ω premains higher at all temperatures, suggesting that the increased scattering and the reduced Drude spectral weightare jointly responsible for the lower conductivity of the filmon STO. In the context of electronic correlations, the higher ω p observed for the film on LAO [Fig. 5(a)] means that Kexp/K LDA is larger for the film on LAO than for the film on STO (see inset in Fig. 4); i.e., the film subject to tensile strain is more strongly correlated than the film with compressive strain.Furthermore, features B and C are centered at lower frequencyfor the film on STO than the one on LAO. This is consistentwith the hypothesis that electronic correlations suppress theenergy of interband transitions (Sec. III A ), as these would be redshifted more in the film with stronger correlations, thefilm on STO. Our finding is in agreement with x-ray lineardichroism data showing that compressive strain suppresses thetendency to charge and bond disproportionation more so thantensile strain, 16which could increase the coherent response of the carriers for the film on LAO. Finally, our results arealso consistent with the data in Ref. 12showing a larger mass enhancement for the films subject to tensile strain than tocompressive strain. IV . CONCLUSIONS Our optical study of strained LNO films reveals that despite the improved metallicity observed in these samplesrelative to the bulk, three canonical properties of correlatedsystems are retained: (i) a suppression in the ratio K exp/K LDA, (ii) a pseudogap, and (iii) a striking increase in ωpat low temperature. This latter effect is particularly relevant for theunderstanding of correlated electron systems, as SW changesof this magnitude have not been previously observed ina correlated system without a phase transition. This resultindicates that the low temperature enhancement of the DrudeSW can be considered an intrinsic property brought on byelectronic correlations, showing that ω pcan be tuned with temperature even in the absence of an IMT. Finally, we briefly discuss our results in the context of the recent theoretical proposal to mimic the gross features ofthe cuprate dbands in nickelates through heterostructuring. 40 Some similarities can be found between LNO and the high- Tc cuprates based on our data. As described above, the temper- ature dependence of the Drude spectral weight is linear withT 2[Fig. 5(c)]. This behavior is consistent with that observed in the normal state of several cuprate superconductors.33–35 Additionally, features A–C present in the optical conductivity (Fig. 3) are also seen in the cuprates.41Furthermore, even though our results show an increased metallicity of thestrained films relative to bulk LNO, the strength of electroniccorrelations, possibly a key ingredient for superconductivity, 37 remains quite high. Reference 40suggests that the presence of tensile strain will be beneficial in the attempt to inducesuperconductivity in LNO heterostructures. While our datashow that strain increases the metallicity of the films, it isthe film subject to compressive strain that has the highestconductivity. Therefore, it cannot be concluded from our datathat tensile strain is more favorable that compressive strain. 075125-7M. K. STEW ART et al. PHYSICAL REVIEW B 83, 075125 (2011) ACKNOWLEDGMENTS J.C. was supported by DOD-ARO under Grant No. 0402- 17291 and by NSF Grant No. DMR-0747808. Work at UCSDis supported by DOE-BES. APPENDIX: SAMPLE CHARACTERIZATION In addition to the RHEED measurements shown in Fig. 1, the samples have been fully characterized using a variety oftechniques to ensure the high quality of the films. The toppanel of Fig. 8shows a 2 μm×2μm AFM image of a 30 ucLNO film on TiO 2-terminated STO.42The image indicates that the sample surface is atomically flat with surface roughnessless than 75 pm and preserved vicinal steps. High-resolutioncross-sectional TEM imaging (bottom panel of Fig. 8)s h o w s a coherent, defect-free film structure and an atomically sharpinterface with no cation interdiffusion. Reciprocal space mapsof the LNO (222) reflections obtained with x-ray diffractionfor 10 uc LNO films on LAO and on STO can be seen inFig. 9. 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PhysRevB.80.045413.pdf

Plasmonic excitations in tight-binding nanostructures Rodrigo A. Muniz *and Stephan Haas Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089-0484, USA A. F. J. Levi Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089-0082, USA Ilya Grigorenko Theoretical Division T-11, Center for Nonlinear Studies and Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA /H20849Received 20 March 2009; revised manuscript received 28 May 2009; published 15 July 2009 /H20850 We explore the collective electromagnetic response in atomic clusters of various sizes and geometries. Our aim is to understand, and hence to control, their dielectric response based on a fully quantum-mechanicaldescription which captures accurately their relevant collective modes. The electronic energy levels and wavefunctions, calculated within the tight-binding model, are used to determine the nonlocal dielectric responsefunction. It is found that the system shape, the electron filling, and the driving frequency of the external electricfield strongly control the resonance properties of the collective excitations in the frequency and spatial do-mains. Furthermore, it is shown that one can design spatially localized collective excitations by properlytailoring the nanostructure geometry. DOI: 10.1103/PhysRevB.80.045413 PACS number /H20849s/H20850: 73.22. /H11002f, 73.20.Mf, 36.40.Gk, 36.40.Vz I. INTRODUCTION Recent advances in nanoscience have created a vast num- ber of experimentally accessible ways to configure atomicand molecular clusters into different geometries withstrongly varying physical properties. Specifically, exquisitecontrol of the shape and size of atomic and molecular clus-ters has made it now possible to investigate the collectiveelectromagnetic response of ultrasmall metal and semicon-ductor particles. 1,2The aims of this study are to model and examine plasmonic excitations in such structures and thus togain an understanding of the quantum-to-classical crossoverof collective modes with increasing cluster size. There isobvious technological relevance to tunable collective modesin nanostructures. For example, surface plasmon resonancesin metallic nanospheres and films have been found to behighly sensitive to nearby microscopic objects and hence arecurrently investigated for potential sensing applications. 3In this context, it is desirable to design customized nanostruc-tures with specifically tailored resonance properties, 4and this study is intended to be a step into this direction. It is natural to expect that in many cases the electromag- netic response of nanoclusters is considerably different fromthe bulk. In particular for very small clusters, the quantumproperties of electrons confined in the structure need to betaken into account. 5Moreover, unlike in the bulk, the cou- pling between single-particle excitations and collectivemodes can be very strongly affected by its system param-eters. This exponential sensitivity opens up excellent oppor-tunities to optimize the dielectric response via tuning thecluster geometry and its electron filling. For example, byproper arrangement of atoms on a surface one can designnanostructures with controllable resonances in the near infra-red or visible frequency range. 2A possible application of such nanostructures is the creation of metamaterials withnegative refractive index at a given frequency. Furthermore, since geometry optimization of bulk resonators has demon-strated minimization of losses in metamaterials, 6it is also interesting to investigate the effect of the nanostructure shapeon the loss function at a given resonance frequency. To approach this problem, in this study we investigate the formation of resonances in generic systems of finite conduct-ing clusters and examine how their frequency and spatialdielectric response depends on the system size and geometry.In particular, the nonlocality of the dielectric response func-tion in these structures is important and will therefore beproperly accounted for. A similar analysis for the case ofsmall metallic nanostructures was performed recently using an effective mass approximation. 7Here we focus on the op- posite limit, namely, we assume that electrons in the clustercan be effectively described using a tight-binding model. 8 Because of the more localized nature of the electronic wavefunctions in this model the overall magnitude of the collec-tive modes is expected to be strongly suppressed as com-pared to metallic clusters. This paper is organized in the following way. In Sec. II we introduce the model and method. For a more extendeddiscussion, the reader is referred to Ref. 7, where the same calculations were done for other models such as effectivemass and particle in a box. Results for the induced energy asa function of the driving frequency of an externally appliedelectric field and the corresponding spatial modulations ofthe charge density distribution function are discussed in Sec.III. Finally, a discussion of possible extensions and applica- tions is given in Sec. IV. II. MODEL The interaction of electromagnetic radiation with nano- scale conducting clusters is conventionally described byPHYSICAL REVIEW B 80, 045413 /H208492009 /H20850 1098-0121/2009/80 /H208494/H20850/045413 /H208496/H20850 ©2009 The American Physical Society 045413-1semiclassical Mie theory.9This is a local continuum-field model which uses empirical values of the linear optical re-sponse of the corresponding bulk material and has been ap-plied in nanoparticles to describe plasmon resonances. 10 However, such a semiempirical continuum descriptionbreaks down beyond a certain degree of roughness intro-duced by atomic length scales and thus cannot be used todescribe ultrasmall systems. In addition, near-field applica-tions, such as surface-enhanced Raman scattering, 11are most naturally described using a real-space theory which includesthe nonlocal electronic response of inhomogeneous struc-tures. Therefore, we will use a recently developed self-consistent and fully quantum-mechanical model which fullyaccounts for the nonlocality of the dielectric responsefunction. 7 Specifically, to identify the plasmonic modes in small clusters we calculate the total induced energy due to an ap-plied external electric field with driving frequency /H9275and scan for the resonance peaks. The induced energy is deter-mined within the nonlocal linear response approximation. To keep the computational complexity of this procedure at a minimum, we use a one-band tight-binding model to obtainthe electronic energy levels E iand wave functions /H9274i/H20849r/H20850as a linear combination of sorbitals, /H9274i/H20849r/H20850=/H20858 i,j/H9251ij/H9272/H20849r−Rj/H20850, /H208491/H20850 where /H9272/H20849r−Rj/H20850is the wave function of an sorbital around an atom localized at position Rjand/H9251ijare the coefficients of the eigenvector /H20849with energy Ei/H20850of the Hamiltonian, which has the matrix elements /H20855/H9272/H20849r−Ri/H20850/H20841H/H20841/H9272/H20849r−Rj/H20850/H20856=/H20902/H9262 fori=j −tfori,jnn. 0 otherwise. /H20903/H208492/H20850 Here tis the tight-binding hopping parameter which deter- mines the width of the electronic band by 4 tand/H9262is the on-site potential that corresponds to the electronic energy atthe center of the band. Throughout this paper we set /H9262=0 andt=1, such that the energy levels are measured relative to the center of the band, and the energy scale is given by thehopping parameter. The Hamiltonian matrix is diagonalizedusing the Householder method to first obtain a tridiagonalmatrix and then a QL algorithm for the final eigenvectors andeigenvalues. 12 Once the electronic wave functions have been obtained, it is possible to calculate the dielectric susceptibility /H9273/H20849r,r/H11032,/H9275/H20850 via /H9273/H20849r,r/H11032,/H9275/H20850=/H20858 i,jf/H20849Ei/H20850−f/H20849Ej/H20850 Ei−Ej−/H9275−i/H9253/H9274i/H11569/H20849r/H20850/H9274i/H20849r/H11032/H20850/H9274j/H11569/H20849r/H11032/H20850/H9274j/H20849r/H20850. /H208493/H20850 The induced charge density distribution function is then ob- tained by/H9267ind/H20849r,/H9275/H20850=/H20885/H9273/H20849r,r/H11032,/H9275/H20850/H20851/H9278ind/H20849r/H11032,/H9275/H20850+/H9278ext/H20849r/H11032,/H9275/H20850/H20852dr/H11032, /H208494/H20850 where in turn the induced potential is given by /H9278ind/H20849r,/H9275/H20850=/H20885/H9267ind/H20849r/H11032,/H9275/H20850 /H20841r−r/H11032/H20841dr/H11032. /H208495/H20850 We avoid the large memory requirement to store /H9273/H20849r,r/H11032,/H9275/H20850 by calculating the induced charge density distribution itera-tively via /H9267ind/H20849r,/H9275/H20850=/H20858 i,jf/H20849Ei/H20850−f/H20849Ej/H20850 Ei−Ej−/H9275−i/H9253/H9274i/H11569/H20849r/H20850 /H11003/H9274j/H20849r/H20850/H20885/H9274i/H20849r/H11032/H20850/H9278tot/H20849r/H11032,/H9275/H20850/H9274j/H11569/H20849r/H11032/H20850dr/H11032, /H208496/H20850 with/H9278tot/H20849r/H11032,/H9275/H20850=/H9278ind/H20849r/H11032,/H9275/H20850+/H9278ext/H20849r/H11032,/H9275/H20850. The integrals are evaluated using a fourth-order formula obtained from a com-bination of Simpson’s rule and Simpson’s 3/8 rule. Equations/H208495/H20850and /H208496/H20850are solved self-consistently by iterating /H9278ind/H20849r,/H9275/H20850 and/H9267ind/H20849r,/H9275/H20850. This procedure typically converges in three to eight steps when starting with /H9278ind/H20849r,/H9275/H20850=0, depending on the proximity to a resonance and on the value of the dampingconstant /H9253, which throughout this paper is chosen as /H9253=0.08 t. A much better performance can be achieved when the initial /H9278ind/H20849r,/H9275/H20850is taken as the solution of a previously solved nearby frequency. Upon its convergence, the fre-quency and spatial dependence of the induced electric fieldand the induced energy are obtained using E ind/H20849r,/H9275/H20850=−/H11612/H9278ind/H20849r,/H9275/H20850/H20849 7/H20850 and Uind/H20849/H9275/H20850=/H20885/H20841Eind/H20849r,/H9275/H20850/H208412dr. /H208498/H20850 The observed resonances in the induced energy and charge density distribution at certain driving frequencies of the ap-plied electric field correspond to collective modes of thecluster. In the following, the local induced charge density distri- bution is used for analyzing the characteristic spatial modu-lation of a given plasmonic resonance. The energy scale isgiven in terms of the tight-binding hopping parameter tand /H6036=1. III. RESULTS Let us first focus on the dielectric response function in linear chains of atoms, with the intent to identify the basicfeatures of their collective excitations. Unless otherwisestated, the interatom spacing is fixed to a=3r B, where rBis the Bohr radius, and the number of atoms in the chain isvaried. The frequency dependence of the induced energy insuch systems, exposed to a driving electric field along thechain direction, is shown in Fig. 1/H20849a/H20850. It exhibits a series of resonances, which increase in number for chains with in-MUNIZ et al. PHYSICAL REVIEW B 80, 045413 /H208492009 /H20850 045413-2creasing length. As observed in the spatial charge density distribution, e.g., shown for the five-atom chain in Fig. 1/H20849b/H20850, the lowest peak corresponds to a dipole resonance. Whenincreasing the system size N, there are more electronic levels available in the spectrum of the system and the spacing be-tween them decreases, i.e., E n+1−En=4tsin/H20873/H9266 2N+2/H20874sin/H208732n/H9266+/H9266 2N+2/H20874/H11008 N/H1127111 N2. /H208499/H20850 Since the dipole resonance frequency is associated with the transition between the highest occupied and the lowest va-cant energy level, it also decreases for larger chains, as ob-served in Fig. 1/H20849a/H20850. The resonances at higher frequencies correspond to higher harmonic charge density distributions.For example, in Fig. 1/H20849c/H20850, we show the charge density distri- bution corresponding to the highest frequency resonance ofthe six-atom chain. In contrast to the dipole resonance, thesemodes show a rapidly oscillating charge density distributionand thus have the potential to provide spatial localization ofcollective excitations in more sophisticated structures. Whilean extension to much larger chains is numerically prohibitivewithin the current method, the finite-size scaling of the ob-served dielectric response of these clusters indicates that thefrequency of the dominant low-energy plasmon mode scales as/H9004E/H110081/N 2, consistent with the discussion above. In order to study the transverse collective modes we apply an external electric field perpendicular to ladder structuresmade of coupled linear chains of atoms. 13Figure 2/H20849a/H20850shows that for every chain size there are two resonance peaks forthe total induced energy, the higher energy is an end mode, as shown in Figs. 2/H20849b/H20850and2/H20849c/H20850for the three- and five-atom double chains, respectively, whereas the lower-energy peakcorresponds to a central mode, as displayed in Fig. 2/H20849d/H20850for the six-atom double chain. It is also confirmed that as thelength of the chain is increased, the central mode gets stron-ger relative to the end mode, which is the expected behaviorfor bulk versus surface excitations. These results are inagreement with the findings in Ref. 14. Let us next examine what happens when the direction of the external electric field is varied. Figure 3shows the di- electric response of a 4 /H110036-atom rectangular structure for different angle incidence directions of the applied field.When the field is parallel to one of the edges /H20849 /H9258=0° orFIG. 1. /H20849Color online /H20850Longitudinal modes in atomic chains. /H20849a/H20850 Decimal logarithm of the total induced energy /H20849artificially offset /H20850as a function of the frequency of an external electric field which isapplied along the direction of the chain. The resonance peaks cor-respond to different modes. The arrows indicate the peaks for whichthe corresponding charge density profiles are shown in /H20849b/H20850and /H20849c/H20850. /H20849b/H20850Induced charge density distribution for the lowest-energy mode at /H9275=0.73 tin the five-atom chain. /H20849c/H20850Induced charge density dis- tribution for the highest-energy mode at /H9275=3.56 tin the six-atom chain. In /H20849b/H20850and /H20849c/H20850the arrows indicate the direction of the external applied electric field.FIG. 2. /H20849Color online /H20850Transverse modes in coupled chain struc- tures. /H20849a/H20850Logarithm of the total induced energy /H20849artificially offset /H20850 as a function of the frequency of an external electric field appliedtransversely to the chain. The low-energy mode is central, the ana-log of a bulk plasmon, and the high-energy mode is located at thesurface, the analog of a surface plasmon. The arrows indicate thepeaks for which the corresponding charge density profiles areshown in the other insets. /H20849b/H20850Induced charge density distribution for the mode at /H9275=4.91 tin the three-atom double chain. /H20849c/H20850In- duced charge density distribution for /H9275=5.41 tin the five-atom double chain. /H20849d/H20850Induced charge density distribution for /H9275=1.93 t in the six-atom double chain. In /H20849b/H20850–/H20849d/H20850the arrow indicates the direction of the external applied electric field.PLASMONIC EXCITATIONS IN TIGHT-BINDING … PHYSICAL REVIEW B 80, 045413 /H208492009 /H20850 045413-3/H9258=90° /H20850, the response is essentially that of a single chain with the same length, shown in Fig. 1/H20849a/H20850. Also the induced spatial charge density modulations are analogous to those of thecorrespondent linear chain, which can be seen in Figs. 3/H20849b/H20850 and3/H20849c/H20850. At intermediate angles the response is a superposi- tion of the two above cases, changing gradually from oneextremum to the other as the angle is changed. Notice, forinstance, that as the angle increases, the peak at the same frequency of the four-atom dipole resonance diminishes,while simultaneously another resonance is formed at the fre-quency of the dipole mode of a six-atom chain when theangle is tuned from /H9258=0° to /H9258=90°. For /H9258=0° there is only the peak at the frequency of the four-atom chain dipole reso-nance, whereas for /H9258=90° only the dipole peak correspond- ing to the six-atom dipole frequency is present. The super-position of the responses from each direction is aconsequence of the linear response approximation employedsince the response is a linear combination of those obtainedfrom each direction component of the external field. Next, let us analyze the dependence of the resonance modes on the number of electrons in the cluster. Figure 4/H20849a/H20850 shows significant changes in the response of a nine-atomchain with the external field applied along its direction. Inparticular, it is observed that the response is stronger whenthere are more electrons in the sample, a quite obvious factsince there are more particles contributing to the collectiveresponse. Moreover the resonance frequencies of lowermodes increase with the number of electrons, which can beunderstood as a consequence of the one-dimensional tight-binding density of states being smallest at the center of theband. Hence the energy levels around the Fermi energy aremore sparse in the finite system, and therefore the excitationsrequire larger frequencies at half-filling. The same does nothold for higher frequency modes since these correspond totransitions between the lowest and highest levels for any number of electrons in the sample. Therefore these modeshave the same frequency, independent of the electronic fill-ing. Higher filling also allows the induced charge density toconcentrate closer to the boundaries of the structure, as acomparison between Figs. 4/H20849b/H20850and4/H20849c/H20850demonstrates. Figure 4/H20849b/H20850shows that a nine-atom chain with N el=1 electron has its induced charge density localized around the center of thechain. In contrast, Fig. 4/H20849c/H20850displays the induced charge den- sity localized at the boundaries of the same structure withN el=9. This concentration closer to the surface happens be- cause higher-energy states have a stronger charge densitymodulation than the lower-energy ones. Therefore the in-duced charge density is more localized for higher fillingsbecause at low fillings the excitations responsible for the induced charge density are between the more homogeneouslower-energy levels. This can be interpreted as a finite-sizerendition of the fact that by increasing the electronic fillingone obtains the classical response with all the induced chargedensity on the surface of the object. Access to high-energy states is very important for achiev- ing spatial localization of the induced charge density, as thenext example shows. In order to find a structure with spa-tially localized plasmons we consider two parallel eight-atomchains connected to each other by an extra atom at the center.When an external electric field is applied transversely to thechains, the electrons are stimulated to hop between them, butthis is only realizable through the connection, therefore theplasmonic excitation is sharply localized around it. Figure5/H20849a/H20850shows the response of two eight-atom chains; Figs. 5/H20849b/H20850 and5/H20849c/H20850show the induced charge density for the dipole and the highest modes, respectively. It is seen that the inducedcharge density of the lowest frequency mode is spread alongthe chains, whereas the high frequency plasmon is more lo-calized since it corresponds to excitations to the highest-energy state that has a large charge modulation as it waspointed out before.FIG. 3. /H20849Color online /H20850Dependence on the direction of the ap- plied electric field. /H20849a/H20850Logarithm of the total induced energy /H20849arti- ficially offset /H20850as a function of frequency of external electric fields applied to a 4 /H110036 rectangle at different incident angles. /H9258=0° when the field is parallel to the four-atom edge and /H9258=90° when it is parallel to the six-atom edge. The arrows indicate the peaks forwhich the corresponding charge density profiles are shown in theother insets. /H20849b/H20850Induced charge distribution for /H9258=0° and /H9275=3.15 t./H20849c/H20850Induced charge density distribution for /H9258=90° and /H9275=2.15 t.I n /H20849b/H20850and /H20849c/H20850the arrow indicates the direction of the external applied electric field.FIG. 4. /H20849Color online /H20850Variation in the number of electrons. /H20849a/H20850 Logarithm of the total induced energy as a function of the externalelectric field frequency. The number of electrons N elin a nine-atom chain is varied. The arrows indicate the modes whose charge den-sity profiles are shown in the other insets. /H20849b/H20850Induced charge den- sity distribution for N el=1 at /H9275=0.36 t./H20849c/H20850Induced charge density distribution for Nel=9 at /H9275=0.53 t.I n /H20849b/H20850and /H20849c/H20850the arrow indicates the direction of the external applied electric field.MUNIZ et al. PHYSICAL REVIEW B 80, 045413 /H208492009 /H20850 045413-4Let us finally analyze the dependence of the various di- electric response modes on the interatomic distance. Now weconsider a fixed number of atoms in the chain but the intera-tom distance is changed. The dipole moment of the chain isproportional to its length and consequently also proportionalto the distance between atoms. Hence one would naivelyexpect that the strength of the dielectric response is strictlyproportional to the atomic spacing. However, higher fre-quency modes require that the electrons are able to hopquickly along the chain in order to produce the fast chargeoscillations of the mode. Hence the oscillator strength of thehigh frequency modes is suppressed for systems where elec-trons cannot move fast enough. In the tight-binding model,the hopping rate is determined by the hopping parameter t and stems from the overlap of the atomic orbitals on differentsites, which decreases with increasing spacing betweenatoms. 15Therefore the high frequency modes are suppressed for chains with large interatom spacing because the hoppingis so weak that it overcomes the gain coming from a largerdipole moment. On the other hand the oscillator strength ofthe slow modes increases for larger spacings because they donot require fast motion of electrons along the chain. In thiscase, the contribution from a larger dipole moment domi-nates over the suppression due to the smaller hopping rates.This fact is demonstrated in Fig. 6where the response of seven-atom chains with different atomic spacings is shown.The tight-binding hopping parameter tchanges with the atomic spacing a, and here we considered a generic 15power- law dependence t/H11011a−3. Comparing the oscillator strength of the slowest mode for all the different spacings, we see thata=4r Bhas the strongest response, while a=2.5 rBhas the weakest. On the other hand, for the fastest mode we see thatthe chain with spacing a=2.5 r Bhas the strongest response, while the chain with a=4rBhas such a small response that we cannot see a peak because it is washed out by otherpeaks.IV. CONCLUSION In conclusion, we have analyzed the evolution of plas- monic resonances in small clusters as a function of the sys-tem shape, applied external fields, electron filling, and atomic separation. Using a fully quantum-mechanical nonlo-cal response theory, we observe that longitudinal and trans-verse modes are very sensitive to these system parameters.This is reflected in their frequency, oscillator strength, andthe spatial modulation of the induced charge density. Specifi-cally, we identify bulk and surface plasmonic excitationswhich can be controlled in amplitude and frequency by thecluster size. Furthermore, we observe a nontrivial filling de-pendence, which critically depends on the electronic levelspacing in a given structure. We also find that changes inatomic spacings have a very different impact on low-energyvs high-energy modes. And we see that changing the positionof a single atom in a nanostructure can completely alter itscollective dielectric response. This strong sensitivity to smallchanges is the key to controlling the modes of ultrasmallstructures, and it can thus become the gateway to a newgeneration of quantum devices which effectively utilizequantum physics for new functionalities. ACKNOWLEDGMENTS We would like to thank Gene Bickers, Richard Thompson, Vitaly Kresin, Aiichiro Nakano, and Yung-Ching Liang foruseful conversations. We also acknowledge financial supportby the Department of Energy /H20849Grant No. DE-FG02- 06ER46319 /H20850. The numerical computations were carried out on the University of Southern California high-performancecomputer cluster.FIG. 5. /H20849Color online /H20850Connection between two chains, Nel=1. /H20849a/H20850Logarithm of the total induced energy as a function of the fre- quency of an external electric field applied to two eight-atom chainswith an extra atom connecting them at the center. The arrows indi-cate the peaks for which the corresponding charge density profilesare shown in the other insets. /H20849b/H20850Induced charge density distribu- tion for /H9275=0.28 t./H20849c/H20850Induced charge density distribution for /H9275=4.36 t.I n /H20849b/H20850and /H20849c/H20850the arrow indicates the direction of the external applied electric field.FIG. 6. /H20849Color online /H20850Variation in the distance between neigh- bor atoms a. Logarithm of the total induced energy as a function of the frequency of an external electric field applied to seven-atomchains with different spacings between atoms ain units of the Bohr radius r B. The damping constant /H9253is kept constant for all the dif- ferent atomic spacings. In this figure the frequency unit is t3, the tight-binding hopping parameter for a=3rB. The thin red lines con- nect corresponding peaks for systems with different interatom spac-ings, indicating that the oscillator strength of the dominant low-energy mode decreases with decreasing spacing, whereas the peaksof the higher-energy modes increase.PLASMONIC EXCITATIONS IN TIGHT-BINDING … PHYSICAL REVIEW B 80, 045413 /H208492009 /H20850 045413-5*rmuniz@usc.edu 1V. V. Kresin, Phys. Rep. 220,1 /H208491992 /H20850; K. D. Bonin and V. V. Kresin, Electric-Dipole Polarizabilities of Atoms, Molecules and Clusters /H20849World Scientific, Singapore, 1997 /H20850. 2G. V. Nazin, X. H. Oiu, and W. Ho, Science 302,7 7 /H208492003 /H20850;G . V. Nazin, X. H. Qiu, and W. Ho, Phys. Rev. Lett. 90, 216110 /H208492003 /H20850; N. Nilius, T. M. Wallis, M. Persson, and W. Ho, ibid. 90, 196103 /H208492003 /H20850; N. Nilius, T. M. Wallis, and W. Ho, Science 297, 1853 /H208492002 /H20850. 3J. Homola, S. S. Yee, and G. Gauglitz, Sens. Actuators B 54,3 /H208491999 /H20850. 4C. R. Moon, L. S. Mattos, B. K. Foster, G. Zeltzer, W. Ko, and H. C. Monoharan, Science 319, 782 /H208492008 /H20850. 5D. J. Bergman and M. I. Stockman, Phys. Rev. Lett. 90, 027402 /H208492003 /H20850. 6A. K. Sarychev and V. M. Shalaev, Electrodynamics of Metama- terials /H20849World Scientific, Singapore, 2007 /H20850. 7I. Grigorenko, S. Haas, and A. F. J. Levi, Phys. Rev. Lett. 97, 036806 /H208492006 /H20850; A. Cassidy, I. Grigorenko, and S. Haas, Phys. Rev. B 77, 245404 /H208492008 /H20850; I. Grigorenko, S. Haas, A. V. Bal- atsky, and A. F. J. Levi, New J. Phys. 10, 043017 /H208492008 /H20850.8J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 /H208491954 /H20850;C .M . Goringe, D. R. Bowler, and E. Hernndez, Rep. Prog. Phys. 60, 1447 /H208491997 /H20850; N. W. Ashcroft and N. D. Mermin, Solid State Physics /H20849Thomson Learning, New York, 1976 /H20850. 9G. Mie, Ann. Phys. 330, 377 /H208491908 /H20850. 10D. M. Wood and N. W. Ashcroft, Phys. Rev. B 25, 6255 /H208491982 /H20850; M. J. Rice, W. R. Schneider, and S. Strassler, ibid. 8, 474 /H208491973 /H20850; Q. P. Li and S. Das Sarma, ibid. 43, 11768 /H208491991 /H20850;D .R . Fredkin and I. D. Mayergoyz, Phys. Rev. Lett. 91, 253902 /H208492003 /H20850. 11S. Nie and S. R. Emory, Science 275, 1102 /H208491997 /H20850. 12W. H. Press, B. P. Flanney, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes /H20849Cambridge University Press, Cambridge, 1988 /H20850. 13Within the current approach, at least two coupled chains are necessary to visualize charge redistributions along the transversedirection since charge fluctuations within the orbitals are notaccounted for. 14J. Yan, Z. Yuan, and S. Gao, Phys. Rev. Lett. 98, 216602 /H208492007 /H20850. 15W. A. Harrison, Electronic Structure and the Properties of Solids /H20849Freeman, San Francisco, 1980 /H20850.MUNIZ et al. PHYSICAL REVIEW B 80, 045413 /H208492009 /H20850 045413-6

PhysRevB.87.035425.pdf

PHYSICAL REVIEW B 87, 035425 (2013) Optical control of magnetization and spin blockade in graphene quantum dots A. D. G ¨uc¸l¨u and P. Hawrylak Quantum Theory Group, Security and Disruptive Technology, Emerging Technologies Division, National Research Council of Canada, Ottawa, Canada and Department of Physics, University of Ottawa, Ottawa, Canada (Received 11 July 2012; published 23 January 2013) We show that the magnetization of triangular graphene quantum dots with zigzag edges can be manipulated optically. When the system is charge neutral, the magnetic moment can be first erased by addition of a singleelectron spin with a gate, then restored by absorption of a photon. The conversion of a single photon to a magneticmoment results in a many-body effect, optical spin blockade. The effect demonstrated here can potentially leadto efficient spin to photon conversion, quantum memories, and single-photon detectors. DOI: 10.1103/PhysRevB.87.035425 PACS number(s): 73 .22.−f, 73.22.Pr, 78.67.−n, 85.75.−d Graphene quantum dots1–31offer the possibility of in- tegrating electronic, photonic, and magnetic functionalitiesin a single material, carbon, realizing a long standing goal of semiconductor spintronics. 32,33Without external doping or spin injection34but rather through edge, shape, and size engineering, graphene quantum dots are predicted to exhibit a finite magnetic moment while retaining good electronic andoptical properties. 17–22 At present, silicon is the material of choice for elec- tronics, compound semiconductors for optoelectronics andphotonics, and ferromagnets for memory. The integration ofthese different functionalities is the goal of semiconductorspintronics, 32,33which attempts to exploit the spin of the electron in addition to its charge. This requires either efficientspin injection and detection, 34or doping of semiconductors with magnetic ions.32,33In semiconductors doped with Mn ions, ferromagnetism can be controlled with gate voltage bycontrolling carrier density. 33However, the doping responsible for ferromagnetism leads to the degradation of optical prop- erties. On the other hand, graphene, when reduced in sizeto a nanoscale island, has an energy gap tunable by the sizefrom terahertz to UV . 21The zigzag edge of graphene quantum dot leads to a shell of degenerate states in the middle of thegap between valence and conduction bands. In particular, intriangular graphene quantum dots (TGQDs) where all the edgeatoms belong to the same sublattice, Coulomb interactionsamong electrons occupying the half-filled degenerate shelllead to a finite magnetic moment. 23–31Since the shell filling can be controlled by the gate, so are the optical transitionsfrom the filled valence band to the degenerate shell allowingfor the gate tunable optical and magnetic properties. 21,27Here, we show that in TGQDs it is possible to optically control the magnetization through optical spin blockade and hence convert a photon to a magnetic moment. Figure 1schematically shows the process of optical ma- nipulation of the magnetic moment S, total spin, in a TGQDwith zigzag edges. The blue balls illustrate carbon atoms heldtogether by sp 2bonds, and red arrows illustrate pzelectron spin density. When the TGQD is charge neutral [Fig. 1(a)] electrons in the vicinity of zigzag edges align their spin throughexchange interaction, giving rise to a net magnetic momentS. If the TGQD is charged with a single additional electron by a gate, the added electron must have spin opposite tothe magnetization S [Fig. 1(b)]. Through electron-electroninteractions, electrons attempt to align their spin with the added electron, inducing spin depolarization as illustrated inFig.1(c). However, the spin polarization can be recovered by absorption of a single photon. The absorbed photon createsa hole in the valence band (thick arrow) and an electron inthe degenerate shell at zero-energy Fermi level, as shown inFig.1(d). The exchange interaction between the valence hole and all the electrons in the degenerate shell aligns the spinof electrons in the degenerate shell and restores the magneticmoment [Fig. 1(e)]. Hence one can erase the magnetic moment with a gate and restore it optically. It is thus possible to controlthe magnetization of a graphene quantum dot with zigzagedges through optical spin blockade. We now briefly describe the theoretical model and com- putational details that underlie the optical spin blockade. Weconsider a TGQD with zigzag edges and Ncarbon atoms as shown in Fig. 1. We use an effective tight-binding Hamiltonian H TB=/summationtext i,l,σtilc† iσclσfor a single electron on carbon pz orbitals, where the operator c† iσcreates a pzelectron on site “i” with spin σ. The hopping terms tilare taken to be t=−2.5 eV for nearest neighbors and t/prime=−0.1 eV for next-nearest neighbours. Edge atoms are assumed to be passivated by asingle hydrogen atom. The stability of zigzag edges passivatedby hydrogen atoms was previously established theoretically 29 and recently confirmed experimentally.35We illustrate our theory on the example of TGQD with N=97 atoms, the single particle spectrum of which, obtained by diagonalizationof the one-electron Hamiltonian, is shown in Fig. 2(a) (left panel). As discussed already, 21,23–27there exists a shell of degenerate states at the Fermi level (zero-energy states) withdegeneracy N d, where Ndis the number of atoms at one edge minus one,28Nd=7i nF i g . 2(a). In the zero-energy shell the number of electrons Neequals the number of zero-energy states Nd. The exchange interaction aligns the spin of all electrons. This is schematically illustrated in Fig. 2(a) by placing an arrow in each state of the degenerate shell. Withthe zero-energy shell partially occupied, optical transitionsfrom valence to zero-energy band can occur, 21with oscillator strength determined by dipole moments |/angbracketlefti|r|j/angbracketright|2where jis the valence-band state, and iis a zero-energy state. In the remainder of the discussion we will be charging the TGQD with additional electrons from a nearby metallic gate.As discussed in Ref. 27, we first empty the zero-energy shell and perform Hartree-Fock calculation for doubly occupied 035425-1 1098-0121/2013/87(3)/035425(5) Published by the American Physical SocietyA. D. G ¨UC¸L¨U AND P. HAWRYLAK PHYSICAL REVIEW B 87, 035425 (2013) gate e-e interactions photon e-e and e-h interactionsMagnetic moment S Erase S Restore S(a) (b) (d)(c) (e) FIG. 1. (Color online) Schematic illustration of optical control of magnetization and origin of optical spin blockade: Creation of magnetic moment S; erasure of S with addition of a single electron, which through e-e interactions destroys S; restoration of S byabsorption of a single photon that creates an exciton, which restores magnetic moment S through e-e and e-h interactions. valence states separated by energy gap from empty zero- energy and conduction band states. Once the self-consistentHartree-Fock quasiparticle levels |q/angbracketrightfor valence and |p/angbracketrightfor zero-energy states are obtained, we proceed with rotatingthe original interacting electron Hamiltonian to the basis ofquasi-electrons in the zero-energy shell interacting with holesin the valence band, H=/summationdisplay p,σ/epsilon1pb† pσbpσ+/summationdisplay p,σ/epsilon1qh† qσhqσ +1 2/summationdisplay pqrs σσ/prime/angbracketleftpq|V|rs/angbracketrightb† pσb† qσ/primebrσ/primebsσ +1 2/summationdisplay pqrs σσ/prime/angbracketleftpq|V|rs/angbracketrighth† pσh† qσ/primehrσ/primehsσ −/summationdisplay pqrs σσ/prime(/angbracketleftrp|V|sq/angbracketright−(1−δσσ/prime)/angbracketleftrp|V|qs/angbracketright) ×b† pσh† qσ/primehrσ/primebsσ, (1) where b† pσ(h† qσ) creates an electron (hole) in the Hartree-Fock state|p/angbracketright(|q/angbracketright). Next, we construct a basis of all configurations correspond- ing to a given number of electrons Nein the zero-energy shell and holes Nhin the valence band, build a Hamiltonian matrix in the space of configurations, and diagonalize the matrix45 50 55-2-1012Energy (eV) -6.8-6.7-6.6-6.5-6.4Energy (eV) 0123-5.9-5.8-5.7-5.6 Energy (eV) 0123-4.4-4.3-4.2 Energy (eV) Total spin Scharge neutral +1e +1e+1X0.5 1.5 2.5 3.5(a) (b) (c)45 50 55-2-1012Energy (eV) 45 50 55-2-1012Energy (eV) Eigenstate index FIG. 2. (Color online) Noninteracting (left panels) and many- body (right panels) energy spectra showing the ground-state total spin of (a) charge neutral, (b) charged, and (c) charged and photoexcited quantum dot with seven zero-energy states. to obtain eigenstates and eigenenergies of the interacting electron-hole system. The left panel of Fig. 2(a)shows the single-particle energy levels of a noninteracting TGQD. The arrows schematicallyshow a single configuration of N e=7 quasi-electrons with all electron spins aligned. The total spin Sof this spin polarized configuration is S=7/2. There are many other configurations possible with total spin varying from S=7/2t oS=1/2. The low-energy spectra for the charge neutral TGQD fordifferent possible total spin Sare shown in Fig. 2(a), right panel. We see that the ground state, indicated by a circle,indeed corresponds to a maximally spin-polarized state withS=3.5. Hence neutral TGQD carries a magnetic moment as shown in Fig. 1(a). Figure 2(b) shows the effect of the additional electron on single-particle (left) and many-particle (right) spectrum ofTGQD. In a single-particle spectrum, an additional electronis added to the spin-polarized configuration, also shown inFig. 1(b). This electron has a spin opposite to the total spin of the TGQD. Such configuration has a total spin of S= 7/2−1/2=3. Figure 2(b), right panel, shows the low-energy spectrum of the interacting system. The ground state, markedwith a circle, has instead a total spin S=0. Addition of a single electron erased the magnetic moment of a TGQD, as wasillustrated in Fig. 1(c)and discussed earlier in Ref. 27. It has been recently shown that the erasure of the magnetic momentby a single charge is possible up to a critical size of TGQD. 30 The robustness of spin depolarization and its stability against 035425-2OPTICAL CONTROL OF MAGNETIZATION AND SPIN ... PHYSICAL REVIEW B 87, 035425 (2013) temperature is controlled by the energy gaps shown in the energy diagram of Fig. 2(b). These energy gaps are controlled by e-e interactions alone. The magnitude depends on the sizeand inversely on the dielectric screening. Reducing the sizefromN=97 (Fig. 2)t oN=22 atoms increases the energy gap to 20 meV . Reducing screening from κ=6 (Fig. 2)t o κ=1 increases the energy gap from 5meV (Fig. 2)t o3 0m e V . Figure 2(c)shows the effect of absorption of a single photon in a charged TGQD of Fig. 2(b). In the left panel, nonin- teracting single-particle states are shown. The photoexcitedconfiguration consists of a spin-polarized shell, one additionalelectron with opposite spin and a photoexcited opposite spinelectron and a hole in the valence band, i.e., an exciton X.T h e right panel of Fig. 2(c)shows the low-energy spectrum of the interacting electron-hole system. We see that the ground statecorresponds to total spin S=6/2. Since the optically excited exciton Xis in a singlet state, i.e., does not carry net spin, the ground-state total spin S=6/2 corresponds to a configuration shown in Fig. 1(b) and left panel of Fig. 2(b). Hence addition of exciton to the charged TGQD restored the maximallypolarized state. We can understand this remarkable effect asfollows. When the system is photoexcited, a valence electronis transferred into the zero-energy shell leaving a hole behind.The addition of an extra electron to the strongly correlated spinS=0 state does not change the spin polarization, resulting in aS=1/2 spin-depolarized ground state, as shown in Fig. 3. However, if this additional electron is accompanied by the valence hole, a significant rearrangement of electroniccorrelations takes place. The introduction of the valence holespin maximizes the exchange energy between the valence holeand electrons in a degenerate zero-energy shell only if theyhave aligned spins. Hence there is a competition betweenelectronic correlations in the charged degenerate shell, whichdestroy spin polarization and exchange interaction with thevalence hole, which favor the spin-polarized state. Exact 0123413579 1 1 1 3 Initial system Final system (photoexcited) Number of electronsElectron total spin Shell filling 1 5/7 3/7 1/7 9/7 11/7 13/7+1e+X +2e+X +1e+2e FIG. 3. (Color online) Ground-state total spin as a function of filling of the zero-energy band of the system described in Fig. 2, with and without optical activation. Magnetization of the systemis stabilized by the presence of an exciton. Optically allowed and blockaded transitions are shown with blue and red arrows, respectively.diagonalization of the interacting electron system shows that the exchange with the valence hole wins and, as a result,for optically excited system, the total spin is maximized: theelectron total spin is S e=|Nd−2|/2 due to the two extra spins in the zero-energy shell. Since the valence hole totalspin is S h=−1/2, the net spin of the system is given by S=|Nd−1|/2(S=3 in our example). The maximal spin polarization of the photoexcited TGQD is observed not only at filling factor ν=1 but at all filling factors. Figure 3shows the calculated ground-state total electronic spin Seof TGQD as a function of the number of electrons (top) and filling fraction νof the zero-energy shell. The black curve shows the total spin of the initial stateand the red curve shows the total spin after absorption of aphoton, i.e., with exciton X. Without the exciton, away from charge neutrality, depolarization occurs for one added electron,ν=(N d+1)/Nd=8/7, and for two added electrons, ν= (Nd+2)/Nd=9/7. By contrast, the zero-energy shell after illumination is spin polarized at all filling factors. Blue and redarrows show the difference between the total spin of the initialand final photoexcited states. The blue arrow corresponds tospin difference equal to a single-electron spin while the redarrow points to a larger difference. As we demonstrate below, 0123-5.70-5.75-5.80-5.85 1.3 1.4 1.5 1.6 1.71E-51E-41E-30.010.1110100 Energy (eV)Absorption/EmissionEmission from S=3Absorption in S=0Spin blockade-4.4-4.3-4.2 0123-5.85-5.80-5.75-5.70-4.4-4.3-4.2Energy (eV) Total spin S Total spin SSpin blockadeEmission Absorption Spin blockade(a) (b) (c) FIG. 4. (Color online) Allowed and blockaded optical transitions due to spin conservation rule in (a) the absorption and (b) emission many-body spectra. Corresponding absorption and emission lines are shown in (c). 035425-3A. D. G ¨UC¸L¨U AND P. HAWRYLAK PHYSICAL REVIEW B 87, 035425 (2013) the large spin difference between the initial and final states, shown by red dashed arrows in Fig. 3., causes an optical spin blockade in absorption and emission spectra. The spectral function A(ω) describing annihilation of a photon and addition of exciton to a TGQD A(ω)=/summationdisplay f|/angbracketleftMf|P†|Mi/angbracketright|2δ[ω−(Ef−Ei)] (2) involves transitions between the initial many-body state |Mi/angbracketrightand all final states |Mf/angbracketrightconnected by the polarization operator P†=/summationtextδσ¯σ/prime/angbracketleftp|r|q/angbracketrightb† pσ/primeh† qσ, creating an electron in the zero-energy shell and a hole in the valenceband. The many-body matrix element contains a term <f,N e+1,Sf e|b† pσ|Si e,Ne,i > in which an electron with spinσ=±1/2 in a single-particle state pis added to Ne electrons in the initial many-body state iwith total spin Si e. The resulting Ne+1 state with spin S=Si e±1/2m u s th a v e a finite overlap with final state with total spin Sf e. The overlap is finite if the total spin difference between initial and finalmany-body states equals the spin of one added electron. Thecomputed spin difference between the initial and final statesin the absorption process is shown with arrows in Fig. 3.B l u e arrows correspond to allowed transitions with spin differenceof 1/2, while blocked transitions are shown as red arrows. We now discuss the effect of optical spin blockade on the exciton addition and emission spectra. Figures 4(a) and4(b) show blocked (red arrows) and allowed (blue solid arrows)optical transitions during the absorption and subsequentemission processes for TGQD charged with a single electron[Figs. 2(b) and2(c)]. Since the ground state has a total spin S=0 [Fig. 2(b)] and photon creates a singlet exciton, final states must have S=0 [Fig. 4(a)]. The TGQD containing an exciton will relax to its ground state, which from Fig. 2(c) has total spin S=3. The emission from the ground state with S=3 to ground state with S=0 is also spin blockaded, thus the system will go through optical transitions ending withexcited states with S=3. As a result, the absorption and emission spectra are shifted, as shown in Fig. 4(c), where the lowest energy absorption occurs at around 1 .57 eV , while the lowest emission line occurs around 1 .37 eV . The resulting shift between the emission and absorption spectra, 0 .2e Vi n this example, is a direct measure of e-e and e-h interactionsand should be experimentally measurable. In conclusion, while in a doped triangular graphene quantum dot depolarization occurs due to electron-electroninteractions, the magnetization can be recovered by absorptionof a photon due to electron-hole interactions. The conversionof the photon to a magnetic moment results in a many-bodyoptical spin blockade that can be observed in absorptionand emission spectra. Hence, we have demonstrated opticalcontrol of magnetization through spin blockade in graphenequantum dots, which can potentially lead to efficient spin-to-photon conversion, quantum memories, and single-photondetectors. The authors thank P. 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PhysRevB.96.041110.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 96, 041110(R) (2017) Giant planar Hall effect in topological metals A. A. Burkov Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (Received 21 April 2017; published 10 July 2017) Much excitement has been generated recently by the experimental observation of the chiral anomaly in condensed matter physics. This manifests as strong negative longitudinal magnetoresistance and has so farbeen clearly observed in Na 3Bi, ZrTe 5, and GdPtBi. In this Rapid Communication, we point out that the chiral anomaly must lead to another effect in topological metals, the giant planar Hall effect (GPHE), which is theappearance of a large transverse voltage when the in-plane magnetic field is not aligned with the current.Moreover, we demonstrate that the GPHE is closely related to the angular narrowing of the negative longitudinalmagnetoresistance signal, observed experimentally. DOI: 10.1103/PhysRevB.96.041110 The recent theoretical [ 1–8] and experimental [ 9–14] discovery of Dirac and Weyl semimetals has extended thenotions of nontrivial electronic structure topology to metals.It has also reinforced the connection that exists between thephysics of materials with topologically nontrivial electronicstructures and the physics of relativistic fermions. Chiral anomaly, which refers to nonconservation of the chiral charge in the presence of collinear external electric and magneticfields, is a particularly important example that highlights sucha connection. Discovered theoretically by Adler [ 15] and by Bell and Jackiw [ 16] in the relativistic particle physics context, it provided the explanation for the observed fast decay of aneutral pion into two photons, naively not allowed by thechiral charge conservation. Very recently, a condensed mattermanifestation of the chiral anomaly was finally observed inDirac semimetals Na 3Bi [17], ZrTe 5[18], and in a half-Heusler compound GdPtBi [ 19]. The chiral anomaly manifests in Weyl and Dirac semimetals as a very unusual large negative longitudinal magnetoresis- tance, quadratic in the applied magnetic field, as predicted theoretically [ 20–22]. While the existence of the effect, and most of its observed features, are in qualitative agreement withthe theory, one puzzling feature has remained unexplained. Aswas first pointed out in Ref. [ 17], the observed dependence of the magnetoresistance on the angle θbetween the current and the applied magnetic field is against the expectations,drawn from the existing theory. Namely, the theory ofRefs. [ 20–22] naively predicts a cos 2θdependence, due to the quadratic dependence of the chiral anomaly contributionto the conductivity on the magnetic field, but the observedangular dependence appears to be much stronger. In this Rapid Communication, we both explain the angular narrowing phenomenon and connect it with another effect, the giant planar Hall effect (GPHE). Note that a closely related explanation of the angular narrowing effect has already beenproposed by us in Ref. [ 23], but the connection with the GPHE was not understood there. We argue that the presenceof both the negative longitudinal magnetoresistance with acharacteristic dependence on the angle between the currentand the magnetic field, and the GPHE, may be regarded as asmoking gun signature of the chiral anomaly. As was argued in Refs. [ 20–22,24], transport in topological (both Weyl and Dirac) metals is distinguished by the existenceof an extra (nearly) conserved quantity, the chiral charge, which is coupled to the electric charge in the presence of anexternal magnetic field. The hydrodynamic transport equationsfor the electric and the chiral charge have the following form[21,22], ∂n ∂t=D∇2(n+gV)+/Gamma1B·∇(nc+gVc), ∂nc ∂t=D∇2(nc+gVc)−nc+gVc τc+/Gamma1B·∇(n+gV). (1) Here,−enis the electric charge density, and −encis the chiral charge density (defined as the difference between the totalright-handed and total left-handed charge); Dis the diffusion coefficient (we take the diffusion coefficients, corresponding tothe electric and the chiral charges to be the same for simplicity,although they may in general be different due to electron-electron interaction effects); gis the density of states at the Fermi energy; /Gamma1=e/2π 2gis a transport coefficient, which characterizes the chiral anomaly induced coupling betweenthe electric and the chiral charge in the presence of an appliedmagnetic field B; andτ cis the chiral charge relaxation time, which is taken to be long, reflecting the near conservation ofthe chiral charge. We will use ¯ h=c=1 units throughout this Rapid Communication, except in some of the final results. The presence of the electrostatic potential Vand the “chiral electrostatic potential” V c(this is a hypothetical external potential that couples antisymmetrically to the right- andleft-handed charge) reflects the presence of both diffusionand drift contributions to the electric and chiral currentscorrespondingly. In equilibrium the two contributions mustcancel each other, which, in particular, implies n c+gVc=0 in this case. We note that a time-independent spatially uniformV cwill always be present in a noncentrosymmetric topological metal [ 25]. Let us consider an experimental setup, shown in Fig. 1.W e will assume a sample of length Lxin thexdirection, attached to current-carrying normal (i.e., nontopological) metallic leadsatx=±L x/2, and a square (for simplicity) cross section of areaL2 y. Suppose electric current Iis injected and extracted uniformly at the attached leads. We want to find the voltage that 2469-9950/2017/96(4)/041110(5) 041110-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS A. A. BURKOV PHYSICAL REVIEW B 96, 041110(R) (2017) LxLyI FIG. 1. Schematics of the sample setup. Ordinary metal elec- trodes (green) are attached along the whole width of the sample crosssection and inject current into the sample uniformly. The sample has a square cross section of area L 2 yand length Lx. develops in response to this current, or the resistivity tensor of our system. It is convenient to introduce “electrochemical potentials” μ=(n+gV)/gandμc=(nc+gVc)/g. The electric current density is then given by j=σ e∇μ+eg/Gamma1μ cB, (2) where σ=e2gDis the Drude conductivity. The second term in Eq. ( 2) expresses the chiral magnetic effect (CME) [ 26]. Sinceμc=0 in equilibrium, CME vanishes in equilibrium, as it should [ 27,28]. We assume that in the steady state the current only flows in thexdirection, which means jx=I L2y,j y=jz=0. (3) We will also assume that the magnetic field is only rotated in the xyplane (the xzplane is identical). This implies that we may take both electrochemical potentials μandμcto be independent of z. Then Eqs. ( 2) and ( 3) allow us to express ∇μ in terms of the current and the chiral electrochemical potentialμ cas ∂μ ∂x=eI σL2y−μc Lacosθ,∂μ ∂y=−μc Lasinθ, (4) where θ=arctan( By/Bx) is the angle between the applied magnetic field and the current and we have introduced amagnetic field related length scale L a=D /Gamma1B, (5) which will play a crucial role in what follows. Note that this length scale is distinct from the usual magnetic length /lscriptB= 1/√ eBand appears due to the chiral anomaly (hence the subscript a). 1/Laquantifies the strength of the chiral anomaly induced coupling between the electric and the chiral charge.The existence of this length scale was first pointed out inRef. [ 29]. Substituting Eq. ( 4) into the equation for the chiral electrochemical potential, we obtain ∂ 2μc ∂x2+∂2μc ∂y2−μc λ2=−eIcosθ σLaL2y, (6) where λ2=L2 aL2c L2a+L2c, (7)and we have introduced another important length scale Lc=√Dτc, which has the meaning of the chiral charge diffusion length. Transport effects due to the chiral anomaly may beexpected to be significant only when L cis a macroscopic length scale, i.e., when the chiral charge is a nearly conservedquantity. More specifically, as will be shown below, theparameter that determines the strength of the chiral anomalyinduced magnetotransport effects in topological metals is theratio of the two length scales L c/La. We may further simplify Eq. ( 6) by noticing that the condition of no charge current in the ydirection, expressed by the second of Eqs. ( 4), may always be satisfied by taking μcto be independent of y, which then implies uniform gradient of the electrochemical potential μin theydirection. Equation (6) then simplifies to d2μc dx2−μc λ2=−eIcosθ σLaL2y. (8) Equation ( 8) needs to be solved with the appropriate boundary conditions. These are naturally not universal and depend onthe details of the experimental setup being modeled. We willassume that the sample is attached to uniform current carryingleads across the whole width of the sample cross section, andthe lead material is a normal nontopological metal. In thiscase, chiral charge must rapidly relax upon entering the normalleads and the most appropriate boundary condition is thus ofthe Dirichlet type, μ c(x=±Lx/2)=0. (9) We note, however, that the boundary conditions do not affect the final results at all for large sample sizes Lx/greatermuchLa,Lc (the scale dependence of transport coefficients is, however, interesting in its own right in this case and may be observabledue to the large size of L c[29]). Solving Eq. ( 8) with the above boundary conditions, we obtain μc(x)=eIλ2cosθ σLaL2y/bracketleftbigg 1−cosh(x/λ) cosh(Lx/2λ)/bracketrightbigg . (10) Substituting Eq. ( 10) into Eq. ( 4), we may now calculate the voltage drops that develop across the sample in the xand ydirections in response to the current in the xdirection, as integrals of the corresponding electrochemical potentialgradients. We obtain V x=1 e/integraldisplayLx/2 −Lx/2dx∂μ ∂x=ILx σL2y/parenleftbigg 1−λ2 L2acos2θ/parenrightbigg +2Iλ3cos2θ σL2yL2atanh(Lx/2λ), (11) and Vy=1 eLx/integraldisplayLx/2 −Lx/2dx/integraldisplayLy/2 −Ly/2dy∂μ ∂y =−Iλ2cosθsinθ σL2aLy/bracketleftbigg 1−2λ Lxtanh(Lx/2λ)/bracketrightbigg ,(12) where we have averaged Vyalong the length of the sample in thexdirection. 041110-2RAPID COMMUNICATIONS GIANT PLANAR HALL EFFECT IN TOPOLOGICAL METALS PHYSICAL REVIEW B 96, 041110(R) (2017) Equations ( 11) and ( 12) then imply the following result for the scale-dependent resistivity tensor, ρxx=1 σ/parenleftbigg 1−λ2 L2acos2θ/parenrightbigg +2λ2cos2θ σL2aLxtanh(Lx/2λ), ρyx=−λ2sinθcosθ σL2a/bracketleftbigg 1−2λ Lxtanh(Lx/2λ)/bracketrightbigg . (13) The resistivity tensor thus contains both diagonal and off- diagonal components, the off-diagonal ones being induced bythe magnetic field. Since λis always dominated by the shortest of the two length scales L a,c, it is clear that the off-diagonal resistivity vanishes when Ladiverges, making it clear that the origin of the off-diagonal component of the resistivity tensoris the chiral anomaly. More specifically, its origin can be easilytraced back to the CME contribution to the electrical currentin Eq. ( 2). Equation ( 13) may be rewritten in a more illuminating form in terms of ρ /bardblandρ⊥, i.e., diagonal components of the resistivity tensor, corresponding to the current flow alongand perpendicular to the direction of the magnetic field. FromEq. ( 13), we have ρ /bardbl=1 σ/parenleftbigg 1−λ2 L2a/parenrightbigg +2λ3 σL2aLxtanh(Lx/2λ),ρ ⊥=1 σ. (14) Then Eq. ( 13) may be written as ρxx=ρ⊥−/Delta1ρcos2θ, ρ yx=−/Delta1ρsinθcosθ, (15) where /Delta1ρ=ρ⊥−ρ/bardblis the chiral anomaly induced resistivity anisotropy. Equation ( 15) has the form of a standard relation between the anisotropic magnetoresistance (AMR), represented by thefirst equation, and the planar Hall effect (PHE), which isexpressed by the second equation [ 30–32]. Note that the name PHE is a bit of a misnomer: The off-diagonal resistivity doesnot satisfy the antisymmetry property of a true Hall effectρ xy=−ρyx, since it does not originate from the Lorentz force. It is, however, the standard name for this phenomenonin the literature and we will thus use it as well. Both AMRand PHE are well-known phenomena in ferromagnetic metals,originating in this case from the interplay of the magnetic orderand the spin-orbit interactions. Both are typically very weak,but can be much stronger in ferromagnets with significant spin-orbit interactions, such as doped magnetic semiconductors[32]. What is remarkable about our result is that neither AMR nor PHE in a topological metal require magneticorder, originating instead from the chiral anomaly, and theirmagnitude can be extremely large (in fact, approaching thetheoretical upper limit at increasing magnetic field), as weshow below. The sign of the effect in our case is also oppositeto what is typically observed in ferromagnets: /Delta1ρ, as defined in Eq. ( 15), is positive in our case, but would typically be negative in a metallic ferromagnet [ 31]. The parallel resistivity ρ /bardblexhibits a nontrivial dependence on the two intrinsic length scales La,cof the material and on the sample size Lx. Let us first consider the regime of weak magnetic fields, corresponding to La/greatermuchLc. In this case, takingthe limit of large sample size Lx/greatermuchLc, we obtain ρ/bardbl≈1 σ/bracketleftBigg 1−/parenleftbiggLc La/parenrightbigg2/bracketrightBigg , (16) which corresponds to a small negative quadratic magnetic- field-dependent correction to the longitudinal resistivity. ThePHE in this case is also small and given by ρ yx=−1 σ/parenleftbiggLc La/parenrightbigg2 sinθcosθ. (17) A more interesting regime is the regime of stronger magnetic field, corresponding to La/lessmuchLc. Assuming the sample size Lx>La, we obtain ρ/bardbl=L2 a σL2c/parenleftbigg 1+2L2 c LaLx/parenrightbigg . (18) Equation ( 18) exhibits an interesting and nontrivial scale dependence. Indeed, suppose that La<Lx<L2 c/La(the upper limit is a third nontrivial length scale in this problem).In this case, ρ /bardbl≈2La σLx=4π2/lscript2 B e2Lx. (19) To understand the meaning of this result, it is convenient to evaluate the corresponding conductance G/bardbl=ρ/bardblL2 y Lx=e2Nφ 2π, (20) where Nφ=L2 y/2π/lscript2 Bis the number of magnetic flux quanta, penetrating the sample cross section. This is identical to theresult of Ref. [ 29], obtained by a different method. Physically, this corresponds to a regime in which the sample conductanceis dominated by the chiral lowest Landau level, which is wherethe chiral anomaly contribution to Eqs. ( 1) and ( 2) comes from [ 21,22].G /bardblthen corresponds to a conductance of e2/h per lowest Landau level orbital state, i.e., is identical to theconductance of an effective one-dimensional system with N φ conduction channels. Most importantly, the resistivity anisotropy and thus the magnitude of the PHE in this regime is given by /Delta1ρ=1 σ/parenleftbigg 1−2La Lx/parenrightbigg . (21) Thus/Delta1ρis starting to approach its maximal possible value of 1/σwhen the sample size is increased. We thus call this the giant planar Hall effect (GPHE) (this name was first used inrelation to PHE in the context of magnetic semiconductors inRef. [ 32]). For larger sample sizes, when L x>L2 c/La, the magnetic field dependence of the longitudinal resistivity crosses overfrom 1 /Bto 1/B 2, ρ/bardbl≈1 σ/parenleftbiggLa Lc/parenrightbigg2 . (22) The GPHE magnitude in this case becomes independent of the sample size and has reached its maximal magnitude, /Delta1ρ=1 σ(Lc/La)2 1+(Lc/La)2, (23) which converges to 1 /σas the magnetic field is increased. 041110-3RAPID COMMUNICATIONS A. A. BURKOV PHYSICAL REVIEW B 96, 041110(R) (2017) Interestingly, there exists a direct connection between the GPHE and the angular narrowing of the negative longitudinalmagnetoconductivity signal [ 17], as we will now demonstrate. Using Eq. ( 15), we obtain ρ −1 xx(B)−ρ−1 xx(0)=σ(/Delta1ρ/ρ /bardbl) cos2θ 1+(/Delta1ρ/ρ /bardbl)s i n2θ =σ(Lc/La)2cos2θ 1+(Lc/La)2sin2θ. (24) What is missing in the standard expressions for the chiral anomaly induced magnetoconductivity [ 20] is the angular dependence in the denominator in Eq. ( 24). This clearly leads to narrowing of the angular dependence: Equation ( 24)a t small angles has the form of a Lorentzian with the angularwidth /Delta1θ∼L a/Lc, which gets narrower as the magnitude of the chiral anomaly induced GPHE increases. This nontrivialconnection between the GPHE and the angular narrowing ofthe negative longitudinal magnetoresistance signal may beregarded as a smoking gun evidence for the chiral anomaly. In order to relate our results to the existing experimental data [ 17,18], it is useful to express the ratio L c/Laexplicitly in terms of the magnetic field. We obtain Lc La=/Gamma1B/radicalbiggτc D∼/parenleftbigg¯hvF//lscriptB /epsilon1F/parenrightbigg2/radicalbiggτc τ, (25) where vFis the Fermi velocity of the Weyl (Dirac) fermions, /epsilon1Fis the Fermi energy, and τis the momentum relaxation time. Taking the values for these parameters from Ref. [ 17], we have vF≈3.5×107cm/s,/epsilon1F≈30 meV, and τc/τ≈50. Assuming the density of states to be g=/epsilon12 F/2π2¯h3v3 F, and substituting the above values in Eq. ( 25), we obtain Lc/La≈ 0.7B/1T . In Fig. 2we plot both the angular dependence of the inverse longitudinal magnetoresistivity and the magnitudeof the GPHE using L c/Lavalues, which correspond to the range of magnetic fields of up to about 10 T, which wasused in Refs. [ 17,18]. Qualitatively, the behavior of the magnetoresistivity appears to agree with the experimental data.In particular, at low magnetic fields, the magnetoconductivitypeak follows a B 2dependence, while the angular width goes roughly as 1 /B, consistent with Eqs. ( 24) and ( 25). At higher fields both appear to saturate, which is likely explainedby the fact that in both experiments the quantum regime/epsilon1 F<¯hvF//lscriptBis reached at magnetic fields of just a few T. In this regime, we expect g∼Band 1/τ∼B,1/τc∼B, thus making the ratio Lc/Laindependent of the magnetic field. Figure 2also shows that the magnitude of the GPHE may approach its maximal value of /Delta1ρ=1/σfor experimentally accessible values of the magnetic field.FIG. 2. (a) Inverse longitudinal magnetoresistivity as a function of the angle between the current and the magnetic field. Different curves correspond to different values of the ratio Lc/La:2( b l u e , solid), 3 (orange, dashed), and 5 (green, dotted). (b) Dependence of the resistivity anisotropy /Delta1ρand thus the magnitude of the GPHE onLc/La. In conclusion, we have described a magnetotransport effect, related to the chiral anomaly, called the giant planar Hall effect.We have also connected this effect to the angular dependence ofthe longitudinal magnetoconductivity, explaining its magnetic-field-dependent narrowing, pointed out in Ref. [ 17]. 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PhysRevB.77.195423.pdf

Electromagnetic response and pseudo-zero-mode Landau levels of bilayer graphene in a magnetic field T. Misumi and K. Shizuya Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan /H20849Received 29 February 2008; published 15 May 2008 /H20850 The electromagnetic response of bilayer graphene in a magnetic field is studied in comparison to that of monolayer graphene. Both types of graphene turn out to be qualitatively quite similar in dielectric and screen-ing characteristics, especially those deriving from vacuum fluctuations, but the effect is generally much moresizable for bilayers. The presence of the zero- /H20849energy- /H20850mode Landau levels is a feature specific to graphene. In bilayers, unlike in monolayers, the effect of the zero-mode levels becomes visible and even dominant indensity response as an externally controllable band gap develops. It is pointed out that the splitting of nearlydegenerate pseudo-zero-mode levels at each valley, which are specific to bilayer graphene, is controlled by anapplied inplane electric field or by an injected current. In addition, a low-energy effective gauge theory ofbilayer graphene is constructed. DOI: 10.1103/PhysRevB.77.195423 PACS number /H20849s/H20850: 73.43./H11002f, 71.10.Pm, 77.22.Ch I. INTRODUCTION Graphene, a monolayer of graphite, has recently been attracting great attention, both experimentally1–3and theoretically,4–8for its unusual electronic transport, which is characteristic of “relativistic” charge carriers that behave likemassless Dirac fermions. Graphene is naturally of interestfrom the viewpoint of relativistic quantum field theory and isa special laboratory to test the particle-hole picture 9of the quantum vacuum and, especially in a magnetic field, to studypeculiar quantum phenomena 10–14tied to the chiral and par- ity anomalies. Actually, the half-integer quantum Hall /H20849QH/H20850 effect and the presence of the zero-energy Landau levelsobserved 1,2in graphene are a manifestation of spectral asym- metry implied by the anomaly. Bilayer /H20849and multilayer /H20850graphene is as interesting and exotic15,16as monolayer graphene. In bilayer graphene, inter- layer coupling modifies the intralayer relativistic spectra toyield a quasiparticle spectrum with a parabolic energydispersion. 15The relativistic feature thereby disappears but the particle-hole structure still remains, leading to a “chiral”Schrödinger Hamiltonian, which has no analog in particlephysics. Bilayer graphene, like monolayer graphene, is in-trinsically a gapless semiconductor but has a notable prop-erty that the energy gap between the conduction and valencebands is controllable 17–21by use of external gates or chemi- cal doping. Theoretical studies show that electronic transport4–6and screening22in graphene are substantially different from those in standard planar systems. The difference becomes evenmore prominent under a magnetic field: 23For graphene, the vacuum state is a dielectric medium with appreciable electricand magnetic susceptibilities over a whole range of wave-lengths. Curiously, the zero-energy Landau levels, althoughcarrying normal Hall conductance e 2/hper level, scarcely contribute to the dielectric effect. The purpose of this paper is to study the electromagnetic response of bilayer graphene in a magnetic field at integerfilling factor /H9263, in comparison to that of monolayer graphene. It turns out that both types of graphene are qualitatively quitesimilar in dielectric and screening characteristics, but the ef- fect is much more sizable for bilayers /H20849because of the differ- ence in the basic cyclotron energy /H20850. In bilayer graphene, there arise two species of zero-energy levels /H20849or zero modes /H20850, which, when an interlayer field is applied, move up or down /H20849oppositely /H20850at the two valleys. Remarkably, the dielectric effects due to the /H20849pseudo- /H20850zero modes, unlike in monolayers, become visible and even growsteadily as the tunable band gap develops. It is pointed outthat the splitting of nearly degenerate pseudo-zero-mode lev-els at each valley, which are specific to bilayer graphene, iscontrolled by an inplane electric field or by an injected cur-rent. In addition, we construct out of the response a low-energy effective gauge theory of bilayer graphene in a mag-netic field and verify that the electric susceptibility of a QHsystem is generally expressed as a ratio of the Hall conduc-tance to the Landau gap. In Sec. II, we briefly review the low-energy effective theory of bilayer graphene and study its Landau-level spec-trum. In Sec. III, we examine the electromagnetic responseand screening properties of bilayer graphene. In Sec. IV, wederive an effective gauge theory. In Sec. V, we study theeffect of an inplane field on the almost degenerate zero-modelevels at each valley and discuss its consequences. SectionVI is devoted to a summary and discussion. II. BILAYER GRAPHENE Bilayer graphene consists of two coupled hexagonal lat- tices of carbon atoms, arranged in Bernal A/H11032Bstacking, with inequivalent sites denoted as /H20849A,B/H20850in the bottom layer and /H20849A/H11032,B/H11032/H20850in the top layer. The electron fields in the bilayer are described by four-component spinors on the four sites and, asin the case of monolayer graphene, their low-energy spec-trum is governed by the electron states near the twoinequivalent Fermi points Kand K /H11032in the Brillouin zone. The intralayer coupling /H92530/H11013/H9253AB/H110152.9 eV is related to the Fermi velocity in monolayer graphene, v0=/H20849/H208813/2/H20850aL/H92530//H6036 /H11015106m/s, with the lattice constant aL=0.246 nm. The interlayer couplings /H92531/H11013/H9253A/H11032Band/H92533/H11013/H9253AB/H11032are 1 order ofPHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 1098-0121/2008/77 /H2084919/H20850/195423 /H208498/H20850 ©2008 The American Physical Society 195423-1magnitude weaker than the intralayer coupling /H92530; numerically,24/H92531/H110150.30 eV and/H92533/H110150.10 eV. Actually, the neighboring sites A/H11032andBform interlayer dimers via/H92531and get shifted to higher energy bands, and the low-energy sector is essentially described by two-componentspinors residing on the AandB /H11032sites. The effective Hamil- tonian is written as15 H=/H20885d2x/H20851/H9274†/H20849H+−eA0/H20850/H9274+/H9273†/H20849H−−eA0/H20850/H9273/H20852, H/H9264=H0+Has, H0=/H9264v3/H20873/H9016 /H9016†/H20874−1 2m/H11569/H20873/H20849/H9016†/H208502 /H90162/H20874, Has=/H9264U 2/H20875/H208731 −1/H20874−1 /H92531m/H11569/H20873/H9016†/H9016 −/H9016/H9016†/H20874/H20876,/H208492.1/H20850 where coupling to electromagnetic potentials /H20849Ai,A0/H20850is in- troduced through /H9016=/H9016x−i/H9016y,/H9016†=/H9016x+i/H9016y, and/H9016i=−i/H11509i +eAi. Here,/H9264=1 refers to the Kvalley with H+=H/H9264=1and /H9274=/H20849/H9274A,/H9274B/H11032/H20850t, while/H9264=−1 refers to the K/H11032valley with H− =H/H9264=−1and/H9273=/H20849/H9273B/H11032,/H9273A/H20850t. We suppress the electron spin, which is simply taken care of via spin degeneracy gs=2. InH0, the first term with a linear dispersion represents a direct interlayer hopping via /H92533, with a characteristic velocity v3=/H20849/H208813/2/H20850aL/H92533//H6036/H11011v0/30, while the second term with a quadratic dispersion represents a A↔B/H11032hopping via the dimer state, which gives /H60362//H208492m/H11569/H20850=v02//H92531. TheHastakes into account a possible asymmetry between the two layers, which leads to a gap Ubetween the conduc- tion and valence bands. An important feature of bilayergraphene is that such a gap is controllable 17–19by use of external gates, U/H11015e/H9004A0with an interlayer voltage /H9004A0 =A0top−A0bottom. The second term in Hasis a layer asymmetry associated with the depleted charge on the AB/H11032dimer sites; we call it a kinetic asymmetry. Let us place graphene in a strong uniform magnetic field B/H110220 normal to the sample plane; to this end, we set Ai/H20849x/H20850 →AB=B/H20849−y,0/H20850. It is convenient to rescale /H9016=/H208812eBa =/H20849/H208812//H5129/H20850aand/H9016†=/H20849/H208812//H5129/H20850a†with the magnetic length /H5129 =1 //H20881eBso that /H20851a,a†/H20852=1. The kinetic terms thereby acquire the scales v3→v3/H208812//H5129/H11013/H92753and 1 //H208492m/H11569/H20850→eB /m/H11569/H11013/H9275c; nu- merically,/H9275c/H110153.9/H11003B/H20851T/H20852meV and/H92753/H110151.2/H11003/H20881B/H20851T/H20852meV, where B/H20851T/H20852stands for the magnetic field in Tesla. The asso- ciated Landau-level spectra scale like /H92753/H20881nand/H9275c/H20881n/H20849n−1/H20850 with the level index n=0,1,..., with ratio /H110110.3 //H20849/H20881n−1/H20881B/H20851T/H20852/H20850. Therefore, the /H92753term is practically negligible, compared to the /H9275cterm, for higher Landau levels n/H113502 under a strong magnetic field. With this in mind, we cast the Hamiltonian H/H9264in the following form: H/H9264=/H9275c/H20873/H9262/H208491−za†a/H20850/H9261a−/H20849a†/H208502 /H9261a†−a2−/H9262/H208491−zaa†/H20850/H20874, /H208492.2/H20850 with/H9261=/H9264/H92753//H9275c/H11015/H110060.3 //H20881B/H20851T/H20852. Here,/H9262=/H9264U//H208492/H9275c/H20850stands for half of the band gap in units of /H9275c, and it seems feasible18 to achieve an interlayer-voltage change of magnitude /H9262/H11011O/H208491/H20850. Note that the kinetic asymmetry /H11011/H9262za†ais very weak, with z=2/H9275c//H92531/H110150.026/H11003B/H20851T/H20852/H112701. Let us for the moment set a tiny parameter z→0. It is, then, generally seen from the structure of H/H9264that its eigen- modes are the same as those of /H20841H/H9264/H20841/H9262→0and that the spec- trum of H/H9264is symmetric about /H9280=0, except for a possible /H9280=/H9275c/H9262spectrum or the zero-energy spectrum of /H20841H/H9264/H20841/H9262→0. We shall call the /H9280=/H11006/H9275c/H20841/H9262/H20841eigenmodes of H/H9264pseudo-zero- modes, or simply “zero” modes. Such spectra are slightly modulated by the tiny O/H20849/H9262z/H20850 kinetic asymmetry. For /H9261=0, it is possible to explicitly write down the eigenmodes. The “nonzero” modes of H/H9264are Lan- dau levels of energy /H9280n=sn/H9275c/H20881/H20841n/H20841/H20849/H20841n/H20841−1/H20850+/H9262ˆn2−1 2/H9275c/H9262z, /H208492.3/H20850 labeled by integers n=/H110062,/H110063,..., and px/H20849ory0/H11013/H51292px/H20850; /H9262ˆn=/H9262/H208531−/H20849/H20841n/H20841−1 2/H20850z/H20854. Here, sn/H11013sgn/H20853n/H20854=/H110061 specifies the sign of the energy /H9280n. The associated eigenmodes are written as /H9274ny0/H20849x/H20850=1 /H208812/H20851cn+/H9278/H20841n/H20841/H20849x/H20850,−sncn−/H9278/H20841n/H20841−2/H20849x/H20850/H20852t, /H208492.4/H20850 where cn/H11006=/H208811/H11006/H9262ˆn//H9280n/H11032and/H9280n/H11032=sn/H20881/H20841n/H20841/H20849/H20841n/H20841−1/H20850+/H9262ˆn2;/H9278n/H20849x/H20850 =/H9278n/H20849y−y0/H20850/H20849eixy0//H51292//H208812/H9266/H51292/H20850are the eigenfunctions for the Landau levels /H20849n,y0/H20850of the usual Hall electron. In bilayer graphene, there arise two nearly degenerate zero-mode levels per valley and spin, with spectrum /H92800=/H9275c/H9262,/H92801=/H9275c/H9262/H208491−z/H20850, /H208492.5/H20850 and eigenfunctions /H92740y0=/H20851/H92780/H20849x/H20850,0/H20852tand/H92741y0=/H20851/H92781/H20849x/H20850,0/H20852t. We take, without loss of generality, U//H9275c=2/H9264/H9262/H110220 and label the Landau levels associated with the /H9264=1 valley /H20849i.e., the/H9274 sector with/H9262/H110220/H20850byn=0+,1,/H110062,... and those associated with the/H9264=−1 valley /H20849the/H9273sector with/H9262/H110210/H20850byn=0−, −1,/H110062,.... Note that the zero-mode spectrum is ordered according to/H92800−/H11021/H9280−1/H110210/H11021/H92801/H11021/H92800+; see Fig. 1. An interlayer voltage U/H110082/H9262thus works to shift the zero modes oppositely at the two valleys, opening a gap U, while the nonzero-modeFIG. 1. /H20849Color online /H20850Landau-level spectrum. The zero-mode levels move almost linearly with the interlayer voltage /H11008/H9262and oppositely at the two valleys while other levels are shifted onlyslightly.T. MISUMI AND K. SHIZUYA PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-2levels get shifted only slightly and remain symmetric, apart from the O/H20849/H9262z/H20850asymmetry. The zero-mode levels at each valley are degenerate for z →0 and the degeneracy remains even in the presence of the linear kinetic term. This is because the zero modes persisteven for/H9261/HS110050; indeed, D /H9278=0 with D=/H9261a†−a2has two in- dependent solutions of power series in /H9261, /H92780/H11032/H20849x/H20850=/H208491/N0/H20850/H20858 n=0/H11009 /H9251n/H92783n/H20849x/H20850, /H92781/H11032/H20849x/H20850=/H208491/N1/H20850/H20858 n=0/H11009 /H9256n/H92783n+1/H20849x/H20850, /H9251n=/H9261n/H20881/H9003/H208492/3/H20850/H9003/H20849n+1 /3/H20850 3nn!/H9003/H20849n+2 /3/H20850/H9003/H208491/3/H20850, /H208492.6/H20850 with normalization factors N02=/H20858n/H9251n2and N12=/H20858n/H9256n2;/H9256nis given by/H9251nwith replacements 2 /3→4/3 and 1 /3→2/3 in the argument of the gamma functions. These n=0+/H11032,1/H11032 solutions have an infinite radius of convergence in /H9261. This verifies that the index of the Dirac Hamiltonian,index /H20851/H20841H /H9264/H20841/H9262→0/H20852=dim ker D-dim ker D†, is 2, apart from the Landau-level degeneracy: Index /H20851/H20841H/H9264/H20841/H9262→0/H20852=2/H110031 2/H9266/H51292/H20885d2x=/H20885d2xeB /H9266./H208492.7/H20850 The zero-mode spectrum, when corrected by the O/H20849/H9262z/H20850 asymmetry, eventually reads /H92800/H11032=/H9275c/H9262/H208531−zb0/H20849/H9261/H20850/H20854, /H92801/H11032=/H9275c/H9262/H208531−zb1/H20849/H9261/H20850/H20854, /H208492.8/H20850 where b0/H20849/H9261/H20850=/H20858n3n/H9251n2//H20858n/H9251n2and b1/H20849/H9261/H20850=/H20858n/H208493n+1/H20850/H9256n2//H20858n/H9256n2, or b0/H20849/H9261/H20850=/H208491/2/H20850/H92612+ 0.05/H92614+ ... , b1/H20849/H9261/H20850=1+ /H208491/2/H20850/H92612− 0.036/H92614+ ... . /H208492.9/H20850 There is a common level shift of O/H20849z/H92612/H20850while the tiny level splitting /H11011/H9275c/H9262z/H11270/H9275cis practically unchanged /H20849for/H9261/H110110.3/H20850. It is thus generally difficult to resolve the almost degeneratezero-mode levels; in a sense, it is nonzero index /H208492.7/H20850of the Hamiltonian H /H9264that underlies this stability in the zero-mode degeneracy. In connection with the index, it would be worth remarking that D†/H9278=0 has no solution; a solution of infinite power series in 1 //H9261, starting with/H92780/H20849x/H20850, fails to converge for finite /H9261. Actually, the/H9261→/H11009limit corresponds to the case of mono- layer graphene with linear dispersion. The index thus jumpsfrom “2” to “−1” as one passes from finite /H9261to infinite/H9261. The linear kinetic term affects the spectrum /H9280nof the /H20841n/H20841 /H113502 levels only to O/H20849/H92612/H20850, which is still negligible in a strong magnetic field. We shall therefore set /H9261→0 in most of our analysis below, except in Sec. V where we discuss possibleresolution of the zero-mode levels.For actual calculations, it is useful to make the Landau- level structure explicit via the expansion 25,26/H9274/H20849x,t/H20850 =/H20858n,y0/H20855x/H20841n,y0/H20856/H9274n/H20849y0,t/H20850./H20849From now on, we shall only display the/H9274sector since the /H9273sector is obtained by reversing the signs of/H9262and/H9261./H20850The Hamiltonian Hthereby is rewritten as H=/H20885dy0/H20858 n=−/H11009/H11009 /H9274n†/H9280n/H9274n, /H208492.10 /H20850 and the charge density /H9267−p/H20849t/H20850=/H20848d2xeip·x/H9274†/H9274as /H9267−p=e−/H51292p2/4/H20858 k,n=−/H11009/H11009 gkn/H20849p/H20850/H20885dy0/H9274k†eip·r/H9274n, /H208492.11 /H20850 with the following coefficient matrix: gkn/H20849p/H20850=1 2/H20851ck+cn+f/H20841k/H20841,/H20841n/H20841/H20849p/H20850+sksnck−cn−f/H20841k/H20841−2,/H20841n/H20841−2/H20849p/H20850/H20852; /H208492.12 /H20850 r=/H20849r1,r2/H20850=/H20849i/H51292/H11509//H11509y0,y0/H20850stands for the center coordinate with uncertainty /H20851r1,r2/H20852=i/H51292. Here, fkn/H20849p/H20850=/H20881n! k!/H20873i/H5129p /H208812/H20874k−n Ln/H20849k−n/H20850/H208731 2/H51292p2/H20874 /H208492.13 /H20850 fork/H11350n, and fnk/H20849p/H20850=/H20851fkn/H20849−p/H20850/H20852†;p=py+ipx; actually, fkn/H20849p/H20850 are the coefficient functions for the charge density of theordinary Hall electrons. In a similar fashion, one can derive the expression for the current operator j/H11011 /H9254H//H9254A. We omit it here, and simply remark that the current has a component coming from theO/H20849 /H9262z/H20850asymmetry as well. III. ELECTROMAGNETIC RESPONSE In this section, we study the electromagnetic response of bilayer graphene. Let us first consider the polarization func-tion P/H20849p, /H9275/H20850/H20849/H11011−i/H20855/H9267/H9267/H20856/H20850in Fourier space P/H20849p,/H9275/H20850=−/H20858 k,n/H208771 /H9280kn−/H9275+1 /H9280kn+/H9275/H20878/H9268nk/H20849p/H20850, /H9268nk/H20849p/H20850=1 2/H9266/H51292e−/H51292p2/2/H20841gkn/H20849p/H20850/H208412, /H208493.1/H20850 where/H9280kn=/H9280k−/H9280n; the sum is taken over occupied levels /H20853n/H20854 and unoccupied levels /H20853k/H20854. In what follows, we focus on the real part of P/H20849p,/H9275/H20850in the static limit /H9275→0, and denote the components coming from the virtual /H20849n→k→n/H20850transitions as Pnk/H20849p/H20850=−1 /H9266/H512921 /H9280kne−/H51292p2/2/H20841gkn/H20849p/H20850/H208412. /H208493.2/H20850 Actually, /H20841gkn/H20849p/H20850/H208412=gnk/H20849−p/H20850gkn/H20849p/H20850are functions of /H51292p2and are thus symmetric in /H20849k,n/H20850, which implies the relation Pnk =−Pkn. For conventional QH systems, the polarization function vanishes for the vacuum since the charge operator triviallyannihilates the vacuum /H9267/H20841/H9263=0/H20856=0. For graphene, in contrast,ELECTROMAGNETIC RESPONSE AND PSEUDO-ZERO-MODE … PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-3/H9267/H20841/H9263=0/H20856/HS110050 because of pair creation from the Dirac sea, and even the vacuum acquires nonzero polarization /H20841P/H20849p,0/H20850/H20841/H9263=0=/H20858 k=0N /H20858 n=2N P−nk/H20849p/H20850. /H208493.3/H20850 Some care is needed in carrying out sums over an infinite number of Landau levels, which are potentially singular. Forregularization, we here choose, as before, 23to truncate the spectrum to a finite interval − N/H11349n,k/H11349Nand let N→/H11009in the very end. Regularization not only keeps calculations un-der control but also refines them: Actually, instead of Eq./H208493.3/H20850, one may first consider, for a given level n, the polar- ization “per level” P n=/H20858k=−NNPnkby summing over all levels k butn, and then obtain P/H20849p,0/H20850=/H20858nPnby summing over filled levels n. This gives the same result as Eq. /H208493.3/H20850owing to the antisymmetry Pnk=−Pkn. Equation /H208493.3/H20850is an expression for the /H9274sector /H20849P →P/H9274/H20850. For the charge operator /H9267/H9273=/H9273†/H9273in the/H9273sector, one may replace, in Eq. /H208492.11 /H20850,gkn/H20849p/H20850by gkn/H9273/H20849p/H20850=g−k,−n/H20849p/H20850, /H208493.4/H20850 i.e., with the sign of /H9262reversed. One can accordingly define /H20849P/H9273/H20850nk/H20849p/H20850and write the vacuum polarization function as P/H9273/H20841/H20849p,0/H20850/H20841/H9263=0=/H20858k=2N/H20858n=0N/H20849P/H9273/H20850−nk/H20849p/H20850. Note that these compo- nents enjoy the following property: /H20849P/H9274/H20850−nk=− /H20849P/H9274/H20850k−n=/H20849P/H9273/H20850−kn. /H208493.5/H20850 This implies, in particular, that the /H9274and/H9273sectors equally contribute to the vacuum polarization /H20841P/H9274/H20849p,0/H20850/H20841/H9263=0 =/H20841P/H9273/H20849p,0/H20850/H20841/H9263=0, and also that P/H20849p,0/H20850=P/H9274/H20849p,0/H20850+P/H9273/H20849p,0/H20850is the same for the charge-conjugate states with /H9263=/H11006integer. /H20849In view of this, we shall focus on the case /H9263/H113500 from now on./H20850 In calculating the density response, we suppose that the almost degenerate zero-mode levels at each valley practicallyremain inseparable and treat them as both occupied orempty; accordingly, we set z→0 below. Let us first look into the leading long-wavelength part /H11011O/H20849p 2/H20850ofP/H20849p,0/H20850,t ob e denoted as P/H208492/H20850/H20849p,0/H20850, which is related to the electric suscep- tibility/H9251e=−/H20849e2/p2/H20850P/H208492/H20850/H20849p,0/H20850. A look into the matrix ele- ments in Eq. /H208492.12 /H20850shows that Pn/H208492/H20850/H20849p,0/H20850derives only from virtual transitions to the adjacent levels /H20849n→n/H110061/H20850and the related ones across the Dirac sea /H20849n→−n/H110061/H20850. A direct cal- culation then yields Pn/H208492/H20850=−p2 2/H9266/H9275 c/H20873/H9252n−/H9262 /H20841n/H20841/H20849/H20841n/H20841−1/H20850/H20874, /H208493.6/H20850 /H9252n=sn/H20841n/H20841−1 /2 /H20881/H20841n/H20841/H20849/H20841n/H20841−1/H20850+/H92622, /H208493.7/H20850 for 2/H11349/H20841n/H20841/H11349N−1, and P1/H208492/H20850+P0+/H208492/H20850=− /H20849p2/2/H9266/H9275 c/H208502/H20841/H9262/H20841. /H208493.8/H20850 The bottom of the Dirac sea yields P−N/H208492/H20850=/H20849P/H208492/H20850/H20850−NN−1 +/H20849P/H208492/H20850/H20850−N−/H20849N−1/H20850/H11008N−1+ O/H208491/N/H20850, which properly makes /H20841P/H208492/H20850/H20841/H9263=0=/H20858n=2NP−n/H208492/H20850finite. This leads to the vacuum electric susceptibility /H20849per valley and spin /H20850/H9251evac=e2 2/H9266/H9275 cF/H20849/H9262/H20850, F/H20849/H9262/H20850=−/H20858 n=2N−1 /H9252n−/H20841/H9262/H20841+N−1 , /H208493.9/H20850 /H110150.87715 − /H20841/H9262/H20841+O/H20849/H92622/H20850. /H208493.10 /H20850 An alternative expression for this /H9251evacis obtained by sum- ming up the virtual /H20849−n→n/H110061/H20850processes, or /H20858n=2NP−nn−1 +/H20858n=2N−1P−nn+1, which yields F/H20849/H9262/H20850=/H20858 n=1N−11 4/H9280n/H11032/H9280n+1/H11032/H208512/H20849/H9280n+1/H11032−/H9280n/H11032/H20850−/H20849/H9280n+1/H11032−/H9280n/H11032/H208503/H20852 =/H20858 n=1N−1/H20849/H20881n+1− /H20881n−1/H208503 4/H20881n−/H20841/H9262/H20841+O/H20849/H92622/H20850,/H208493.11 /H20850 where/H9280n/H11032=/H20881n/H20849n−1/H20850+/H92622. Equations /H208493.6/H20850and /H208493.8/H20850tell us that the susceptibility carried by a positive-energy level of the same nslightly dif- fers by O/H20849/H9262/H20850terms at the two valleys. Such O/H20849/H9262/H20850corrections are visible only for the zero-mode levels since the two val-leys are practically indistinguishable for n/H113502/H20851up to O/H20849 /H9262z/H20850 splitting /H20852. Let us suppose that the electrons fill up an integral number/H9263of Landau levels, with uniform density /H20855/H9267/H20856/H11013/H9267¯ =/H9263//H208492/H9266/H51292/H20850. We write/H9263=/H20858n/H9263nin terms of the filling factors /H9263nof the nth level /H208510/H11349/H9263n/H113494 for n/H113502 and 0/H11349/H9263/H208531,0/H20854 /H113494;/H9263/H208531,0/H20854/H11013/H92631+/H92630+/H20852, with both valley and spin taken into account. The electric susceptibility at integer filling /H9263is then written as /H9251e=e2 2/H9266/H9275 c/H208534F/H20849/H9262/H20850+/H9263/H208531,0/H20854/H20841/H9262/H20841+/H20858 n/H113502/H9263n/H9252n/H20854. /H208493.12 /H20850 /H20849This/H9251eis even in/H9263, and applies to the case of holes with /H9263/H113490 equally well. /H20850 It is amusing to note here that some expressions greatly simplify for a special value /H9262=1 /2: The spectrum is equally spaced,/H9280n=sn/H9275c/H20849/H20841n/H20841−1 /2/H20850for /H20841n/H20841/H113502 and/H92801=/H92800+=/H9275c/2, apart from the O/H20849/H9262z/H20850splitting. For the susceptibility, one finds/H9252n=1 for n/H113502, and F/H208491/2/H20850=1 /2. This yields /H20841/H9251e/H20841/H9262=1 /2=/H208492/H9254/H92630+/H9263/H20850e2//H208492/H9266/H9275 c/H20850, /H208493.13 /H20850 which rises linearly with /H9263=4,8,12,.... In Fig. 2/H20849a/H20850, we plot/H9251eas a function of the band gap 2/H20841/H9262/H20841/H9275cfor/H9263=0, 4, 8, and 12. The vacuum susceptibility /H9251evac=/H20841/H9251e/H20841/H9263=0is almost comparable to the contribution of the filled n=2 level for/H9262/H110110. The/H9251evacdecreases gradually as the band gap/H11008/H9262develops; this is intuitively clear. In con- trast, when the zero-mode levels get filled, i.e., at /H9263=4,/H9251e starts to grow almost linearly with /H9262, and such characteristic behavior persists for higher /H9263as well. This is somewhat un- expected but is easy to understand: The zero-mode levelsmove linearly with the band gap /H11008/H20841 /H9262/H20841while the n/H113502 levels are shifted only slowly, as seen in Fig. 1. The growth of /H9251eis therefore due to a decrease in activation gap from the zero-mode levels. This implies that the dielectric effect due to theT. MISUMI AND K. SHIZUYA PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-4zero modes, although negligible for /H20841/H9262/H20841/H112701, becomes domi- nant for large gaps /H20841/H9262/H20841/H110111. This is also the case with the full expression for the po- larization P/H20849p,0/H20850in Eq. /H208493.2/H20850, which one can evaluate nu- merically. In Fig. 2/H20849b/H20850, we plot the susceptibility function /H9251e/H20851p/H20852=−/H20849e2/p2/H20850P/H20849p,0/H20850at/H9263=0,4,8,12 for zero band gap and for a finite gap. There, as in the monolayer case, thezero-mode levels scarcely contribute to /H9251e/H20851p/H20852for zero gap, but one clearly sees that as the band gap /H11008/H9262develops, they make/H9251e/H20851p/H20852distinct between the vacuum and the /H9263=4 state. The susceptibility is also related to screening properties of graphene. Let us now turn on the Coulomb interaction v =/H9251//H20849/H9280b/H20841x/H20841/H20850orvp=2/H9266/H9251//H20849/H9280b/H20841p/H20841/H20850with/H9251=e2//H208494/H9266/H92800/H20850/H110151/137 and the substrate dielectric constant /H9280b, and study its effects in the random-phase approximation /H20849RPA /H20850. The RPA dielec- tric function is written as27 /H9280/H20849p,/H9275/H20850=1− vpP/H20849p,/H9275/H20850. /H208493.14 /H20850 Figure 2/H20849c/H20850shows the static function /H9280/H20849p,0/H20850−1 for/H9263= 0 ,4 ,8 , 12, plotted in units of /H208812/H9251//H20849/H9280b/H9275c/H5129/H20850. Note first that there is no screening at long distances, /H9280/H20849p,0/H20850→1 for p→0, as is typi- cal of two-dimensional systems. As wave vector /H20841p/H20841is in- creased,/H9280/H20849p,0/H20850grows rapidly, becomes sizable for /H20841p/H20841/H5129/H110111,and then decreases only gradually for larger /H20841p/H20841. Such profiles of/H9280/H20849p,0/H20850and/H9251e/H20851p/H20852in Fig. 2are qualitatively quite similar to those of monolayer graphene studied earlier.23 Still, there are some clear differences: /H208491/H20850Note that the basic Landau gap of bilayer graphene, /H9275c/H1101545B/H20851T/H20852K, is about 1 order of magnitude smaller than the monolayer gap /H9275cmono/H11015400/H20881B/H20851T/H20852Ka t B=1 T, or /H9275c//H9275cmono/H110150.1/H20881B/H20851T/H20852. /H208493.15 /H20850 Numerically, it turns out that, for both monolayer and bilayer graphene, the vacuum susceptibility /H9251evacis around 3 in units ofe2//H208492/H9266/H9275 c/H20850and the peak value of /H20841/H9280/H20849p,0/H20850/H20841/H9263=0−1 is around 1.8 in units of /H208812/H9251//H20849/H9280b/H9275c/H5129/H20850. This actually means that /H9280/H20849p,0/H20850 and/H9251e/H20851p/H20852are numerically more sizable for bilayers than for monolayers. In particular, for the bilayer, the peak value of /H9280/H20849p,0/H20850in Fig. 2/H20849c/H20850would range from /H9280/H20849p,0/H20850/H110159.5–18 for /H9263=0/H1101112, with the choices /H9280b/H110154 and B=1 T, or/H208812/H9251//H20849/H9280b/H9275c/H5129/H20850/H110155.1. /H20849The value of the substrate dielectric constant/H9280bdepends on the structure of the sample; with SiO 2 on both sides of the bilayer, e.g., one can set /H9280b/H11015/H9280SiO2/H110154./H20850 This implies that the Coulomb interaction is very efficiently screened /H20849and weakened /H20850in bilayer graphene. /H208492/H20850For graphene, unlike standard QH systems, even the /H9263=0 vacuum state has an appreciable amount of polarization /H9280/H20849p,0/H20850−1 over a wide range of wavelengths, which reflects the quantum fluctuations or “echoes” of the Dirac sea inresponse to an applied field. The echoes are in a sense“harder” for monolayer graphene 23for which/H9280/H20849p,0/H20850appears almost constant over the wave-vector range in Fig. 2/H20849c/H20850. The rise of the peak values of /H9280/H20849p,0/H20850−1 with filling factor /H9263is more prominent for monolayers than bilayers; see Eq. /H208494.6/H20850. These features reflect the difference in the underlyingLandau-level structures. /H208493/H20850In bilayer graphene, unlike in monolayers, the effects of the zero-mode Landau levels become visible as the bandgap/H110082/H20841 /H9262/H20841develops and this makes the /H9263=0 and/H9263=4 states distinguishable in /H9280/H20849p,0/H20850and/H9251e/H20851p/H20852. In the RPA, the response function is written as PRPA/H20849p,/H9275/H20850=P/H20849p,/H9275/H20850//H9280/H20849p,/H9275/H20850, from which one can derive28 the inter-Landau-level excitation spectra corrected by the Coulomb interaction. Isolating from P/H20849p,/H9275/H20850one of its poles at/H9275/H11011/H9280k−/H9280nand setting/H9280/H20849p,/H9275/H20850=1− vpP/H20849p,/H9275/H20850→0 fix the pole position of PRPA/H20849p,/H9275/H20850,/H9280k,nRPA=/H9280k−/H9280n+/H17005/H9280k,n/H20849p/H20850with /H17005/H9280k,n/H20849p/H20850/H11015/H9251 /H9280b/H5129/H5129/H20841p/H20841 2/H9280/H20849p,0/H20850/H9263g/H20849p/H20850, /H9263g/H20849p/H20850=/H20858gnk/H20849−p/H20850gkn/H20849p/H20850e−x/x, /H208493.16 /H20850 where x=/H51292p2/2. The vacuum state at /H9263=0 supports excitons associated with the n=−2→/H208491,0 +/H20850or/H20849−1,0 −/H20850→2 transitions with the excitation gap/H9280/H11015/H9275c/H20849/H208812+/H92622+/H9262/H20850and /H9263g/H20849p/H20850=4/H208491−/H9262//H208812+/H92622/H20850h/H20849x/H20850, /H208493.17 /H20850 where h/H20849x/H20850=e−x/H208531+1 4x/H20849x−3/H20850/H20854. The/H9263=4 state supports excitons with energy /H9280/H11015/H9275c/H20849/H208812+/H92622/H11006/H9262/H20850and/H9263g/H20849p/H20850 =2/H208491/H11007/H9262//H208812+/H92622/H20850h/H20849x/H20850, respectively; see Fig. 1. Similarly, at68 1 02468 1 00.51.01.52.02.53.03.5 2.02.53.03.5 240.51.01.5 1.5 2.0 3 40.5 1.02 1 224681012 468101214 FIG. 2. /H20849Color online /H20850/H20849a/H20850Electric susceptibility /H9251edecreases with the band gap 2 /H20841/H9262/H20841/H9275cfor/H9263=0 and rises for /H9263=4, 8, and 12. /H20849b/H20850 Static susceptibility function /H9251e/H20851p/H20852, in units of e2//H208492/H9266/H9275 c/H20850,a t/H9263=0, 4, 8, 12 for zero band gap /H11008/H9262=0 /H20849thick curves /H20850and for a finite gap /H20841/H9262/H20841=0.5 /H20849thin red curves /H20850./H20849c/H20850Static dielectric function /H9280/H20849p,0/H20850−1, in units of /H208812/H9251//H20849/H9280b/H9275c/H5129/H20850,a t/H9263=0/H1101112 for a band gap /H11008/H9262=0 and /H20841/H9262/H20841 =0.5.ELECTROMAGNETIC RESPONSE AND PSEUDO-ZERO-MODE … PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-5/H9263=4n/H20849n/H113502/H20850, the n→n+1 transition with a gap /H17005/H9275c/H20849n/H20850 /H11015/H9280n+1−/H9280ngives rise to excitons with /H9263g/H208490/H20850=/H20849/H20881n−1 +/H20881n+1/H208502. As a result of screening, /H17005/H9280k,n/H20849p/H20850/H110081//H9280/H20849p,0/H20850, one would observe a prominent reduction in magnitude of theexciton spectra; see Fig. 3. IV . EFFECTIVE GAUGE THEORY In this section, we study low-energy response of bilayer graphene and construct an effective gauge theory. Let us con-sider the Hall conductance that is read from a response of the form1 2/H9268H/H20849Ax/H11509tAy−Ay/H11509tAx/H20850; it is calculated from the current- current correlation function and from Berry’s phase aswell. 14A direct calculation similar to the one in the mono- layer case23shows that/H9268H→e2/H51292per electron so that /H9268H →e2//H208492/H9266/H6036/H20850=e2/hper filled level. The result is independent of/H9261andz. Care is needed to determine /H9268Hcarried by the vacuum state. As before, we truncate the spectrum and find that thebottom of the Dirac sea contributes /H11011−/H20849N−1/H20850e 2/h. This yields the following vacuum Hall conductance: /H9268Hvac/H20849/H9264=/H110061/H20850=/H11007e2/h /H208494.1/H20850 per valley and spin. This implies that nonzero current and charge are induced12–14in the vacuums at the two valleys, but they combine to vanish in the vacuum, leaving no ob-servable effect. In general, the Hall conductance /H9268His cast in the form of a spectral asymmetry and the vacuum Hall con- ductance/H9268Hvacis related to half of the index /H208492.7/H20850. The long-wavelength response of bilayer graphene with uniform electron density /H9267¯is now summarized by the follow- ing Lagrangian: LA=/H9267¯eA0−e2/H51292/H9267¯1 2/H9280/H9262/H9263/H9267A/H9262/H11509/H9263A/H9267+1 2/H9251eE/H206482−1 2/H9251m/H20849A12/H208502. /H208494.2/H20850 Here, A0detects the charge density /H9267¯,/H9251eis the electric sus- ceptibility in Eq. /H208493.12 /H20850, and/H9251m/H11008e2/H51292/H9267¯/m/H11569stands for the magnetic susceptibility probed by a local variation A12 =/H11509xAy−/H11509yAxabout B. This response is essentially the same as that for mono- layer graphene,23except for the values of /H9251eand/H9251m. Accord- ingly, the effective theory also takes the same form, i.e., atheory of a vector field b /H9262=/H20849b0,b1,b2/H20850, with the Lagrangian toO/H20849/H115092/H20850Leff/H20851b/H20852=−eA/H9262/H9280/H9262/H9263/H9261/H11509/H9263b/H9261+1 /H51292b0+1 2/H51292/H9267¯b/H9262/H9280/H9262/H9263/H9261/H11509/H9263b/H9261 +1 2/H51292/H9267¯/H9275eff/H20849bk0/H208502−1 2/H9254b12v/H9254b12 /H208494.3/H20850 and the effective cyclotron frequency: /H9275eff=e2/H51292/H9267¯//H9251e=/H9275cg/H20849/H9263/H20850, g/H20849/H9263/H20850=/H9263 4F/H20849/H92622/H20850+/H9263/H208531,0/H20854/H20841/H9262/H20841+/H20858n/H113502/H9263n/H9252n, /H208494.4/H20850 where b/H9262/H9263=/H11509/H9262b/H9263−/H11509/H9263b/H9262;/H51292/H9267¯=/H9263//H208492/H9266/H20850and/H9263=/H20858/H9263n. An advan- tage of bosonization29,30is that it allows one to handle the Coulomb interaction exactly; it is included in Eq. /H208494.3/H20850with shorthand notation /H9254b12v/H9254b12=/H20848d2y/H9254b12/H20849x/H20850v/H20849x−y/H20850/H9254b12/H20849y/H20850 and/H9254b12=b12−/H9267¯./H20849We have omitted the /H9251mterm from Leff/H20851b/H20852 since the Coulomb interaction overtakes it at long wave-lengths. /H20850 This effective Lagrangian not only reproduces the original response /H208494.2/H20850but also shows that the Coulomb interaction vp=2/H9266/H9251//H20849/H9280b/H20841p/H20841/H20850substantially modifies the dispersion of the cyclotron mode at long wavelengths p→0, /H9275/H20849p/H20850/H11015/H9275eff+1 2/H20849/H51292/H9267¯vp+¯/H20850p2, /H208494.5/H20850 where¯involves/H9251m//H20849/H9251e/H9275eff/H20850,i f/H9251mis recovered. This ex- citation spectrum is in good agreement with the RPA result /H208493.16 /H20850at long wavelengths; the filling factor /H9263=2/H9266/H51292/H9267¯in Eq. /H208494.5/H20850corresponds to /H9263g/H208490/H20850in Eq. /H208493.16 /H20850. For graphene, the Landau levels are not equally spaced and the excitation gaps depend on the level index nor/H9263. At/H9263=4n/H20849n/H113501/H20850, e.g., the minimum gap is /H17005/H9275c/H20849n/H20850 =/H20849/H20881n/H20849n+1/H20850+/H92622−/H20881n/H20849n−1/H20850+/H92622/H20850/H9275c, and/H9275effin Eq. /H208494.4/H20850rep- resents such an activation gap. In particular, at /H9263=4 where the zero-mode levels are filled, the effective gap /H11008g/H208494/H20850=1 //H20851F/H20849/H9262/H20850+/H9262/H20852solely derives from the vacuum fluctuations; it deviates from the true gap /H17005/H9275c/H208491/H20850=/H20849/H208812+/H92622−/H9262/H20850/H9275cby about 20% for /H9262=0 and by 5% or less for/H9262/H110220.4. At/H9263=8,/H9275effand/H17005/H9275c/H208492/H20850agree within 1.1% for all/H9262. The agreement is almost exact for higher /H9263.I ti s somewhat surprising that, as in the monolayer case, an effec-tive theory constructed from the long-wavelength responsealone gives an excellent description of the excitation spec-trum. Here, we verify again from Eq. /H208494.4/H20850that the suscep- tibility /H9251eof a QH system is generally given by a ratio of the Hall conductance to the Landau gap,23 /H9251e=/H9263/H20849e2/h/H20850//H9275eff/H11015/H9268H//H17005/H9275c/H20849n/H20850. /H208494.6/H20850 This implies that /H9251edepends nontrivially on /H9263and/H9262while it grows like/H9251e/H11008/H92633/2for monolayer graphene. V . ZERO-MODE LANDAU LEVELS In this section, we take a close look into the properties of the zero-mode levels at each valley, with a tiny splitting /H9275c/H9262z/H11270/H9275c. Special care is needed in considering the re-0 1 2 3 4 5 60.00.20.40.60.8 FIG. 3. /H20849Color online /H20850Exciton spectra at /H9263=0 and 4 for /H20841/H9262/H20841 =0.1, in units of /H9251//H9280b/H5129. The spectra are reduced from the bare ones /H20849dashed curves /H20850via screening. At /H9263=4, there are two spectra with excitation gaps /H11011/H9275c/H20849/H208812+/H92622/H11007/H20841/H9262/H20841/H20850.T. MISUMI AND K. SHIZUYA PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-6sponse from such an almost degenerate sector since even a weak field applied as a probe may affect the true eigen-modes. The first task, therefore, is to resolve the degeneracy of the /H208490 +/H11032,1/H11032/H20850sector by diagonalizing the external probe plus theO/H20849/H9262z/H20850asymmetry, /H9254H=−/H20885d2xeA0/H9267+/H20885dy0/H9274†/H9004Has/H9274, /H208495.1/H20850 where/H9004Has=/H9275c/H9262zdiag /H20851−a†a,aa†/H20852. We suppose that A0is slowly varying in space and retain only terms up to the firstorder in /H11509x, i.e., E=−/H11509xA0. In practice, it is convenient, with- out loss of generality, to take A0/H20849x,t/H20850→A0/H20849y,t/H20850,o rE/H20648Ey. We take the linear kinetic term /H11008/H9261into account /H20849since it may potentially be important at such a low-energy scale /H20850. Within the /H208490+/H11032,1/H11032/H20850sector,/H9254Heffectively turns31into the fol- lowing matrix Hamiltonian /H20849iny0space /H20850: −eA0/H20849y0,t/H20850−/H9260b+/H20849/H9261/H20850+/H20873/H9260b−/H20849/H9261/H20850Ec/H20849/H9261/H20850 Ec/H20849/H9261/H20850−/H9260b−/H20849/H9261/H20850/H20874, /H208495.2/H20850 with E/H11013e/H5129Ey//H208812,/H9260=/H208491/2/H20850/H9275c/H9262z; c/H20849/H9261/H20850 =/H20858n=0/H11009/H208813n+1/H9251n/H9256n//H20849N0N1/H20850. Here, b/H11006/H20849/H9261/H20850=b1/H20849/H9261/H20850/H11006b0/H20849/H9261/H20850in terms of b0andb1, defined in Eq. /H208492.9/H20850. Numerically, c/H20849/H9261/H20850=1−3/H1100310−5/H92614+ ... , b+/H20849/H9261/H20850=1+/H92612+ 0.086/H92614+ ... , b−/H20849/H9261/H20850= 1 − 0.014/H92614+ ... . /H208495.3/H20850 The O/H20849/H92614/H20850corrections are practically negligible for /H9261/H110110.3. We therefore set c/H20849/H9261/H20850/H11015b−/H20849/H9261/H20850/H110151 and may keep the O/H20849/H92612/H20850 correction to b+/H20849/H9261/H20850. Hamiltonian /H208495.2/H20850leads to the level splitting /H11006/H20881/H92602+E2 with the new eigenmodes 0+/H11033and 1 /H11033related to 0+/H11032and 1 /H11032via the unitary transformation /H92740+/H11033=/H92740+/H11032cos/H9258−/H92741/H11032sin/H9258, with tan 2/H9258=e/H5129Ey//H20849/H208812/H9260/H20850. Here, we see that an inplane electric field enhances the splitting of the zero-mode levels, with thegap: /H9004 /H9280=/H9275c/H20881/H20849/H9262z/H208502+2e2/H51292E/H206482//H9275c2. /H208495.4/H20850 This shows, unexpectedly, that the zero-mode level gap is controllable with an injected Hall current. For a rough estimate, let us note the following: If we take /H9262/H110110.05, the level gap is /H9275c/H9262z/H110115/H1100310−3meV at B=1 T, with/H9262z/H110111.3/H1100310−3B/H20851T/H20852; an applied field of strength E/H20648 /H110111V /cm leads to a level gap of roughly the same magni- tude, as seen from /H208812e/H5129/H20841E/H20841//H9275c/H1101110−3/H11003E/H20851V/cm/H20852/B/H20851T/H208523/2. /H208495.5/H20850 One can calculate the spectral weight /H11011/H20841/H208550/H11033/H20841/H9267p/H208411/H11033/H20856/H208412and derive the susceptibility function associated with the 1 /H11033 →0+/H11033transition, /H20849/H9251e/H20851p/H20852/H208501/H110330+/H11033=gse2 2/H9266/H9004/H9280e−x1+ /H20849x/4/H20850E2//H92602 1+E2//H92602, /H208495.6/H20850 with x=/H51292p2/2 and spin degeneracy gs=2. /H20849Here, we have set/H9261→0 for clarity; there is no appreciable change for /H9261/H110110.3. /H20850This/H9251e/H20851p/H20852, with its scale set by the tiny gap /H9004/H9280 /H11270/H9275c, essentially governs the dielectric property of the /H9263=2 state. Figure 4shows/H9251e/H20851p/H20852and the associated dielectric func- tion/H9280/H20849p,0/H20850−1 at/H9263=/H92631/H11033=2. It is seen that both of them decrease rapidly as E/H20648becomes strong in the sequence/H208812e/H5129/H20841E/H20648/H20841//H20849/H9275c/H20841/H9262/H20841z/H20850=0,1,2,5; this is because the 1 /H11033and 0+/H11033 modes are chosen so as to diagonalize /H20848d2xA0/H9267/H20849for/H9260→0/H20850. In Fig. 4, both/H9251e/H20851p/H20852and/H9280/H20849p,0/H20850−1 are plotted in units 1//H20841/H9262z/H20841/H11011/H2084940 //H20841/H9262/H20841/H20850/H208491/B/H20851T/H20852/H20850times as large as those in Fig. 2. Numerically, for /H9262/H110110.05, the level gap /H1101160 mK at B =1 T would become as large as /H11011300 mK /H20849/H110116.2 GHz h/H20850 by an inplane field of E/H20648/H110115V /cm. This would lead to /H9251e/H208510/H20852/H1101114 and /H20849the peak value of /H9280/H20849p,0/H20850−1/H20850/H1101130 in com- mon units, i.e., 1 order of magnitude larger than those in Fig.2. This suggests that the dielectric effect would show a marked enhancement around /H9263/H110112 if the level gap is re- solved by an inplane field at very low temperatures. Actually, a larger gap, which leads to a better level reso- lution, tends to suppress the effect. Alternatively, it will bemore practical to observe the field-induced level gap /H9004 /H9280/H20849E/H20648/H20850 via the QH effect with an injected current; one would be ableto resolve the /H9263=/H110062 Hall plateaus by using a suitably strong current /H20849that may even suppress the dielectric effect /H20850. It will also be possible to detect the level gap directly by micro-wave absorption or via conductance modulations 32associ- ated with it. VI. SUMMARY AND DISCUSSION In this paper, we have studied the electromagnetic re- sponse of bilayer graphene in a magnetic field at integerfilling factor /H9263, with emphasis on clarifying the similarities and differences in quantum features between monolayers andbilayers. Bilayer graphene has a unique feature that its bandgap is externally controllable; this makes bilayers richer inelectronic properties.12340.51.01.52.0 12 3 40.20.40.60.8 FIG. 4. /H20849Color online /H20850/H9251e/H20851p/H20852and/H9280/H20849p,0/H20850−1 at/H9263=2, in units of e2//H208492/H9266/H9275 c/H20841/H9262/H20841z/H20850and/H208812/H9251//H20849/H9280b/H5129/H9275c/H20841/H9262/H20841z/H20850, respectively, for inplane elec- tric fields of strength /H208812e/H5129/H20841E/H20648/H20841//H20849/H9275c/H20841/H9262/H20841z/H20850=0, 1, 2, and 5. The inset depicts an enhancement of level splitting by an inplane field.ELECTROMAGNETIC RESPONSE AND PSEUDO-ZERO-MODE … PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-7The particle-hole picture of the vacuum state is one of the basic features specific to graphene, both monolayers andmultilayers, and is not shared with conventional QH systems.In graphene, even the vacuum state responds to an externalfield and acts as a dielectric medium, with the Coulomb in-teraction being efficiently screened over the scale of themagnetic length. Graphene bilayers and monolayers arequalitatively quite similar in the vacuum dielectric character-istics /H20849apart from some differences that reflect the underlying Landau-level structures /H20850, as we have seen in Sec. III, but, numerically, the dielectric effect /H20849for the vacuum and for /H9263 /HS110050 as well /H20850is generally much more sizable for bilayers because of the difference in the basic Landau gap, /H9275cbi//H9275cmono /H110150.1/H20881B/H20851T/H20852. The presence of the zero-energy Landau levels is another basic feature specific to graphene. The monolayer supportsfour zero-mode levels /H20849one per valley and spin /H20850, while the bilayer supports eight such levels /H20849two per valley and spin /H20850. The zero-mode levels carry normal Hall conductance e 2/h per level, but in monolayers their effect is hardly visible indensity response. In contrast, in bilayer graphene, a gate-controlled interlayer field acts to open a band gap betweenthe zero-mode levels at the two valleys and this valley gap, in a sense, activates them: The dielectric effect due to thepseudo-zero-mode levels grows linearly with the band gapand becomes dominant for large gaps, as we have seen inSec. III.A finite band gap introduces an asymmetry in the zero- mode spectrum while it leaves other levels practically sym-metric. The two zero-mode levels /H20849per spin /H20850at each valley thereby remain degenerate, apart from a tiny kinetic splitting.This robustness in the zero-mode degeneracy at each valleyas well as the presence of the zero modes itself are conse-quences of nonzero index /H208492.7/H20850of the basic bilayer Hamil- tonian /H20849with A 0→0 and z→0/H20850. In Sec. V, we have pointed out that this tiny zero-mode level gap at each valley is en-hanced /H20849and even controlled /H20850by an inplane electric field or by an injected current. Such a gap, if properly enhanced byan injected current, may be detected via the QH effect ordirectly via microwave absorption. Finally, in Sec. IV, we have noted that the low-energy characteristics of bilayer graphene are neatly summarized byan effective Chern-Simons gauge theory, which accommo-dates graphene and standard QH systems equally well. ACKNOWLEDGMENTS The authors wish to thank A. Sawada for useful discus- sions, especially on detection of field-induced level splitting.This work was supported in part by a Grant-in-Aid for Sci-entific Research from the Ministry of Education, Science,Sports and Culture of Japan /H20849Grant No. 17540253 /H20850. 1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,Nature /H20849London /H20850438, 197 /H208492005 /H20850. 2Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature /H20849Lon- don /H20850438, 201 /H208492005 /H20850. 3Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W. Tan, M. Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L. Stormer, and P.Kim, Phys. Rev. Lett. 96, 136806 /H208492006 /H20850. 4Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 /H208492002 /H20850. 5V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 /H208492005 /H20850. 6N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys. Rev. B 73, 125411 /H208492006 /H20850. 7K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602 /H208492006 /H20850. 8J. Alicea and M. P. A. Fisher, Phys. Rev. B 74, 075422 /H208492006 /H20850. 9Note in this connection a proposal to simulate the Klein paradox or tunneling in graphene: M. I. Katsnelson, K. S. Novoselov,and A. K. Geim, Nat. Phys. 2, 620 /H208492006 /H20850. 10R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 /H208491976 /H20850;A .N . Redlich, Phys. Rev. Lett. 52,1 8 /H208491984 /H20850; R. Jackiw, Phys. Rev. D 29, 2375 /H208491984 /H20850. 11A. J. Niemi and G. W. Semenoff, Phys. Rev. Lett. 51, 2077 /H208491983 /H20850. 12G. W. Semenoff, Phys. Rev. Lett. 53, 2449 /H208491984 /H20850. 13F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 /H208491988 /H20850. 14N. Fumita and K. Shizuya, Phys. Rev. D 49, 4277 /H208491994 /H20850. 15E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805 /H208492006 /H20850. 16M. Koshino and T. Ando, Phys. Rev. B 73, 245403 /H208492006 /H20850. 17T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg,Science 313, 951 /H208492006 /H20850. 18E. McCann, Phys. Rev. B 74, 161403 /H20849R/H20850/H208492006 /H20850. 19E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J. M. B. Lopes dos Santos, J. Nilsson, F. Guinea, A. K. Geim, andA. H. Castro Neto, Phys. Rev. Lett. 99, 216802 /H208492007 /H20850. 20H. Min, B. Sahu, S. K. Banerjee, and A. H. MacDonald, Phys. Rev. B 75, 155115 /H208492007 /H20850. 21J. B. Oostinga, H. B. Heersche, X. Liu, A. F. Morpurgo, and L. M. K. Vandersypen, Nat. Mater. 7, 151 /H208492008 /H20850. 22T. Ando, J. Phys. Soc. Jpn. 75, 074716 /H208492006 /H20850; E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418 /H208492007 /H20850; B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J. Phys. 8, 318 /H208492006 /H20850. 23K. Shizuya, Phys. Rev. B 75, 245417 /H208492007 /H20850;77, 075419 /H208492008 /H20850. 24L. M. Malard, J. Nilsson, D. C. Elias, J. C. Brant, F. Plentz, E. S. Alves, A. H. Castro Neto, and M. A. Pimenta, Phys. Rev. B 76, 201401 /H20849R/H20850/H208492007 /H20850. 25S. M. Girvin and T. Jach, Phys. Rev. B 29, 5617 /H208491984 /H20850. 26K. Shizuya, Phys. Rev. B 45, 11143 /H208491992 /H20850;52, 2747 /H208491995 /H20850. 27G. D. Mahan, Many-Particle Physics /H20849Kluwer, Dordrecht/ Plenum, New York, 2006 /H20850. 28C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655 /H208491984 /H20850. 29E. Fradkin and F. A. Schaposnik, Phys. Lett. B 338, 253 /H208491994 /H20850. 30D.-H. Lee and S.-C. Zhang, Phys. Rev. Lett. 66, 1220 /H208491991 /H20850. 31Note that the matrix element /H20855k,y0/H11033/H20841/H20858p/H20849A0/H20850p/H9267−p/H20841n,y0/H11032/H20856is rewritten ase−/H51292p2/4gkn/H20849p/H20850/H20855y0/H11033/H20841A0/H20849r,t/H20850/H20841y0/H11032/H20856, with p=−i/H11509//H11509racting on A0/H20849r,t/H20850; note also that g00/H11015g11/H110151,g1,0+=e/H5129p//H208812, and g0+,1 =e/H5129p¯//H208812t o O/H20849p/H20850. 32R. S. Deacon, K.-C. Chuang, R. J. Nicholas, K. S. Novoselov, and A. K. Geim, Phys. Rev. B 76, 081406 /H20849R/H20850/H208492007 /H20850.T. MISUMI AND K. SHIZUYA PHYSICAL REVIEW B 77, 195423 /H208492008 /H20850 195423-8

PhysRevB.78.054445.pdf

Kondo effect in single-molecule magnet transistors Gabriel González,1,2Michael N. Leuenberger,1,2,*and Eduardo R. Mucciolo2,† 1NanoScience Technology Center, University of Central Florida, Orlando, Florida 32826, USA 2Department of Physics, University of Central Florida, P .O. Box 162385, Orlando, Florida 32816-2385, USA /H20849Received 27 March 2008; revised manuscript received 9 July 2008; published 26 August 2008 /H20850 We present a careful and thorough microscopic derivation of the anisotropic Kondo Hamiltonian for single- molecule magnet /H20849SMM /H20850transistors. When the molecule is strongly coupled to metallic leads, we show that by applying a transverse magnetic field it is possible to topologically induce or quench the Kondo effect in theconductance of a SMM with either an integer or a half-integer spin S/H110221/2. This topological Kondo effect is due to the Berry-phase interference between multiple quantum tunneling paths of the spin. We calculate therenormalized Berry-phase oscillations of the two Kondo peaks as a function of the transverse magnetic field bymeans of the poor man’s scaling. In particular, we show that the Kondo exchange interaction between itinerantelectrons in the leads and the SMM pseudospin 1/2 depends crucially on the SMM spin selection rules for theaddition and subtraction of an electron and can range from antiferromagnetic to ferromagnetic. We illustrateour findings with the SMM Ni 4, which we propose as a possible candidate for the experimental observation of the conductance oscillations. DOI: 10.1103/PhysRevB.78.054445 PACS number /H20849s/H20850: 75.50.Xx, 75.45. /H11001j, 72.10.Fk, 03.65.Vf I. INTRODUCTION Single-molecule magnets /H20849SMMs /H20850, such as Mn 12/H20849see Refs. 1and2/H20850and Fe 8/H20849see Refs. 3and4/H20850, have become the focus of intense research since experiments on bulk samplesdemonstrated the quantum tunneling of a single magneticmoment on a macroscopic scale. These molecules are char-acterized by a large total spin, a large magnetic anisotropybarrier, and a weak in-plane anisotropy, which allow the spinto tunnel through the barrier. Electronic transport throughSMMs offers several unique features with potentially largeimpact on applications such as high-density magnetic storageas well as quantum computing. 5Recent experiments have pointed out the importance of the interference between spintunneling paths in molecules. For instance, measurements ofthe magnetization in bulk Fe 8have observed oscillations in the tunnel splitting /H9004m,−mbetween states Sz=mand − mas a function of a transverse magnetic field at temperatures be-tween 0.05 and 0.7 K /H20849see Ref. 6/H20850. This effect can be ex- plained by the interference between Berry phases associatedwith spin tunneling paths of opposite windings. 7,8Theoreti- cally, a coherent spin-state path-integral formulation is usedto account for the coherence of the virtual states over whichthe spin tunnels, although the initial and final spin states donot retain their coherence. A different approach to the study of SMMs opened up recently with the first observation of quantized electronictransport through an isolated Mn 12molecule.9One expects a rich interplay between quantum tunneling, phase coherence,and electronic correlations in the transport properties ofSMMs. In fact, it has been argued that the Kondo effectwould only be observable for SMMs with half-integerspin 10,11and therefore absent for SMMs, such as Mn 12,F e 8, and Ni 4, where the spin is integer. Later, two of us showed that this is not the case.12Remarkably, a transverse magnetic field H/H11036can be tuned to topologically quench the two lowest levels of a full-integer spin SMM, making them degenerate.In fact, the same Berry-phase interference also affects trans-port for SMMs with half-integer spin. In that case, sweeping H /H11036can lead not only to one but to a series of Kondo reso- nances. In the case of SMMs, as we show below, the Berry-phase oscillations of the tunnel splitting /H9004 m0,m0/H11032lead to an oscillation of the Kondo effect as a function of H/H11036. This means that, at zero bias, the Kondo effect is observable forall values of the magnetic field H /H11036,0such that /H9004m0,m0/H11032/H20849H/H11036,0/H20850 =0. It is interesting to note that, at a finite bias, the Kondo effect in a quantum dot in the presence of a magnetic fieldcan be restored by tuning the bias to eV=/H11006g /H9262BH/H11036/H20849see Ref. 13/H20850. For SMMs, however, the interference between the Berry phases of the molecule total spin makes the distance betweenthe split Kondo peaks, which is equal to eV=/H11006/H9004 m0,m0/H11032and oscillates as a function of H/H11036. A necessary condition for observing these oscillations is a large enough tunnel split-ting. Recently a different SMM based on tetranuclear nickelclusters Ni 4with a S=4 ground state has been synthesized.14 Our motivation for studying this particular nanomagnet stems partly from its high symmetry /H20849S4/H20850but, more impor- tantly, from the large tunnel splittings /H9004m0,−m0/H110110.01 K or larger /H20849depending on the transverse field H/H11036/H20850between the /H20841m/H20856=/H208414/H20856and /H20841m/H11032/H20856=/H20841−4/H20856ground states. Recently, some authors have argued that the Kondo effect is absent at the diabolic points of the Berry-phaseinterference. 11This conclusion came from considering a Kondo Hamiltonian,10,11which was not derived microscopi- cally. Moreover, in recent analysis of the sequential tunnel-ing regime, an exchange Hamiltonian mixed with anAnderson-type Hamiltonian was also used without a micro-scopic derivation. 15In this paper, we provide a careful mi- croscopic derivation of the Kondo Hamiltonian suitable forfull- and half-integer spin single-molecule magnets by meansof a Schrieffer-Wolff transformation. 16By using the exact eigenstates of the positively and negatively charged single-molecule magnet, it is sufficient to apply the Schrieffer-Wolff transformation to the second order in the tunnelingmatrix element. The resulting Kondo exchange parametersPHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 1098-0121/2008/78 /H208495/H20850/054445 /H2084912/H20850 ©2008 The American Physical Society 054445-1exhibit the interference between the second-order transition paths going over the two virtual charged states. We show thatthis very same interference phenomenon is also responsiblefor the Berry-phase blockade of the current through theSMM in the cotunneling regime, which extends our previousresults obtained for the Berry-phase blockade in the sequen-tial tunneling regime. 17 Our derivation of the Kondo Hamiltonian reveals an im- portant detail. The Anderson-type Hamiltonian of the SMMcan be mapped onto a spin-1/2 anisotropic Kondo Hamil-tonian in two different ways. Let S q=0be the total spin of the uncharged single-molecule magnet. /H20849i/H20850If the total spins Sq=1 andSq=−1in the ground state of the positively /H20849q=−1 /H20850and negatively /H20849q=1/H20850charged single-molecule magnets are equal toSq=/H110061=Sq=0−1 /2, then the anisotropic Kondo Hamiltonian exhibits an antiferromagnetic exchange coupling , which cor- responds to the Kondo problem for spin-1/2 impurities orspin-1/2 quantum dots. /H20849ii/H20850IfS q=/H110061=Sq=0+1 /2, then the an- isotropic Kondo Hamiltonian exhibits a ferromagnetic ex- change coupling that leads to a vanishing renormalized transverse exchange coupling /H20849Ising interaction /H20850, in which case the Kondo effect is absent.18 This dependence of the Kondo exchange coupling on the change of the spin was found long ago for the isotropic case,19where the Berry-phase interference is not present. Here, we show that this result also holds true for the aniso-tropic case relevant to SMMs. Thus, in SMMs, the Kondoeffect depends crucially on the spin selection rules for theaddition/subtraction of an electron to/from the molecule.This result is in contrast to the Kondo effect seen in lateralquantum dots, where the exchange coupling is always anti-ferromagnetic due to the fact that spin states are degeneratein the absence of anisotropies. 20 In the following, we provide a complete and detailed de- scription of the Kondo effect in SMMs. Starting from a mi-croscopic model /H20849Sec. II /H20850, we derive the effective Kondo Hamiltonian for a SMM attached to metallic leads throughtunneling barriers /H20849Sec. III /H20850. In Sec. IV , we derive expres- sions for the conductance through a SMM for both zero andfinite bias as a function of a transverse magnetic field and usethe SMM Ni 4to comment the experimental significance of our theoretical results. Our conclusions are summarized inSec. VI. II. MICROSCOPIC HAMILTONIAN The total Anderson-impurity-like Hamiltonian of a system formed by a SMM attached to two metallic leads can beseparated into three terms /H20849see Fig. 1/H20850, H tot=HSMM+Hlead+HSMM-lead . /H208491/H20850 The first term on the right-hand side of Eq. /H208491/H20850denotes the SMM part, which can be broken into spin, orbital, charging,and gate contributions, H SMM=Hspin/H20849q/H20850+Horbital +q2 2U−qeV g, /H208492/H20850 where Udenotes the charging energy, qis the number of excess electrons /H20849the charge state of the molecule /H20850, and Vgisthe electric potential due to an external gate voltage.21In the presence of an external magnetic field, the spin Hamiltonianof the SMM reads, H spin/H20849q/H20850=−AqSq,z2+B2,q 2/H20849Sq,+2+Sq,−2/H20850+B4,q 3/H20849Sq,+4+Sq,−4/H20850 +1 2/H20849h/H11036/H11569Sq,++h/H11036Sq,−/H20850+h/H20648Sq,z, /H208493/H20850 where the easy axis is taken along the zdirection and Sq,/H11006 =Sq,x/H11006iSq,y. The magnetic-field components were rescaled toh/H11036=g/H9262B/H20849Hx+iHy/H20850andh/H20648=g/H9262BHzfor the transversal and longitudinal parts, respectively, where gdenotes the electron gyromagnetic factor. Note that the transverse magnetic fieldlies in the xyplane. In this Hamiltonian, the dominant lon- gitudinal anisotropy term creates a ladder structure in themolecule spectrum where the /H20841/H11006m q/H20856eigenstates of Sq,zare degenerate. The weak transverse anisotropy terms couplethese states. The total spin, as well as the coupling param-eters, depend on the charging state of the molecule. For ex-ample, it is known that Mn 12changes its easy-axis anisotropy constant /H20849and its total spin /H20850from A0=56/H9262eV /H20849Sq=0=10 /H20850to A−1=43/H9262eV /H20849Sq=1=19 /2/H20850and A−2=32/H9262eV /H20849Sq=2=10 /H20850 when singly and doubly charged, respectively.22 The orbital contribution to the SMM energy is given by Horbital =/H20858 n,/H9268/H9255n/H9268/H9274n/H9268†/H9274n/H9268, /H208494/H20850 where /H9274n/H9268†/H20849/H9274n/H9268/H20850creates /H20849annihilates /H20850electrons in the molecular-orbital state nwith spin orientation /H9268and energy /H9255n/H9268. Here we neglect any diamagnetic response to external magnetic fields. The second and third terms on the right-hand side of Eq. /H208491/H20850read, Hlead=/H20858 a,k,/H9268/H9264k/H20849a/H20850/H9274k/H9268,a†/H9274k/H9268,a+1 2/H20849h/H11036/H11569sa,++h/H11036sa,−/H20850+h/H20648sa,z, /H208495/H20850 and HSMM-lead =−/H20858 a,k,/H9268,n/H20851tn,k/H20849a/H20850/H9274k/H9268,a†/H9274n/H9268+ H.c. /H20852, /H208496/H20850 respectively, where sa=/H20858k,k/H11032,/H9268,/H9268/H11032/H20858a=R,L/H9274k/H9268,a†/H20849/H9268/H9268/H9268/H11032/2/H20850/H9274k/H11032/H9268/H11032,a andtn,k/H20849a/H20850is the lead-molecule tunneling amplitude. The opera-VLVR VgqS source drain gatemolecule FIG. 1. /H20849Color online /H20850Schematic illustration of a single- molecule field-effect transistor formed by a SMM attached to twometallic leads /H20849source and drain /H20850and controlled by a back gate voltage.GONZÁLEZ, LEUENBERGER, AND MUCCIOLO PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-2tor/H9274k/H9268,a†/H20849/H9274k/H9268,a/H20850creates /H20849annihilates /H20850electronic states in the a lead /H20849a=R,L/H20850with linear momentum k, spin orientation /H9268, and energy /H9264k/H20849a/H20850. III. KONDO HAMILTONIAN The weak tunneling amplitudes between the leads and the molecule and the large charging energy cause an effectiveexchange interaction between electrons in the leads and thespin of the molecule. Although this interaction at first glanceseems to have the familiar Kondo s−dform, it is actually unusual because the transversal coupling involves only asubspace of the spin of the molecule. Below, we use pertur-bation theory to carefully derive an expression for the effec-tive Kondo Hamiltonian of a SMM. We begin by considering only the SMM and SMM-lead terms in Eq. /H208491/H20850. We divide the Hilbert space into subspaces corresponding to distinct charge sectors of the SMM. Using ablock matrix representation, we have H SMM+HSMM-lead =/H20898/GS Hq−1V 0 V†HqW 0W†Hq+1 /GS/H20899, /H208497/H20850 where Hqis the SMM Hamiltonian for the charge sector q /H20851see Eq. /H208492/H20850/H20852, while VandWrepresent the lead-SMM tun- neling Hamiltonian /H20851Eq. /H208496/H20850/H20852. For certain values of the gate voltage, the charging energy is compensated and the Cou-lomb blockade is lifted. Away from these resonant points,there is an energy gap of order Ubetween consecutive diag- onal elements in Eq. /H208497/H20850. Since U/H11271/H20841t n,k/H20841, we can assume that the off-diagonal elements VandWare small perturbations and use a Schrieffer-Wolff transformation16to decouple dis- tinct charge sectors up to terms of order O/H20849/H20841tn,k/H208412/U2/H20850/H20849see Appendix A /H20850. In the Coulomb blockade valley, when Vg=0 in Eq. /H208492/H20850, the eigenstates of the Hamiltonian Hqare expressed in terms of symmetric and antisymmetric combinations of the eigen-states of the S q,zoperator, namely, /H20841s,a/H20856mq=1 /H208812/H20849/H20841mq/H208560/H11006/H20841−mq/H208560/H20850, /H208498/H20850 with mq=0,1,2,..., SqifSqis an integer and mq =1 2,3 2,..., SqifSqis a half integer.23Note that since the matrices WandVrepresent the addition /H20849subtraction /H20850of an electron to /H20849from /H20850the SMM, only q=0 states that differ in spin projection by one are coupled. Conservation of angularmomentum upon electron tunneling requires that the inter-mediate states in the q=1 or q=−1 sectors obey /H20841S /H110061−S0/H20841 =1 /2. For instance, when the SMM total spin is lowered by the addition or subtraction of an electron, longitudinal spincomponents satisfy m /H110061=S/H110061=S0−1 /2 and − m/H110061=−S/H110061= −S0+1 /2. Therefore, we define the spin states of the ground state to be /H20841↑q/H20856=/H20841mq/H20856and /H20841↓q/H20856=/H20841−mq/H20856. The corresponding eigenenergies are presented in Fig. 2.Let us first consider intermediate states involving the q =1 sector. This situation corresponds to a small positive gatevoltage /H20849V g/H110220/H20850. Using Eq. /H20849A12 /H20850we can write the matrix elements of the reduced Hamiltonian of the q=0 sector in terms of a product of energy denominators and matrix ele-ments of W. To obtain the effective Kondo Hamiltonian, we use the following definition for the operator W: W=−/H20858 a,k,/H9268,ntn,k/H20849a/H20850/H9274k/H9268,a†/H9274n/H9268, /H208499/H20850 with nbeing an occupied molecular orbital for a SMM in the charge state q=1. We will now consider the case when add- ing or subtracting an electron always decreases the SMM total spin, namely, Sq=/H110061=Sq=0−1 /2. This selection rule can be enforced through the adoption of the following matrixelements: 0/H20855↑/H20841/H9274n/H9268/H20841↑/H208561=/H9254m0,m1−/H9268/H9254/H9268,↓, /H2084910/H20850 0/H20855↓/H20841/H9274n/H9268/H20841↓/H208561=/H9254−m0,−m1−/H9268/H9254/H9268,↑, /H2084911/H20850 1/H20855↑/H20841/H9274n/H9268†/H20841↑/H208560=/H9254m0,m1−/H9268/H9254/H9268,↓, /H2084912/H20850 and 1/H20855↓/H20841/H9274n/H9268†/H20841↓/H208560=/H9254−m0,−m1−/H9268/H9254/H9268,↑. /H2084913/H20850 /H20849For the sake of simplicity, we will assume that m0,m1/H110221 2 and/H9268=/H110061 2hereafter. /H20850By using the above selection rules and calculating all the matrix elements of the reduced Hamil-tonian of the q=0 sector /H20849see Appendix A /H20850, one finds the following effective Kondo Hamiltonian: H˜0=H0+/H20858 k/H11032,a/H11032/H20858 k,a/H20851Jk,a;k/H11032,a/H11032z/H90180z/H20849/H9274k↑,a†/H9274k/H11032↑,a/H11032−/H9274k↓,a†/H9274k/H11032↓,a/H11032/H20850 −Jk,a;k/H11032,a/H11032/H11036/H20849/H90180+/H9274k↓,a†/H9274k/H11032↑,a/H11032+/H90180−/H9274k↑,a†/H9274k/H11032↓,a/H11032/H20850 −jk,a;k/H11032,a/H11032/H11036/H20849/H9274k↓,a†/H9274k/H11032↑,a/H11032+/H9274k↑,a†/H9274k/H11032↓,a/H11032/H20850/H20852, /H2084914/H20850 where the last term in Eq. /H2084914/H20850is a scattering term that does∆∆1 0E E E EE (0) (0)(1) (−1) (−1) a,m s,ma,m a,ms,m∆−1 q=−1 q=0 q=1U 2U 2 0 01 −1−1 Es,m1(1) FIG. 2. /H20849Color online /H20850Energy spectra of consecutive charging sectors of the SMM Hamiltonian /H20849q=−1,0,1 /H20850./H9004q/H11013/H9004mq,−mqis the tunnel splitting due to the in-plane anisotropy and Es/a,mq/H20849q/H20850denote energy eigenstates corresponding to symmetric and antisymmetric combinations of the ground eigenstates /H20841/H11006sq/H20856of the longitudinal component of the SMM total spin Sq,z.KONDO EFFECT IN SINGLE-MOLECULE MAGNET … PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-3not affect the dynamics of the SMM and can be neglected. The effective exchange coupling constants that appear in Eq./H2084914/H20850are given by the following expressions: J k,a;k/H11032,a/H11032z=2tk/H20849a/H20850tk/H11032/H20849a/H11032/H20850/H20875U+/H90040 /H20849U+/H90040/H208502−/H900412+U−/H90040 /H20849U−/H90040/H208502−/H900412/H20876, Jk,a;k/H11032,a/H11032/H11036=4tk/H20849a/H20850tk/H11032/H20849a/H11032/H20850/H20875/H90041 /H20849U+/H90040/H208502−/H900412+/H90041 /H20849U−/H90040/H208502−/H900412/H20876, Jk,a;k/H11032,a/H11032/H11036=4tk/H20849a/H20850tk/H11032/H20849a/H11032/H20850/H20875/H90041 /H20849U+/H90040/H208502−/H900412+/H90041 /H20849U−/H90040/H208502−/H900412/H20876. The longitudinal exchange coupling Jk,a;k/H11032,a/H11032zis positive, which indicates an antiferromagnetic Kondo exchange. We have neglected the dependence of the hopping matrix ele-ments on the SMM molecular-orbital number n. This is jus- tified when the addition or subtraction of an electron bringsthe molecule to the electronic ground state of the particularcharge sector. In this sense, only one orbital state can befilled /H20849emptied /H20850when an electron is added /H20849removed /H20850. A diagrammatic representation of the longitudinal and transverse exchanging interactions is shown in Fig. 3. These diagrams differ from the usual Kondo effect in the sense thatthe transverse spin-flipping interaction in a SMM requires aquantum tunneling of the total magnetization during its vir- tual state. If we consider the case where adding or subtracting an electron always increases the total spin in the SMM, namely, S q=/H110061=Sq=0+1 /2, we have to modify Eqs. /H2084910/H20850–/H2084913/H20850by adopting the following matrix elements instead: 0/H20855↑/H20841/H9274n/H9268/H20841↑/H208561=/H9254m0,m1+/H9268/H9254/H9268↑, /H2084915/H20850 0/H20855↓/H20841/H9274n/H9268/H20841↓/H208561=/H9254−m0,−m1+/H9268/H9254/H9268↓, /H2084916/H208501/H20855↑/H20841/H9274n/H9268†/H20841↑/H208560=/H9254m0,m1+/H9268/H9254/H9268↑, /H2084917/H20850 and 1/H20855↓/H20841/H9274n/H9268†/H20841↓/H208560=/H9254−m0,−m1+/H9268/H9254/H9268↓. /H2084918/H20850 Using these selection rules, we arrive at a Kondo Hamil- tonian with exactly the same form as that in Eq. /H2084914/H20850. The expressions for the exchange coupling constants are the same as before, except that the longitudinal coupling Jk,a;k/H11032,a/H11032z changes its overall sign and becomes negative , signaling a ferromagnetic Kondo exchange interaction. However, thestrong anisotropy remains, with the bare longitudinal cou-pling dominant over the transversal one. This result makesdirect contact with the Bethe ansatz study of Aligia et al. , 19 where it was shown that, for an isotropic intermediate- valence rare-earth impurity embedded in a metal, the sign ofthe Kondo exchange coupling depends on the change in theimpurity spin upon charging. Therefore, in our derivation ofthe Kondo Hamiltonian for SMMs, we have generalized thisearly result to the anisotropic case. We can summarize our results so far by stating that; /H20849i/H20850if S q=/H110061=Sq=0−1 /2 then Jk,a;k/H11032,a/H11032z/H11271Jk,a;k/H11032,a/H11032/H11036/H110220/H20849antiferromag- netic exchange coupling /H20850; and /H20849ii/H20850ifSq=/H110061=Sq=0+1 /2 then Jk,a;k/H11032,a/H11032z/H11270−Jk,a;k/H11032,a/H11032/H11036/H110210/H20849ferromagnetic exchange coupling /H20850. IV. CONDUCTANCE AND THE KONDO EFFECT IN A SMM Strong evidence now exists for the Berry-phase interfer- ence between spin paths of opposite windings in molecularnanomagnets. 4Different quantum spin tunneling trajectories can combine and give rise to constructive or destructive in-terference effects, which can be confirmed by measuring thetunnel splitting /H9004as a function of the transverse magnetic field applied along the hard axis of the SMM. It turns out thata magnetic field along the hard anisotropy direction of theSMM can periodically change the tunnel splitting /H20849Berry- phase oscillations /H20850. 6In this section we are going to point out the importance of the topological interference term of theBerry phase for the problem of the conductance and theKondo effect in a single-molecule magnet /H20849SMM /H20850transistor. In order to evaluate the conductance of the SMM in the equilibrium regime subjected to the Hamiltonian /H2084914/H20850,w e make use of the standard poor man’s scaling approach torenormalize the effective exchange coupling constants J zand J/H11036and the gfactor. This will allow us to qualitatively de- scribe the important features of the system above the Kondotemperature. If we want to go below the Kondo temperatureor to address the question of nonequilibrium transport whenT→0, one should use more sophisticated techniques such as numerical renormalization-group techniques 24that are be- yond the scope of this paper. We start by calculating the renormalization flow at the points where the Kondo effect is observable at zero bias, namely, where the tunnel splitting vanishes— /H90040/H20851h/H11036/H20849l/H20850/H20852=0. For half-integer spins, it is reasonable to assume that h/H11036/H20849l/H20850/H11271TK, except for the first zero /H20849l=0/H20850. The total Hamiltonian readstt U1∆z ~z t2 U(S = 0 , s= 0 )∆zS= mzS= m zS= m+ s0 0 0z szsz szszt~ 1 t1 U UUt2 ∆S= − mz zz S = −m +sz 2S= m− sz 0 0S= m0 0 ∆ 1z z 1∆( S = −2m ,0s = +1)z∆ (b)(a) FIG. 3. /H20849Color online /H20850Diagrams representing the /H20849a/H20850longitudi- nal and /H20849b/H20850transversal exchange interactions between the SMM magnetization and the electrons in the leads.GONZÁLEZ, LEUENBERGER, AND MUCCIOLO PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-4Htot=/H20858 m/H20875/H9280m+1 2/H9257/H20849h/H11036/H11569/H9018++h/H11036/H9018−/H20850/H20876+/H20858 k,s/H9264k/H9274ks/H9274ks+H˜0, /H2084919/H20850 where /H9280mis the eigenvalue of /H20841m/H20856forh/H11036=0 and, due to the Knight shift, /H9257=1−/H92670J/H11036/2, with /H92670denoting the density of states of the itinerant electrons at the Fermi energy.20We do not include the Zeeman term for the itinerant electrons in Eq. /H2084919/H20850because at finite values of h/H11036/H20849l/H20850, one has to cut the edges of the spin-up and spin-down bands in the leads to makethem symmetric with respect to the Fermi energy. 20We call Dthe resulting bandwidth. The Hamiltonian /H2084919/H20850remains invariant under renormalization-group transformations. We obtain the fol-lowing flow equations: dJ /H11036 d/H9256=2/H92670J/H11036Jz, /H2084920/H20850 dJz d/H9256=2/H92670/H20849J/H11036/H208502, /H2084921/H20850 and d/H9257 d/H9256=−/H926702J/H11036Jz, /H2084922/H20850 where /H9256=ln /H20849D˜/D/H20850andD˜is the rescaled bandwidth. Dividing Eq. /H2084920/H20850by Eq. /H2084921/H20850and integrating by parts gives /H20849Jz/H208502 −/H20849J/H11036/H208502=C2, where Cis a positive constant.25 We have to distinguish between two cases; Jzis either positive or negative. If Jzis positive, then the exchange cou- pling constants remain antiferromagnetic during the flow butthe exchange interaction becomes increasingly isotropic.Solving Eqs. /H2084920/H20850and /H2084921/H20850yields, 1 2/H92670Carctan h/H20873C Jz/H20874=l n/H20873D˜ TK/H20874. /H2084923/H20850 The solution for J/H11036is determined by Jz=/H20881/H20849J/H11036/H208502+C2. The flow of /H9257is shown in Fig. 4. The flow stops at D˜/H11015/H9275/H11011T /H11022TK. In the antiferromagnetic case the Berry-phase oscilla-tions get strongly renormalized by the scaling of the Knight shift. Since /H20841Jz/H20841/H11271J/H11036, when Jzis negative, the transverse ex- change coupling J/H11036renormalizes to zero. Therefore, in this case the Kondo resonance cannot form and the interactionbecomes Ising-like. The interesting feature of J /H11036=0 is that the Knight shift vanishes. A. Linear conductance In order to calculate the linear conductance through the SMM, we use the following well-known expression valid inthe weak-coupling regime when T K/H11270T:20 G/H20849T/H20850=G0/H20885 −/H11009/H11009 d/H9275/H20873−df d/H9275/H20874/H92662/H926702 16/H20841A/H20849/H9275/H20850/H208412, /H2084924/H20850 where G0is the classical /H20849incoherent /H20850conductance of the molecule, df /d/H9275is the derivative of the Fermi function, and A/H20849/H9275/H20850is the electron-scattering amplitude related to the trans- mission through the SMM with an energy /H6036/H9275above the leads’ Fermi energy. At the end of the scaling flow, the tran-sition amplitude can be calculated in the first-order perturba-tion theory as A D˜/H11015/H9275=J/H9275/H11036=C/H20875/H20849/H9275/TK/H208502/H92670C /H20849/H9275/TK/H208504/H92670C−1/H20876. /H2084925/H20850 The Knight shift is related to the scattering amplitude by /H9257D˜/H11015/H9275=1−/H92670AD˜/H11015/H9275 2. /H2084926/H20850 By making the substitution /H9275→Tinto Eq. /H2084925/H20850, one finds that the linear conductance diverges when T→TK, signaling the onset of the Kondo effect. Since C/H110220, the singularity in Eq. /H2084925/H20850differs from the usual logarithmic behavior found for isotropic exchange interactions. We note, however, that inreality, the conductance does not diverge but is ratherstrongly enhanced near the Kondo temperature. The poorman’s scaling breaks down near the Kondo temperature andmore accurate nonperturbative methods, such as the density-matrix renormalization group, 27have to be employed for ob- taining a quantitative description of the conductance depen-dence on temperature. Using Eq. /H2084924/H20850, we get for the linear conductance, G/H20849T/H20850 G0=/H92662/H926702 16/H20849JD˜/H11015T/H11036/H208502, /H2084927/H20850 which has the same functional form as the result of Ref. 28 for the resistivity of bulk metals in the presence of Kondoimpurities. All the zero points of the Berry-phase oscillation are res- caled by the g-factor renormalization b /H11036/H20849l/H20850=h/H11036/H20849l/H20850//H9257D˜/H11015T. Thus, the zero points become dependent on the contributing states /H20841m/H20856and /H20841−m/H20856. This result indicates that the period of the Berry-phase oscillations becomes temperature dependent atT/H11022T K/H20849see Fig. 5/H20850. This fact allows us to conclude that the scaling equations can be checked experimentally by measur-ing the renormalized zero points of the Berry phase. Further-/CID16/CID18/CID20 /CID22 /CID24/CID17 /CID16 /CID52/CID15/CID52/CID43/CID13/CID17/CID13/CID16/CID14/CID21/CID16/CID16/CID14/CID21/CID17/CID17/CID14/CID21/CID72 FIG. 4. The renormalization of the gfactor due to the Knight shift. As an estimate, we use /H92670J/H11006=/H92670Jz=0.15, where /H92670=9.45 /H110031020J−1/H20849see Ref. 26/H20850.KONDO EFFECT IN SINGLE-MOLECULE MAGNET … PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-5more, due to the scale invariance of the Kondo effect, the distance between the zeros should follow a universal func-tion of T/T K/H20849see Fig. 5/H20850. B. Nonlinear conductance Let us now study the conductance for nonzero bias, V /HS110050. We will assume that eV/H11271kBTKand thus set T/H11270TK.I n this limit, we can make use of the results obtained from thepoor man’s scaling to get a qualitative description of trans-port through the SMM. We begin from the general expres-sion of the current generated in response to an appliedvoltage, 29 I=e h/H20858 /H9268/H20885 0/H11009 dE/H9003L/H9268/H9003R/H9268 /H9003L/H9268+/H9003R/H9268/H9267/H20849E/H20850/H20851fL/H20849E/H20850−fR/H20849E/H20850/H20852, /H2084928/H20850 where /H9267/H20849E/H20850is the energy dependent density of states, /H9003L/H9268 /H20849/H9003R/H9268/H20850is the escape rate for the left /H20849right /H20850lead, and fL/H20849fR/H20850is the Fermi function for the left /H20849right /H20850lead. For the sake of simplicity, we assume /H9003L/H9268=/H9003R/H9268=/H9003. Since, at low tempera- tures, fL/H20849E/H20850−fR/H20849E/H20850/H11015/H208771ifEF−eV /2/H11349E/H11349EF+eV /2, 0 otherwise, /H20878 /H2084929/H20850 where EFis the Fermi energy of the leads, we get the fol- lowing expression for the differential conductance: G=dI dV=/H9266e2 2h/H20885 0/H11009 dE/H20841A/H20849E/H20850/H208412/H20851/H9254/H20849E−EF−eV /2/H20850 −/H9254/H20849E−EF+eV /2/H20850/H20852. /H2084930/H20850 Consider the situation where one moves from the zero point b/H11036/H20849n/H20850to the magnetic-field value b/H11036=b/H11036/H20849l/H20850+/H9004b/H11036, where /H9004b/H11036=/H9004h/H11036//H9257.I f /H20841eV/H20841/H11270/H9004 0/H20849b/H11036/H20850/H11270TK, the transmission ampli- tude is well approximated by Eq. /H2084925/H20850. On the other hand, for /H20841eV/H20841/H11271TK/H11271/H9004 0/H20849b/H11036/H20850, the transmission amplitude is given by AeV=JeV/H11036. For the case /H20841eV/H20841/H11011/H90040/H20849b/H11036/H20850/H11271TK, we can expand AD˜/H11015max /H20851T,/H90040/H20849b/H11036/H20850/H20852up to the second order in perturbation theory at the end of the flow, yielding, AD˜/H20849/H9275/H20850=JD˜/H11036+/H92670/H20885 −D˜+eV /2D˜−eV /2 d/H9280/H11032JD˜/H11036JD˜z /H9275−/H9280/H11032 =JD˜/H11036+/H92670JD˜/H11036JD˜zln/H20879/H9275+D˜−eV /2 /H9275−D˜+eV /2/H20879, /H2084931/H20850 where the integration limits account for the asymmetric cut of the bands /H20849see Fig. 6/H20850. For /H20841eV/H20841/H11011/H90040/H20849b/H11036/H20850/H11022T, the renor- malization flow stops at D˜=/H90040/H20849b/H11036/H20850. Substituting Eq. /H2084931/H20850 into Eq. /H2084930/H20850and setting EF=0, we obtain the differential conductance up to third order in JD˜for both positive and negative biases eV, G G0=/H92662/H926703 16JD˜/H110362JD˜zln/H20873/H90040 /H20648eV/H20648/H20841−/H90040/H20841/H20874, /H2084932/H20850 which agrees with the corresponding expression found in Ref. 28. Equation /H2084932/H20850represents the conductance of the 01234560123456 10 10 0 1 2 3 4 5 60123456 10 10 0 1 2 3 4 5 60123456 10 10 0 1 2 3 4 5 60123456 10 10(b)(a) (c) (d) FIG. 5. The graph shows the temperature dependence of the zeros for the spin ground state of the SMM Ni 4due to the Berry- phase oscillation as a function of the transverse magnetic field forthe tunnel splittings between /H208414/H20856and /H20841−4/H20856states /H20849S 0=4/H20850and the following temperature values: /H20849a/H20850T/TK=1.5, /H20849b/H20850T/TK=1.6, /H20849c/H20850 T/TK=1.7, and /H20849d/H20850T/TK=1.8. We used the following values for the anisotropy constants Aq=0.000 11 eV, B2,q=2.96 /H9262eV, and B4,q= −0.25 /H9262eV.GONZÁLEZ, LEUENBERGER, AND MUCCIOLO PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-6SMM as a function of temperature and tunnel splittings be- tween the /H208414/H20856and /H20841−4/H20856states. In Fig. 7one can see how the conductance depends on the temperature and takes a mini-mum value for those points where the tunnel splitting is zero,which corresponds to the zeros in Fig. 5. Note that our con- ductance formula depends only on the bias voltage and noton the individual chemical potentials of the leads, i.e., ourconductance formula is gauge invariant, which is a result ofthe asymmetric band cutting shown in Fig. 6. The two split Kondo peaks appear at /H20841eV/H20841=/H9004 0/H20849b/H11036/H20850. Thus, the distance between the two peaks oscillates with the mag-netic field, following the renormalized periodic oscillationsof the tunnel splitting /H9004 0/H20849b/H11036/H20850. V. CONDUCTANCE AND THE KONDO EFFECT IN A SMM IN THE COTUNNELING REGIME At zero temperature, the current in a single-electron tran- sistor can be understood as a sequential process of singleelectrons tunneling in and out of the SMM, where the trans-port channel is in between the electrochemical potentials ofthe source and drain reservoir. The electron transport throughthe SMM is also possible for any off-resonant energy, whichis commonly called the cotunneling regime, but it is ex-pected to be very small compared to the sequential tunneling.The cotunneling contribution can be calculated by Fermi’sgolden rule in the second-order perturbation theory, i.e., w i→f=2/H9266 /H6036/H20879/H20855f/H20841H/H20841i/H20856+/H20858 m/H20855f/H20841H/H20841m/H20856/H20855m/H20841H/H20841i/H20856 Em−Ef/H208792 /H9267/H20849E/H20850,/H2084933/H20850 and plays a dominant role whenever the sequential tunneling is suppressed, i.e., /H20855f/H20841H/H20841i/H20856=0, where iandfdenote the initial and final states of the SMM, respectively. Perturbation theorycan be applied when the coupling between the SMM and theleads is weak, i.e., t/H11270U. 20In contrast to the Kondo effect, where the tunnel coupling tof the leads is very large, in the cotunneling regime, tis typically small and, thus, it is nec- essary to apply a gate voltage Vgin order to get close to the resonance condition of, for example, the q=1 charged state. Thus, we need to take only the electron scattering into ac-count, thereby neglecting the hole scattering contribution.Using the incoherent spin states for temperatures around 1K, 17we can calculate the cotunneling contribution by means of Eq. /H2084933/H20850in the following form: w↓→↑=2/H9266 /H6036/H208790/H20855↓/H20841Htot/H20841s/H20856/H20855s/H20841Htot/H20841↑/H208560 U˜/2−/H90041/2 +0/H20855↓/H20841Htot/H20841a/H20856/H20855a/H20841Htot/H20841↑/H208560 U˜/2+/H90041/2/H208792 /H9267/H20849E/H20850, /H2084934/H20850 FIG. 6. Diagrams showing the asymmetric cut of the left- and right-contact bands when a finite bias is applied. 01234560123456 10 10 01234560123456 10 10 0123456012345610 10 01234560123456 10 10(b)(a) (c) (d) FIG. 7. Plots showing the temperature dependence of the con- ductance and the zeros for the tunnel splittings between the spinground states /H208414/H20856and /H20841−4/H20856of the SMM Ni 4/H20849S0=4/H20850due to the Berry-phase oscillation as a function of the transverse magneticfield for the following temperature values: /H20849a/H20850T/T K=1.5, /H20849b/H20850 T/TK=1.6, /H20849c/H20850T/TK=1.7, and /H20849d/H20850T/TK=1.8. We used a bias volt- age of Vb=VL−VR=2.7/H9262V and the following values for the aniso- tropy constants Aq=0.000 11 eV, B2,q=2.96 /H9262eV, and B4,q= −0.25 /H9262eV.KONDO EFFECT IN SINGLE-MOLECULE MAGNET … PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-7where U˜=U+Vgand the factor of 1/2 in the denominator is because of the definition of the energy spectra for the SMMeigenstates given in Fig. 2. Using Eq. /H208498/H20850in Eq. /H2084934/H20850we get, w ↓→↑=2/H9266 /H6036/H208790/H20855↓/H20841H˜/H20841↓/H2085600/H20855↑/H20841H˜/H20841↑/H208560 U˜−/H90041 −0/H20855↓/H20841H˜/H20841↓/H2085600/H20855↑/H20841H˜/H20841↑/H208560 U˜+/H90041/H208792 /H9267/H20849E/H20850. /H2084935/H20850 After some simple algebra, we arrive at the following expres- sion for the tunnel rate process: w↓→↑=2/H9266 /H6036t4/H208792/H90041 U˜2−/H900412/H208792 /H9267/H20849E/H20850, /H2084936/H20850 where tis the lead-molecule tunneling amplitude. Comparing Eq. /H2084936/H20850with Jk,a;k/H11032,a/H11032/H11036, we see that the second contribution to Fermi’s golden rule is proportional to the exchange coupling constant, which is to be expected since the Schrieffer-Wolfftransformation is a perturbative approach of the second or-der. The advantage of using the Schrieffer-Wolff transforma-tion is that you can apply the formalism of the renormaliza-tion theory to get a better description of the physical systemnear the Kondo temperature. Focusing on the Ni 4single-molecule magnet, we use Eq. /H2084936/H20850to calculate the total cotunneling rate between states /H20841↑/H20856=/H208414/H20856and /H20841↓/H20856=/H20841−4/H20856that will contribute to the current flowing through the single-electron transistor, W4,−4=2/H9266t4/H900412 /H6036/H20885 −eV /2−U˜eV /2−U˜dE /H20849E2−/H900412/H208502, /H2084937/H20850 where we integrate over all initial and final states that are available within the range of the bias voltage V. Performing the integration in Eq. /H2084937/H20850yields, W4,−4=4/H9266t4 /H6036/H20875U˜−eV /2 /H20849U˜−eV /2/H208502−/H900412−U˜+eV /2 /H20849U˜+eV /2/H208502−/H900412 +1 2/H90041ln/H20879/H20849/H90041+eV /2/H208502−U˜2 /H20849/H90041−eV /2/H208502−U˜2/H20879/H20876. /H2084938/H20850 The total current flowing through the SMM can be calculated in terms of the density matrix by using the master equation.Following the same procedure as in Ref. 17, we obtain the coupled differential equations, /H9267˙4=/H20873/H90041 /H6036/H2087422/H92534,−4 Vg2//H60362+/H92534,−42/H20849/H9267−4−/H92674/H20850+W4,−4/H9267−4−W−4,4/H92674, /H2084939/H20850 and/H9267˙−4=/H20873/H90041 /H6036/H2087422/H92534,−4 Vg2//H60362+/H92534,−42/H20849/H92674−/H9267−4/H20850+W−4,4/H92674−W4,−4/H9267−4, /H2084940/H20850 where /H92534,−4is the incoherent tunneling rate from the lead to the molecule. Solving the set of differential equations for /H92674 and/H9267−4for the stationary case, we obtain the current flowing through the SMM for the case where the source and the drainleads are oppositely spin polarized, as described in Ref. 17, I=eW 4,−4/H9267−4=2e/H92534,−4/H900412W4,−4 W4,−4/H20849Vg2+/H92534,−42/H60362/H20850+4/H92534,−42/H900412. /H2084941/H20850 Figure 8shows the cotunneling current as a function of the transverse magnetic field. Interestingly, the current is sup-pressed at the zeros of the tunnel splittings /H9004 1and/H90040, ex- actly as in the sequential tunneling regime.17 VI. CONCLUSIONS The main contribution of this paper is to show how the total Hamiltonian of a SMM transistor can be mapped intothe Kondo Hamiltonian by means of a Schrieffer-Wolfftransformation. While the derivation of the effective KondoHamiltonian in other contexts /H20851such as quantum dots and ordinary single /H20849nonmagnetic /H20850molecule coupled to leads /H20852is well known, the case is different for SMM. The dominantKondo effect is unusual for a SMM since it involves a pseu-dospin of the molecule rather than its total spin. We showthat if the total spin of the molecule is reduced /H20849increased /H20850 for the charged states, then the Kondo Hamiltonian exhibitsan antiferromagnetic /H20849ferromagnetic /H20850coupling, which leads to the screening /H20849antiscreening /H20850of the total spin of the SMM. In the case of antiferromagnetic coupling, the renormaliza-tion leads to a Kondo effect, i.e., the conductance through theSMM exhibits a resonance at the Fermi energy. In the case ofthe ferromagnetic coupling, the transverse exchange is renor-malized to zero, in which case the Kondo resonance is ab-100200300 100200300-40-35-30 100200300405 Hy[T]Hx [T] FIG. 8. /H20849Color online /H20850Graph showing log 10Iversus the trans- verse magnetic field for Vb=VL−VR=4/H1100310−3eV, Vg=0.01 eV, and/H92534,−4=1012s−1. One can see that the current is suppressed at the zeros of the tunnel splittings /H90041and/H90040.GONZÁLEZ, LEUENBERGER, AND MUCCIOLO PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-8sent. This result is in contrast to the case of the Kondo effect in lateral quantum dots exhibiting only antiferromagnetic ex-change coupling, which is due to the fact that all the spinstates are degenerate in the absence of anisotropies. 20The standard Kondo screening of the molecule magnetization byitinerant electrons in the leads is a very weak effect in thiscontext, given the large spin of a SMM /H20849its onset is therefore likely to occur only at exceedingly small temperatures,which are inaccessible to current experiments /H20850. A careful derivation of the effective Kondo Hamiltonian shows the strong dependence of this phenomenon on theamplitude and orientation of an external magnetic field. Thestrong uniaxial magnetic anisotropy of the SMM combinedwith the weaker in-plane anisotropy creates a pseudospin 1/2involving states with opposite magnetization orientation. Atransverse magnetic field modulates the tunnel barrier be-tween these states through a Berry-phase interference effect.That, in turn, modulates periodically the Kondo effect inSMMs. We have calculated the conductance of the single- molecule transistor in the presence of the Kondo effect byusing the standard poor man’s scaling approach. We haveshown that in the case of antiferromagnetic Kondo exchangecoupling by applying a transverse magnetic field to a SMMwith a large full- or half-integer spin S/H110221/2, it is possible to topologically induce or quench the Kondo effect of the con-ductance of a current through the SMM that is sufficientlywell coupled to metallic leads. We have also shown how thezero points of the Berry-phase oscillation become tempera-ture dependent above the Kondo temperature and how theychange direction within the plane /H20849Fig. 5/H20850. The latter indi- cates that the parity of the Berry-phase oscillations 7,8 changes from an integer spin S=4 to a half-integer spin S =7 /2. We have also shown how this motion affects the tem- perature dependence of the conductance /H20849see Fig. 7/H20850. Inter- estingly, the maximum value of the conductance encirclesthe zeros of the Berry-phase oscillation, providing a mecha-nism for establishing the location of these zeros when theorientation of the molecule symmetry axis with respect to themetallic contacts is not known. We illustrate these features ofthe conductance of a SMM using, as an example, the SMMNi 4. In our view, due to its large ground-state tunnel splitting, this is currently the best SMM available for the experimentalobservation of the Berry-phase oscillations of the Kondoresonance.ACKNOWLEDGMENTS The authors gratefully acknowledge useful discussions with George Christou, Enrique del Barco, Leonid Glazman, Chris Ramsey, and Peter Schmitteckert. E.R.M. acknowl-edges partial support from the NSF under Grant No. CCF0523603 and from the I 2Lab at UCF. He also thanks the Max-Planck Institute for the Physics of Complex Systemsfor their hospitality. M.N.L. acknowledges partial supportfrom the NSF under Grant No. ECCS 0725514. APPENDIX A: DERIVATION OF THE ANISOTROPIC KONDO HAMILTONIAN For simplicity, let us consider the sectors q=−1,0,+1 only, where qdenotes the number of excess electrons and write,30 H=A+B, /H20849A1/H20850 where A=/H20898H−1 00 0H00 00 H+1/H20899,B=/H208980V 0 V†0W 0W†0/H20899./H20849A2/H20850 Using the similarity transformation H˜=eTHe−T, where Tis anti-Hermitian, we have, H˜=H+/H20851T,H/H20852+1 2†T,/H20851T,H/H20852‡+¯. /H20849A3/H20850 We want to determine Tsuch that B+/H20851T,A/H20852=0. For that purpose, it is sufficient to assume that Thas the form T=/H208980 C 0 −C†0D 0 −D†0/H20899, /H20849A4/H20850 withCandDsatisfying H−1C−CH 0=V /H20849A5/H20850 and H0D−DH +1=W, /H20849A6/H20850 respectively. Thus, H˜=A+1 2/H20851T,B/H20852+O/H20849B3/H20850, /H20849A7/H20850 where /H20851T,B/H20852=/H20898CV†+VC†0 CW−VD 0 DW†+WD†−C†V−V†C 0 W†C†−D†V†0 −D†W−W†D/H20899. /H20849A8/H20850KONDO EFFECT IN SINGLE-MOLECULE MAGNET … PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-9Note that the neutral sector /H20849q=0/H20850has been decoupled from the charged sectors at the expense of adding two contribu-tions of order O/H20849V,W/H20850 2toH0. To specify the form of these contributions, we use the eigenbasis /H20853/H20841/H9251/H20856q/H20854ofHq, namely, q/H20855/H9251/H20841Hq/H20841/H9252/H20856q=/H20851Hq/H20852/H9251/H9252=/H9254/H9251,/H9252E/H9251/H20849q/H20850. /H20849A9/H20850 Equations /H20849A5/H20850and /H20849A6/H20850can be solved in this representation to yield, −1/H20855/H9251/H20841C/H20841/H9252/H208560=/H20851C/H20852/H9251/H9252=/H20851V/H20852/H9251/H9252 E/H9251/H20849−1/H20850−E/H9252/H208490/H20850, /H20849A10 /H20850 and 0/H20855/H9251/H20841D/H20841/H9252/H20856+1=/H20851D/H20852/H9251/H9252=/H20851W/H20852/H9251/H9252 E/H9251/H208490/H20850−E/H9252/H20849+1/H20850, /H20849A11 /H20850 respectively, where /H20851V/H20852/H9251/H9252=−1/H20855/H9251/H20841V/H20841/H9252/H208560and /H20851W/H20852/H9251/H9252 =0/H20855/H9251/H20841W/H20841/H9252/H20856−1. This allows us to write the following matrix elements for the neutral sector: 0/H20855/H9251/H20841DW†/H20841/H9252/H208560=/H20858 /H9253/H20851W/H20852/H9251/H9253/H20851W†/H20852/H9253/H9252 E/H9251/H208490/H20850−E/H9253/H20849+1/H20850, /H20849A12 /H20850 and 0/H20855/H9251/H20841V†C/H20841/H9252/H208560=/H20858 /H9253/H20851V†/H20852/H9251/H9253/H20851V/H20852/H9253/H9252 E/H9253/H20849−1/H20850−E/H9252/H208490/H20850. /H20849A13 /H20850 In order to evaluate the matrix elements of the reduced Hamiltonian of the q=0 sector /H20851Eqs. /H20849A12 /H20850and /H20849A13 /H20850/H20852,w e insert complete eigenvector sets for each sector, i.e., I/H20849q/H20850 =/H20841s/H20856qq/H20855s/H20841+/H20841a/H20856qq/H20855a/H20841, in the following way: 0/H20855↑/H20841DW†+WD†/H20841↑/H208560=0/H20855↑/H20841s/H2085600/H20855s/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↑/H208560 +0/H20855↑/H20841a/H2085600/H20855a/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↑/H208560 +0/H20855↑/H20841s/H2085600/H20855s/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↑/H208560 +0/H20855↑/H20841a/H2085600/H20855a/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↑/H208560+ H.c., /H20849A14 /H20850 0/H20855↓/H20841DW†+WD†/H20841↓/H208560=0/H20855↓/H20841s/H2085600/H20855s/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↓/H208560 +0/H20855↓/H20841a/H2085600/H20855a/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↓/H208560 +0/H20855↓/H20841s/H2085600/H20855s/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↓/H208560 +0/H20855↓/H20841a/H2085600/H20855a/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↓/H208560+ H.c., /H20849A15 /H20850 and 0/H20855↑/H20841DW†+WD†/H20841↓/H208560=0/H20855↑/H20841DW†/H20841↓/H208560+/H208490/H20855↓/H20841DW†/H20841↑/H208560/H20850†, /H20849A16 /H20850 where, 0/H20855↑/H20841DW†/H20841↓/H208560=0/H20855↑/H20841s/H2085600/H20855s/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↓/H208560 +0/H20855↑/H20841a/H2085600/H20855a/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↓/H208560 +0/H20855↑/H20841s/H2085600/H20855s/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↓/H208560+0/H20855↑/H20841a/H2085600/H20855a/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↓/H208560,/H20849A17 /H20850 and 0/H20855↓/H20841DW†/H20841↑/H208560=0/H20855↓/H20841s/H2085600/H20855s/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↑/H208560 +0/H20855↓/H20841a/H2085600/H20855a/H20841D/H20841s/H2085611/H20855s/H20841W†/H20841↑/H208560 +0/H20855↓/H20841s/H2085600/H20855s/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↑/H208560 +0/H20855↓/H20841a/H2085600/H20855a/H20841D/H20841a/H2085611/H20855a/H20841W†/H20841↑/H208560./H20849A18 /H20850 Using Eq. /H20849A11 /H20850and the fact that Ea/H20849+1/H20850=Es/H20849+1/H20850+/H90041, 0/H20855↑/H20841s/H208560=0/H20855↓/H20841s/H208560=0/H20855↑/H20841a/H208560=1 //H208812, and0/H20855↓/H20841a/H208560=−1 //H208812, we find, after some algebra, 0/H20855↑/H20841DW†+WD†/H20841↑/H208560=1 /H208812/H9004s/H9004sa/H208510/H20855s/H20841/H20849/H9004sWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↑/H208560/H20852 +1 /H208812/H9004a/H9004as/H208510/H20855a/H20841/H20849/H9004asWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↑/H208560/H20852+ H.c., /H20849A19 /H20850 0/H20855↓/H20841DW†+WD†/H20841↓/H208560=1 /H208812/H9004s/H9004sa/H208510/H20855s/H20841/H20849/H9004sWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↓/H208560/H20852 −1 /H208812/H9004a/H9004as/H208510/H20855a/H20841/H20849/H9004asWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↓/H208560/H20852+ H.c., /H20849A20 /H20850 0/H20855↑/H20841DW†/H20841↓/H208560=1 /H208812/H9004s/H9004sa/H208510/H20855s/H20841/H20849/H9004sWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↓/H208560/H20852 +1 /H208812/H9004a/H9004as/H208510/H20855a/H20841/H20849/H9004asWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↓/H208560/H20852, /H20849A21 /H20850 and 0/H20855↓/H20841DW†/H20841↑/H208560=1 /H208812/H9004s/H9004sa/H208510/H20855s/H20841/H20849/H9004sWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↑/H208560/H20852 −1 /H208812/H9004a/H9004as/H208510/H20855a/H20841/H20849/H9004asWI/H208491/H20850W† −/H90041W/H20841s/H2085611/H20855s/H20841W†/H20850/H20841↑/H208560/H20852, /H20849A22 /H20850 where /H9004s=Es/H208490/H20850−Es/H20849+1/H20850,/H9004sa=Es/H208490/H20850−Ea/H20849+1/H20850,/H9004a=Ea/H208490/H20850−Ea/H20849+1/H20850, and /H9004as=Ea/H208490/H20850−Es/H20849+1/H20850. To calculate the matrix elements in Eqs. /H20849A19 /H20850and /H20849A15 /H20850–/H20849A17 /H20850, we use the following definition forGONZÁLEZ, LEUENBERGER, AND MUCCIOLO PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-10the operator Wthat originates from the lead-SMM Hamil- tonian in Eq. /H208496/H20850: W=−/H20858 a,k,/H9268,ntn,k/H20849a/H20850/H9274k/H9268,a†/H9274n/H9268, /H20849A23 /H20850 with nbeing an occupied molecular orbital for a SMM in the charge state q=1. Similarly, we define, W†=/H20858 a,k,/H9268,ntn,k/H20849a/H20850/H9274k/H9268,a/H9274n/H9268†, /H20849A24 /H20850 which leads to WI/H208491/H20850W† =−/H20858 a,k,/H9268,n/H20858 a/H11032,k/H11032,/H9268/H11032,n/H11032tn,k/H20849a/H20850tn/H11032,k/H11032/H20849a/H11032/H20850/H9274k/H9268,a†/H9274k/H11032/H9268/H11032,a/H11032/H20849/H9274n/H9268I/H208491/H20850/H9274n/H11032/H9268/H11032†/H20850, /H20849A25 /H20850 and W/H20841s/H2085611/H20855s/H20841W† =−/H20858 a,k,/H9268,n/H20858 a/H11032,k/H11032,/H9268/H11032,n/H11032tn,k/H20849a/H20850tn/H11032,k/H11032/H20849a/H11032/H20850/H9274k/H9268,a†/H9274k/H11032/H9268/H11032,a/H11032/H20849/H9274n/H9268/H20841s/H2085611/H20855s/H20841/H9274n/H11032/H9268/H11032†/H20850. /H20849A26 /H20850 Then, substituting Eqs. /H20849A25 /H20850and /H20849A26 /H20850into Eqs. /H20849A19 /H20850and /H20849A15 /H20850–/H20849A17 /H20850, we arrive at 0/H20855↑/H20841DW†+WD†/H20841↑/H208560=−/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032z/H20849+1/H20850/H9274k↓,a†/H9274k/H11032↓,a/H11032 −/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032/H11036/H20849+1/H20850/H20849/H9274k↑,a†/H9274k/H11032↓,a/H11032 +/H9274k↓,a†/H9274k/H11032↑,a/H11032/H20850, /H20849A27 /H20850 0/H20855↓/H20841DW†+WD†/H20841↓/H208560=−/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032z/H20849+1/H20850/H9274k↑,a†/H9274k/H11032↑,a/H11032 −/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032/H11036/H20849+1/H20850/H20849/H9274k↑,a†/H9274k/H11032↓,a/H11032 +/H9274k↓,a†/H9274k/H11032↑,a/H11032/H20850, /H20849A28 /H20850 0/H20855↑/H20841DW†+WD†/H20841↓/H208560=−/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032/H11036/H20849+1/H20850/H9274k↓,a†/H9274k/H11032↑,a/H11032 −/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032z/H20849+1/H20850/H20849/H9274k↑,a†/H9274k/H11032↑,a/H11032 +/H9274k↓,a†/H9274k/H11032↓,a/H11032/H20850, /H20849A29 /H20850 and 0/H20855↓/H20841DW†+WD†/H20841↑/H208560=−/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032/H11036/H20849+1/H20850/H9274k↑,a†/H9274k/H11032↓,a/H11032 −/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032z/H20849+1/H20850/H20849/H9274k↑,a†/H9274k/H11032↑,a/H11032 +/H9274k↓,a†/H9274k/H11032↓,a/H11032/H20850, /H20849A30 /H20850 where we have used the spin selection rules given in the Eqs./H2084910/H20850–/H2084913/H20850. Using a similar procedure, we also find, 0/H20855↑/H20841C†V+V†C/H20841↑/H208560=/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032z/H20849−1/H20850/H9274k/H11032↑,a/H11032/H9274k↑,a† +/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032/H11036/H20849−1/H20850/H20849/H9274k/H11032↓,a/H11032/H9274k↑,a† +/H9274k/H11032↑,a/H11032/H9274k↓,a†/H20850, /H20849A31 /H20850 0/H20855↓/H20841C†V+V†C/H20841↓/H208560=/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032z/H20849−1/H20850/H9274k/H11032↓,a/H11032/H9274k↓,a† +/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032/H11036/H20849−1/H20850/H20849/H9274k/H11032↓,a/H11032/H9274k↑,a† +/H9274k/H11032↑,a/H11032/H9274k↓,a†/H20850, /H20849A32 /H20850 0/H20855↑/H20841C†V+V†C/H20841↓/H208560=/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032/H11036/H20849−1/H20850/H9274k/H11032↑,a/H11032/H9274k↓,a† +/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032z/H20849−1/H20850/H20849/H9274k/H11032↑,a/H11032/H9274k↑,a† +/H9274k/H11032↓,a/H11032/H9274k↓,a†/H20850, /H20849A33 /H20850 and 0/H20855↓/H20841C†V+V†C/H20841↑/H208560=/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032Jk,a;k/H11032,a/H11032/H11036/H20849−1/H20850/H9274k/H11032↓,a/H11032/H9274k↑,a† +/H20858 k,a,n/H20858 k/H11032,a/H11032,n/H11032jk,a;k/H11032,a/H11032z/H20849−1/H20850/H20849/H9274k/H11032↑,a/H11032/H9274k↑,a† +/H9274k/H11032↓,a/H11032/H9274k↓,a†/H20850, /H20849A34 /H20850 where, Jk,a;k/H11032,a/H11032z/H20849/H110061/H20850=2tn,k/H20849a/H20850tn/H11032,k/H11032/H20849a/H11032/H20850/H20875U+/H90040 /H20849U+/H90040/H208502−/H9004/H1100612+U−/H90040 /H20849U−/H90040/H208502−/H9004/H1100612/H20876, /H20849A35 /H20850 Jk,a;k/H11032,a/H11032/H11036/H20849/H110061/H20850=/H110062tn,k/H20849a/H20850tn/H11032,k/H11032/H20849a/H11032/H20850/H20875/H9004/H110061 /H20849U+/H90040/H208502−/H9004/H1100612 +/H9004/H110061 /H20849U−/H90040/H208502−/H9004/H1100612/H20876, /H20849A36 /H20850 jk,a;k/H11032,a/H11032z/H20849/H110061/H20850=tn,k/H20849a/H20850tn/H11032,k/H11032/H20849a/H11032/H20850/H20875U+/H90040 /H20849U+/H90040/H208502−/H9004/H1100612−U−/H90040 /H20849U−/H90040/H208502−/H9004/H1100612/H20876, /H20849A37 /H20850 and jk,a;k/H11032,a/H11032/H11036/H20849/H110061/H20850=/H11006tn,k/H20849a/H20850tn/H11032,k/H11032/H20849a/H11032/H20850/H20875/H9004/H110061 /H20849U+/H90040/H208502−/H9004/H1100612−/H9004/H110061 /H20849U−/H90040/H208502−/H9004/H1100612/H20876. /H20849A38 /H20850 In order to arrive at each matrix element expressed in Eqs. /H20849A27 /H20850–/H20849A30 /H20850and Eqs. /H20849A31 /H20850–/H20849A34 /H20850, we have only taken into account the leading terms in the small ratio /H9004q/U. For in- stance, in Eqs. /H20849A27 /H20850and /H20849A28 /H20850we have neglected spin-KONDO EFFECT IN SINGLE-MOLECULE MAGNET … PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-11flipping terms, which carry an amplitude smaller than the direct non-spin-flipping terms by /H90040/U. Note that the longi- tudinal exchange coupling is positive, i.e., antiferromagnetic.Moreover, since U/H11271/H9004 0,/H9004/H110061, we find that the exchange cou- plings are strongly anisotropic with, /H20841Jk,a;k/H11032,a/H11032/H11036/H20849/H110061/H20850/H20841/H11015/H9004/H110061 UJk,a;k/H11032,a/H11032z/H20849/H110061/H20850. /H20849A39 /H20850 Finally, introducing pseudospin operators that act solely on theq=0 sector of the SMM, namely, /H90180z=/H20841↑/H2085600/H20855↑/H20841−/H20841↓/H2085600/H20855↓/H20841, /H20849A40 /H20850 /H90180+=/H20841↑/H2085600/H20855↓/H20841, /H20849A41 /H20850/H90180−=/H20841↓/H2085600/H20855↑/H20841, /H20849A42 /H20850 and assuming /H20849for brevity /H20850that/H90041=/H9004−1, we can write the effective Hamiltonian of the q=0 charge sector as H˜0=H0+/H20858 k/H11032,a/H11032/H20858 k,a/H20851Jk,a;k/H11032,a/H11032z/H90180z/H20849/H9274k↑,a†/H9274k/H11032↑,a/H11032−/H9274k↓,a†/H9274k/H11032↓,a/H11032/H20850 −Jk,a;k/H11032,a/H11032/H11036/H20849/H90180+/H9274k↓,a†/H9274k/H11032↑,a/H11032+/H90180−/H9274k↑,a†/H9274k/H11032↓,a/H11032/H20850 −jk,a;k/H11032,a/H11032/H11036/H20849/H9274k↓,a†/H9274k/H11032↑,a/H11032+/H9274k↑,a†/H9274k/H11032↓,a/H11032/H20850/H20852. /H20849A43 /H20850 Following a similar procedure, one can obtain the anisotropic Kondo Hamiltonian with the ferromagnetic exchange cou-pling constants. *mleuenbe@mail.ucf.edu †mucciolo@physics.ucf.edu 1C. 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Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, and A. Cornia, Europhys. Lett. 50, 552 /H208492000 /H20850; M. N. Leuenberger and D. Loss, Phys. Rev. B 61, 12200 /H208492000 /H20850. 7D. Loss, D. P. DiVincenzo, and G. Grinstein, Phys. Rev. Lett. 69, 3232 /H208491992 /H20850; J. von Delft and C. L. Henley, ibid. 69, 3236 /H208491992 /H20850; A. Garg, Europhys. Lett. 22, 205 /H208491993 /H20850. 8M. N. Leuenberger and D. Loss, Phys. Rev. B 63, 054414 /H208492001 /H20850. 9H. B. Heersche, Z. de Groot, J. A. Folk, H. S. J. van der Zant, C. Romeike, M. R. Wegewijs, L. Zobbi, D. Barreca, E. Tondello,and A. Cornia, Phys. Rev. Lett. 96, 206801 /H208492006 /H20850. 10C. Romeike, M. R. Wegewijs, W. Hofstetter, and H. Schoeller, Phys. Rev. Lett. 96, 196601 /H208492006 /H20850. 11M. R. Wegewijs, C. Romeike, H. Schoeller, and W. Hofstetter, New J. Phys. 9, 344 /H208492007 /H20850. 12M. N. Leuenberger and E. R. Mucciolo, Phys. Rev. Lett. 97, 126601 /H208492006 /H20850. 13D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch- Madger, U. Meirav, and M. A. Kastner, Nature /H20849London /H20850391, 156 /H208491998 /H20850. 14A. Sieber, C. Boskovic, R. Bircher, O. Waldmann, S. T. Ochsen-bein, G. Chaboussant, H.-U. Guedel, N. Kirchner, J. van Slageren, W. Wernsdorfer, A. Neels, H. Stoeckli-Evans, S. Jan-ssen, F. Jurannyi, and H. Mutka, Inorg. Chem. 44, 4315 /H208492005 /H20850. 15C. Timm and F. Elste, Phys. Rev. B 73, 235304 /H208492006 /H20850; F. Elste and C. Timm, ibid. 73, 235305 /H208492006 /H20850; F. Elste and C. Timm, ibid. 75, 195341 /H208492007 /H20850; C. Timm, ibid. 76, 014421 /H208492007 /H20850. 16J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 /H208491966 /H20850. 17G. González and M. N. Leuenberger, Phys. Rev. Lett. 98, 256804 /H208492007 /H20850. 18A. C. Hewson, The Kondo Problem to Heavy Fermions /H20849Cam- bridge University Press, Cambridge, 1997 /H20850. 19A. A. Aligia, C. A. Balseiro, and C. R. Proetto, Phys. Rev. B 33, 6476 /H208491986 /H20850. 20L. I. Glazman and M. Pustilnik, in Nanophysics: Coherence and Transport , edited by H. Bouchiat, Y . Gefen, S. Guéron, G. Mon- tambaux, and J. Dalibard /H20849Elsevier, New York, 2005 /H20850. 21The actual applied gate voltage is typically much larger than Vg due to screening and the setup geometry. 22R. Basler, A. Sieber, G. Chaboussant, H.-U. Guedel, N. E. Cha- kov, M. Soler, G. Christou, A. Desmedt, and R. Lechner, Inorg.Chem. 44, 649 /H208492005 /H20850. 23The state with mq=0 can be easily included in this description. However, since it typically requires a relatively high temperatureor bias to be accessed, it will be neglected here. 24R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 /H208492008 /H20850. 25P. W. Anderson, J. Phys. C 3, 2436 /H208491970 /H20850. 26J. Park, A. N. Pascupathy, J. I. Goldsmith, C. Chang, Y . Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D. Abruna, P. L. McEuen,and D. C. Ralph, Nature /H20849London /H20850417, 722 /H208492002 /H20850. 27S. R. White, Phys. Rev. Lett. 69, 2863 /H208491992 /H20850. 28J. Appelbaum, Phys. Rev. 154, 633 /H208491967 /H20850. 29Y . Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 /H208491992 /H20850; A. C. Hewson, J. Bauer, and A. Oguri, J. Phys.: Condens. Matter 17, 5413 /H208492005 /H20850. 30Generalizations to include additional charge sectors are straight- forward.GONZÁLEZ, LEUENBERGER, AND MUCCIOLO PHYSICAL REVIEW B 78, 054445 /H208492008 /H20850 054445-12

PhysRevB.48.11427.pdf

PHYSICAL REVIEW8 VOLUME 48,NUMBER 15 15OCTOBER 1993-I Electronic structure ofperiodically b-doped GaAs:Si L.Chico,F.Garcia-Moliner, andV.R.Velasco Instituto deCiencia deMateriales, Consejo Superior deInvestigaciones Cientificas, Serrano 128,28008Madrid, Spain (Received 29December 1992;revisedmanuscript received 5May1993) Theelectronic structure ofperiodically dopedGaAs:Si systems hasbeenself-consistently calcu- latedwithaHedin-Lundqvist local-density functional forexchange andcorrelation. Thein6uence oftheperiodic spacing d,thearealimpurity concentration¹,andthespreadoftheimpurity distribution havebeeninvestigated. Miniband widthsandgaps,potential-well depths, andFermi- levelposition havebeenstudied between d=100and500A.,thusfollowing thetransition from superlattice behavior toindependent wellregime. Theresultsareusedtointerpret someobserved photoluminescence spectra. I.INTRODUCTION Asimprovements inepitaxial growth haveallowed for abettercontrolofimpurity doping insemiconductors therehasbeenincreasing interest intheb-doped sys- tems,thuscalledbecauseoftheultimate targetofachiev- ingastrictlocalization oftheimpurities inoneatomic layer.'Raman spectroscopy,'infrared absorption, magnetotransport, andphotoluminesc ence'have beenusedinexperimental studiesofthesesystems. Anewtypeofsuperlattice isproduced inwhichthereis onlyonebulkmaterial butthepotential ismodulated by periodic bdoping.Ifonlyonetypeofdopant isused,e.g., donors,thenintheuncompensated sampletheextrinsic electrons screentheionizeddonorsandproduce inhomo- geneous potentials abouttheplaneofthedonorsheets whicharetobeobtained self-consistently. Someself- consistent calculations havebeenrecently reported. Thepurposeofthispaperistodescribe somefurthercal- culations whichcomplement theresultsobtained inthese studies inasignificant wayandtheyalsoallowforthe interpretation ofsomephotoluminescence experiments, givingstrongsupporttotheclaimspresented there. II.MODEL ANDCALCULATION Wefi.rststudyperiodical bdopingofSiinGaAswith adonorimpurity distribution oftheform n(z)=Nd)8(z—nd) andtheninvestigate theeffectofasmearing outtheim- puritydensity intheformofGaussian distributions: n(z)=)—(znd)z/2oz— z ()2&2ln2 Thehalf-width Lzwaschosentobebetween 20and40 A.,whichappeartoberealistic figuresforcurrent ex- perimental devices'oncetheeffectofthesubstrate temperature T,onthesegregation ofSihasbeenclari- fied.HighvaluesofT„oforder600C(Ref.10)seemto beconvenient toensurethegrowthofgoodquality, fairly defect-free structures. However withsuchtemperatures Sitendstodiffuseconsiderably, withaspreading ofthedopedzonewhichcanreachvaluesoforder100A,io'ii soacompromise mustbesoughtbetween thequalityof thestructure andthecontroloftheimpurity segregation. ItisfoundthatforT,(530Cthereisasignificant reduction inSidiffusion andvaluesofAzoftheorder quotedabovecanbeachieved, soweshalltaketheseval- uesasrealistically representative inordertoassessthe effectsofimpurity spreading. Forarealimpurity densities below10cmitwould bequiteunrealistic toassumeacontinuous positive chargedistribution oftheionized impurities, butthis canbeassumed beyond1.3x10cm.Ontheother hand,beyond 8x10cmasaturation oftheelec- tronicdensity isobserved,''thatis,Halleffectmea- surements yieldanarealelectron concentration of8x 10cmwhichstaysconstant evenwhenthenominal impurity dopingisincreased. Inthisstudyweshallcon- siderarealimpurity concentrations between 1and7x 10cm,whichcorrespond totheexperimental data weshalllaterdiscuss. Inthisrangewecanassumethat allthedonorsareionized andthattheirarealdensity iscontinuous. Thevaluesofthesuperlattice periodd herestudied varybetween 100and500A,whichcover therangefromasuperlattice toasystemofeffectively independent quantum wells. Weuseaone-band effective-mass modelandcalculate self-consistently theelectronic statesandtheone-electron potential U(z)asthesumoftheHartree-Poisson po- tentialandanexchange andcorrelation potential which weapproximate bytheHedin-Lundqvist local-density functional. Otherparametrizations oftheexchange andcorrelation densityfunctional arepossible andyield rathersimilarresultsfortheone-electron densityofstates fromwhichtheelectronic chargedensityp(z)isself- consistently calculated. Wehaveusedthesamescheme employed inarecentstudyofmodulation dopedquan- tumwells,whichappears toworkwell,although any useofalocal-density functional istobetakenofcourse asjustanapproximation. Fortheintegration oftheSchrodinger equation weuse thefulltransfer matrixdiscussed inRef.15,whichtrans- fersbothamplitude andderivative, theamplitude being inthiscasetheenvelope function E~~(z)forthesubbandjwithone-dimensional (1D)momentum qassociated to 0163-1829/93/48(15)/11427(4)/$06. 00 4811427 1993TheAmerican Physical Society 11428 BRIEFREPORTS theperioddofthesuperlattice. Bydefinition F(") ~~~()~M(z,zo)~,()~= ~~,()E(meV) 2 whereM(z,zo)isthe2x2fulltransfer matrix which transfers fromachosenreference pointzotoavariable endpointz.Therelationship betweenMandtheGreen function, fromwhichthelocaldensityofstatesandthe electron chargedensitycanbecalculated, isfullydis- cussedinRefs.16and17.Furthermore, forthesystem withperiodd:100- 50-Vo M(+d) ~(o) ~ ~AA A17 ~~(o)("))&»D~)(F'(") (zo)) (P'(zo))' wheremisthefulltransfer matrixacrossoneentirepe- riod,whichleadstothewell-known eigenvalue equation cosqd——trm (5) intermsoftheenergy-dependent fulltransfer matrix. TheBloch-Floquet property (4)allowsustoobtain thederivative atsomechosenpointzobymeansofthe algebraic relationship(4) III.RESULTS ANDCOMMENTS Wehavestudiedsuperlattices withd=l00,300,and 500A.withvaluesofNg=ndx10cmwith ng——1,3,5,7,aswellassomespecific casesforwhichex- perimental information isavailable. Besidesstrictbdop- ingwehavealsostudiedGaussian distributions withAz equalto20and40A..Theparameters usedforGaAsare m*=0.0665nlo anda=12.5inGaussian units. Figure1displays theresultsforasuperlattice withnp 5,d=300A,andAz=20A,andwillserveustoshow thenotation. Theshadedareasaretheallowed energies forr=0whenqvariesoverthe1DBrillouin zone,i.e., thesuperlattice minibands. ThedashedlineistheFermi levelofthesystem, andVoisthemaximum potentialm+D whence wecanobtainthewave-function amplitudes from thefirstrowof(3)andthenuseastandard normalization scheme. Alternatively, wecanobtaintheGreenfunction forwhichthereisnoneedtonormalize. Thecalculation alsoyieldsself-consistently thepositionoftheFermilevel E~,fromthecondition thatthetotalnumberofelectrons mustequalthetotalnumberofdonors.Inpractical terms wehavestopped thecalculation whenthemeansquare deviation ofthetotalone-electron potential intwocon- secutive cyclesis&0.1meV. Theschemejustoutlined allowsustotreatonthe samefooting longandshortperiodswithout resorting to aseparate treatment forwideminibands, typicalofshort periods. Thenumerical laborinvolved isalsoinsensitive tochanges ind,whichisasubstantial practical advan- tageoverschemes basedonaplane-wave representation, wherelongperiods requireasubstantial increase inthe numberofcomponents inordertoachieve anadequate description oftheelectronic wavefunctions.050 100 150 200 250 300(") FIG.1.Self-consistent one-electron potential profilefora systemwithperiodic dopingwithanimpurity spreading Az=20A.,impurity density nd,=5,andsuperlattice periodd= 300A.Theshaded zonesaretheminiband energies, i.e.,the allowed energies forK=0.ThedashedlineistheFermilevel andVodenotesthemaximum depthofthewells.Allenergies arereferredtothebottomofthepotential well. height. Wehavecalculated V(z),p(z),miniband edges,and Fermi-level position studying theeKectsofchanging the keyparameters %pandd.Thegeneralaspectofthe potential andchargedensity profilesdoesnotchangein anysignificant manner. Itismoreinteresting tolookat thechanges inVo,E~,andminiband edges,shownin Fig.2asafunction ofd(varying between 100and500 A.)fordiferent valuesofNg,alwaysintherangeof10cm,asindicated. Theseresultscontain theinforma- tionwhichisphysically significant inpractice. Obviously Voincreases withincreasing Ng,andalso,forfixedK~, asdincreases. Whenthewellsarewideapartandtheir potentials havenegligible overlap, thesystembehaves as asetofindependent isolated wells,whileasddecreases theircoupling increases andthesystemtendstoshow superlattice behavior. Oneofthepurposes ofthisstudy isprecisely toascertain whenthistakesplace,asthisis apractical issueintheanalysisoftheexperimental data tobediscussed below. Generally speaking ourresultscanbedescribed asfol- lows. (i)Ford=100 Aallcasesshowdefinitesuperlattice be- havior.Forthehighimpurity densities allminibands still showawidthwhichshouldbeexperimentally detectable: Thelowestone,whichisalwaysthenarrowest, reaches valuesoforder30meVatleast.Thetotalnumber of occupied minibands isneverlargerthantwo. (ii)Ford=500 A.allcasesshowseparate wellbehavior. Theoccupied miniband. widthsarenegligible forallval- uesof%gandE~andVotendtobeveryclose.There arebetween threeandfouroccupied minibands. (iii)Forintermediate valuesofd=250Athelow- estminiband alwaysoccupied isverynarrow, soin thisenergyrangethewellsareefI'ectively independent on thisaccount. However, thenextoneissignificantly wide. Population ofthisminiband startsforXg&3x10 cmwhenitswidthisatleast=20meV,sointhisen- ergyrangesuperlattice behavior setsin. 48 BRIEFREPORTS 11429 TheefI'ectofspreading outtheimpurity concentration istochangeconsiderably theV(z)andp(z)profilesand theposition oftheminibands relativetothebottomof thewells(Fig.3),inagreement withtheconclusions reached inRef.9.However, thisneednotmeanthat theelectronic properties ofthesystemchangeapprecia- bly.Wehavestudiedthesubband populations andfind thesetobeverylittleafFected, alsoinagreement with Ref.9.Furthermore, andmoresignificantly, wehave studiedthevaluesofthegapsandminiband widths, as E(meV) 300 200 100,E(mev) 150- 100 50 -50 I -100 -150- FIG.3.Potential profiles, miniband energies(r=0, shaded), andFermilevel(dashed line)forsuperlattices with nz=3,d=100Aandimpurity spreadings of0,20,and40 A,fromlefttoright.Allenergies arereferredtotheFermi level. 0 E(meV) 300 200 100 0200 300 400 E(meV) 300. 200 100ooooo o 0200 300 400 E(meV) 300oro\oroooooro oorrrooro 200 300 400d(A)50 500'(") (c)wellastheirrelativepositions, whicharethemagnitudes moredirectly relatedtoexperimental results, andfind thesetoberatherinsensitive toplausible changes inthe impurity distribution. Photoluminescence spectroscopy appearstoprovidea suitable experimental toolfortheinvestigation ofthese superlattices. Thetechnique isratherunsuitable for isolated wellsbecause thepotential whichisattractive fortheelectrons isrepulsive forthephotocreated holes, whichreduces considerably theoverlapbetween theelec- tronandholewavefunctions. ThisdiKculty iscircum- ventedinthesuperlattice duetotheperiodic modulation ofthepotential, whence anenhancement oftherecom- bination rate.Quiterecently thesesystems werestudied bymeansofphotoluminescence techniques anditwas pointed outthatthedifFerence between the3DFermi energyE&andtheobserved widthofthemainpho- toluminescence peakTVpI,provides strongevidence that electrons areconfined inthespacechargepotential and thatsubbands broaden intominibands asthelayersepa- rationisreduced. VJehavestudiedthreeofthesamples experimentally studied byphotoluminescence andsome parameters andresultsaresummarized inTableI.The presence ofdefectsinthesamples mayrelaxthemomen- tumconservation selection ruleforopticaltransitions, sothatforfreeelectron behavior onemightexpectthe equivalent 3DFermienergyE&togiveafairaccount ofWpz,.Firstlyacomparison ofcolumns threeandfour 200 100 0oor ~r 200 300 400dA500TABLEI.d,superlattice period. Np,arealimpurity den- sity.EF,3DFermienergyestimated fromtheequiva- lentuniform carrierdensity.EF,Fermienergycalculated self-consistently. Wpz,,experimental widthofthedominant PLpeak(Ref.3).A~,widthofthehighest miniband with significant population intheself-consistent calculation.FIG.2.Fulllines:miniband edges (K,=0,qspanning the 1DBrillouin zone).Dottedline:valuesofthewelldepth Vo (seeFig.1).Dashed line:R~.Allenergies arereferredto thebottomofthepotential wellsandperiodic bdoping is assumed. Valuesofnd,areasfollows: (a)1;(b)3;(c)5;(d)7.d (A) 100 200 500Ng (10'cm') 1.4 1.2 1.0E3D (meV) 68 39 18EF (meV) 69 57 33Wpz, (meV) 43 25 7Ap (meV) 51 32 3 11430 BRIEFREPORTS showsasignificant disagreement betweenE+andthe actualself-consistent resultE~forlarged,andverygood agreement ford=100 A..PromtheresultsshowninFig. 2wewouldexpectthatmostofthefreeelectron popula- tionisinenergylevelsabove Voandistherefore largely unlocalized. However, although thismightaccount for thenumerical agreement betweenEFandE~,thereal behavior isnotactually kee-electron-like. Moreover, as alreadynotedinRef.3,E+disagrees withtheobserved valuesofWpL(column five).Infacttheassumption that onecanidentify theobserved widthofthephotolumines- centpeakwiththeFermienergyisstrongly dependent upontheassumption offreeelectron behavior anditis clearthatthisisnotthecaseevenford=l00A. Theresultsofourcalculations areinlinewiththein- terpretation putforward inRef.3.Thusitseemsmore plausible totryandinterpret peakwidthsasgivingan approximate measureofthewidthsoftheoccupied mini- bands.SincedifI'erent peaksaresometimes observed we shouldassociate thedistances between peakswithsome approximate meandistance between minibands. Thisexpectation isfairlyborneoutbythecompari- sonbetween columns fiveandsix.ItisseenthatTVpz, agreesindeedsubstantially betterwith4„thanitdoes withEJ;.Forthesamplewithd=200 Aaccording toour self-consistent calculations thereshouldbetwooccupied minibands butthepopulation ofthesecondoneisnegli- giblecompared withthatofthefirstone,whichistheone forwhichwecalculate4„.Forthesamplewithd=500 AthevaluegivenintheMthcolumnofthetablecorre- spondstothedominant peak.Another twopeakshave alsobeenobserved, whichwecanassociate withanother twopopulated subbands which,according toourcalcu- lation,shouldhavesomesignificant population. Dueto poorspectral resolution itisextremely difIiculttoobtain anyreliableestimate oftheirwidths, butthedistances between thepeakscanbeestimated as20and15meV, infairagreement withourestimated meandistances be- tweenoccupied minibands of20and8meV,respectively. Another photoluminescence spectruxn hasbeenre- ported forasuperlattice withd=300Aandng=5,whichcanbelikewise interpreted. Inourcalculation wefind thatthereshouldbefouroccupied minibands, thetwo lowestonesbeingsubstantially narrower. Twopeakscan beclearlyidentified intheexperimental spectrum, which wecanassociate withthethirdandfourthminibands, andalsosomestructure isobserved atlowerenergies whichonecouldinterpret asindicative oftwopeaksat 1.430and1.495eV.Onthisbasisthethreedistances between thefourpeaksare65,15,and15meV.Ouresti- mateforthemeandistances between thecorresponding minibands yields70,25,and10meV,whichisagainas fairasonecouldexpect. Thequantitative agreement withexperimental datais onlyapproximate, ascorresponds totheavowedapprox- imations madeintheproposed interpretation. Themea- sureofthe"meandistance" between bands,forinstance, israthervague.Butthewholeinterpretation seemsquite plausible andinthatsenseeverything fitswithinapic- turewhichdoesunmistakeably bearoutthequasi-2D natureofthespectrum, evenford=100A,whenmostof theelectron population isabovethetopofthepotential wells.Theseelectrons arehighlydelocalized butthereis nowaytoidentify WpLwiththeFermienergy, evenif thisisself-consistently calculated, sinceagapexistsbe- tweenoccupied delocalized stateswhichdonottherefore havefree-electron-like behavior. Whileallowing ustoseetheinfIuence ofthekeyparam- eters,Ngandd,thisalsoprovides areasonable basisto attempt anactualcalculation ofthephotoluminescence spectrum. ACKNOWLEDGMENTS ThisworkwascarriedwithpartialsupportoftheSpan- ishCICYT (GrantNo.MAT91-0738) andtheResearch Scholarships Programme oftheSpanish Ministry ofEd- ucation andScience(L.Ch.).Wewishtoexpress our warmest gratitude toProfessor R.Perez-Alvarez forhis generous helpwithinnumerable discussions whichpro- videdagreatdealofhelpandstimulation. E.F.Schubert, A.Fischer, andK.Ploog,IEEETrans. Electron DevicesED-33, 625(1986). K.Ploog,J.Cryst.Growth81,304(1987). A.C.Maciel,M.Tatham,J.F.Ryan,J.M.Worlock, R. E.Nahory,J.P.Harbison, andL.T.Florez,Surf.Sci.228, 251(1990).J.Wagner, A.Fischer, andK.Ploog,Phys.Rev.B42, 7280(1990). E.F.Schubert, Surf.Sci.228,240(1990). A.Zrenner,F.Koch,andK.Ploog,Surf.Sci.196,671 (1988). M.-L.Ke,J.S.Rimmer,B.Hamilton, M.Missous,B. Khamsehpour, J.H.Evans,K.E.Singer, andP.Zalm, Surf.Sci.267,65(1992). F.A.Reboredo andC.R.Proetto, SolidStateCommun. 81,163(1992).M.H.Degani,J.Appl.Phys.70,4362(1991). M.Santos,T.Sajoto,A.-M.Lanzillotto, A.Zrenner, and M.Shayegan, Surf.Sci.228,255(1990). HCNutt~R.S.Smith,M.Towers,P.K.Rees,andD.J. James,J.Appl.Phys.70,821(1991). E.I~Levin,M.E.Raikh,andB.I.Shklovskii, Phys.Rev. B44,ll281(1991). L.HedinandB.I.Lundqvist,J.Phys.C4,2064(1971). L.Chico,W.Jaskolski, andF.Garcia-Moliner, Phys.Scr. 47,284(1993). M.E.Mora,R.Perez,andCh.B.Sommers,J.Phys.46, 1021(1985). F.Garcia-Moliner, R.Perez-Alvarez, H.Rodriguez- Coppola, andV.R.Velasco,J.Phys.A23,1405(1989). F.Garcia-Moliner andV.R.Velasco, TheoryofSingleand Multiple Interfaces (WorldScientific, Singapore, 1992).

PhysRevB.95.245315.pdf

PHYSICAL REVIEW B 95, 245315 (2017) Gate control of the spin mobility through the modification of the spin-orbit interaction in two-dimensional systems M. Luengo-Kovac,1F. C. D. Moraes,2G. J. Ferreira,3A. S. L. Ribeiro,2G. M. Gusev,2 A. K. Bakarov,4V . Sih,1and F. G. G. Hernandez2,* 1Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 2Instituto de Física, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil 3Instituto de Física, Universidade Federal de Uberlândia, Uberlândia, MG 38400-902, Brazil 4Institute of Semiconductor Physics and Novosibirsk State University, Novosibirsk 630090, Russia (Received 24 March 2017; published 30 June 2017) Spin drag measurements were performed in a two-dimensional electron system set close to the crossed spin helix regime and coupled by strong intersubband scattering. In a sample with an uncommon combination oflong spin lifetime and high charge mobility, the drift transport allows us to determine the spin-orbit field andthe spin mobility anisotropies. We used a random walk model to describe the system dynamics and foundexcellent agreement for the Rashba and Dresselhaus couplings. The proposed two-subband system displays alarge tuning lever arm for the Rashba constant with gate voltage, which provides a new path towards a spintransistor. Furthermore, the data show large spin mobility controlled by the spin-orbit constants setting the fieldalong the direction perpendicular to the drift velocity. This work directly reveals the resistance experienced inthe transport of a spin-polarized packet as a function of the strength of anisotropic spin-orbit fields. DOI: 10.1103/PhysRevB.95.245315 I. INTRODUCTION The pursuit for a new active electronic component based on flow of spin, rather than that of charge, strongly motivates re-search in semiconductor spintronics [ 1–5]. Since the Datta-Das proposal for a ballistic spin transistor, full electrical control ofthe spin state was suggested using the gate-tunable Rashbaspin-orbit interaction (SOI) [ 6–10]. Further studies, including the Dresselhaus SOI [ 11], were made to assure a nonballistic transistor robust against spin-independent scattering [ 12–14]. For example, it has been demonstrated that SU(2) spin rotationsymmetry, preserving the spin polarization, can be obtained inthe persistent spin helix (PSH) formed when the strengths ofthe Rashba and Dresselhaus SOI are equal ( α=β)[15–19]. This is possible because the uniaxial alignment of the spin-orbit field suppresses the relaxation mechanism when thespins precess about this field while experiencing momentumscattering [ 20]. Gate control of this symmetry point was experimentally observed [ 21–23] and allowed to produce a transition to the PSH −(α=−β) in the same subband [ 24]. Drift in those systems showed surprising properties [ 25,26] such as the current control of the temporal spin-precessionfrequency [ 27]. Although the helical spin-density texture could be even transported without dissipation under certainconditions [ 15], the spin transport suffers additional resistance from the spin Coulomb drag [ 28–32]. These frictional forces appear as a lower mobility for spins than for charge and studiesin new systems are still necessary to understand this importantconstraint for future devices. A two-dimensional electron gas (2DEG) hosted in a quantum well (QW) with two occupied subbands offersunexplored opportunities for the study of spin transport[33,34]. Theoretically, the inter- and intrasubband spin-orbit couplings (SOCs) have been extensively studied [ 35–38]. In *Corresponding author: felixggh@if.usp.brterms of a random walk model (RWM) [ 39], the spin drift and diffusion was recently developed for these systems displayingtwo possible scenarios regarding the intersubband scattering(ISS) rate [ 40]. The interplay between the two subbands may introduce new features to the PSH dynamics, for example, acrossed persistent spin helix [ 41] may arise when the subbands are set to orthogonal PSHs (i.e., α 1=β1andα2=−β2)i n the weak ISS limit. In this report we experimentally studyspin drag in a system with the two subbands individually setclose to the PSH +and PSH−, but with strong ISS, where the dynamics is given by the averaged SOCs of both subbands. Thecombination of long spin lifetime and high charge mobilityallows us to determine the spin mobility and the spin-orbitfield anisotropies with the application of an acceleratingin-plane voltage. We are able to control the SOCs in bothsubbands and to show a linear dependence for the sum of theRashba constants with gate voltage. Finally, we determine aninverse relation for the spin mobility dependence on the SOCsdirectly revealing the resistance experienced in the transportof a spin-polarized packet as a function of the strength ofanisotropic spin-orbit fields. II. MATERIALS The sample consists of a single 45-nm-wide GaAs QW grown in the [001] ( z) direction and symmetrically doped. Due to the Coulomb repulsion of the electrons, the charge dis-tribution experiences a soft barrier inside the well. Figure 1(a) shows the calculated QW band profile and charge densityfor both subbands. The electronic system has a configuration with symmetric and antisymmetric wave functions for the two lowest subbands with subband separation of /Delta1 SAS=2m e V . The subband density ( n1=3.7,n2=3.3×1011cm−2)w a s obtained from the Shubnikov–de Hass (SdH) oscillations asshown in Fig. 1(a) and the low-temperature charge mobility was 2.2×10 6cm2/Vs[ 42]. A device was fabricated in a cross-shaped configuration with width of w=270μm and 2469-9950/2017/95(24)/245315(6) 245315-1 ©2017 American Physical SocietyM. LUENGO-KOV AC et al. PHYSICAL REVIEW B 95, 245315 (2017) FIG. 1. (a) Longitudinal ( Rxx) and Hall ( Rxy) magnetoresistance of the two-subband QW. From the SdH periodicity, one can obtainthe subbands density n νin the lower inset. The top inset shows the potential profile and subbands charge density calculated from the self-consistent solution of Schrödinger and Poisson equations for Ez=0. (b) Subband energy levels and (c) electron concentration dependence on VgandEz. (d) Geometry of the device and contacts configuration. channels along the [1 ¯10] (x) and [110] ( y) directions. Lateral Ohmic contacts deposited l=500μm apart were used to apply an in-plane voltage ( Vip) in order to induce drift transport. For the fine tuning of the subband SOCs, a semitransparentcontact on top of the mesa structure ( V g) was used to modify structural symmetry and subband occupation. The effect of Vg on the subband energy levels ( /epsilon1ν) and densities ( nν)i ss h o w n in Figs. 1(b)and1(c)as a function of the out-of-plane electric field (Ez). Note that the total density changes linearly with Vgand that Vg=0 corresponds to a built-in electric field of 0.15 V /μm. Figure 1(d) displays the experimental scheme with the connection of VipandVg[43]. III. MODEL To describe the magnetization dynamics and the measured SO fields for our two-subband system, we combine thecalculated SOCs with RWM [ 39,40,44]. For a [001] GaAs 2DEG, the xandycomponents of the SO fields for each subband ν={1,2}are B SO,ν(k)=2 gμB⎛ ⎜⎜⎜⎜⎝/parenleftBigg +αν+β1,ν+2β3,νk2 x−k2 y k2/parenrightBigg ky /parenleftBigg −αν+β1,ν−2β3,νk2 x−k2 y k2/parenrightBigg kx⎞ ⎟⎟⎟⎟⎠,(1) plus corrections due to the intersubband SOCs [ 23,35–38,41]. Above, g=−0.44 is the electron gfactor for GaAs and μB is the Bohr magneton. The SOCs are the usual Rashba αν, linear β1,ν, and cubic β3,νDresselhaus terms. We consider the strong intersubband scattering (ISS) regime of the RWM[40], for which both the inter- and intrasubband scattering rates are much faster than the spin precession, thus yielding arandomization of both the momenta k(within the Fermi circle k=k F) and the subband ν. Consequently, the dynamics is governed by an averaged SOC field /angbracketleftBSO/angbracketright=(/angbracketleftBx SO/angbracketright,/angbracketleftBy SO/angbracketright), where /angbracketleft ···/angbracketright labels averages over kandν. Namely, the fieldcomponents read /angbracketleftBx SO/angbracketright=/bracketleftBigg m ¯hgμ B2/summationdisplay ν=1(+αν+β∗ ν)/bracketrightBigg vy dr, (2) /angbracketleftBy SO/angbracketright=/bracketleftBigg m ¯hgμ B2/summationdisplay ν=1(−αν+β∗ ν)/bracketrightBigg vx dr, (3) which are transverse to the drift velocity vdr=(vx dr,vy dr). Here β∗ ν=β1,ν−2β3,ν,m=0.067m0is the effective electron mass for GaAs, and ¯ his Planck’s constant. Since Bx(y) SO∝vy(x) dr, it is convenient to analyze the linear coefficients bx(y)= By(x) SO/vx(y) dr, which are given by the terms between square brackets above. The intra- and intersubband SOCs are calculated within the self-consistent Hartree approximation [ 35–38] for GaAs quantum wells tilted by Ez. The chemical potential is set to return the density n=n1+n2=7×1011cm−2forEz=0, leading to the linear dependence of nwithEzshow in Fig.1(c).F o l l o w i n gR e f s .[ 35–38,45], the SOCs are defined from the matrix elements ην,ν/prime=/angbracketleftν|ηwV/prime+ηHV/prime H|ν/prime/angbracketrightand /Gamma1ν,ν/prime=γ/angbracketleftν|k2 z|ν/prime/angbracketright, where |ν/angbracketrightis the eigenket for subband ν,ηw=3.47˚A2andηH=5.28˚A2are bulk coefficients, V/prime=∂zV(z) and V/prime H=∂zVH(z) are the derivatives of the heterostructure and Hartree potentials along z,γ=11 eV ˚A3 is the bulk Dresselhaus constant, and kzis the zcomponent of the momentum. The usual intrasubband Rashba and linearDresselhaus SOCs are α ν=ην,νandβ1,ν=/Gamma1ν,ν. The nondi- agonal terms are the intersubband SOCs η=η12and/Gamma1=/Gamma112. The calculated SOCs, plotted in Figs. 2(a)–2(c)as a function ofEz, show agreement with previous studies [ 46,47]. The high-density nmakes the cubic Dresselhaus β3,ν≈γπn ν/2 comparable with β1,ν, strongly affecting the PSH tuning [ 17] αν=βν, with βν=β1,ν−β3,ν. NearEz≈0.04 V/μm, the SOCs reach almost simultane- ously the balanced condition for the PSH+in the first subband (α1/β1=+1) and for the PSH−in the second subband (α2/β2=−1), as shown by the ratio αν/βνin Fig. 2(d).T h e expected magnetization patterns for the single-subband PSH isshown in the inset of Fig. 2(d). The PSH −shows more stripes than the PSH+due to the higher value of α, which grows quickly within the Ezrange. However, the ratio of the averaged SOCs (/summationtextαν)/(/summationtextβν) approaches the PSH regimes only for |Ez|>0.3V/μm. As we will see next, the experimental data matches well the strong ISS regime of the RWM, therefore the dynamics is governed by the averaged SOCs. In this case, theexpected magnetization patterns are shown in Fig. 2(e). With increasing E zthe system transitions from isotropic ( Ez=0) to uniaxial ( Ez>0.3V/μm), as indicated by the formation of stripes and the orientation of the first harmonic componentof the total field/summationtextB SO,ν(k) [arrows in Fig. 2(e)]. IV . EXPERIMENT AND RESULTS We are interested in the determination of the anisotropy for the coefficients bx(y), estimated in one order of magnitude in Figs. 3(a) and3(b). We measured the spin polarization using time-resolved Kerr rotation as function of the space 245315-2GATE CONTROL OF THE SPIN MOBILITY THROUGH THE . . . PHYSICAL REVIEW B 95, 245315 (2017) FIG. 2. (a)–(c) Calculated SOCs for the Rashba ( αν), linear ( β1,ν), and cubic ( β3,ν) Dresselhaus for each subband ν={1,2}, as well as intersubband SOCs ηand/Gamma1as a function of Ez. The purple lines give the sum of ανandβ∗ ν. (d) The ratio αν/βν=±1w h e n the subband νis set to the PSH±regime. The insets show the single-subband magnetization maps on the xyplane for the PSH± regimes, and the self-consistent potentials and subband densities for the respective Ez. (e) Two-subband magnetization maps in the strong ISS regime for different Ez.A tEz=0 the well is symmetric (αν=0) and the magnetization shows an isotropic Bessel pattern. For finite Ezthe broken symmetry leads to the stripped PSH pattern in accordance with the positive ratio/summationtextαν//summationtextβν[purple line in (d)]. The arrows in the Fermi circle show the first harmonic component of/summationtextBSO,ν(k), illustrating the transition from isotropic to uniaxial field with increasing Ez. All the xymaps are frames of the spin pattern at t=13 ns. and time separation of pump and probe beams. All optical measurements were performed at 10 K. A mode-lockedTi:sapphire laser with a repetition rate of 76 MHz tunedto 816.73 nm was split into pump and probe pulses. Thepolarization of the pump beam was controlled by a photoelasticmodulator and the intensity of the probe beam was modulatedby an optical chopper for cascaded lock-in detection. An electromagnet was used to apply an external magnetic field in the plane of the QW. The spatial positioning of the pumprelative to the probe ( d) was controlled using a scanning mirror. We defined the spin injection point to be x=y=0a tt=0. The application of an in-plane electric field ( E ip=Vip/l), in thex-o ry-oriented channel, adds a drift velocity to the 2DEG electrons and allows us to determine the spin mobility and the spin-orbit field components [ 48–50]. FIG. 3. Calculated coefficients bwith vdrparallel to (a) xand (b)yfor each subband (colored) and total field (black). (c) Amplitude of the drifting spin polarization in space showing, for example, the center of the packet dcfor 75 mV . (d) Linear dependence of vdrwith the channel Vip. The slope gives the spin mobility along vdrinxor y. (e) Field scan of φKfor several Vipmeasured at dc. (f)By(x) SOas a function of vx(y) drand the current flowing in that channel. The slopes bx(y)give the strength of the SOCs that generate the field along y(x) for drift in x(y). The solid lines are Gaussian (c) and linear [(d) and (f)] fittings. Scans taken at t=13 ns. The sample was rotated such that each channel under study was oriented parallel to the external magnetic field Bext/bardblvdr for all measurements reported here. From the SOI form in kspace, we expected BSO⊥vdrimplying that the observable BSOdirection will be BSO⊥Bext. Considering this orientation, we can model the Kerr rotation signal as φK(Bext,d)= A(d) cos (ωt) with the precession frequency given by ω=(gμB/¯h)/radicalBig B2 ext+B2 SO, where A(d) is the amplitude at a given pump-probe spatial separation and BSOis the internal SO field component perpendicular to Bext(and to vdr). Figure 3shows the results of the spin drag experiment with the gate contact open. Scanning the pump-probe separationin space at fixed long time delay (13 ns), we determined thecentral position d cof the spin packet amplitude for several Vipin a given crystal orientation. From the values of dcin Fig. 3(c), we calculated the drift velocity as vdr=dc/tand plotted it as a function of Vipin Fig. 3(d). The slope of the linear fit give us spin mobilities ( μx,y s) in the range of 105cm2/V s. Values in the same order of magnitude have been measured by Doppler velocimetry for the transport insingle subband samples [ 32]. Nevertheless, in those systems the spin lifetimes were restricted to the picosecond range andthe transport was limited to the nanometer scale. Following the drifting spin packet in space, Fig. 3(e) dis- plays a B extscan from where changes in the amplitude of zeroth 245315-3M. LUENGO-KOV AC et al. PHYSICAL REVIEW B 95, 245315 (2017) FIG. 4. (a) Spin mobility and BSOas a function of the gate-tunable Ez. (b) Ratio bx(y)from (a), showing a crossing at Ez=0. (c) SOCs obtained from the addition and subtraction of bxandbyin (b). (d) Spin mobility as function of the SOCs that define the BSOstrength along the direction perpendicular to vdr. The solid lines are linear fittings and the dashed lines [(b) and (c)] are the theoretical results from the RWM combined with the self-consistent calculation of theSOCs. resonance determined BSOstrength at dc. As explained above, the data confirmed the perpendicular orientation between BSO andvdrand did not show a component parallel to Bextwithin the experimental resolution [ 51]. From the Lorenztian shape of the Bextscan [ 52,53], we evaluated a spin lifetime of 7 ns atVip=0. This experiment was only possible due to the nanosecond spin lifetime in our sample that extends the spintransport to several tens of micrometers [ 54,55]. Figure 3(f) shows the fitted values of B SOfor several Vipapplied along xandy. We observed highly anisotropic spin-orbit fields in the range of several mT as expectedfrom Figs. 3(a) and3(b).T h e B SOorientation was aligned primary with the xaxis in agreement with the simulation in Fig2(e). The slopes bx(y)=By(x) SO/vx(y) drgive the strength of the SOCs that generate the field according to Eqs. ( 2) and (3). For this condition of the sample as-grown, we found/summationtextαν=0.57 meV ˚A and/summationtextβ∗ ν=0.75 meV ˚A. Note the inverse behavior on Vipfor the mobility and for BSOstrength in perpendicular directions. In Figs. 3(c)and3(d), the axis with the largest mobility is also the axis with smallerspin-orbit field in the perpendicular direction. This result maybe related to the spin Coulomb drag observed previously inthe transport of spin-polarized electrons [ 31,32]. Next, we demonstrate the direct control of the spin mobility through thegate modification of the subband SOCs. Figure 4(a) shows that the magnitude and the orientation with the largest μ scan be tuned by Ez.BSOdisplays anisotropiccomponents with Bx SObeing larger in all the studied range, which confirms the preferential alignment towards the PSH+ in Fig. 2(e). The variation of Bx SOhas a minimum (indicated by an arrow) close to position when the second subband attainsthe PSH −(with BSOalong y). Dividing Fig. 4(a) panels, the values for bare plotted in Fig. 4(b). The lines plotted together with the data are the expected values using Eqs. ( 2) and ( 3) with the SOCs from Figs. 2(a)–2(c). When the QW approaches the symmetric condition ( Ez=0),bx(y)decreases removing the anisotropy of BSOas simulated in Fig. 2(e). The addition and subtraction of bxandbygive the sum of the Rashba and Dressselhaus SOCs displayed in Fig. 4(c). Dashed lines corresponding to the purple curves in Figs. 2(a) and2(c)are plotted together displaying excellent agreement. The slope for the Rashba SOI indicates a tuning lever armof 35 e˚A 2. This value is considerably larger than those reported in recent studies for single subband samples, typically below 10 e˚A2[17,23]. Finally, Fig. 4(d) presents μx(y) s[from Fig. 4(a)] against the SOCs defining By(x) SO:/summationtext(−αν+β∗ ν) and/summationtext(αν+β∗ ν), respectively. This last plot illustrates the inverse dependence, with negative slope, for the spin mobilityand strength of the SOCs perpendicular to the drift direction.The different slopes for xandychannels give us a hint that this effect depends not only on how B SOchanges with vdr (given by the SOCs) but also in the magnitude of the fields. A common maximum value μ0 s=3×105cm2/V s was found independent of vdrorientation. V . CONCLUSIONS In conclusion, we have studied a 2DEG system with two subbands set close to the crossed PSH regime under strongintersubband scattering and successfully described it using arandom walk model. In the spin transport with nanosecondlifetimes over micrometer distances, we demonstrate thecontrol of the subbands spin-orbit couplings with gate voltageand observed spin mobilities in the range of 10 5cm2/Vs . Specifically, the sum of the Rashba SOCs presents a linearbehavior with remarkably large tunability lever arm withgate voltage. We tailored the spin mobility by controlling thestrength of the spin-orbit interaction in the direction perpen-dicular to the drift velocity. Our findings provided evidenceof the rich physical phenomena behind multisubband systemsand experimentally demonstrated relevant properties requiredfor the implementation of a nonballistic spin transistor. ACKNOWLEDGMENTS This work is a result of the collaboration initiative SPRINT No. 2016/50018-1 of the São Paulo Research Foundation(FAPESP) and the University of Michigan. F.G.G.H alsoacknowledges financial support from FAPESP Grants No.2009/15007-5, No. 2013/03450-7, No. 2014/25981-7, and No.2015/16191-5. 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PhysRevB.77.115353.pdf

Interaction effects in a two-dimensional electron gas in a random magnetic field: Implications for composite fermions and the quantum critical point T. A. Sedrakyan and M. E. Raikh Department of Physics, University of Utah, Salt Lake City, Utah 84112, USA /H20849Received 14 December 2007; published 27 March 2008 /H20850 We consider a clean two-dimensional interacting electron gas subject to a random perpendicular magnetic field h/H20849r/H20850. The field is nonquantizing in the sense that Nh, a typical flux into the area /H9261F2in the units of the flux quantum /H20849/H9261Fis the de Broglie wavelength /H20850, is small, Nh/H112701. If the spatial scale /H9264of change of h/H20849r/H20850is much larger than/H9261F, the electrons move along semiclassical trajectories. We demonstrate that a weak-field-induced curving of the trajectories affects the interaction-induced electron lifetime in a singular fashion: it gives rise tothe correction to the lifetime with a very sharp energy dependence. The correction persists within the interval /H9275/H11011/H92750=EFNh2/3much smaller than the Fermi energy EF. It emerges in the third order in the interaction strength; the underlying physics is that a small phase volume /H11011/H20849/H9275/EF/H208501/2for scattering processes involving twoelectron-hole pairs is suppressed by curving. An even more surprising effect that we find is that disorder- averaged interaction correction to the density of states /H9254/H9263/H20849/H9275/H20850exhibits oscillatory behavior periodic in /H20849/H9275//H92750/H208503/2. In our calculations of interaction corrections, a random field is incorporated via the phases of the Green functions in the coordinate space. We discuss the relevance of the new low-energy scale for realizationsof a smooth random field in composite fermions and in disordered phase of spin-fermion model of ferromag-netic quantum criticality. DOI: 10.1103/PhysRevB.77.115353 PACS number /H20849s/H20850: 73.40.Gk, 71.10.Pm, 71.10.Ay, 71.70.Di I. INTRODUCTION Electron-electron interactions are strongly modified when electrons move diffusively.1The resulting enhancement of the interactions leads in two dimensions to a divergent cor-rection to the density of states 1,2/H9254/H9263/H20849/H9275/H20850. When electrons move ballistically and are scattered by point impurities, theanomaly persists, although it has a different underlyingscenario. 3 Within this scenario, individual impurities /H20849unlike the dif- fusive case1,2/H20850are responsible for the ballistic zero-bias anomaly by virtue of the following process. Static screeningof each impurity by the Fermi sea creates a Friedel oscilla-tion of the electron density with a period /H9261 F/2, where/H9261Fis the de Broglie wavelength. Then, the amplitude of combinedscattering from the impurity and the Friedel oscillation,which it created, exhibits anomalous behavior 3when the scattering angle is either 0 or /H9266. The energy /H9275of the scat- tered electron measured from the Fermi level EFdefines the angular interval, /H11011/H20849/H9275/EF/H208501/2, within which the scattering is enhanced. This enhancement translates into /H9254/H9263/H20849/H9275/H20850/H11008ln/H9275cor- rection to the density of states. In a diagrammatic language, creation of the Friedel oscil- lation is described by a static polarization bubble. We note inpassing that the same polarization bubble at finite frequency /H9275is responsible for the lifetime of electron of energy /H11011/H9275 with respect to creation of an electron-hole pair. It is known4that in perfectly clean electron gas, finite- range interactions do not cause anyanomaly in/H9254/H9263/H20849/H9275/H20850. Then, a natural question to ask is whether or not the anomalousbehavior of /H9254/H9263/H20849/H9275/H20850holds when a weak disorder is not point- like, as in Ref. 3, but is instead smooth. Finding an answer to this question is the main objective of the present paper. Forconcreteness, we choose a particular case of two-dimensional/H208492D/H20850electron gas in a smooth random magnetic field, al-though our main results apply to the arbitrary smooth disor- der. Historically, the interest to the problem of 2D electron motion in a random static magnetic field first emerged inconnection with a gauge field description of the correlated spin systems. 5–7Later, this interest was stimulated by the notion that electron density variations near the half-filling ofthe lowest Landau level reduces to random magnetic fieldacting on composite fermions. 8,9Another motivation was the possibility to artificially realize an inhomogeneous magneticfield acting on 2D electrons. 10–18For noninteracting elec- trons, this motion has been studied theoretically in Refs.19–32. In this paper, we trace how the perturbation of elec- tron motion by a smooth random field affects the interactioncorrections to the single-particle characteristics of the elec-tron gas. In Refs. 5–7, the averaging over static random field was carried out with the help of the path integral approach origi-nally employed for diffusively moving electrons in a noisyenvironment 33/H20849see also Refs. 34and35/H20850. A crucial fact that ensures the effectiveness of this approach is that the field isassumed to be /H9254correlated. In fact, the correlation radius must be even smaller than /H9261F. However, in realizations8–18 mentioned above, the spatial scale of change of the random field in much bigger than /H9261F. This leads to a completely different, semiclassical picture of the electron motion, whenonly the paths close to the classical trajectories are relevant.In this paper, we consider only this limit. The semiclassicalcharacter of motion suggests the way in which to perform theaveraging over disorder realizations. Namely, the equation ofmotion can be first solved for a given realization, while av- eraging over realizations is carried out at the last step. Thisorder is opposite to Refs. 5–7, where averaging was carried out in the general expression for the Green function after itwas cast in the form of a path integral.PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 1098-0121/2008/77 /H2084911/H20850/115353 /H2084928/H20850 ©2008 The American Physical Society 115353-1It might seem counterintuitive that any smooth disorder could generate a low-frequency scale for the interaction ef-fects. Indeed, a smooth random field /H20849including magnetic /H20850 does not produce Friedel oscillations, which are required forthe anomaly 3to develop. In a formal language, there are no static bubbles in the diagrams for the interaction correctionto the self-energy. More precisely, in the smooth randomfield, they are exponentially suppressed. We will, however,demonstrate that the low-frequency scale emerges from dy- namic bubbles after they are modified by a smooth disorder. The new low- /H9275scale shows up in the virtual processes involving more than one electron-hole pair, i.e., two or more bubbles. This is because the momenta of states involved inthese processes are strongly correlated, as was first pointedout in Ref. 36. Namely, these momenta are either almost parallel or almost antiparallel to each other. It is this corre-lation in momenta directions that is affected by the smoothrandom magnetic field. By suppressing the correlation, therandom field gives rise to the low- /H9275feature in/H9254/H9263/H20849/H9275/H20850. Clearly, both the height and the width of the feature dependon the magnitude of the random field. The above argumentmakes it clear why the low- /H9275scale does not emerge on the level of a single bubble modified by the random field. Thereason is that the single bubble describes excitation of asingle pair; there is no strong restriction on the momentumdirections in this process. Once the mechanism of nontrivial interplay of smooth disorder and interactions is identified, the following ques-tions arise: What is the shape of the anomaly in /H9254/H9263/H20849/H9275/H20850? How does it depend on the strength and the correlation radius ofthe random field? To address these questions, we develop asystematic approach to the calculation of interaction correc-tions in a smooth random field. The key element of our ap-proach is incorporating the action along the curved semiclas- sical trajectories into the phases of the Green functions. Ourcalculation reveals a surprising fact, which could not be ex-pected on the basis of the above qualitative consideration. Itturns out that disorder-averaged correction /H20855 /H9254/H9263/H20849/H9275/H20850/H20856exhibits anoscillatory behavior. Oscillations emerge when two pairs participating in one of the possible processes giving rise to /H9254/H9263are strongly correlated with each other. As an example, consider the process involving creation of the electron-holepair, rescattering within the pair, and its subsequent annihi-lation. In this process, oscillations come from electron-electron scattering events that happen at the points located onastraight line and at equal distances. This is an example when disorder does not suppress but, on the contrary, brings about the oscillations. Therefore, as we demonstrate in this paper, anomaly in the density of states is created by smooth spatial variation ofthe magnetic field, even though this variation does not pro-duce Friedel oscillations. Although modification of the Frie-del oscillations from a pointlike impurity by a smooth ran-dom field is not directly related to our situation with noimpurities, this problem is still useful for gaining a qualita-tive understanding. Indeed, the relevant random-field-induced length scales, in our clean case, emerge in this prob-lem as well. For this reason, we start with the study ofsuppression of the Friedel oscillations by the random fieldbefore the analysis of the interaction corrections in the ran-dom field.We are not aware of literature on disorder-induced smear- ing of the Friedel oscillations. 37However, a closely related issue of smearing of Ruderman-Kittel-Kasuya-Yoshida/H20849RKKY /H20850interaction between the localized spins by the dis- order has a long history. 38–44It is easy to see38that a short- range disorder suppresses exponentially the average RKKY interaction. However,39–41the average interaction does not represent the actual value of exchange in a given realization . This is due to the fast oscillations of the exchange with dis-tance. The typical magnitude of the exchange can be inferredfrom the averaging of the square of the RKKY interaction; 39–41this average is suppressed by the disorder only as a power law. In this paper, we demonstrate that the decay of the aver- aged Friedel oscillations in the presence of a smooth disorder is quite nontrivial. In particular, when the field is strongenough, the average, in contrast to Ref. 38, falls off with distance as a power law. We would like to note that, recently,the notion of averaged Friedel oscillations became meaning-ful. This is because the possibility of visualization of asingle-impurity-induced oscillation had been demonstratedexperimentally. 45–53The role of averaging can then be played by slow temporal fluctuations of the environment. Since ex-perimental advances 45–53were reported for correlated sys- tems, recent theoretical studies54–57addressed the Friedel os- cillations created by a single impurity in such systems. The paper is organized as follows. In Sec. II, possible regimes of electron motion in a random magnetic field areidentified. In Sec. III, we summarize our results on Friedeloscillations and interaction correction to the density of statesfor a weak random field, i.e., for the field in which the straight-line electron trajectories are weakly perturbed by thefield. Subsequent Secs. IV–IX are devoted to the derivationof the results, outlined in Sec. III. Finally, in Sec. X, wetranslate our results into predictions for experimentally ob-servable quantities in two prominent situations: compositefermions in half-field Landau level and electrons interactingwith critical magnetic fluctuations near the quantum criticalpoint. Details of some of the calculations are presented inAppendixes A–F. II. REGIMES OF ELECTRON MOTION IN A RANDOM MAGNETIC FIELD Letr/H11013/H20849x,y/H20850be the coordinates of the 2D electron. The random magnetic field along the zdirection is characterized by the correlator /H20855h/H20849r/H20850h/H20849r/H11032/H20850/H20856=h02K/H20849/H20841r−r/H11032/H20841//H9264/H20850,K/H208490/H20850/H110131, /H208492.1/H20850 where h0is the rms magnetic field and /H9264is the correlation radius. Throughout this paper, we will assume that the ran-dom field is slow fluctuating in the sense that /H9264is much bigger than the de Broglie wavelength /H9261F, the case opposite to the limit/H9264→0 considered in Refs. 5–7. In terms of semi- classical description, different regimes of motion are classi-fied according to the classical electron trajectory, which be- gins at the origin and ends at point r. One should distinguish three different regimes, as illustrated in Fig. 1. /H20849i/H20850Short-distance regime /H20849regime I in Fig. 1/H20850. The trajec-T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-2tory is of the arc-type. For this regime to be realized, two conditions must be met. Firstly, the change of magnetic fieldover the distance rshould be negligible, i.e., r/H11270 /H9264. Secondly, the curving of electron trajectory in the locally constant mag- netic field must be relatively small. The measure of this curv-ing is r/R L, where RL=/H6036ckF/eh0is the Larmour radius in the field h0, and kF=2/H9266//H9261Fis the Fermi momentum. Thus, the short-distance regime corresponds to r/H11270/H9264,RL. /H20849ii/H20850“Weak-field” long-distance regime /H20849regime II in Fig. 1/H20850. The trajectory is of the snake type. One condition for this regime is that the magnetic field changes sign many timeswithin the distance r, i.e., r/H11271 /H9264. The other is that within each interval of length /H11011/H9264, the curving of the trajectory is weak, i.e.,/H9264/H11270RL. /H20849iii/H20850“strong-field” long-distance regime /H20849regime III in Fig. 1/H20850. Electron executes many full Larmour circles before arriv- ing to the point r. The conditions for this regime are RL/H11270r and RL/H11270/H9264. Note that the last two regimes correspond to the “semi- classical” and “strong” random magnetic field regimes in thelanguage of Ref. 26. In order to accommodate all three re- gimes within a single diagram, it is convenient to introducethe dimensionless parameters u=k FRL=/H20849c/H6036kF2/eh0/H20850=Nh−1, v=r/RL/H11011kFrNh, /H208492.2/H20850 where Nh/H110211 is the flux of the field h0into the area /H9261F2/H20849in the unites of the flux quantum /H20850. Then, regime I is defined by the lines u=kF/H9264andv=kF/H9264/u, see Fig. 2. Regime III is sepa- rated from Regime I by the line v=1 and from regime II by the line u=kF/H9264. Finally, the dashed region u/H110211 in Fig. 2 corresponds to a quantizing magnetic field. The diagram inFig. 2is compiled for k F/H9264/H112711, so it does not reflect the white-noise regime, kF/H9264/H112701, of Refs. 5–7.III. MAIN RESULTS A. Friedel oscillations The simplest manifestation of the interplay of external field and electron-electron interactions shows up in spatialresponse of the electron gas to a pointlike impurity or, inother words, in Friedel oscillations. Denote with U imp/H20849r/H20850the short-range potential of the impurity. In the presence of in-teraction V/H20849r−r 1/H20850, the effective electrostatic potential in a clean electron gas falls off with rasVH/H20849r/H20850/H11008sin/H208492kFr/H20850/r2in a zero field. In Ref. 58, we had demonstrated that in a constant magnetic field h=h0, this behavior modifies to VH/H20849r/H20850=−/H92630gV/H208492kF/H20850 2/H9266r2sin/H208752kFr−/H20849p0r/H208503 12/H20876, /H208493.1/H20850 where the characteristic momentum p0is defined as p0=kF /H20849kFRL/H208502/3=/H20873h0 kF1/2/H90210/H208742/3 , /H208493.2/H20850 where/H90210is the flux quantum. In Eq. /H208493.1/H20850/H92630=m//H9266/H60362is the free electron density of states, V/H208492kF/H20850is the Fourier compo- nent of V/H20849r/H20850, and the parameter gis defined as g =/H20848Uimp/H20849r/H20850dr. Equation /H208493.5/H20850is valid within the domain kF−1 /H11351r/H11351RL, so that /H20849p0r/H208503/12 in the argument of sine does not exceed the main term 2 kFr. As follows from Eq. /H208493.2/H20850, the characteristic length scale rI=1 p0=kF1/3/H20873/H90210 h0/H208742/3 , /H208493.3/H20850 defined by p0is intermediate between RLand/H9261F, so that RL/H11271rI/H112711/kF. /H208493.4/H20850 We see from Eq. /H208493.1/H20850that only the phase of the Friedel oscillations is affected by the constant field, while the mag-nitude still falls off as 1 /r 2. The randomness of h/H20849x,y/H20850re- sults in randomness of the field-induced phase of the oscil-lations. This, in turn, translates into a faster decay ofdisorder-averaged oscillations. To quantify the behavior ofthe average /H20855V H/H20849r/H20850/H20856, we rewrite it in the formΙΙΙ ξξΙ r r ξξΙΙξ ξ r/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc /PaintProc/PaintProc FIG. 1. /H20849Color online /H20850Types of semiclassical trajectories be- tween two points separated by a distance rin a random magnetic field: in regime I, the trajectories are of arc type; in regime II, thetrajectories are of snake type; regime III corresponds to a driftingLarmour circle.LR 1 k F u= LR Fkr Fk=v 1ΙΙΙ ΙΙΙ ξ uv=Fk ξ ()3/2ξ FIG. 2. /H20849Color online /H20850Parametric regions for regimes I, II, and III. The dashed line, v=u−1 /3, separates slow and fast power-law decays of the averaged Friedel oscillations within regime I: theoscillations fall off as 1 /r 2to the left from the dashed line and as 1/r7/2to the right from the dashed line.INTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-3/H20855VH/H20849r/H20850/H20856=−/H92630gV/H208492kF/H20850 2/H9266r2F/H20849r/H20850sin/H208512kFr+/H9278/H20849r/H20850/H20852, /H208493.5/H20850 so that F/H20849r/H20850describes the decay of the magnitude of the disorder-averaged oscillations. For a given distance r, the character of the phase randomization is different in regimes Iand II. In regime I, we have /H9264/H11271r, and thus the relevant scale for the decay of /H20855VH/H20849r/H20850/H20856isrI/H20849see Fig. 3/H20850. In Sec. V , we find that in this regime, the magnitude FIand the phase /H9278Iare the following functions of the dimensionless ratio x=r/rI: FI/H20849x/H20850=1 /H208491+x6/H208501/4, /H9278I/H20849x/H20850= − arctan /H20875/H208811+x6−1 x3/H20876. /H208493.6/H20850 In regime II, with snakelike trajectories /H20849Fig.1/H20850, the sign of the random field changes many, /H11011r//H9264/H112711, times within the distance r. As demonstrated in Sec. V , in this regime, we have FII/H20849x/H20850=/H208812x /H208511+4 9x4/H208521/2/H20881cosh2x− cos2x, /H208493.7/H20850 /H9278II/H20849x/H20850= − arctan /H208751−2 1 − cot xtanh x/H20876− arctan/H208752 3x2/H20876, /H208493.8/H20850 where x=r/rII, with rIIdefined as rII=/H9257/H20873kF /H9264/H208741/2/H90210 h0. /H208493.9/H20850 In Eq. /H208493.9/H20850, the numerical factor /H9257depends on the functional form of the correlator Eq. /H208492.1/H20850and will be defined in Sec. V . In conclusion of this subsection, we point out that the actual character of the decay of Friedel oscillations with dis-tance is governed by the following dimensionless combina-tion of parameters, h 0and/H9264, in the correlator Eq. /H208492.1/H20850of the random field:/H9255=h02/H92643 /H902102kF. /H208493.10 /H20850 For/H9255/H112711, i.e., for strong random field, the averaged oscilla- tions decay with raccording to Eq. /H208493.6/H20850in regime I. This is because for/H9255/H112711, we have p0/H9264/H112711. In the opposite limit of a weak random field /H9255/H112701, we have p0/H9264/H112701, so that the scale p0−1is irrelevant, and, also, no dephasing takes place within the distance /H9264. Thus, the characteristic decay length rII /H11011/H9264//H92551/2is much larger than /H9264. This automatically guarantees that kFrII/H112711. B. Tunnel density of states Two spatial scales, rIand rII, define two energy scales, /H92750=vF rI/H11011EF/H20873h0 /H90210kF2/H208742/3 /H11011EFNh2/3, /H92751=vF rII/H11011EF/H20873/H92641/2h0 kF2/3/H90210/H20874/H11011EF/H20849kF/H9264/H208501/2Nh. /H208493.11 /H20850 As shown below, these scales manifest themselves in the anomalous behavior of the density of states in the third order in the electron-electron interaction parameter /H92630V. More spe- cifically, in regime I, the bare density of states /H92630acquires a correction/H9254/H9263I/H20849/H9275/H20850/H11011/H92630/H20849/H92630V/H208503/H20849/H92750/EF/H208503/2I/H20849/H9275//H92750/H20850, where EFis the Fermi energy. In regime II, the correction has a similarform /H9254/H9263II/H20849/H9275/H20850/H11011/H92630/H20849/H92630V/H208503/H20849/H92751/EF/H208503/2J/H20849/H9275//H92751/H20850. Both functions I/H20849z/H20850andJ/H20849z/H20850have characteristic magnitude and scale /H110111. Moreover, they exhibit a quite “lively” behavior. In particu-lar, a zero-bias anomaly /H9254/H9263I/H20849/H9275/H20850falls off at/H9275/H11271/H92750with ape- riodic oscillations , i.e., I/H20849z/H20850has a contribution /H11008sin/H2084928/3/H208813z/H20850z−3 /4exp /H20853−28/3z/H20854forz/H112711. The origin of the os- cillations is the power-law decay of FI/H20849x/H20850, given by Eq. /H208493.6/H20850, and the brunch point, x=ei/H9266/6. The contribution /H9254/H9263II/H20849/H9275/H20850also has a nonmonotonic behav- ior despite the fact that FII/H20849x/H20850falls off exponentially as exp /H20849−r/rII/H20850/H20851see Eq. /H208493.7/H20850/H20852. It is instructive to trace the evolution of the zero-bias anomaly upon increasing the magnitude of the random fieldh 0. This evolution is governed by parameter /H9255, Eq. /H208493.10 /H20850. While/H9255remains smaller than 1, where regime II applies, the anomaly is described by the function J/H20849/H9275//H92751/H20850and broadens with h0asvF/rII/H20849h0/H20850/H11008h0. Upon further increasing h0, when/H9255 exceeds 1, the crossover to regime I takes place. Zero-biasanomaly is then described by I/H20849 /H9275//H92750/H20850; it broadens with h0as vF/rI/H20849h0/H20850/H11008h02/3and develops oscillations . The fact that oscil- lations in/H9254/H9263/H20849/H9275/H20850emerge upon strengthening disorder might seem counterintuitive. This issue will be discussed in detailin Sec. VII. In a zero magnetic field, an intimate relation between impurity-induced Friedel oscillations and the zero-biasanomaly was first established in Ref. 3. Namely, it was dem- onstrated that for short-range interaction, /H9254/H9263/H20849/H9275/H20850//H92630 /H11011/H20849/H92630V/EF/H9270/H20850ln/H9275, where 1 //H9270=/H92630/H9266g2nimpis the electron scat- tering rate by the impurities, and nimpis the impurity concen- tration. This anomaly is of the first order in V. A nontrivial question is whether or not the modification, Eq. /H208493.1/H20850,i naFIG. 3. /H20849Color online /H20850Friedel oscillations of the potential cre- ated by an impurity located at the origin. Thick line: averaged Frie-del oscillations in regime I is plotted from Eqs. /H208493.5/H20850and /H208493.6/H20850. Thin line: oscillations in the absence of the random field /H20851Eq. /H208493.1/H20850with p 0=0/H20852.T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-4constant magnetic field results in field dependence of the density of states in this order. In other words, whether or nota weak magnetic field introduces a cutoff of ln /H9275at small/H9275. The answer to this question is negative. In Ref. 58,i tw a s demonstrated that the sensitivity of /H9254/H9263/H20849/H9275/H20850to a weak mag- netic field indeed emerges but in the second order in/H92630V/H20849but still in the first order in 1 //H9270/H20850. The field-dependent correction /H20851/H9254/H9263/H20849/H9275,h/H20850−/H9254/H9263/H20849/H9275,0/H20850/H20852//H92630has a characteristic frequency scale /H9275=/H92750. It is interesting to note that at /H9275/H11271/H92750, this impurity- induced correction has an oscillating character: /H9254/H9263/H20849/H9275,h/H20850−/H9254/H9263/H20849/H9275,0/H20850 /H92630=/H20849/H92630V/H208502 EF/H9270/H20873/H92750 EF/H208741/2 P/H20873/H9275 /H92750/H20874./H208493.12 /H20850 The dimensionless function Phas the following large- xas- ymptote: P/H20849x/H20850/H110081 x3/4cos/H208758 3/H208813x3/2−/H9266 4/H20876. /H208493.13 /H20850 In Fig. 4, we show the oscillating correction to the density of states; the form of the function P/H20849/H9275//H92750/H20850is addressed in Sec. VII. Technically, the derivation of Eqs. /H208493.12 /H20850and /H208493.13 /H20850is quite analogous to the derivation of the oscillatory /H9254/H9263in the random field in regime I. For this reason, we will outline thisderivation in Sec. VII. IV . POLARIZATION OPERATOR IN A RANDOM MAGNETIC FIELD Friedel oscillations VH/H20849r/H20850created by a pointlike impurity and the ballistic zero-bias anomaly originating from theseoscillations 3are intimately related to the Kohn anomaly in the polarization operator /H9016/H20849q/H20850of a clean electron gas near q=2kF. In two dimensions, this anomaly behaves as59 /H20849q−2kF/H208501/2, which translates into 1 /r2decay of the Friedel oscillations and /H11008ln/H9275correction to the density of states. Suppression of the Friedel oscillations VH/H20849r/H20850in a random field is a result of smearing of the Kohn anomaly in themomentum space. However, since the momentum is not agood quantum number in the presence of the random field, itis much more convenient to study the field-induced suppres-sion of V H/H20849r/H20850directly in the coordinate space. A. Evaluation in the coordinate space The polarization operator /H9016/H9024/H20849r,r/H11032/H20850is defined in a stan- dard way as /H9016/H20849r,r/H11032,/H9024/H20850=−i/H20885d/H9024/H11032 2/H9266G/H9024/H11032/H20849r,r/H11032/H20850G/H9024−/H9024/H11032/H20849r/H11032,r/H20850./H208494.1/H20850 Here, G/H9024/H20849r,r/H11032/H20850denotes causal Green function, which coin- cides with the retarded, G/H9024R/H20849r,r/H11032/H20850, or advanced, G/H9024A/H20849r,r/H11032/H20850, Green functions for /H9024/H110220 and/H9024/H110210, respectively. At dis- tances /H20841r−r/H11032/H20841/H11271kF−1, the polarization operator in coordinate space represents the sums /H90160/H20849r,/H9275/H20850and/H90162kF/H20849r,/H9275/H20850of slow and rapidly oscillating parts: /H90160/H20849r,/H9275/H20850=−i/H9266/H926302/H60364 2kFr/H20841/H9275/H20841exp/H20877i/H20841/H9275/H20841r vF/H20878, /H208494.2/H20850 /H90162kF/H20849r,/H9275/H20850=−/H92630/H60363 2r2sin/H208492kFr/H20850A/H208732/H9266rT vF/H20874exp/H20877i/H20841/H9275/H20841r vF/H20878. /H208494.3/H20850 Subindices 0 and 2 kFemphasize that these parts come from small momenta and momenta close to 2 kFin/H9016/H20849q/H20850, respec- tively. Equation /H208494.2/H20850emerges if one of the Green functions in Eq. /H208494.1/H20850is retarded and the other is advanced. Equation /H208494.3/H20850corresponds to the case when the Green functions in Eq. /H208494.1/H20850are both advanced or both retarded.60,61Derivation of Eqs. /H208494.2/H20850and /H208494.3/H20850is presented in Appendix A. In Eq. /H208494.3/H20850, the function A/H20849x/H20850=x sinh x/H208494.4/H20850 in/H90162kFdescribes the temperature damping. B. Qualitative derivation for the constant field For a constant magnetic field h/H20849x,y/H20850/H11013h0, the phase/H9278/H20849r/H20850 in the argument of Eq. /H208493.5/H20850can be inferred from the follow- ing simple qualitative consideration. Classical trajectory of an electron in a weak magnetic field is curved due to the Larmour motion even at the spatialscales much smaller than R L. As a result of this curving, the electron propagator G/H20849r1,r2/H20850between the points r1andr2 contains in the semiclassical limit a phase kFL, where Lis the length of the arc of a circle with the radius RLthat con- nects the points r1andr2, see Fig. 5/H20849a/H20850. Since the Friedel oscillations are related to the propagation from r1tor2and back, it is important that two arcs, corresponding to the op- posite directions of propagation, define a finite areaAso that the product G/H20849r1,r2/H20850G/H20849r2,r1/H20850should be multiplied by the Aharonov–Bohm phase factor exp /H20851ih0A//H90210/H20852. Then, the phase of this product is equal to 2kFr+/H9278/H20849r/H20850=2kFL−h0A/H20849r1,r2/H20850 /H90210. /H208494.5/H20850 Simple geometrical relations, see Fig. 5/H20849a/H20850, yieldFIG. 4. /H20849Color online /H20850Magnetic-field-induced contribution Eq. /H208497.25 /H20850to the ballistic zero-bias anomaly Eq. /H208493.12 /H20850. Field- dependent correction /H20851/H9254/H9263/H20849/H9275,h/H20850−/H9254/H9263/H20849/H9275,0/H20850/H20852//H92630in the units /H20849/H92630V/H208502/H20849/H92750/EF/H208501/2/H20849EF/H9270/H20850−1is plotted versus dimensionless energy 22/3/H20849/H9275//H92750/H20850, where/H92750=/H208492EF/H208501/3/H9275c2/3/H11271/H9275c, and/H9275cis the cyclotron frequency.INTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-5r=/H20841r1−r2/H20841=2RLsin/H20849/H9254/2/H20850, L=RL/H9254,A=2RL2/H20849/H9254− sin/H9254/H20850. /H208494.6/H20850 Using this relation and assuming r/H11270RL,w efi n d /H9278/H20849r/H20850=−h02r3 12kF/H902102=−/H20849p0r/H208503 12. /H208494.7/H20850 At this point, we would like to note that the conventional way62of incorporating a magnetic field into the semiclassical zero-field Green function amounts to multiplying it byexp /H20851/H208491//H9021 0/H20850/H20848a·dl/H20852, where the phase factor is the integral of the vector potential aalong the straight line connecting the points r1andr2. Such an incorporation neglects the field- induced curvature of the electron trajectories, and thus doesnot capture the modification Eq. /H208493.5/H20850of the Friedel oscilla- tions in the magnetic field. Indeed, the magnetic phase fac-tors, introduced following Ref. 62,cancel out in the polar- ization operator. With phase /H9278/H20849r/H20850given by Eq. /H208494.7/H20850, Friedel oscillations in a constant magnetic field acquire the form58of Eq. /H208493.1/H20850.T o see this, we notice that with accuracy of a factor of g/2/H9266 the potential, VH/H20849r/H20850coincides with /H90162kF/H20849r,0/H20850. Then, the additional phase Eq. /H208494.7/H20850transforms sin /H208492kFr/H20850into sin/H208512kFr−/H20849p0r/H208503/12/H20852,a si nE q . /H208493.1/H20850. In Appendix B, we present a rigorous derivation of Eq. /H208493.1/H20850starting from exact electronic states in a constant mag- netic field, as in Ref. 63. C. Field-induced phase of the Green function: Analytical derivation in a spatially inhomogeneous field An additional semiclassical phase /H9254/H92720→rof the Green function due to the random magnetic field h/H20849x,y/H20850is given by the following generalization of Eq. /H208494.7/H20850: /H9254/H92720→r=kF 2/H20885 0r dx/H20873dy dx/H208742 −1 /H90210/H20885 0r dxy /H20849x/H20850h/H20849x,0/H20850,/H208494.8/H20850 where the first term comes from the elongation of the trajec- tory in the magnetic field. The second term describes theAharonov–Bohm flux into the area restricted by the curve y/H20849x/H20850and the xaxis. In Eq. /H208494.8/H20850, we assumed that the field does not change along the yaxis. This is the case when the maximal yis smaller than the correlation radius /H9264of the random field. The condition y/H11021/H9264is met in the regime of the arcs and the regime of the snakes, see Figs. 1and2. In Eq. /H208494.8/H20850, we have also assumed that the magnitude of the de Broglie wavelength of the electron does not changealong the trajectory. This can be justified from the equationsof motion md 2y dt2=e ch/H20849x,0/H20850dx dt, md2x dt2=−e ch/H20849x,0/H20850dy dt. /H208494.9/H20850 It follows from Eq. /H208494.9/H20850that the energy of electron m 2/H20851/H20849dx /dt/H208502+/H20849dy /dt/H208502/H20852is conserved even if the magnetic field changes with coordinates. The most important step that allows to find /H9254/H92720→ranalyti- cally is that in regimes I and II in Fig. 1, we can replace dx /dtbyvFand set t=x/vFin the right-hand side of Eq. /H208494.9/H20850. This allows to replace d2y/dt2byvF2d2y/dx2. Then, the first of the equations yields mvF2d2y dx2=evF ch/H20849x,0/H20850. /H208494.10 /H20850 By integrating this equation, we obtain dy dx=e mcvF/H20885 0x dx/H11032h/H20849x/H11032,0/H20850+C. /H208494.11 /H20850 The constant Cshould be found from the conditions y/H208490/H20850 =0 and y/H20849r/H20850=0, leading to C=−e mcvFr/H20885 0r dx/H11032/H20885 0x/H11032 dx/H11033h/H20849x/H11033,0/H20850=−e mcvFr/H20885 0r dx/H11032/H9011/H20849x/H11032/H20850, /H208494.12 /H20850 where we have introduced an auxiliary function /H9011/H20849x/H20850=/H20885 0x dx/H11032h/H20849x/H11032,0/H20850. /H208494.13 /H20850 The meaning of /H9011/H20849x/H20850is the yprojection of the vector poten- tial. By substituting Eq. /H208494.12 /H20850back into Eq. /H208494.11 /H20850,w efi n d dy dx=e mcvF/H20875/H20885 0x dx/H11032h/H20849x/H11032,0/H20850−1 r/H20885 0r dx/H11032/H20885 0x/H11032 dx/H11033h/H20849x/H11033,0/H20850/H20876 =e mcvF/H20875/H9011/H20849x/H20850−1 r/H20885 0r dx/H9011/H20849x/H20850/H20876. /H208494.14 /H20850 With the help of Eq. /H208494.14 /H20850, one can express the first term in the additional phase Eq. /H208494.8/H20850in terms of/H9011/H20849x/H20850. It turns out that the second term in Eq. /H208494.8/H20850exceeds twice the first term. To see this, one should multiply the first of the equations inEq. /H208494.9/H20850byy/H20849x/H20850and integrate over x:1 01 r2 rr2 0 r2r r A δ a) b)R L R L1 FIG. 5. /H20849a/H20850Origin of the net “magnetic” phase Eq. /H208494.7/H20850: two arcs, corresponding to the opposite directions of propagation, definea finite area A. Aharonov–Bohm flux through this area makes the net phase negative ./H20849b/H20850Schematic illustration of the scattering pro- cesses giving rise to the additional phases Eqs. /H208497.6/H20850and /H208497.7/H20850in the product Eq. /H208497.5/H20850.T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-6/H20885 0r dxy /H20849x/H20850d2y dx2=1 /H90210kF/H20885 0r dxh /H20849x,0/H20850y/H20849x/H20850. /H208494.15 /H20850 The right-hand side of Eq. /H208494.15 /H20850is the second term in Eq. /H208494.8/H20850. The left-hand side of Eq. /H208494.15 /H20850can be related to the first term in Eq. /H208494.8/H20850upon integration by parts: /H20885 0r dxy /H20849x/H20850d2y dx2=−/H20885 0r dx/H20873dy dx/H208742 . /H208494.16 /H20850 Finally, we get /H9254/H92720→r=−kF 2/H20885 0r dx/H20873dy dx/H208742 =−1 /H902102kF/H20873/H20885 0r dx/H90112/H20849x/H20850 −1 r/H20875/H20885 0r dx/H9011/H20849x/H20850/H208762/H20874. /H208494.17 /H20850 It is convenient to rewrite the final result Eq. /H208494.17 /H20850directly in terms of the random field h/H20849r/H20850. By substituting Eq. /H208494.13 /H20850 into Eq. /H208494.17 /H20850, we obtain /H9254/H92720→r=1 /H902102kF/H9264/H20885dr1/H20885dr2h/H20849r1/H20850R/H20849r1,r2/H20850h/H20849r2/H20850, /H208494.18 /H20850 where the dimensionless kernel R/H20849r1,r2/H20850is defined as R/H20849r1,r2/H20850=/H9264/H9254/H20849y1/H20850/H9254/H20849y2/H20850/H20875r−x1x2 r−x2/H9258/H20849x2−x1/H20850 −x1/H9258/H20849x1−x2/H20850/H20876. /H208494.19 /H20850 Note that for the constant field h/H20849x,y/H20850=h0, evaluation of Eq. /H208494.18 /H20850using the kernel Eq. /H208494.19 /H20850reproduces the result Eq. /H208494.7/H20850, as expected. V . DISORDER-SMEARED FRIEDEL OSCILLATIONS IN DIFFERENT REGIMES Smearing of the Friedel oscillations in the random field h/H20849r/H20850originates from the randomness of the phase /H92720→r, which is related to h/H20849r/H20850via Eqs. /H208494.18 /H20850and /H208494.19 /H20850. Quantita- tively, the magnitude F/H20849r/H20850and the phase /H9278/H20849r/H20850of smeared Friedel oscillations Eq. /H208493.5/H20850are determined by the following averages: /H90201/H20849r/H20850=I m /H20855e2i/H9254/H92720→r/H20856h/H20849r/H20850,/H90202/H20849r/H20850=R e /H20855e2i/H9254/H92720→r/H20856h/H20849r/H20850. /H208495.1/H20850 Then, F/H20849r/H20850and/H9278/H20849r/H20850are related to the functions /H90201/H20849r/H20850and /H90202/H20849r/H20850as F/H20849r/H20850=/H20881/H20851/H90201/H20849r/H20850/H208522+/H20851/H90202/H20849r/H20850/H208522, /H9278/H20849r/H20850= arctan/H20875/H90201/H20849r/H20850 /H90202/H20849r/H20850/H20876. /H208495.2/H20850 In this section, the averages in Eq. /H208495.1/H20850will be calculated separately for the regime of arcs and the regime of snakes.A. Regime I In the regime of arcs, we have r/H11270/H9264, so that the field is almost constant within the interval /H208490,r/H20850and is equal to its “local” value. For this reason, we can perform the averagingof exp /H208512i /H92720→r/H20852over realizations of the random field h/H20849x,y/H20850 explicitly without specifying the form of the correlatorK/H20849r/ /H9264/H20850. This is because we can first set h/H20849x,y/H20850/H11013const in exp /H208512i/H92720→r/H20852and then make use of the fact that the distribu- tion function of the local field is Gaussian.64The character- istic spatial scale rIforF/H20849r/H20850and/H9278/H20849r/H20850immediately follows from Eq. /H208494.18 /H20850upon setting h/H208490,r/H20850=h0and requiring 2/H9254/H92720→r=1. This yields rI=22/331/3/p0, where p0is given by Eq. /H208493.2/H20850. 1. Random magnetic field As discussed above, we start with Friedel oscillations in a constant local magnetic field hfor which we know that FI/H20849r,h/H20850=1 ,/H9278I/H20849r,h/H20850=−/H9280r/H20873h h0/H208742 , /H208495.3/H20850 where p0=kF/H20849/H9275c/EF/H208502/3, and/H9275c=eh0/mcis the cyclotron fre- quency in the field h0. In Eq. /H208495.3/H20850, the parameter /H9280ris de- fined as /H9280r=h02r3 12/H902102kF=/H20849p0r/H208503 12. /H208495.4/H20850 To find the form of the averaged Friedel oscillation in regime I, in which p0/H9264/H112701, we have to simply substitute the local value hof the magnetic field into Eq. /H208493.1/H20850, i.e., replace p03by p03h2/h02, and perform the Gaussian averaging over the distri- bution of the local field. This averaging can be carried outanalytically with the use of the identity /H20885 −/H11009/H11009dx /H20881/H9266e−x2cos/H20849/H9280rx2+/H9252/H20850=/H90201/H20849/H9280r/H20850cos/H9252−/H90202/H20849/H9280r/H20850sin/H9252, /H208495.5/H20850 where the functions /H90201and/H90202for this case assume the fol- lowing forms: /H90201→/H20873/H9266 2/H208741/2/H20881/H208491+/H9280r2/H208501/2+1 1+/H9280r2, /H208495.6/H20850 /H90202→/H20873/H9266 2/H208741/2/H20881/H208491+/H9280r2/H208501/2−1 1+/H9280r2. /H208495.7/H20850 Using Eq. /H208495.2/H20850, we recover from Eqs. /H208495.6/H20850and /H208495.7/H20850the final result Eq. /H208493.6/H20850for the magnitude FI/H20849r/rI/H20850and the phase /H9278I/H20849r/rI/H20850of the Friedel oscillations in regime I. In terms of variables uandvin the parametric space /H20849Fig. 2/H20850, the condition /H9280r=1 can be presented as v=1 u1/3, where u=kFRL,v=r RL. /H208495.8/H20850 The dependence Eq. /H208495.8/H20850is shown in Fig. 2with a dashed line within regime I. To the left of this line, we have /H9280r/H110211, so that 1 /r2decay of the Friedel oscillations is unchanged inINTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-7the random field. To the right of the dashed line, /H9280ris bigger than 1. Then, the dependence F/H20849r/H20850/H11008/H9280r−1 /2, which follows from Eq. /H208493.6/H20850, translates into faster but still power-law de- cay,/H110081/r7/2, of the Friedel oscillations. Note also that the phase of the oscillations also changes as /H9280rcrosses over from small to large values. Indeed, as follows from Eq. /H208493.6/H20850,w e have/H9278/H20849r/H20850→−/H9266/4+1 //H208492r3/H20850in the limit/H9280r/H112711. 2. Periodic magnetic field Consider a particular case of a spatially periodic magnetic field h/H20849x,y/H20850=h˜0cos/H20849qx/H20850. For small enough q, the local de- scription applies. The corresponding condition reads q/H11270p˜0=kF/H20873h˜0 kF2/H90210/H208742/3 . /H208495.9/H20850 Under this condition, the averaged Friedel oscillation can be found by averaging Eq. /H208493.1/H20850, in which p0is replaced by p˜0/H20849h/h˜0/H208502/3, over the distribution P/H20849h/H20850of the local values of the magnetic field rather than over the Gaussian distributionEq. /H208495.5/H20850. This distribution has the form P/H20849h/H20850=1 /H9266/H20881h˜ 02−h2, /H208495.10 /H20850 so that instead of Eq. /H208495.5/H20850, we have 1 /H9266/H20885 −11 dxcos/H20849/H9280˜rx2+/H9252/H20850 /H208811−x2=J0/H20849/H9280˜r/2/H20850cos/H20873/H9280˜r 2+/H9252/H20874 =/H9020˜1/H20849/H9280˜r/H20850cos/H9252−/H9020˜2/H20849/H9280˜r/H20850sin/H9252, /H208495.11 /H20850 where J0is the Bessel function, /H9255˜r=/H20849p˜0r/H208503/12, and /H9020˜1/H20849/H9280˜r/H20850=J0/H20873/H9280˜r 2/H20874cos/H20873/H9280˜r 2/H20874, /H9020˜2/H20849/H9280˜r/H20850=J0/H20873/H9280˜r 2/H20874sin/H20873/H9280˜r 2/H20874, /H208495.12 /H20850 so that in a periodic field, instead of Eq. /H208493.6/H20850, we have F˜/H20849r/H20850=/H20851/H9020˜ 12/H20849/H9280˜r/H20850+/H9020˜ 22/H20849/H9280˜r/H20850/H208521/2=/H20879J0/H20873/H9280˜r 2/H20874/H20879, /H9278˜/H20849r/H20850= − arctan/H20875/H9020˜2/H20849/H9280˜r/H20850 /H9020˜1/H20849/H9280˜r/H20850/H20876=−/H9280˜r 2. /H208495.13 /H20850 It is instructive to present the results Eq. /H208495.13 /H20850in a different form by simply showing how the Friedel oscillation Eq. /H208493.1/H20850 gets modified on average in the presence of a periodic mag- netic field. By substituting Eq. /H208495.13 /H20850into Eq. /H208493.5/H20850,w eg e t /H20855VH/H20849r/H20850/H20856=−/H92630gV/H208492kF/H20850 2/H9266r2J0/H20873p˜03r3 24/H20874sin/H208752kFr−/H20849p˜0r/H208503 24/H20876. /H208495.14 /H20850 Equation /H208495.14 /H20850is a quite remarkable result. It suggests that,due to the periodic smooth magnetic field, the averaged Frie- del oscillations do not get smeared. Rather, they acquire an oscillatory envelope ,J0/H20849p˜03r3 24/H20850. This envelope oscillates with “period”much larger than the de Broglie wavelength butmuch smaller than the period 1 /qof change of the magnetic field. Note that this effect provides a unique possibility to mea- sure experimentally the amplitude of a periodic modulation. The reason is the following. The envelope Eq. /H208495.14 /H20850due to periodic magnetic field /H20849or electric field, i.e., due to the lat- eral superlattice /H20850translates into a distinct low-frequency be- havior of the tunnel density of states . Namely, the tunnel density of states would exhibit an oscillatorybehavior with a period /H9275/H11011p˜0vF. This period in /H9275depends only on the mag- nitude of the modulation h˜0but not on the spatial period of modulation, 2 /H9266/q. Therefore, the magnitude of modulation, which, unlike the period, is hard to measure otherwise, canbe inferred from the bias dependence of the tunneling con-ductance. B. Friedel oscillations in a random magnetic field: Regime II As the magnitude h0of the random field decreases, the character of semiclassical motion changes from arclike /H20849re- gime I in Fig. 1/H20850to the snakelike /H20849regime II in Fig. 1/H20850.T o estimate for the “widths” /H9254yof the snakelike trajectories, we use Eq. /H208494.14 /H20850and set x/H11011/H9264. This yields /H9254y /H9264/H11011eh0/H9264 mcvF/H11011/H20873/H9280 kF/H9264/H208741/2 . /H208495.15 /H20850 Since kF/H9264/H112711 and/H9280/H112701 in regime II, we confirm that /H9254y /H11270/H9264, i.e., that the snake is “narrow.” It is clear that at large enough distances r, the magnitude F/H20849r/H20850of the averaged Friedel oscillations falls off exponen- tially with r. The prime question is what is the characteristic decay length. As stated in Sec. III, this length rIIis given by Eq. /H208493.9/H20850. Below, we derive this length qualitatively and then establish the form of the magnitude FII/H20849r/H20850as well as the phase/H9278II/H20849r/H20850for the average Friedel oscillations within the entire domain of rby performing the functional averaging of exp /H208492i/H9254/H92720→r/H20850. 1. Qualitative consideration To recover qualitatively the scale rIIfrom Eq. /H208494.18 /H20850,w e consider the following toy model. Let us divide the interval/H208490,r/H20850into small intervals of a fixed length /H9264/H20849overall, r//H9264 intervals /H20850. Assume now that the random field takes only two values, h0and − h0, each with probability 1 /2 within a given interval/H9264. Under this assumption, we find from Eq. /H208494.13 /H20850 /H9011/H20849r/H20850=h0/H20851m/H20849r/H20850−n/H20849r/H20850/H20852/H9264, where m/H20849r/H20850and n/H20849r/H20850are the numbers of small intervals within the length r, with h=h0and h= −h0, respectively /H20849obviously, m+n=r//H9264/H20850. From Eq. /H208494.17 /H20850, we get for/H9254/H92720→rT. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-8/H9254/H92720→r=h02/H92642 /H902102kF/H20877/H20885 0r dx/H20851m/H20849x/H20850−n/H20849x/H20850/H208522 −1 r/H20873/H20885 0r dx/H20851m/H20849x/H20850−n/H20849x/H20850/H20852/H208742/H20878. /H208495.16 /H20850 The second term in Eq. /H208495.16 /H20850is the square of the difference /H20855m/H20856−/H20855n/H20856of coordinate /H20849not statistical /H20850average values of m/H20849x/H20850and n/H20849x/H20850. Rewriting m/H20849x/H20850as/H20855m/H20856+/H9254m/H20849x/H20850and n/H20849x/H20850as /H20855n/H20856+/H9254n/H20849x/H20850and taking into account that /H9254m/H20849x/H20850+/H9254n/H20849x/H20850=0, one can cast Eq. /H208495.16 /H20850into the form /H9254/H92720→r=4h02/H92642 /H902102kF/H20885 0r dx/H20851/H9254m/H20849x/H20850/H208522. /H208495.17 /H20850 Since the typical value of /H20851/H9254m/H20849x/H20850/H208522is/H20855m/H20849x/H20850/H20856=x/2/H9264,w ea r - rive at the estimate /H9254/H92720→r/H11011h02/H9264r2//H902102kF. Equating this addi- tional phase to unity yields r=/H90210kF1/2/h0/H92641/2, which coincides with rIIdefined by Eq. /H208493.9/H20850within a numerical factor. 2. Evaluation of the functional integral Below, we present the analytical derivation of Eqs. /H208493.7/H20850 and /H208493.8/H20850. The averaging of exp /H208492i/H9254/H92720→r/H20850required to calcu- late FII/H20849r/H20850and/H9278II/H20849r/H20850from Eqs. /H208495.1/H20850and /H208495.2/H20850reduces to the functional integral /H20855e2i/H9254/H92720→r/H20856=/H20848D/H20853h/H20849r/H20850/H20854exp /H208512i/H9254/H9272/H20849r/H20850−W/H20853h/H20849r/H20850/H20854/H20852 /H20848D/H20853h/H20849r/H20850/H20854exp /H20851−W/H20853h/H20849r/H20850/H20854/H20852,/H208495.18 /H20850 where/H9254/H9272/H20849r/H20850=/H9254/H92720→ris given by Eq. /H208494.17 /H20850, and exp /H20849−W/H20853h/H20854/H20850with W/H20853h/H20849r/H20850/H20854given by W/H20853h/H20854=1 /H92644h02/H20885 0r2/H20885 −/H11009/H11009 dx1dy1/H20885 0r2/H20885 −/H11009/H11009 dx2dy2 /H11003h/H20849x1,y1/H20850h/H20849x2,y2/H20850/H9260/H20849x1−x2,y1−y2/H20850,/H208495.19 /H20850 is the statistical weight of the realization h/H20849x,y/H20850. The dimen- sionless function /H9260/H20849r,r/H11032/H20850is related to the correlator Eq. /H208492.1/H20850 in a standard way: /H20885dr/H11032/H9260/H20849r,r/H11032/H20850K/H20849r/H11032,r/H11033/H20850=/H92644/H9254/H20849r−r/H11033/H20850. /H208495.20 /H20850 The reason why the functional integral Eq. /H208495.18 /H20850can be evaluated explicitly is that both W/H20853h/H20854and/H9254/H92720→rarequa- dratic in the random field h/H20849x,y/H20850. The fact that we integrate over realizations of h/H20849x,y/H20850defined on the interval which is finite ,0/H11021x/H11021r,i nt h e xdirection and infinite in the ydirec- tion suggests the following expansion of h/H20849x,y/H20850: h/H20849x,y/H20850=h0/H20858 n=−/H11009/H11009/H20885 −/H11009/H11009 dqAn,qeiqy //H9264exp/H208732/H9266inx r/H20874./H208495.21 /H20850 The asymmetry between xand yis quite significant in the calculation below, namely, for r/H11271/H9264, the characteristic values ofxturn out to be much larger than the characteristic values ofyif/H11011/H9254y/H11270/H9264, see Eq. /H208495.15 /H20850. This allows to replace K/H20849x,y,x/H11032,y/H11032/H20850in Eq. /H208495.20 /H20850by/H9253/H9264K/H208490,y−y/H11032/H20850/H9254/H20849x−x/H11032/H20850, where the dimensionless constant /H9253is defined by the relation/H9253=/H208480/H11009dx/H20848−/H11009/H11009dyK /H20849x,y/H20850 /H9264/H20848−/H11009/H11009dyK /H208490,y/H20850=/H20873/H9266 2/H20874/H208480/H11009dzzK /H20849z/H20850 /H208480/H11009dzK /H20849z/H20850, /H208495.22 /H20850 where in the second identity, we used the fact that K/H20849x,y/H20850is isotropic. Substituting Eq. /H208495.21 /H20850into Eq. /H208495.19 /H20850, we obtain W/H20853h/H20854=r /H9253/H9264/H20858 n=−/H11009/H11009/H20885dq/H20841An,q/H208412 K˜/H20849q/H20850, /H208495.23 /H20850 where K˜/H20849q/H20850is the Fourier transform of the correlator, more precisely, K˜/H20849q/H20850=1 /H208812/H9266/H20885dy /H9264eiqy //H9264K/H208490,y/H20850. /H208495.24 /H20850 The expression for /H9254/H9272/H20849r/H20850in terms of the coefficients An,q follows upon substitution of Eq. /H20849B3/H20850into Eq. /H208494.18 /H20850: /H9254/H9272/H20849r/H20850=h02 /H902102kF/H20858 n1=−/H11009/H11009 /H20858 n2=−/H11009/H11009/H20885dq1An1,q1/H20885dq2An2,q2/H20885 0r dx1 /H11003/H20885 0r dx2/H20875r−x1x2 r−x2/H9008/H20849x2−x1/H20850 −x1/H9008/H20849x1−x2/H20850/H20876exp/H208772/H9266i r/H20849n1x1+n2x2/H20850/H20878. /H208495.25 /H20850 Performing the integration, we obtain /H9254/H9272/H20849r/H20850=/H9255r3 /H92643/H208771 12/H20885dqA0,q2+/H20858 n/H110220cn/H20879/H20885dqAn,q/H208792 +/H20885dqA0,q/H20885dq/H20858 n/H110220/H20851bnAn,q+bn*An,q*/H20852/H20878, /H208495.26 /H20850 where numerical coefficients bnand cnare defined as bn=−1 2/H92662n2+i 2/H9266n,cn=1 2/H92662n2. /H208495.27 /H20850 In writing the result of integration in the form of Eq. /H208495.26 /H20850, we have used the dimensionless parameter /H9255defined by Eq. /H208493.10 /H20850. The meaning of this parameter is the additional phase Eq. /H208494.17 /H20850, acquired by the electron traveling the distance /H11011/H9264 in a constant magnetic field h0. Since our calculation pertains to the limit r/H11271/H9264, the relevant values of /H9255are small. The functional integration reduces now to the infinite product of the ratios of integrals over An,qandAn,q*. The details of calculation are given in Appendix C. Here, we present only the final result for r/H11271/H9264: /H20855e2i/H9254/H9272/H20849r/H20850/H20856=1 1−2i 3/H20873r rII/H208742/H20863 n=1/H11009n2 n2−2i/H20849r/rII/H208502//H92662,/H208495.28 /H20850 where the characteristic length rIIis defined asINTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-9rII=2/H9264 /H20851/H208812/H9266/H9253/H9255/H208521/2=/H208814kF/H902102 /H208492/H9266/H208501/2/H9253/H9264h02. /H208495.29 /H20850 The above definition specifies the numerical coefficient /H9257in Eq. /H208493.9/H20850of Sec. III as /H9257=2 //H208492/H9266/H208501/4/H92531/2. This coefficient depends on the explicit form of the correlator via the factor /H9253, given by Eq. /H208495.22 /H20850. It is seen that rII/H11011/H9264//H92551/2is indeed much larger than /H9264. This means that, in regime II, Friedel oscillations survive well beyond the correlation radius ofrandom magnetic field. Note also a distinctive dependencer II/H110081/h0of the characteristic scale on the magnitude of the random field. In fact, the infinite product in Eq. /H208495.28 /H20850can be evaluated for arbitrary r/rIIusing the identity sinx x=/H20863 n/H208731−x2 /H92662n2/H20874, /H208495.30 /H20850 which yields /H20855e2i/H9254/H9272/H20849r/H20850/H20856 =1 1−2i 3/H20873r rII/H208742/H208491+i/H20850/H20849r/rII/H20850 sin/H20849r/rII/H20850cosh /H20849r/rII/H20850+icos/H20849r/rII/H20850sinh /H20849r/rII/H20850. /H208495.31 /H20850 With the help of Eq. /H208495.31 /H20850, we can calculate the magnitude FII/H20849r/H20850and the phase /H9278II/H20849r/H20850of the Friedel oscillations in re- gime II. Corresponding expressions are given by Eqs. /H208493.7/H20850 and /H208493.8/H20850. 3. Limiting cases It is not surprising that Friedel oscillations in regime II are smeared more efficiently than in regime I. The small- rand the large- rasymptotes of FII/H20849r/H20850are the following: FII/H20849r/H20850=1−11 45/H20873r rII/H208744 ,r/H11270rII, /H208495.32 /H20850 FII/H20849r/H20850=3/H208812/H20873rII r/H20874exp/H20873−r rII/H20874,r/H11271rII. /H208495.33 /H20850 We see from Eq. /H208495.33 /H20850that Friedel oscillations decay expo- nentially as rexceeds rII. This should be contrasted to Eq. /H208493.6/H20850for regime I, where the FI/H20849r/H20850falls off slowly as r−3 /2 with r. On the qualitative level, the strong difference be- tween regimes I and II, which is reflected in the differentcharacters of decay of F I/H20849r/H20850and FII/H20849r/H20850, is that in regime I, the random field does not change within the characteristic spatialinterval r I, while in regime II, the sign of the random field changes many times within the characteristic spatial intervalr II. VI. DENSITY OF STATES: QUALITATIVE DISCUSSION In the previous consideration, we had demonstrated that in two regimes of electron motion in random magnetic field,i.e., regime of arcs, I, and regime of snakes, II, there are twolength scales, r Iand rII, respectively, that govern the interac-tion effects. In this section, we demonstrate that the density of states/H9254/H9263/H20849/H9275/H20850exhibits an anomalous behavior within the frequency range /H9275/H11011vF/rIin the regime of arcs and /H9275 /H11011vF/rIIin the regime of snakes. The process underlying the interaction corrections to the density of states is creation /H20849and annihilation /H20850of the virtual electron-hole pairs by an electron moving in the randomfield. Our central finding is that, unlike the case of pointlikeimpurities, 3the low-/H9275structure in the density of states emerges as a result of electron-electron scattering processesinvolving more than one pair . We start with a three-scattering process in the regime of arcs and demonstrate qualitatively how the frequency scale, vF/rI, emerges. Three-scattering process involves twovirtual pairs. Consider first this process in the absence of the randomfield. It is illustrated in Fig. 6. In the analysis of this process, 36,65,66it was established that the directions of mo- menta of the participating electrons are strongly correlated,namely, they are either almost parallel or almost antiparallel.A quantitative estimate for the degree of alignment of themomenta can be obtained from inspection of Fig. 6.I ft h e scattering acts take place at points 0, r 1, and r2, then the corresponding matrix element contains a phase factor exp /H208512ikF/H20849r1−r2+/H20841r1−r2/H20841/H20850/H20852. /H208496.1/H20850 This phase factor does not oscillate if the angle between the vectors r1andr2is smaller than /H208491/kFr/H208501/2, where ris the typical length of r1,r2. The above angular restriction constitutes the origin of a zero-bias anomaly in the regime of arcs. Zero-bias anomalyemerges as a result of the suppression of the three-scatteringprocess in the field h 0. This suppression is due to curving of the electron trajectory by the angle /H11011r/RL/H20849see Fig. 5/H20850, and it occurs when the curving angle exceeds the allowed angle ofalignment. Therefore, upon equating /H208491/k Fr/H208501/2tor/RL,w e find r=rI, which leads us to the conclusion that /H9275/H11011vF/rIis the energy scale at which /H9254/H9263/H20849/H9275/H20850exhibits a feature. Note that in considering the Friedel oscillations, we inferred the scaler Ifrom a different condition, namely, that the additional phase /H11011/H20849p0rI/H208503due to the elongation of a trajectory in mag- netic field is /H113511. Thus, we conclude that in the regime ofω−Ωωω ω−Ω E0 r2 Fr1 FIG. 6. /H20849Color online /H20850Third-order process describing creation of a pair by initial electron at point r=0, rescattering within the pair at point r=r1, and annihilation of the pair at point r=r2. The dia- gram corresponding to this process is shown later in the text /H20849the first diagram is in Fig. 8/H20850.T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-10arcs, the same spatial scale rIwhich governs the “dephasing” of/H90162kF/H20849r/H20850/H20849a polarization bubble /H20850also governs the suppres- sion of the three-scattering process, which involves three loops. The above analysis of phases in the matrix element of the three-scattering process can be extended to the regime ofsnakes. This analysis yields that three-scattering process isefficient at distances r/H11351r II, see Eq. /H208493.9/H20850. Analysis of phases similar to the phase, given by Eq. /H208496.1/H20850, also suggests that two-scattering processes are insensitive to the magnetic field. This insensitivity can be explained as follows. Calculation ofthe contribution to the density of states from the three-scattering process with matrix element Eq. /H208496.1/H20850involves integration over positions of r 1andr2,with respect to the origin ,r=0, which reveals the angular restriction on their orientations. Similar integration for a two-scattering processinvolves only the orientation of the interaction point rwith respect to the origin. Then, the angular restriction, and itslifting by magnetic field, does not emerge. In the next sub-section, the above qualitative arguments are supported by arigorous calculation. VII. DENSITY OF STATES: ANALYTICAL DERIV ATION A. Absence of a zero-bias anomaly in the second order in the interaction strength We start from a general expression for the average density of states: /H9254/H9263/H20849/H9275/H20850=−1 /H9266/H20855ImG/H9275/H20849r,r/H20850/H20856h/H20849x,y/H20850, /H208497.1/H20850 where /H20855¯/H20856denotes disorder averaging defined by Eq. /H208495.18 /H20850. In the second order in interaction strength, the random-field-induced correction to the density of states are determined bytwo diagrams shown in Fig. 7. The corresponding analytical expressions read /H9254/H92631/H20849/H9275/H20850=4I m2 /H9266/H20885d/H9024 2/H9266/H20885drdr1dr2G/H9275/H20849r,r1/H20850G/H9024/H20849r1,r2/H20850 /H11003/H20853V2/H208492kF/H20850/H90162kF/H20849r1,r2,/H9275−/H9024/H20850+V2/H208490/H20850 /H11003/H9016 0/H20849r1,r2,/H9275−/H9024/H20850/H20854G/H9275/H20849r2,r/H20850, /H208497.2/H20850 /H9254/H92632/H20849/H9275/H20850=−2I m2 /H9266/H20885d/H9024 2/H9266/H20885drdr1dr2G/H9275/H20849r,r1/H20850 /H11003G/H9024/H20849r1,r2/H20850G/H9275/H20849r2,r/H20850/H20853V/H208490/H20850/H208512V/H208492kF/H20850−V/H208490/H20850/H20852 /H11003/H9016 2kF/H20849r1,r2,/H9275−/H9024/H20850+V2/H208490/H20850/H90160/H20849r1,r2,/H9275−/H9024/H20850/H20854, /H208497.3/H20850where V/H208490/H20850and V/H208492kF/H20850are the Fourier components of the interaction potential V/H20849r/H20850with momenta zero and 2 kF, re- spectively. Three Green functions in Eqs. /H208497.2/H20850and /H208497.3/H20850de- scribe the propagation of electron between the points /H20849r,r1/H20850, /H20849r1,r2/H20850, and /H20849r2,r/H20850/H20849Fig. 7/H20850. The polarization bubble de- scribes the creation of electron-hole pair at point r1and an- nihilation at point r2. The difference in signs in Eqs. /H208497.2/H20850 and /H208497.3/H20850is due to the fact that the first diagram contains two closed fermionic loops, whereas the second diagram containsonly one. Numerical factors 4 and 2 in Eqs. /H208497.2/H20850and /H208497.3/H20850 come from summation over the spin indices. The differencebetween them is due the fact the spin of electron-hole pair isnot fixed in the first diagram, but it is fixed in the seconddiagram. The factor of 2 in the product 2 V/H208490/H20850V/H208492k F/H20850in Eq. /H208497.3/H20850is related to the annihilation of the electron-hole pair since the hole is annihilated with initial electron. Then, the momentum transfer can be 2 kFin the course of creation and zero in the course of annihilation, and vice versa. It is important to emphasize that the Green functions and polarization operators in Eqs. /H208497.2/H20850and /H208497.3/H20850contain the in- formation about the random field h/H20849x,y/H20850via their additional phases:/H9272r1→r2inG/H9275/H20849r1,r2/H20850and 2/H9272r1→r2in/H90162kF/H20849r1,r2,/H9275/H20850. The phase/H9272r1→r2always enters in combination with a main term, kF/H20841r1−r2/H20841. Obviously, /H90160/H20849r1,r2,/H9275/H20850does not contain a field-induced phase. Thus, only the terms containing /H90162kFin Eqs. /H208497.2/H20850and /H208497.3/H20850should be considered. Now, it is easy to see that /H9254/H92631and/H9254/H92632do not exhibit a field-induced anomaly at small /H9275. This is because the field dependence is canceled out in the integrands of Eqs. /H208497.2/H20850 and /H208497.3/H20850. To see this, we first note that the integration over r in Eqs. /H208497.2/H20850and /H208497.3/H20850can be easily performed using the fact that /H20848drG/H9275/H20849r1,r/H20850G/H9275/H20849r,r2/H20850is equal to the derivative /H11509G/H9275/H20849r1,r2/H20850//H11509/H9275. Then, we note that the contribution to /H9254/H92631, /H9254/H92632comes only from “slow” terms in the product of two Green functions, G/H9275/H20849r1,r2/H20850,G/H9024/H20849r1,r2/H20850, and/H90162kF. These slow terms do not contain rapidly oscillating factors exp /H208532ikF/H20841r1−r2/H20841/H20854. On the other hand, cancellation of the rapid terms in the product automatically results in the can- cellation of the field-dependent terms. As it was explained in qualitative discussion, the situation changes in the third order in the interactions. The corre-sponding expression for /H9254/H9263/H20849/H9275/H20850is derived in the next subsec- tion.rr1 r2 r rr1 r2 FIG. 7. Diagrams for the second-order corrections Eq. /H208497.2/H20850 /H20849left/H20850and Eq. /H208497.3/H20850/H20849right /H20850to the density of states. FIG. 8. Third-order diagrams contributing to the zero-bias anomaly in the density of states. The random field enters via thephases of the Green functions.INTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-11B. General expression for the third-order interaction correction to the density of states Relevant diagrams for the third-order correction to the density of states are shown in Fig. 8. The same eight dia- grams were considered in Ref. 36in the momentum space. In Ref. 36, the analysis of these diagrams was restricted to small momenta. In our coordinate representation, this meansthat only/H9016 0/H20849r/H20850parts of the polarization operators was kept, whereas/H90162kF/H20849r/H20850parts were neglected. As explained above, to reveal the sensitivity to the random field, we will keep only the/H90162kF/H20849r/H20850parts. Then, the correction to the Green function corresponding to the sum of the eight diagrams in Fig.8acquires the form /H9254/H9263/H20849/H9275/H20850=2V/H208490/H20850V/H208492kF/H20850/H208512V/H208492kF/H20850−V/H208490/H20850/H20852 /H11003Imi 2/H92662/H20885d/H9024 2/H9266/H20885drdr1dr2G/H9275/H20849r,r1/H20850G/H9024/H20849r1,r2/H20850 /H11003/H9016 2kF/H20849r1,0,/H9275−/H9024/H20850/H90162kF/H208490,r2,/H9275−/H9024/H20850G/H9275/H20849r2,r/H20850. /H208497.4/H20850 All the diagrams reduce to the same integrals. Concerning the difference in numerical coefficients, it comes from thenumber of closed fermionic loops and the spin degrees offreedom. Taking this into account, interaction coefficient cor-responding to the first two diagrams will be 2 /H20849−2/H20850 2V3/H208492kF/H20850. The coefficient of the third diagram is /H20849−2/H208503V3/H208492kF/H20850. Thus, we see that the contributions /H11008V3/H208492kF/H20850cancel each other. The first and the second diagrams in the second row are equal to each other, and each of them has a coefficient/H20849−2/H20850 2V/H208490/H20850V2/H208492kF/H20850. The coefficient of the last diagram in the second row is /H20849−2/H20850V2/H208490/H20850V/H208492kF/H20850since it has only one closed fermionic loop. Finally, the first diagram in the third row hasonly one closed fermionic loop and is equal to the seconddiagram on the third row. Each of these diagrams contributeswith the coefficient /H20849−2/H20850V/H208490/H20850V 2/H208492kF/H20850. On the physical level, the eight diagrams in Fig. 8de- scribe different electron-electron three-scattering processes.For example, the first diagram corresponds to creation ofelectron-hole pair by the initial electron followed by rescat-tering within a created pair and, finally, its annihilation. Three stages of this process are illustrated in Fig. 6. How- ever, creation, rescattering, and annihilation of a pair canfollow a different scenario, namely, the rescattering processcan involve the initial electron . This scenario is captured by the second diagram in the first row in Fig. 8. At this point, we note that the diagrams in Fig. 8donot exhaust all possible three-scattering processes. In fact, alldiagrams in Fig. 8have an identical structure in the sense that they can be combined into a single generalized diagram, as shown in Fig. 9/H20849a/H20850. There are also eight other diagrams combined into a single generalized diagram, as shown in Fig.9/H20849b/H20850, that are not sensitive to the random field. This is be- cause, in the absence of the random field, the phase factorcorresponding to Fig. 9/H20849b/H20850is large, namely, 2 /H208512k F/H20849r1+r2/H20850/H20852. The crucial difference between the contributions Eqs. /H208497.2/H20850and /H208497.3/H20850and Eq. /H208497.4/H20850is that the cancellation of the rapid-oscillating terms in the integrand of Eq. /H208497.4/H20850preservesthe field dependence. To see this, we first replace /H20848drG/H9024/H20849r1,r/H20850G/H9024/H20849r,r2/H20850by/H11509G/H9024/H20849r1,r2/H20850//H11509/H9024, as discussed above, and then consider the phase of the product: G/H9024/H20849r1,r2/H20850G/H9275/H20849r1,r2/H20850/H90162kF/H208490,r2,/H9275−/H9024/H20850/H90162kF/H20849r1,0,/H9275−/H9024/H20850. /H208497.5/H20850 Figure 5/H20849b/H20850illustrates this product graphically. It is seen from Fig. 5/H20849b/H20850that, when the fast oscillating terms exp /H208532ikF/H20841r1 −r2/H20841/H20854, exp /H208532ikFr1/H20854, and exp /H208532ikFr2/H20854cancel each other out, the additional phase enters into the product either in combination 2/H9254/H9272/H9018/H20849+/H20850=2/H9254/H9272r1→0+2/H9254/H9272r2→0−2/H9254/H9272r1→r2, /H208497.6/H20850 or in combination /H20851see Fig. 5/H20849b/H20850/H20852 2/H9254/H9272/H9018/H20849−/H20850=2/H9254/H9272r1→0−2/H9254/H9272r2→0+2/H9254/H9272r1→r2. /H208497.7/H20850 Since additional phases defined by Eqs. /H208494.18 /H20850and /H208494.19 /H20850 arecubic in distance, the combinations Eqs. /H208497.6/H20850and Eq. /H208497.7/H20850arenonzero . This is in contrast to the two-scattering processes, where the cancellation occurs identically for arbi- trary dependence of /H9254/H9272/H20849r/H20850onr. In turn, noncancellation of additional phases in Eqs. /H208497.6/H20850and /H208497.7/H20850means that the ran- dom field causes a zero-bias anomaly, more specifically, afeature in /H9254/H9263/H20849/H9275/H20850at small/H9275. The final form of /H9254/H9263/H20849/H9275/H20850emerges upon integration of Eq. /H208497.4/H20850over azimuthal angles of r1andr2, which can be per- formed analytically, using the relation /H20855eip/H20849r1+r2/H20850/H20856/H9272p,/H9272r1,/H9272r2=sin/H20851p/H20849r1/H11006r2/H20850+/H9266/4/H20852 p/H20849r1r2/H208501/2. /H208497.8/H20850 Upon combining rapidly oscillating terms in the integrand of Eq. /H208497.4/H20850into slow terms, we obtain /H9254/H9263/H20849/H9275/H20850=/H9254/H9263/H20849+/H20850/H20849/H9275/H20850+/H9254/H9263/H20849−/H20850/H20849/H9275/H20850, /H208497.9/H20850 where /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc = + c0/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc r1 r2 /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProcr a drr0/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc r1 r2 b /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc/PaintProc /PaintProc/PaintProc/PaintProcr FIG. 9. /H20849a/H20850Eight diagrams for /H9254G/H9275/H20849r,r/H20850, which are shown in Fig. 8, are combined into one generalized diagram. Electron- electron scattering processes take place at points 0, r1, and r2/H20849b/H20850 Eight third-order diagrams that do not contribute to the zero-bias anomaly are combined into one generalized diagram. /H20849c/H20850Two types of four-leg interaction vertices are combined into big dots. /H20849d/H20850An example of a third-order diagram of type /H20849b/H20850.T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-12/H9254/H9263/H20849+/H20850/H20849/H9275/H20850 /H92630=−/H20849/H92630V/H208503 2EF/H92663/2kF1/2/H20885 r2/H11022r1dr1dr2 /H20849r1r2/H208503/2/H20849r1+r2/H208501/2 /H11003/H20885 0/H9275 d/H9024sin/H20851vF−1/H20849/H9275−/H9024/H20850/H20849r1+r2/H20850/H20852 /H11003sin/H208772/H9254/H9272/H9018/H20849+/H20850+/H9266 4−/H20849/H9275+/H9024/H20850 vF/H20849r1+r2/H20850/H20878 /H208497.10 /H20850 and /H9254/H9263/H20849−/H20850/H20849/H9275/H20850 /H92630=−/H20849/H92630V/H208503 2EF/H92663/2kF1/2/H20885 r2/H11022r1dr1dr2 /H20849r1r2/H208503/2/H20849r2−r1/H208501/2 /H11003/H20885 0/H9275 d/H9024sin/H20851vF−1/H20849/H9275−/H9024/H20850/H20849r1+r2/H20850/H20852 /H11003sin/H208772/H9254/H9272/H9018/H20849−/H20850+/H9266 4+/H20849/H9275+/H9024/H20850 vF/H20849r2−r1/H20850/H20878, /H208497.11 /H20850 where we had assumed that the interaction is short ranged and set V/H208490/H20850=V/H208492kF/H20850=/H92630V. The two contributions in Eq. /H208497.9/H20850correspond to the locations of the points r1andr2on the opposite and the same sides from the origin, respectively,see Fig. 5/H20849b/H20850. We note that the phases /H9254/H9272/H9018/H20849+/H20850,/H9254/H9272/H9018/H20849−/H20850, which enter into the argument of sine in Eqs. /H208497.10 /H20850and /H208497.11 /H20850, are quadratic in the random field h/H20849x,y/H20850, as seen from Eqs. /H208494.18 /H20850and /H208494.19 /H20850. This suggests that the averaging over realizations of h/H20849x,y/H20850 can be carried out analytically in the integrands of Eqs. /H208497.10 /H20850and /H208497.11 /H20850. Similarly to the case of Friedel oscilla- tions, it is convenient to perform this averaging separatelyfor regimes I and II. This is done in Secs. VIII and IX below.In the remainder of this section, we will evaluate the inter-action correction /H9254/H9263/H20849/H9275/H20850for two particular cases: /H20849i/H20850constant magnetic field h/H20849x,y/H20850/H11013h0in a clean electron gas and /H20849ii/H20850 h/H20849x,y/H20850/H11013h0in electron gas with a small concentration of pointlike impurities. C. Case of constant magnetic field: Oscillations of /H9254/H9263(/H9275) In a constant magnetic field h/H20849x,y/H20850/H11013h0, the characteristic scale of frequency in Eqs. /H208497.10 /H20850and /H208497.11 /H20850is/H92750=vF/rI. This was stated in Sec. III. Now, this scale of frequenciesemerges naturally upon substituting in Eqs. /H208497.10 /H20850and /H208497.11 /H20850 the phases 2 /H9254/H9272/H9018/H20849+/H20850,2/H9254/H9272/H9018/H20849+/H20850, calculated from Eq. /H208494.18 /H20850in a constant magnetic field: 2/H9254/H9272/H9018/H20849/H11006/H20850=/H11007p03 4r1r2/H20849r1/H11006r2/H20850, /H208497.12 /H20850 where p0is defined by Eq. /H208493.2/H20850. The integrals in Eqs. /H208497.10 /H20850 and /H208497.11 /H20850converge at distances r1,r2/H11011p0−1=rI. As a result, /H9254/H9263/H20849+/H20850and/H9254/H9263/H20849−/H20850are certain universal functions of /H9275rI/vF =/H9275//H92750. The plot of /H9254/H9263/H20849+/H20850+/H9254/H9263/H20849−/H20850vs dimensionless ratio x =22/3/H9275//H92750is presented in Fig. 10. To isolate the frequency dependence, in addition to x, we had introduced the dimen-sionless variables r1/rIand r2/rIafter which/H9254/H9263/H20849/H9275/H20850acquires the form /H9254/H9263/H20849/H9275/H20850 /H92630=−/H20849/H92630V/H208503 22/3/H20849/H9266kFrI/H208503/2B/H20849x/H20850. /H208497.13 /H20850 The integral over /H9024in Eqs. /H208497.10 /H20850and /H208497.11 /H20850can be evalu- ated analytically. The remaining dimensionless double inte-grals were calculated numerically. While the characteristicscale, x/H110111, of change of the function B/H20849x/H20850follows from qualitative consideration, Fig. 10indicates that B/H20849x/H20850also ex- hibits sizable oscillations. These oscillations come only fromthe contribution /H9254/H9263/H20849+/H20850. They owe their existence to the pecu- liar structure of the argument of sine in Eq. /H208497.10 /H20850. Namely, this argument has saddle points with respect to both r 1and r2 atr1=r2=21/3rI/H20849/H9275//H92750/H208501/2/31/2. Oscillatory behavior of B/H20849x/H20850 is governed by the value of the argument at the saddle point,which is /H11011/H20849 /H9275//H92750/H208503/2. Strictly speaking, the saddle point de- termines the value of the integral only when /H9275/H11271/H92750. How- ever, numerics shows that oscillations in Fig. 10set in start- ing already from x/H110111. These oscillations reflect the distinguished contribution from the three-scattering process,shown in Fig. 5/H20849b/H20850, in which scattering events occur at r 1 =r2=21/3rI/H20849/H9275//H92750/H208501/2/31/2. Equation /H208497.13 /H20850and Fig. 10constitute an experimentally verifiable prediction. Correction Eq. /H208497.13 /H20850describes the fea- ture in the tunneling conductance of a clean two-dimensionalelectron gas as a function of bias that emerges in a weakmagnetic field h 0. It follows from the prefactor in Eq. /H208497.13 /H20850 that the magnitude of /H9254/H9263scales with h0asrI−3 /2/H11008h0.W e emphasize that the correction /H9254/H9263/H20849/H9275/H20850remains distinguishable even when the structure in the density of states due to theLandau quantization is completely smeared out, e.g., due tofinite temperature. This follows from the above relation be-tween /H92750and the cyclotron frequency /H9275c, namely, /H20849/H9275c//H92750/H20850 /H11011/H20849/H9275c/EF/H208501/3/H112701. In discussing the relevance to the experiment, one should have in mind that realistic samples always contain a certaindegree of disorder. Therefore, the question remains as towhether the oscillations of /H9254/H9263/H20849/H9275/H20850in a constant magnetic field survive in the presence of the short-range impurities. Thisquestion is nontrivial since impurities themselves give rise toFIG. 10. /H20849Color online /H20850Dimensionless correction Eq. /H208497.13 /H20850to the tunnel density of states in a weak constant magnetic field isplotted vs dimensionless energy x=2 2/3/H9275//H92750. The plot is obtained upon numerical integration in Eqs. /H208497.10 /H20850and /H208497.11 /H20850.INTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-13the singular correction to /H9254/H9263/H20849/H9275/H20850/H20849zero-bias anomaly /H20850even in a zero field. Then, the above question can be reformulated asfollows: whether the field-induced oscillations are distin-guishable on the background of a zero-bias anomaly. It turnsout that by introducing the Friedel oscillations, pointlike im-purities actually enhance the oscillatory part of /H9254/H9263/H20849/H9275/H20850. This question is addressed in the next subsection. D. Ballistic zero-bias anomaly in a constant magnetic field Conventional ballistic zero-bias anomaly3caused by pointlike impurities is described by two second-order dia-grams, shown Fig. 7, in which one of two interaction lines is replaced by an impurity line. As was shown in Ref. 3, these diagrams with one interaction line and one impurity lineyield a singular correction /H9254/H9263/H20849/H9275/H20850//H92630/H11011/H20849/H92630V/EF/H9270/H20850ln/H20849/H9275/H20850to the density of states. Here, /H9270−1is the scattering rate proportional to the impurity concentration. Qualitatively, the singular cor-rection originates from the combined scattering of electronby the impurity and the Friedel oscillation /H11008sin/H208492k Fr/H20850/r2, created by the same impurity . This Friedel oscillation is rep- resented by the polarization loop in Fig. 7. In the presence of the impurity, this loop describes static response of the elec- tron gas, and thus the polarization operator /H90162kF/H20849/H9275,r/H20850corre- sponding to the loop should be taken at /H9275=0. As was men-tioned in Sec. III, a weak perpendicular magnetic field h leaves the logarithmic correction unchanged. To reveal thesensitivity to h, one should calculate /H9254/H9263to the next /H20849second /H20850 order in V. Corresponding diagrams with one impurity and two interaction lines are shown in Figs. 11–13. It is easy to see that there are overall 24 different diagrams. Indeed, thegeneralized diagram, Fig. 9/H20849a/H20850, for the third-order interaction correction contains three generalized four-leg vertices shownin Fig. 9/H20849c/H20850. Hence, Fig. 9/H20849a/H20850represents 2 3=8 different dia- grams. In each of these eight diagrams, the impurity line canreplace the interaction line in three places, generating one ofthe 24 different diagrams that are shown in Figs. 11–13. All these diagrams are divided into three groups according totheir dependence on /H9275. Namely, all12 diagrams in Fig. 11 have the same /H9275dependence. Similarly, the /H9275dependence of alleight diagrams in Fig. 12is the same. This also applies to the four diagrams in Fig. 13. However, the corresponding /H9275 dependencies are slightly different from each other. The ori- gin of this difference can be traced from comparison of thediagrams in Figs. 11/H20849a/H20850,12/H20849a/H20850, and 13/H20849b/H20850. The diagram in Fig. 11/H20849a/H20850contains two polarization loops separated by the impurity line. As a result, the expression corresponding tothis diagram contains two static polarization operatorsab c d f e g h j i l k FIG. 11. Second-order diagrams contributing to the oscillating part /H20849see Fig. 4/H20850of the ballistic zero-bias anomaly in a weak con- stant magnetic field. The magnetic field enters through the phasesEq. /H208494.7/H20850of the Green functions. The dashed line represents the impurity scattering. All 12 diagrams /H20849a/H20850–/H20849l/H20850contain two static po- larization operators.ab cd ef g h FIG. 12. 8 out of the total 24 second-order diagrams for ballistic zero-bias anomaly in a weak constant magnetic field, which containone dynamic polarization operator. da b c FIG. 13. 4 out of the total 24 second-order diagrams for the ballistic zero-bias anomaly in a weak constant magnetic field, whichcontain a polarization loop crossed by the impurity line.T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-14/H90162kF/H208490,r/H20850. The diagram in Fig. 12/H20849a/H20850contains onefinite-/H9275 polarization loop /H90162kF/H20849/H9275,r/H20850. Finally, the diagram in Fig. 13/H20849b/H20850does not contain polarization operators at all but rather a different object, namely, a polarization loop crossed by the impurity line. Important is that the expression correspondingto this object: /H20863/H20849/H9275−/H9024,/H20841r1−r2/H20841/H20850=−i/H20885d/H90241 2/H9266G/H90241/H208490,r1/H20850G/H9275−/H9024+/H90241/H20849r1,0/H20850 /H11003G/H9275−/H9024+/H90241/H208490,r2/H20850G/H90241/H20849r2,0/H20850, /H208497.14 /H20850 contains a “fast” part, /H208632kF/H20849/H9275,r/H20850, which oscillates as exp /H208492ikF/H20841r1−r2/H20841/H20850, i.e., in the same way as a polarization op- erator. The full analytical expression corresponding to the dia- gram in Fig. 11/H20849a/H20850reads /H9254/H92631/H20849/H9275,h/H20850=I m4V2/H208492kF/H20850 2/H92662/H92630/H9270/H20885drdr1dr2G/H9275/H20849r,r1/H20850G/H9275/H20849r1,r2/H20850 /H11003/H9016 2kF/H208490,r1/H20850/H90162kF/H208490,r2/H20850G/H9275/H20849r2,r/H20850 =I m6V2/H208492kF/H20850 /H92662/H92630/H9270/H20885dr1dr2/H11509/H9275G/H9275/H20849r1,r2/H20850G/H9275/H20849r1,r2/H20850 /H11003/H9016 2kF/H208490,r1/H20850/H90162kF/H208490,r2/H20850, /H208497.15 /H20850 where in the second identity, we had performed integration over r. The analytical expression for the diagram in Fig. 12/H20849a/H20850 has the form /H9254/H92632/H20849/H9275,h/H20850=−I m2V2/H208492kF/H20850 2/H92662/H92630/H9270 /H11003/H20885drdr1dr2G/H9275/H20849r,r1/H20850G/H9275/H20849r1,0/H20850G/H9275/H20849r2,r/H20850 /H11003/H20885d/H9024 2/H9266G/H9024/H208490,r1/H20850G/H9024/H20849r1,r2/H20850/H90162kF/H20849/H9275−/H9024,/H20841r1 −r2/H20841/H20850. /H208497.16 /H20850 Finally, the expression for the diagram in Fig. 13/H20849b/H20850is the following: /H9254/H92633/H20849/H9275,h/H20850=−I m2V2/H208492kF/H20850 2/H92662/H92630/H9270 /H11003/H20885drdr1dr2G/H9275/H20849r,r1/H20850G/H9275/H20849r2,r/H20850 /H11003/H20885d/H9024 2/H9266/H20885d/H90241 2/H9266G/H9024/H20849r1,r2/H20850G/H90241/H208490,r1/H20850G/H9275−/H9024+/H90241 /H11003/H20849r1,0/H20850G/H9275−/H9024+/H90241/H208490,r2/H20850G/H90241/H20849r2,0/H20850. /H208497.17 /H20850 Upon integration over r, it can be expressed through /H208632kF/H20849r/H20850, defined by Eq. /H208497.14 /H20850,a s/H9254/H92633/H20849/H9275,h/H20850=−V2/H208492kF/H20850 /H92664/H92630/H9270 /H11003/H20885dr1dr2/H11509/H9275ImG/H9275/H20849r1,r2/H20850/H20885 0/H9275d/H9024 2/H9266 /H11003ImG/H9024/H20849r1,r2/H20850Im/H208632kF/H20849/H9275−/H9024,/H20841r1−r2/H20841/H20850. /H208497.18 /H20850 Despite that all 12 diagrams in Fig. 11have the same frequency dependence, their prefactors represent differentcombinations of V 2/H208492kF/H20850,V2/H208490/H20850, and V/H208492kF/H20850V/H208490/H20850. The same applies to the eight diagrams in Fig. 12and to the four dia- grams in Fig. 13. Taking into account the numerical factors in these combinations amounts to the following replace-ments: in /H9254/H92631, 4V2/H208492kF/H20850→3V2/H208490/H20850, /H208497.19 /H20850 in/H9254/H92632, −2V2/H208492kF/H20850→4/H20851V/H208490/H20850V/H208492kF/H20850−V2/H208492kF/H20850−V2/H208490/H20850/H20852, /H208497.20 /H20850 and in/H9254/H92633, −2V2/H208492kF/H20850→2/H20851V/H208490/H20850V/H208492kF/H20850−V2/H208492kF/H20850−V2/H208490/H20850/H20852. /H208497.21 /H20850 These replacements must be taken into account when calcu- lating the full correction /H9254/H9263/H20849/H9275/H20850from/H9254/H92631,/H9254/H92632, and/H9254/H92633. Below, we demonstrate that all three contributions /H9254/H92631, /H9254/H92632, and/H9254/H92633are oscillatory functions of /H9275. Detailed deriva- tion will be presented only for /H9254/H92631. Analogously to the derivation of Eqs. /H208497.10 /H20850and /H208497.11 /H20850, we can perform the integration over the azimuthal angles ofr 1andr2analytically using Eq. /H208497.8/H20850. Then, extracting a slow term from the product of trigonometrical functions, we ob- tain/H9254/H92631/H20849/H9275/H20850=/H208516/H926303V2/H208492kF/H20850/EF/H9270/H20852/H20849/H92750/EF/H208501/2P1/H2084922/3/H9275//H92750/H20850, with /H92750=21/3/H9275c2/3EF1/3, where the function P1/H20849x/H20850is defined as P1/H20849x/H20850=P1+/H20849x/H20850+P1−/H20849x/H20850, /H208497.22 /H20850 where P1+/H20849x/H20850=/H9268/H20885 /H92672/H11022/H92671d/H92671d/H92672 /H20849/H92671/H92672/H208503/2/H20877/H20849/H92671+/H92672/H208501/2/H20877cos/H20875x/H20849/H92671+/H92672/H20850−/H9266 4 −/H92671/H92672/H20849/H92671+/H92672/H20850/H20876− cos/H20875x/H20849/H92671+/H92672/H20850−/H9266 4/H20876/H20878/H20878,/H208497.23 /H20850 P1−/H20849x/H20850=−/H9268/H20885 /H92672/H11022/H92671d/H92671d/H92672 /H20849/H92671/H92672/H208503/2/H20877/H20849/H92672−/H92671/H208501/2/H20877cos/H20875x/H20849/H92672−/H92671/H20850 +/H9266 4+/H92671/H92672/H20849/H92672−/H92671/H20850/H20876− cos/H20875x/H20849/H92672−/H92671/H20850+/H9266 4/H20876/H20878/H20878. /H208497.24 /H20850 Here, the constant factor /H9268is given by/H9268=/H208513/H2084921/6/H20850/H20852//H92663/2.I n Appendix E, we demonstrate how the function P1/H20849x/H20850can be cast in the form that is convenient for numerical evaluationINTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-15and extracting asymptotes. This form is given by the follow- ing double integral: P1/H20849x/H20850=4/H9268/H20885 0/H11009dz z3/2/H20885 −40dv /H20881v+4 /H11003/H20873cos/H20875xz+/H9266 4+z3 v/H20876− cos/H20875xz+/H9266 4/H20876/H20874. /H208497.25 /H20850 The fact that P1/H20849x/H20850oscillates at large x/H112711 follows from the observations that /H20849i/H20850the first cosine in the brackets in Eq. /H208497.25 /H20850has a saddle point z=/H20849x/H20841v/H20841/3/H208501/2and /H20849ii/H20850the major contribution to the integral over vcomes from the lower limit v=−4 /H20849corresponding steps are outlined in Appendix D/H20850. This yields /H20841P1/H20849x/H20850/H20841x/H112711=25/339/4 /H92661/21 x7/4sin/H208754/H20873x 3/H208743/2 +/H9266 4/H20876./H208497.26 /H20850 The argument x3/2in the cosine in Eq. /H208497.26 /H20850can be pre- sented as/H92753/2//H2084921/2/H9275cEF1/2/H20850, so that the period in /H9275is much bigger than the cyclotron energy /H9275c, as was discussed above. The analysis of the contributions /H9254/H92632/H20849/H9275/H20850and/H9254/H92633/H20849/H9275/H20850can be carried out in a similar way. They exhibit the same oscil-lations as Eq. /H208497.26 /H20850. The difference is that, due to integration over/H9024in Eqs. /H208497.16 /H20850and /H208497.18 /H20850, both /H9254/H92632/H20849/H9275/H20850and/H9254/H92633/H20849/H9275/H20850 contain an extra factor /H9275//H92750, see Eq. /H208493.13 /H20850, and thus their contribution to the net correction /H9254/H9263is dominant at /H9275/H11271/H92750. VIII. ZERO-BIAS ANOMALY IN THE A VERAGED DENSITY OF STATES IN REGIME I With the help of the identity Eq. /H208495.5/H20850, the integrand in the average/H9254/H9263/H20849/H9275/H20850can be expressed in terms of functions U1,2/H20851r1r2/H20849r2/H11006r1/H20850p03/4/H20852, where the functions U1,2are defined as U1/H20849x/H20850=/H20873/H9266 2/H208741/2/H20881/H208491+x2/H208501/2+1 1+x2, /H208498.1/H20850 U2/H20849x/H20850=/H20873/H9266 2/H208741/2/H20881/H208491+x2/H208501/2−1 1+x2. /H208498.2/H20850 Upon introducing dimensionless variables /H92671,2 =p0r1,2 /22/3, we present the final result in the form /H9254/H9263/H20849/H9275/H20850 /H92630=CI/H20873/H9275 /H92750/H20874, /H208498.3/H20850 with /H92750=vFp0=2EF/H20873h0 kF2/H90210/H208742/3 , /H208498.4/H20850 and with constant Cdefined as C=−/H20849/H92630V/H208503 2/H9266/H20873h0 kF2/H90210/H20874=−/H20849/H92630V/H208503 4/H208812/H9266/H20873/H92750 EF/H208743/2 . /H208498.5/H20850 The dimensionless function I/H20849z/H20850describing the shape of the anomaly is given by the following double integral over /H92671, /H92672:I/H20849z/H20850=I+/H20849z/H20850+I−/H20849z/H20850=/H20885 /H92672/H11022/H92671d/H92671d/H92672 /H20849/H92671/H92672/H208503/2/H20885 0z dz/H11032sin/H20851/H20849z−z/H11032/H20850 /H11003/H20849/H92671+/H92672/H20850/H20852/H20853S+/H20849/H92671,/H92672/H20850+C+/H20849/H92671,/H92672/H20850+S−/H20849/H92671,/H92672/H20850 +C−/H20849/H92671,/H92672/H20850/H20854, /H208498.6/H20850 where the functions S+,S−,C+, and C−are defined as S/H11006/H20849/H92671,/H92672/H20850=/H20849/H92671/H11006/H92672/H208501/2sin/H20875/H9266 4/H11007/H20849z+z/H11032/H20850/H20849/H92671/H11006/H92672/H20850/H20876 /H11003/H20853U1/H20851/H92671/H92672/H20849/H92671/H11006/H92672/H20850/H20852−/H20881/H9266/H20854, /H208498.7/H20850 C/H11006/H20849/H92671,/H92672/H20850=/H20849/H92671/H11006/H92672/H208501/2cos/H20875/H9266 4/H11007/H20849z+z/H11032/H20850 /H11003/H20849/H92671/H11006/H92672/H20850/H20876U2/H20851/H92671/H92672/H20849/H92671/H11007/H92672/H20850/H20852./H208498.8/H20850 In the definitions of S+and S−, we had subtracted from the function U1/H20849/H9251/H20850the zero-field value U1/H208490/H20850=/H20881/H9266. Integration over z/H11032in Eq. /H208498.6/H20850can be easily carried out analytically. The remaining integrals over /H92671,/H92672were evaluated numerically. Direct numerical integration encounters difficulties due tovery fast oscillations of the integrand in Eq. /H208498.6/H20850. These difficulties can be overcome by a proper change of variablesin the integrand. This procedure is described in Appendix D.The resulting shape of the zero-bias anomaly is shown inFig. 14. The small- z/H112701 behavior of I/H20849z/H20850is 8 ln z, i.e., it diverges logarithmically. The cutoff is chosen from the con-dition that I/H20849z/H20850approaches zero at large z. Note that I/H20849z/H20850 exhibits a pronounced feature around z=1. The origin of this feature lies in strong oscillations of the integrand in Eq./H208497.9/H20850. The “trace” of these oscillations survives after averag- ing over the magnitude of the random field. In fact, the os-cillations persist beyond z=3. This is reflected in the z/H112711 asymptote of the function I/H20849z/H20850, /H20841I +/H20849z/H20850/H20841z/H112711/H11015−23/4/H20881/H9266sin/H2084928/3/H208813z/H20850 z3/4exp /H20853−28/3z/H20854./H208498.9/H20850 To derive this asymptote, it is more convenient to first take the limit of large /H9275in Eq. /H208497.9/H20850and perform the averagingFIG. 14. /H20849Color online /H20850Dimensionless function I/H20849z/H20850describing the shape of a zero-bias anomaly in regime I is plotted from Eq./H208498.6/H20850versus dimensionless energy, z= /H9275//H92750. Inset in the lower-right corner: enlarged plot of I/H20849z/H20850in the domain 3 /H11021z/H110215.T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-16over the random field only as a last step . In the limit /H9275 /H11271/H92750, the following simplifications of Eq. /H208497.9/H20850become pos- sible. Firstly, the second term in the square brackets can beneglected, since it does not produce oscillatory contributionto /H9254/H9263. Secondly, one can set /H9024=0 in the integrand, so that the integration over /H9024reduces to multiplying by /H9275. Lastly, upon converting the product of sines into the difference ofcosines, one finds that the /H9275dependence is present only in the term corresponding to the difference of arguments. As aresult, the oscillatory part of /H9254/H9263/H20849/H9275/H20850at/H9275/H11271/H92750acquires the form /H20883/H9254/H9263/H20849/H9275/H20850 /H92630/H20884=−/H20849/H92630V/H208503/H9275/H927501/2 215 /6/H92663/2EF3/2/H20885 /H92672/H11022/H92671d/H92671d/H92672 /H20849/H92671/H92672/H208503/2/H20849/H92671+/H92672/H208501/2 /H11003/H20883/H20873h h0/H20874cos/H20875/H92671/H92672/H20849/H92671+/H92672/H20850+/H9266 4 −25/3/H9275 /H92750/H20873h0 h/H208742/3 /H20849/H92671+/H92672/H20850/H20876/H20884 h/H20849x,y/H20850. /H208498.10 /H20850 The steps leading from this expression to the asymptote Eq. /H208498.9/H20850are outlined in Appendix E. IX. ZERO-BIAS ANOMALY IN A VERAGED DENSITY OF STATES IN REGIME II A. Three polarization operators: Averaging of the net magnetic phase factor over realizations of random magnetic field To derive analytical expressions for /H9254/H9263/H20849+/H20850/H20849/H9275/H20850and/H9254/H9263/H20849−/H20850/H20849/H9275/H20850, one has to perform averaging of Eqs. /H208497.10 /H20850and /H208497.11 /H20850over realizations of the random field. Such an averaging has al-ready been carried out for the Friedel oscillations. In thelatter case, we had averaged /H20855exp /H208492i /H9254/H92720→r/H20850/H20856. In the case of the density of states, the exponents to be averaged are /H20855exp /H208492i/H9254/H9272/H9018/H20849/H11006/H20850/H20850/H20856, defined by Eqs. /H208497.6/H20850and /H208497.7/H20850. Our most important observation is that the netphase/H9254/H9272/H9018/H20849−/H20850=/H9254/H92720→r1+/H9254/H9272r1→r2+/H9254/H9272r2→0does not contain integrals of /H90112/H20849x/H20850since they cancel out. This can be clearly seen from Eq. /H208494.17 /H20850. Instead,/H9254/H9272/H9018/H20849−/H20850is expressed via integrals of /H9011/H20849x/H20850in the first power as follows: /H9254/H9272/H9018/H20849−/H20850=1 /H902102kF/H208751 r1/H20873/H20885 0r1 dx/H9011/H20849x/H20850/H208742 +1 r2−r1/H20873/H20885 r1r2 dx/H9011/H20849x/H20850/H208742 −1 r2/H20873/H20885 0r2 dx/H9011/H20849x/H20850/H208742/H20876. /H208499.1/H20850 This cancellation, as we demonstrate below, has a dramatic consequence for the average /H20855exp /H20849i/H9254/H9272/H9018/H20850/H20856. It turns out that while /H20855exp /H20849i/H9254/H92720→r/H20850/H20856decays with r exponentially , the average /H20855exp /H20849i/H9254/H9272/H9018/H20850/H20856falls off only as apower law. This, in turn, leads to a slow decay of a zero-bias anomaly, /H9254/H9263/H20849/H9275//H92751/H20850, with/H9275. On the technical level, cancellation of /H20848dx/H90112/H20849x/H20850terms leads to a drastic simplification of the disorder averaging ofEqs. /H208497.10 /H20850and /H208497.11 /H20850in regime II, as compared to the aver- aging of the Friedel oscillations in Sec. V B, since the aver-aging of exp /H208492i /H9254/H9272/H9018/H20850can be performed with the help of theHubbard–Stratonovich transformation. For the purpose of functional averaging, it is convenient to rewrite Eq. /H208499.1/H20850in a slightly different form: /H9254/H9272/H9018/H20849−/H20850=1 /H902102kF/H20849r2−r1/H20850/H20875/H20881r2 r1/H20885 0r1 dx/H9011/H20849x/H20850 −/H20881r1 r2/H20885 0r2 dx/H9011/H20849x/H20850/H208762 . /H208499.2/H20850 Subsequent integration by parts yields the further simplifica- tion of Eq. /H208499.2/H20850: /H9254/H9272/H9018/H20849−/H20850=1 /H902102kF/H20849r2−r1/H20850/H20875/H20881r2 r1/H20885 0r1 dx/H20849r1−x/H20850h/H20849x,0/H20850 −/H20881r1 r2/H20885 0r2 dx/H20849r2−x/H20850h/H20849x,0/H20850/H208762 . /H208499.3/H20850 Now, the averaging over realizations of h/H20849x,y/H20850can be per- formed by a sequence of standard steps outlined below. 1. AVERAGING PROCEDURE Using Eq. /H208499.1/H20850, we rewrite the definition of average /H20855exp /H208492i/H9254/H9272/H9018/H20850/H20856by introducing the auxiliary integration vari- able c: /H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856=/H20885 −/H11009/H11009 dcexp /H20849−ic2/H20850/H20883/H9254/H20873c−/H208812 /H90210kF1/2/H20881r2−r1 /H11003/H20875/H20881r2 r1/H20885 0r1 dx/H20849r1−x/H20850h/H20849x,0/H20850 −/H20881r1 r2/H20885 0r2 dx/H20849r2−x/H20850h/H20849x,0/H20850/H20876/H20874/H20884 h/H20849x,y/H20850, /H208499.4/H20850 where the averaging /H20855¯/H20856h/H20849x,y/H20850is defined by Eq. /H208495.18 /H20850. Next, we use the following integral representation of the /H9254function in Eq. /H208499.4/H20850 /H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856=/H20885 −/H11009/H11009 dcexp /H20849−ic2/H20850/H20885 −/H11009/H11009dt 2/H9266eict/H20883exp /H11003/H20877−it/H208812/H20875/H20881r2 r1/H20885 0r1dx/H20849r1−x/H20850h/H20849x,0/H20850 /H90210kF/H20881r2−r1 −/H20881r1 r2/H20885 0r2dx/H20849r2−x/H20850h/H20849x,0/H20850 /H90210kF/H20881r2−r1/H20876/H20878/H20884 h/H20849x,y/H20850. /H208499.5/H20850 Now, the integration over ccan be performed explicitly, yieldingINTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-17/H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856=/H20881/H9266 2e−i/H9266/4/H20885 −/H11009/H11009dt 2/H9266eit2/4 /H11003/H20883exp/H20877−it/H208812/H20875/H20881r2 r1/H20885 0r1dx/H20849r1−x/H20850h/H20849x,0/H20850 /H90210kF/H20881r2−r1 −/H20881r1 r2/H20885 0r2dx/H20849r2−x/H20850h/H20849x,0/H20850 /H90210kF/H20881r2−r1/H20876/H20878/H20884 h/H20849x,y/H20850. /H208499.6/H20850 It follows from Eq. /H208499.6/H20850that the evaluation of /H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856 reduces to the Gaussian averaging of the exponent of a linear inh/H20849x/H20850functional, which is standard: /H20883exp/H20877−it/H20885 0r2 dx/H20885 −/H11009/H11009 dyh /H20849x,y/H20850f/H20849x/H20850/H9254/H20849y/H20850/H20878/H20884 h/H20849x,y/H20850 = exp/H20877−t2 4/H20885 0r2 dx1/H20885 0r2 dx2f/H20849x1/H20850K/H20849x1,0,x2,0/H20850f/H20849x2/H20850/H20878, /H208499.7/H20850 where K/H20849x1,0,x2,0/H20850is related to the correlator of the random field Eq. /H208492.1/H20850asK/H20849x1,0,x2,0/H20850=h02K/H20849/H20841x1−x2/H20841//H9264/H20850. Subsequent integration over tyields the final result /H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856=1 /H208811+i/H208480r2/H208480r2dx1dx2f/H20849x1/H20850K/H20849x1,0,x2,0/H20850f/H20849x2/H20850. /H208499.8/H20850 As seen from Eq. /H208499.6/H20850, the function f/H20849x/H20850in Eq. /H208499.7/H20850has the form f−/H20849x/H20850=/H208812 /H90210kF1/2/H20881r2−r1/H20875/H20881r2 r1/H20849r1−x/H20850/H9258/H20849r1−x/H20850 −/H20881r1 r2/H20849r2−x/H20850/H20876. /H208499.9/H20850 Averaging of exp /H20853i/H9254/H9272/H9018/H20849+/H20850/H20854is performed similarly and also yields Eq. /H208499.7/H20850with f/H20849x/H20850having the form f+/H20849x/H20850=/H208812 /H90210kF1/2/H20881r2/H20875/H20881r1+r2 r1/H20849r1−x/H20850/H9258/H20849r1−x/H20850 −/H20881r1 r1+r2/H20849r1+r2−x/H20850/H20876. /H208499.10 /H20850 We emphasize that expression Eq. /H208499.8/H20850isgeneral and is valid for arbitrary h0and/H9264, i.e., in both regimes I and II. For regime I, we had already performed the averaging over real-izations of the random field. With regard to Eq. /H208499.8/H20850, regime I corresponds to replacement of the correlator by unity .I n regime II, the distances r 1,r2are much larger than /H9264. For this reason, in regime II, the correlator in Eq. /H208499.8/H20850can be re- placed by /H208812/H9266/H9253/H9264/H9254/H20849x1−x2/H20850, with/H9253defined by Eq. /H208495.22 /H20850. Then the averages /H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856and /H20855exp /H208532i/H9254/H9272/H9018/H20849+/H20850/H20854/H20856can be expressed in terms of dimensionless ratios/rho11=r1 /H208816rII,/rho12=r2 /H208816rII, /H208499.11 /H20850 where the characteristic length rIIis defined by Eq. /H208495.29 /H20850. Equation /H208499.8/H20850and analogous expression for /H20855exp /H208532i/H9254/H9272/H9018/H20849+/H20850/H20854/H20856are sufficient to perform the averaging over realizations of random magnetic field in Eqs. /H208497.10 /H20850and /H208497.11 /H20850. However, averaged Eqs. /H208497.10 /H20850and /H208497.11 /H20850contain the real and imaginary parts: /H20855exp /H208532i/H9254/H9272/H9018/H20849/H11006/H20850/H20854/H20856=U1/H11006/H20849/rho11,/rho12/H20850+iU2/H11006/H20849/rho11,/rho12/H20850, /H208499.12 /H20850 of the average exponents separately . The expressions for U1/H11006 andU2/H11006readily follow after replacing the correlator by the delta function and performing integrations over x1and x2in Eq. /H208499.8/H20850: U1−=/H20881/H20881/rho112/H20849/rho12−/rho11/H208502+1+1+ /H20881/H20881/rho112/H20849/rho12−/rho11/H208502+1−1 /H208812/H20881/rho112/H20849/rho12−/rho11/H208502+1, /H208499.13 /H20850 U2−=/H20881/H20881/rho112/H20849/rho12−/rho11/H208502+1+1− /H20881/H20881/rho112/H20849/rho12−/rho11/H208502+1−1 /H208812/H20881/rho112/H20849/rho12−/rho11/H208502+1, /H208499.14 /H20850 U1+=/H20881/H20881/rho112/rho122+1+1+ /H20881/H20881/rho112/rho122+1−1 /H208812/H20881/rho112/rho122+1, /H208499.15 /H20850 U2+=/H20881/H20881/rho112/rho122+1+1− /H20881/H20881/rho112/rho122+1−1 /H208812/H20881/rho112/rho122+1. /H208499.16 /H20850 Final expressions for the contributions /H20855/H9254/H9263−/H20849/H9275/H20850/H20856and /H20855/H9254/H9263+/H20849/H9275/H20850/H20856to the averaged density of states in the second re- gime are obtained by performing integration over /H9024in Eqs. /H208497.10 /H20850and /H208497.11 /H20850and using Eqs. /H208499.13 /H20850–/H208499.16 /H20850. We present this expression in the form similar to Eq. /H208498.3/H20850: /H20883/H9254/H9263/H11006/H20849/H9275/H20850 /H92630/H20884=DJ/H11006/H20873/H9275 /H92751/H20874, /H208499.17 /H20850 where the prefactor Dis defined as D=−/H20849/H92630V/H208503 63/4/H20849/H9266kFrII/H208503/2, /H208499.18 /H20850 and the dimensionless functions J/H11006are the following inte- grals over /rho11,/rho12: J1−/H20849z/H20850=1 4/H20885 /rho12/H11022/rho11d/rho11d/rho12 /H20849/rho11/rho12/H208503/2/H20849/rho12−/rho11/H208501/2/H20851U1−/H20849/rho11,/rho12/H20850−1/H20852 /H11003/H20873/H92671+/H92672 /H92671/H92672sin/H20875/H9266 4+2z/H20849/rho12−/rho11/H20850/H20876−1 /H92671sin/H20875/H9266 4 +2z/rho12/H20876−1 /H92672sin/H20875/H9266 4−2z/rho11/H20876/H20874, /H208499.19 /H20850T. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-18J2−/H20849z/H20850=1 4/H20885 /rho12/H11022/rho11d/rho11d/rho12 /H20849/rho11/rho12/H208503/2/H20849/rho12−/rho11/H208501/2/H20851U2−/H20849/rho11,/rho12/H20850−1/H20852 /H11003/H20873/H92671+/H92672 /H92671/H92672cos/H20875/H9266 4+2z/H20849/rho12−/rho11/H20850/H20876−1 /H92671cos/H20875/H9266 4 +2z/rho12/H20876−1 /H92672cos/H20875/H9266 4−2z/rho11/H20876/H20874, /H208499.20 /H20850 J1+/H20849z/H20850=1 4/H20885 /rho12/H11022/rho11d/rho11d/rho12 /H20849/rho11/rho12/H208503/2/H20849/rho11+/rho12/H208501/2/H20877/H20851U1+/H20849/rho11,/rho12/H20850−1/H20852 /H11003/H20873cos/H20875/H9266 4+2z/H20849/rho11+/rho12/H20850/H20876−1 /H208812/H20874+/H20851U2+/H20849/rho11,/rho12/H20850−1/H20852 /H11003/H20873sin/H20875/H9266 4+2z/H20849/rho11+/rho12/H20850/H20876−1 /H208812/H20874/H20878, /H208499.21 /H20850 J2+/H20849z/H20850=z 2/H20885 /rho12/H11022/rho11d/rho11d/rho12 /H20849/rho11/rho12/H208503/2/H20849/rho11+/rho12/H208501/2/H20877/H20851U1+/H20849/rho11,/rho12/H20850 −1/H20852sin/H20875/H9266 4+2z/H20849/rho11+/rho12/H20850/H20876−/H20851U2+/H20849/rho11,/rho12/H20850−1/H20852sin/H20875/H9266 4 −2z/H20849/rho11+/rho12/H20850/H20876/H20878, /H208499.22 /H20850 where z=/H9275//H92751is the dimensionless frequency. The new en- ergy scale is related to the characteristic length rIIin the second regime in the usual way: /H92751=vF /H208816rII. /H208499.23 /H20850 The second regime corresponds to long distances, rII/H11022/H9264, traveled by electron. This is reflected in the fact that thefrequency /H92751is smaller than /H92750, the characteristic frequency for the first regime. Using Eq. /H208495.29 /H20850, we can establish the relation between /H92750and/H92751, namely,/H92751/H11011/H92750/H92551/6, where/H9255is the small parameter, defined by Eq. /H208493.10 /H20850. We emphasize that the second regime exists only if the condition /H9255/H112701i s met. It is important to compare the scale /H92751to the “diffusive” energy scale /H9275diff/H11011vF/ltr, where ltris the transport mean free path. In regime II, we have26 ltr/H11011vF/H20849kF/H9264/H208502/H20875vFh02/H92643 /H902102/H20876−1 =kF2/H902102 h02/H9264. /H208499.24 /H20850 In this estimate, the combination h02/H92643vF//H902102stands for a single-particle scattering rate, calculated from the golden rule, with h02/H92642//H902102coming from the square of the matrix element; the factor /H20849kF/H9264/H208502accounts for the small-angle scat- tering. Equation /H208499.24 /H20850leads to the following relation be- tween the transport mean free path and rII: /H20873ltr kF/H208741/2 /H11011rII/H11011/H9264 /H20881/H9255. /H208499.25 /H20850 As follows from Eq. /H208499.25 /H20850, the distance rII, over which the phase of the Friedel oscillations is preserved, is intermediatebetween ltrand/H9264. Indeed, the ratio ltr/rIIis/H11011kFrII/H11011kF/H9264//H20881/H9255. This ratio is large both because kF/H9264/H112711 and because /H9255/H112701. Thus, we conclude that the energy scale /H92751is much larger than/H9275diffsince/H9275diff //H92751is/H11011rII/ltr/H112701, i.e., the conventional diffusive zero-bias anomaly develops at frequencies muchsmaller than the width of the zero-bias anomaly in regime II. B. Discussion Dimensionless density of states, J=J−+J+, is plotted in Fig.15. It is seen that the function J/H20849z/H20850exhibits pronounced minimum at z/H110150.75, which is followed by a monotonous decay. This behavior should be contrasted to the dimension-less density of states in regime I, plotted in Fig. 14. The difference is that the function Iexhibits damped oscillations with alternating maxima and minima, while Jcontains only a single minimum. This difference is not unexpected onqualitative grounds. Indeed, the distance /H11011r I, at which the oscillations are formed in regime I, is much smaller than thecorrelation radius /H9264, while the characteristic distance /H11011rIIin regime II is much bigger than /H9264. Therefore, it is remarkable thatJ/H20849z/H20850exhibits even a single minimum. However, qualita- tive difference between I/H20849z/H20850andJ/H20849z/H20850at large zis much harder to trace from their respective representations asdouble integrals over /H92671and/H92672/H20851see Eqs. /H208498.6/H20850and /H208499.19 /H20850– /H208499.22 /H20850/H20852. The structure of one of several contributions to I/H20849z/H20850 andJ/H20849z/H20850can be loosely rewritten as /H20885 0/H11009/H20885 0/H11009d/H92671d/H92672 /H20849/H92671/H92672/H208503/2/H20881/H92671+/H92672sinz/H20849/H92671+/H92672/H20850 /H208811+/H926712/H926722/H20849/H92671+/H92672/H208502,/H20849I/H20850, /H208499.26 /H20850 /H20885 0/H11009/H20885 0/H11009d/H92671d/H92672 /H20849/H92671/H92672/H208503/2/H20881/H92671+/H92672sinz/H20849/H92671+/H92672/H20850 /H208811+/H926712/H926722. /H20849II/H20850 /H208499.27 /H20850 The integrands in Eqs. /H208499.26 /H20850and /H208499.27 /H20850differ only by the structure of the denominators. This difference can be tracedto Eq. /H208499.8/H20850in which the correlator is set either constant /H20849regime I /H20850or a /H9254function /H20849regime II /H20850. From the form of theFIG. 15. /H20849Color online /H20850Dimensionless density of states in re- gime II, J/H20849z/H20850=J+/H20849z/H20850+J−/H20849z/H20850, is plotted in the units of /H20849/H92630D/H20850from Eqs. /H208499.19 /H20850–/H208499.22 /H20850versus dimensionless frequency z=/H9275//H92751, where /H92751is defined by Eq. /H208499.23 /H20850andDis defined by Eq. /H208499.18 /H20850.INTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-19contribution Eq. /H208499.26 /H20850, it is not obvious at all that the large- zbehavior is determined by well-defined values /H92671=/H92672=/H92670in the complex plane, with /H92670satisfying 1+ /H926706=0, so that the contribution is oscillatory Eq. /H208498.9/H20850. This fact was established above by taking the large- zasymptote before the averaging over realizations. It is also supported by numerics in Fig. 14. Monotonous behavior of J/H20849z/H20850at large zimplies that the integral Eq. /H208499.27 /H20850is not dominated by distinct complex /H92671 =/H92672=/H9267˜0, such that 1+ /H9267˜04=0. The only vague explanation of this is that the denominator, /H208811+/H926712/H926722/H20849/H92671+/H92672/H208502,i nE q . /H208499.26 /H20850 fixes/H92671/H11015/H92672/H11015/H92670much more efficiently that the denomina- tor,/H208811+/H926712/H926722,i nE q . /H208499.27 /H20850fixes/H92671,/H92672near/H9267˜0. X. IMPLICATIONS A. Half-filled Landau level The experimental situation of a two-dimensional electron gas placed in an inhomogeneous magnetic field can be cre-ated artificially, see, e.g., Refs. 10–18. This situation also emerges in electron gas in a strong constant magnetic field,when the filling factor of the lowest Landau level is close to1/2. In the latter case, the constant field transforms electrons into composite fermions, 8,9with well defined Fermi surface,67–71while the randomness of the magnetic field is due to spatial inhomogeneity of the electron density. Trans-port properties of noninteracting gas of composite fermions under these conditions were considered theoretically in Refs.22–30. With regard to the tunnel density of states near the half- filling, for the case of homogeneous gas, it was addressedtheoretically in Refs. 72–74both for tunneling into the bulk and into the edge. Unlike interacting homogeneous electron gas, 4composite fermions are expected to exhibit a zero-bias anomaly even without inhomogeneity .72–74This difference between composite fermions and free electrons can be tracedto the form of density-density correlator of composite fermi-ons at small momenta. 9Namely, the pole of this correlator defines the mode of neutral excitations with dispersion /H9275 /H11008iq3, even slower than the diffusive mode in the presence of disorder. The resulting suppression of tunneling into the edgeof homogeneous electron gas at half-filling, predicted inRefs. 73and 74, turned out to be stronger than in the experiment. 75,76 It is convenient to express the random static magnetic field originating from spatial inhomogeneity with magnitude /H9254nin the units of the cyclotron frequency: /H9254/H9275c /H90241/2=2/H9254n n1/2, /H2084910.1 /H20850 where n1/2is the concentration of electrons at which the fill- ing factor in the field, /H90241/2, is equal to 1 /2. Density fluctua- tions not only smear out the “intrinsic” zero-bias anomalybut also give rise to the smooth-disorder-induced zero-biasanomaly, studied in this paper. Quantitatively, we predict thefollowing relation between the width of zero-bias anomalyand the magnitude /H9254nof the density fluctuations:/H92750/H11011/H90241/2/H20873/H9254n n1/2/H208742/3 . /H2084910.2 /H20850 This relation follows directly from Eq. /H208493.11 /H20850and applies for smooth fluctuations with spatial scale /H9264satisfying the condi- tion n1/2/H92642/H11022/H20873n1/2 /H9254n/H208744/3 . /H2084910.3 /H20850 This condition is equivalent to the condition /H9255/H110221, where the parameter/H9255is defined by Eq. /H208493.10 /H20850. In the opposite case of “fast” fluctuations, the width /H92751is given by /H92751/H11011/H90241/2/H20851n1/2/H92642/H208521/4/H20873/H9254n n1/2/H20874, /H2084910.4 /H20850 as follows from Eq. /H208493.11 /H20850. Concerning the magnitude of the anomaly, Eqs. /H208498.5/H20850and /H208499.18 /H20850predict/H9254/H9263//H92630/H11011/H20849/H9254n/n1/2/H20850 for slow fluctuations Eq. /H2084910.2 /H20850and/H9254/H9263//H92630 /H11011/H20849/H9254n/n1/2/H208503/2/H20851n1/2/H92642/H208523/8for the fast fluctuations Eq. /H2084910.4 /H20850, respectively. A qualitative difference between the “intrinsic” zero-bias anomaly72–74and inhomogeneity-induced zero-bias anomaly, considered in this paper, is that the latter necessarily involveselectron-electron scattering processes with momentum trans-fer /H110152k F. As was mentioned above, the intrinsic anomaly gets stronger toward the edge.73,74We would like to empha- size that the anomaly due to the 2 kFprocesses also gets stronger toward the edge. The reason is that the average elec-tron concentration decreases monotonically upon approach-ing the edge. This decrease translates into a nonfluctuating magnetic field, acting on composite fermions, 23which in- creases toward the edge. Correction /H9254/H9263/H20849/H9275/H20850to the density of states in this case is given by Eq. /H208497.13 /H20850and is plotted in Fig. 10. Then, we conclude that the ratio of magnitudes, /H9254/H9263bulk //H9254/H9263edge, is simply /H11011/H20849/H9254nbulk //H9254nedge/H20850/H112701, where/H9254nbulk and/H9254nedgeare the deviations of electron density from n1/2in the bulk and at the edge, respectively. The widths of /H9254/H9263bulk/H20849/H9275/H20850and/H9254/H9263edge/H20849/H9275/H20850are related as /H11011/H20849/H9254nedge //H9254nbulk/H208502/3 /H112701. B. Spin-fermion model Similarly to composite fermions, the dispersion of neutral excitations right at the critical point in the spin-fermionmodel is dominated by a slow mode, 77,78/H9275/H11008iq3. Outside the critical region, the propagator of the neutral excitations/H20849bosons /H20850in the spin-disordered phase has the conventional Ornstein–Zernike form /H9273/H20849q/H20850/H110081//H20849q2+/H9264−2/H20850, where/H9264is the correlation radius, which diverges at the critical point. Inter-action of electrons with slow critical fluctuations can beviewed as scattering by the smooth disorder. The questionthat we will discuss below is how the growth of /H9264, upon approaching the critical point, manifests itself in the behaviorof the averaged /H20849over the fluctuations of the order parameter /H20850 density of states. Our calculations demonstrate that the di-mensionless parameter /H9255, defined by Eq. /H208493.10 /H20850, plays a cru- cial role. Traditionally, in the studies of the response functions, such as spin susceptibility, of two-dimensional electrons nearT. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-20the quantum critical point, see, e.g., Refs. 77–82, electrons are treated as ballistic. More specifically, they interact onlywith critical fluctuations but not with each other . Transport at the quantum critical point was also considered for noninter-acting ballistic 83or diffusive84electrons that are scattered by bosonic excitations. In all theoretical treatments of the spin-fermion model, modification of the response of the electron gas due to inter-action with bosons was governed by the processes with small momentum transfer . Our main point is that incorporating di- rect electron-electron interactions into the spin-fermion model gives rise to a novel feature in the response of theelectron gas near the critical point in spin-disordered phase.The underlying reason is that, while critical bosonic fluctua-tions are “smooth,” so that their momenta are /H11270k F, electron- electron interactions allow 2 kFprocesses. Then, the physics, discussed in this paper, emerges in the following way: /H20849i/H20850interaction with slow bosonic fluctuations, curves slightly the electron trajectories; /H20849ii/H20850interaction between the electrons, moving along slightly curved trajectories, generates a small energy scale,which reflects the “degree” of curving; and /H20849iii/H20850the degree of curving grows with correlation radius /H9264 of the bosonic excitations. As a result, the character of critical fluctuations is re- flected in the density of states /H9254/H9263/H20849/H9275/H20850in a very nontrivial fashion. Namely, they give rise to the lively low-frequencyfeature and even aperiodic oscillations in /H9254/H9263/H20849/H9275/H20850, as was dem- onstrated above. This suggests that information about prox-imity to the critical point can be inferred from tunnelingexperiments. To quantify the above scenario, we will assume for simplicity 85that bosonic critical fluctuations of magnetiza- tion, S/H20849r/H20850, interact with electron spins not as /H9268·S, where /H9268 are the Pauli matrices, but via the position-dependent Zee- man energy EZ/H20849r/H20850with characteristic magnitude E0. Assum- ing that the fluctuations S/H20849r/H20850are static, we get for correlator of random Zeeman energy EZ/H20849r/H20850the standard expression /H20855EZ/H20849r/H20850EZ/H20849r/H11032/H20850/H20856=E02/H20885dq 2/H9266eiq/H20849r−r/H11032/H20850 q2+/H9264−2=E02K0/H20849/H20841x1−x2/H20841//H9264/H20850, /H2084910.5 /H20850 where K0is the Macdonald function. As the next step, we notice that the force /H11612EZ/H20849r/H20850curves the electron trajectories in the same way as random magnetic field h/H20849x,y/H20850. This allows us to use general expressions Eqs. /H208497.10 /H20850and /H208497.11 /H20850for the interaction correction to the density of states. We can also employ the result Eq. /H208499.8/H20850for the general averaging procedure, i.e., to treat critical fluctuationsas a disorder. With the help of Eq. /H2084910.5 /H20850, the result Eq. /H208499.8/H20850 assumes the form /H20855exp /H208532i /H9254/H9272/H9018/H20849−/H20850/H20854/H20856=/H208751+iE02/H20885 0r2/H20885 0r2 dx1dx2f−/H20849x1/H20850f−/H20849x2/H20850/H11509x1/H11509x2 /H11003K0/H20873/H20841x1−x2/H20841 /H9264/H20874/H20876−1 /2 , /H2084910.6 /H20850 where the function f−is defined by Eq. /H208499.9/H20850for the case ofrandom magnetic field. For the case of random Zeeman en- ergy, the prefactor 1 //H90210kF1/2should be replaced by kF1/2/EF. Characteristic energy scales can now be inferred from Eq./H2084910.6 /H20850on the basis of the following reasoning. Characteristic distances r 1,r2in Eq. /H2084910.6 /H20850are determined by the condition /H20885 0r2/H20885 0r2 dx1dx2K0/H20873/H20841x1−x2/H20841 /H9264/H20874/H11509 /H11509x1f−/H20849x1/H20850/H11509 /H11509x2f−/H20849x2/H20850/H110111 E02, /H2084910.7 /H20850 where we performed integration by parts in Eq. /H2084910.6 /H20850. Then, the characteristic width of a zero-bias anomaly is equal to /H9275/H11011vF/r1/H11011vF/r2. Recall now that in the case of random magnetic field, double integral in the left-hand side of Eq. /H2084910.7 /H20850did not contain derivatives and was /H11008r23in regime I and /H11008r22/H9264in regime II. This is because the function f−/H20849x1/H20850is/H11011r21/2atx1 /H11011r2, see Eq. /H208499.9/H20850. Due to the fact that the effective “force” in the spin-fermion model is /H11008/H11612EZ/H20849r/H20850, the left-hand side in Eq. /H2084910.7 /H20850is/H11011kFr2/EF2for/H9264/H11271r2. In this limit, Eq. /H2084910.7 /H20850 yields /H20849with logarithmic in /H9264/r2accuracy /H20850 r2/H11011kF−1/H20873EF E0/H208742 =/H9264c,/H9275/H11011E02 EF=Ec. /H2084910.8 /H20850 Note that Ecis independent of /H9264. We conclude that, upon approaching the critical point, as the correlation radius ex-ceeds the value /H9264c, the zero-bias anomaly “freezes.” Its form is shown in Fig. 14, and its magnitude is /H11011/H20849E0/EF/H208503.A n alternative way to recover the scales Eq. /H2084910.8 /H20850is to notice that parameter /H9255, which is defined by Eq. /H208493.10 /H20850in the con- text of random magnetic field, in the situation with randomZeeman energy acquires the form /H9255=/H20849k F/H9264/H20850/H20849E0/EF/H208502. Then,/H9264c given by Eq. /H2084910.8 /H20850corresponds to /H9255=1, i.e., to the boundary of regime I. For/H9264/H11021/H9264c, the integral in the left-hand side of Eq. /H2084910.7 /H20850 is proportional to /H9264and is independent of r2. Then Eq. /H2084910.7 /H20850 does not have a solution. Therefore, characteristic r1and r2 in the expression for the density of states are /H11011/H9264, and the width of the anomaly is simply /H11011vF//H9264=Ec/H20849/H9264c//H9264/H20850. Concern- ing the magnitude of the anomaly at /H9264/H11021/H9264c, it should be estimated with the account that the integral in the right-handside of Eq. /H2084910.6 /H20850is smaller than 1 for all r 2. Therefore, /H20855exp /H208532i/H9254/H9272/H9018/H20849−/H20850/H20854/H20856in Eq. /H2084910.6 /H20850can be approximately replaced by /H208531−/H20849i/2/H9264c/H20850/H20851r2/H9008/H20849/H9264−r2/H20850+/H9264/H9008/H20849r2−/H9264/H20850/H20852/H20854, where the second term is a small correction. However, only this correctioncauses a zero-bias anomaly. By substituting this correctioninto Eq. /H208497.11 /H20850, we find the estimate for the magnitude, /H9254/H9263 /H92630/H11011/H20873E0 EF/H208743/H20873/H9264c /H9264/H208741/2 /H11011/H20873E0 EF/H2087421 /H20849kF/H9264/H208501/2. /H2084910.9 /H20850 We conclude that, as /H9264grows and approaches /H9264c, the magni- tude of the anomaly falls off as 1 //H20881/H9264, and the anomaly nar- rows as 1 //H9264. The remaining issue to discuss is whether or not the as- sumption that fluctuating Zeeman energy EZ/H20849r/H20850is static ap- plies at relevant frequency and spatial scales, Ecand/H9264c. For this purpose, we recall that the correlator of Zeeman energiesin the momentum space does not have a simple Ornstein–INTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-21Zernike form but is rather /H20855/H20841EZ/H20849q/H20850/H208412/H20856/H110081//H20849q2+/H9264−2+/H9269/H9275/q/H20850, where the dynamic term /H9269/H9275/qdescribes the damping of bosons due to creation of electron-hole pairs. The prefactor /H9269 /H20849the Landau damping coefficient /H20850is thus quadratic in cou- pling of electrons to the spin density fluctuations, i.e., /H9269 /H11008E02. For characteristic frequencies, the dynamic term /H9269/H20849/H9275/q/H20850/H11011/H9269Ec/H9264c/H11011/H9269vF. Therefore, it is negligible only if the condition/H9264c−2=kF2/H20849E0/EF/H208504/H11271/H9269vFholds. With/H9269being propor- tional to E02, the above condition is met for large enough coupling, E0. In the opposite case, when the dynamic part of correlator dominates at /H9275/H11011Ecand q/H11011/H9264c−1, the zero-bias anomaly develops only away from the critical point when /H9264 becomes smaller than /H20849/H9269vF/H20850−1 /2. Upon further departure from the critical point, our prediction /H9254/H9263//H92630/H11008/H9264−1 /2and/H9275/H11011vF//H9264 should apply. Note finally that directly at the critical point , the slow mode /H9275/H11015iq3//H9269gives rise to the intrinsic zero-bias anomaly,81similar to the composite fermions. ACKNOWLEDGMENTS The authors acknowledge the support of NSF /H20849Grant No. DMR-0503172 /H20850and of the Petroleum Research Fund /H20849Grant No. 43966-AC10 /H20850. The authors are grateful to E. G. Mish- chenko and O. A. Starykh for numerous discussions. APPENDIX A: POLARIZATION OPERATOR IN THE COORDINATE SPACE Here, we derive Eqs. /H208494.2/H20850and /H208494.3/H20850for polarization op- erator in coordinate space using the known expression59for /H9016/H20849q,/H9275/H20850in the momentum space. Since we are interested in the behavior of /H9016/H20849r,/H9275/H20850at distances /H20841r/H20841/H11271kF−1, it is sufficient to perform the Fourier transform /H9016/H20849r,/H9275/H20850=1 2/H9266/H20885dqeiqr/H9016/H20849q,/H9275/H20850/H20849 A1/H20850 using the asymptotes of /H9016/H20849q,/H9275/H20850at small q/H11270kFand at q close to 2 kF. The small- qasymptote of /H9016/H20849q,/H9275/H20850has the form /H90160/H20849q,/H9275/H20850=−/H92630/H208751+i/H9275/H9008/H20849qvF−/H9275/H20850 /H20881q2vF2−/H92752+/H9275/H9008/H20849/H9275−qvF/H20850 /H20881/H92752−q2vF2/H20876, /H20849A2/H20850 where/H9008/H20849x/H20850is the step function. The easiest way to perform the integration Eq. /H20849A1/H20850is to first Fourier transform Eq. /H20849A2/H20850 with respect to frequency : −/H9008/H20849qvF−/H9275/H20850 /H20881q2−/H20849/H9275/vF/H208502+i/H9008/H20849/H9275−qvF/H20850 /H20881/H20849/H9275/vF/H208502−q2=/H20885 0/H11009 dsJ 0/H20849qs/H20850exp/H20877i/H9275s vF/H20878. /H20849A3/H20850 Substituting Eq. /H20849A2/H20850into Eq. /H20849A3/H20850and using the orthogo- nality relation /H208480/H11009dqqJ 0/H20849qs/H20850J0/H20849qr/H20850=/H9254/H20849r−s/H20850/r, we readily ob- tain /H90160/H20849r,/H9275/H20850=−i/H92630/H9275 vFrexp/H20877i/H9275r vF/H20878. /H20849A4/H20850 In order to calculate /H90162kF/H20849r,/H9275/H20850, we use the form of the po- larization operator in momentum space for /H20841q−2kF/H20841/H11270kFand /H9275/H11270EF:/H90162kF/H20849q,/H9275/H20850=/H92630/H208751−1 /H208814kF/H20849/H20881q−2kF+/H9275/vF +/H20881q−2kF−/H9275/vF/H20850/H20876, /H20849A5/H20850 where the square roots should be understood as /H20881x →sign /H20849x/H20850/H20881x. Then, the integral over qin Eq. /H20849A1/H20850assumes the form /H90162kF/H20849r,/H9275/H20850=−/H92630/H20885 0/H11009 dqqJ 0/H20849qr/H20850/H20851/H20881q−2kF+/H9275/vF +/H20881q−2kF−/H9275/vF/H20852 /H11003/H11015/H208814kF /H9266r/H20885 0/H11009 dqcos/H20873qr−/H9266 4/H20874 /H11003/H20851/H20881q−2kF+/H9275/vF+/H20881q−2kF−/H9275/vF/H20852, /H20849A6/H20850 where we used the fact that kFr/H112711 and replaced the Bessel function by its large- qasymptotics. Integration over variable qin Eq. /H20849A6/H20850is performed with the use of the identity /H20885 a/H11009 dzcosz/H20881z−a=/H20881/H9266 2sin/H20873a+/H9266 4/H20874 /H20849A7/H20850 and yields the zero-temperature limit of Eq. /H208494.3/H20850. APPENDIX B: POLARIZATION OPERATOR IN A CONSTANT MAGNETIC FIELD We start from the general expression63for the polarizabil- ity in arbitrary magnetic field: /H9016/H20849q/H20850=−2m /H9266/H20858 n1=0/H11009 /H20858 n2=0/H11009/H20849−1/H20850/H20849n2−n1/H20850/H20849fn1−fn2/H20850 n2−n1 /H11003exp /H20849−q2l2/2/H20850Ln1n2−n1/H20873q2l2 2/H20874Ln2n1−n2/H20873q2l2 2/H20874, /H20849B1/H20850 where Ln1n2−n1/H20849x/H20850and Ln2n1−n2/H20849x/H20850are the Laguerre polynomials, and fn=/H20853exp /H20851/H20849n−NF/H20850/H6036/H9275c/T/H20852+1/H20854−1is the Fermi distribution. At small q/H11270kF, Eq. /H20849B1/H20850yields63/H9016/H20849q/H20850=−/H20849m//H9266/H20850/H208511 −J02/H20849qRL/H20850/H20852, i.e., the characteristic scale is q/H11011RL−1. For /H20849q −2kF/H20850/H11270kF, it is convenient to perform the summation over the Landau levels with the help of the following integralrepresentation of the Laguerre polynomial: L mn/H20849x/H20850=1 2/H9266/H20885 02/H9266d/H9258 /H208491−ei/H9258/H20850n+1exp/H20877xei/H9258 ei/H9258−1−im/H9258/H20878./H20849B2/H20850 In the vicinity q=2kF, Eq. /H20849B2/H20850contains a small factor exp /H20849−q2l2/2/H20850. This factor is compensated by the product of Laguerre polynomials since each of them is /H11008exp /H20849x/2/H20850, which comes from the exponent in Eq. /H20849B2/H20850taken at/H9258=/H9266. With contribution from the vicinity /H9258=/H9266dominating the in- tegral /H20849B2/H20850, we can expand the integrand around this point asT. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-22exp /H20851x/2+i/H9266m+i/H9278/H20849/H9274/H20850/H20852/2n+1, where/H9274=/H20849/H9258−/H9266/H20850, and the phase /H9278/H20849/H9274/H20850is equal to /H9278/H20849/H9274/H20850=/H20873x 4−m−n+1 2/H20874/H9274+x/H92743 48. /H20849B3/H20850 Now, we make use of the fact that only a relatively small number /H11011/H20849kFl/H208502/3/H11270NFof Landau levels around EFcontrib- ute to the sum Eq. /H20849B2/H20850. This suggests that we can present n1 and n2asn1=NF+m1and n2=NF−m2, respectively, and ex- tend the sum over m1,m2from −/H11009to +/H11009. After that, the summation over Landau levels can be easily carried out withthe help of the following identity: /H20858 m1,m2=−/H11009/H11009fNF−m1−fNF+m2 m1+m2cos/H20851/H20849m1−m2/H20850/H9251+/H9252/H20852 =2/H92662Tcos/H9252 /H6036/H9275csinh /H208492/H9266/H20841/H9251/H20841T//H6036/H9275c/H20850. /H20849B4/H20850 As the next step, we substitute the representation Eq. /H20849B2/H20850of Laguerre polynomials with the integrand expanded accord-ing to Eq. /H20849B3/H20850into Eq. /H20849B1/H20850. Upon this substitution, we perform the summation over Landau levels using the relationEq. /H20849B4/H20850. Then, the double integral, which emerges in Eq. /H20849B1/H20850as a result of representing the two Laguerre polynomi- als Eq. /H20849B2/H20850, assumes the form /H20885 −/H11009/H11009/H20885 −/H11009/H11009d/H92741d/H92742 /H20841/H92741+/H92742/H20841cos/H20875/H20849/H927413+/H927423/H20850NF 12−/H20849/H92741+/H92742/H20850/H9254qRL 2/H20876, /H20849B5/H20850 where/H9254q=q−2kF. Note that integration over the difference /H20849/H92741−/H92742/H20850in Eq. /H20849B5/H20850can be performed explicitly. It is con- venient to present the final result not for /H9016/H20849q/H20850but rather for the derivative /H9016/H11032/H20849q,T/H20850=/H11509/H9016/H20849q,T/H20850//H11509q. Knowledge of /H9016/H11032/H20849q,T/H20850is sufficient for finding the large-distance behavior of the potential, created by the short-range impurity. Indeed,this potential can be expressed directly through /H9016 /H11032/H208492kF+Q/H20850 as follows: VH/H20849r/H20850=V/H208492kF/H20850g 2/H20849/H9266kFr/H208503/2/H20885 −/H11009/H11009 dQsin/H20875/H208492kF+Q/H20850r−/H9266 4/H20876 /H11003/H9016 /H11032/H208492kF+Q,T/H20850. /H20849B6/H20850 At zero temperature and in a zero magnetic field, we have /H9016/H11032/H20849q,0/H20850/H11008/H9258/H20849/H9254q/H20850//H20881/H9254q. At finite magnetic field and finite tem- perature, taking the derivative of Eq. /H20849B5/H20850with respect to /H9254q, we arrive at the result /H9016/H11032/H20849q,T/H20850=−21/3mT /H20849/H9266kFp0/H208501/2/H92800/H20885 0/H11009dxx1/2 sinh /H208492/H9266xT //H92800/H20850 /H11003sin/H2087322/3/H9254q p0x+1 3x3+/H9266 4/H20874. /H20849B7/H20850 In the limit T→0, substitution of Eq. /H20849B7/H20850into Eq. /H20849B6/H20850and integration over Qreproduces Eq. /H208493.1/H20850. Interestingly, for T=0, the integral Eq. /H20849B7/H20850can be evalu- ated analytically:/H9016/H11032/H20849q/H20850=−m /H20849kFp0/H208501/2Ai/H20873/H9254q p0/H20874Bi/H20873/H9254q p0/H20874, /H20849B8/H20850 where Ai /H20849z/H20850is the Airy function and Bi /H20849z/H20850is another solution of the Airy equation defined, e.g., in Ref. 86. It is seen that the singularity at q=2kFis smeared by the magnetic field in a rather peculiar way: for positive /H9254q/H11271p0, the /H20849/H9254q/H20850−1 /2zero- field behavior /H20851see Eq. /H20849A6/H20850/H20852is restored. However, for large negative/H9254q/p0, the derivative /H9016/H11032/H20849q/H20850approaches zero with oscillations , namely, as cos /H208514/H20849/H20841/H9254q/H20841/p0/H208503/2/3/H20852//H20849/H20841/H9254q/H20841/H208501/2. As the difference 2 kF−qincreases and becomes comparable to kF, these oscillations cross over to the “classical” oscillations63 /H9016/H11032/H20849q/H20850/H11008J0/H20849qRL/H20850J1/H20849qRL/H20850/H11008cos/H208492qRL/H20850. APPENDIX C: EV ALUATION OF THE FUNCTIONAL INTEGRAL Upon combining Eqs. /H208495.24 /H20850and /H208495.26 /H20850, the quadratic form in the exponent in the numerator of the functional in-tegral Eq. /H208495.18 /H20850assumes the form 2i /H9254/H9272/H20849r/H20850−W/H20853h/H20854=2i/H9255r3 /H92643/H208771 12/H20875/H20885dqA0,q/H208762 +/H20858 n/H110220cn/H20879/H20885dqAn,q/H208792 +/H20885dqA0,qG/H20853An/H20854/H20878 −2r /H9253/H9264/H20858 n/H110220/H20885dq/H20841An,q/H208412 K˜/H20849q/H20850−r /H9253/H9264/H20885dq/H20841A0,q/H208412 K˜/H20849q/H20850, /H20849C1/H20850 with numerical coefficients cn=1 /2/H92662n2and bn=−cn +i/2/H9266ndefined by Eq. /H208495.27 /H20850. In the above expression, we had introduced the shorthand notation G/H20853An,q/H20854=/H20858 n/H110220/H20875bn/H20885dqAn,q+bn*/H20885dqAn,q*/H20876. /H20849C2/H20850 We adopt the following sequence of integration over the variables An,q. First, we integrate over A0,qusing the follow- ing decoupling: H/H20853G/H20854=/H20885/H20863 qdA0,qexp/H20877−r /H9253/H9264/H20885dq/H20841A0,q/H208412 K˜/H20849q/H20850 +i/H9255r3 6/H92643/H20875/H20885dqA0,q/H208762 +iG/H20885dqA0,q/H20878 =e−i/H9266/4/H208813r3 2/H9266/H9255/H92643/H20885/H20863 qdA0,qdB0exp/H20877−3ir3B02 2/H9255/H92643 +iB0/H20885dqA0,q−r /H9253/H9264/H20885dq/H20841A0,q/H208412 K˜/H20849q/H20850+iG/H20885dqA0,q/H20878, /H20849C3/H20850 where we had introduced an auxiliary variable B0. The func- tion H/H20853G/H20854combines all integrals in Eq. /H20849C1/H20850containing A0,q. Subsequent integration first over the variables A0,qand then over the auxiliary variable B0yieldsINTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-23H/H20853G/H20854=/H20881/H9266/H9264/H9253 ir/H20885dqK˜/H20849q/H20850exp /H20853iF/H20849r/H20850G2/H20854 /H208811−2i 3/H20873r rII/H208742, where we had used the definition rII=2/H9264//H20849/H208812/H9266/H9253/H9255/H208501/2.I nE q . /H20849C4/H20850, the complex function F/H20849r/H20850is defined as F/H20849r/H20850=−3/H92551/2/H20875/H9253/H20885dqK˜/H20849q/H20850/H208763/2 16/H20873r rII/H208743 +2 4 i/H20873r rII/H20874. /H20849C4/H20850 As a result of integration over A0,qthe exponent in the func- tional integral Eq. /H20849C1/H20850assumes the form i/H20858 n/H110220c˜n/H20879/H20885dqAn,q/H208792 −2r /H9253/H9264/H20858 n/H110220/H20885dq/H20841An,q/H208412 K˜/H20849q/H20850+iF/H20849r/H20850G2, /H20849C5/H20850 where c˜nis related to cnvia a dimensionless factor: c˜n=2/H9255r3 /H92643cn. /H20849C6/H20850 The first and third terms in Eq. /H20849C5/H20850contain squares of the linear combinations of An,q. To decouple these squares, we introduce a set of auxiliary variables /H9251n,/H9251n*for the first term and one auxiliary variable /H92510for the third term as follows: eiV/H20841/H20885dqAn,q/H208412=1 2/H9266/H20885d/H9251nd/H9251n*exp/H20877−i/H20841/H9251n/H208412 +V1/2/H9251n*/H20885dqAn,q−V1/2/H9251n/H20885dqAn,q*/H20878, /H20849C7/H20850 eiF/H20849r/H20850G2=1 /H208814/H9266F/H20849r/H20850/H20885 −/H11009/H11009 d/H92510exp/H20877−i/H925102 4F/H20849r/H20850+i/H92510G/H20878. /H20849C8/H20850 Note that Im /H208511/F/H20849r/H20850/H20852/H110210, so that the decoupling Eq. /H20849C8/H20850of the quadratic in Gterm in the exponent of Eq. /H20849C4/H20850is justi- fied. As the next step, we perform Gaussian integration over the infinite set of variables /H20853An,q/H20854: /H20885dAn,qdAn,q*exp/H20877/H20885dq/H20875−2r/H20841An,q/H208412 /H9253/H9264K˜/H20849q/H20850 +An,q/H20849c˜n1/2/H9251n*+i/H92510bn/H20850+An,q*/H20849−c˜n1/2/H9251n+i/H92510bn*/H20850/H20876/H20878 =2i/H9266 /H208492r//H9253/H9264/H20850/H20885dq/H20851K˜/H20849q/H20850/H20852−1exp/H20877−/H20841−ic˜n1/2/H9251n* +/H92510bn/H208412/H9253/H9264 2r/H20885dqK˜/H20849q/H20850/H20878. /H20849C9/H20850As follows from Eqs. /H20849C7/H20850and /H20849C9/H20850, the integrals over all /H9251n are Gaussian and can be easily evaluated: /H20885d/H9251nd/H9251n* 2/H9266exp/H20877−i/H20841/H9251n/H208412−/H20841−ic˜n1/2/H9251n*+/H92510bn/H208412/H9253/H9264 2r/H20885dqK˜/H20849q/H20850/H20878 =2r 2r−i/H9253/H9264c˜n/H20885dqK˜/H20849q/H20850exp/H20902−/H9253/H9264/H92510/H20841bn/H208412 2r/H20885dqK˜/H20849q/H20850 −c˜n/H20875/H92510/H20841bn/H20841/H9253/H9264/H20885dqK˜/H20849q/H20850/H208762 4ir2+2r/H9253/H9264c˜n/H20885dqK˜/H20849q/H20850/H20903. /H20849C10 /H20850 The remaining integral over /H92510is also Gaussian. Note now that the denominator in Eq. /H208495.18 /H20850, responsible for the nor- malization, can be evaluated by performing the same steps as above. This evaluation amounts to setting c˜n=0 in Eq. /H20849C10 /H20850 and taking the limit rII→/H11009in Eq. /H20849C4/H20850. As a result, the functional integral reduces to the ratio of the ordinary inte-grals: /H20855e 2i/H9254/H9272/H20849r/H20850/H20856=1 /H208811−2i 3/H20873r rII/H208742/H20875/H20863 n=1/H11009n2 n2−2i/H20849r/rII/H208502//H92662/H20876 /H11003/H20885 −/H11009/H11009 d/H92510exp /H20853−w/H92510−u1/H925102/H20854 /H20885 −/H11009/H11009 d/H92510exp /H20853−w/H92510−u0/H925102/H20854, /H20849C11 /H20850 where the coefficients wand u0are defined as w=/H9253/H9264 2r/H20875/H20858 n/H110220/H20841bn/H208412/H20876/H20885dqK˜/H20849q/H20850, u0=r /H9264/H20873/H20885dqK˜/H20849q/H20850/H20874−3 /2 , /H20849C12 /H20850 while the definition of the coefficient u1is the following: u1=i 4F/H20849r/H20850+/H20875/H9253/H9264/H20885dqK˜/H20849q/H20850/H208762 /H20858 n/H110221c˜n/H20841bn/H208412 4ir2+2r/H9253/H9264c˜n/H20885dqK˜/H20849q/H20850. /H20849C13 /H20850 For characteristic r/H11011rII, the first term in Eq. /H20849C13 /H20850is/H11011/H9255−1 /2, as follows from Eq. /H20849C4/H20850. On the other hand, the product r/H9264c˜nin the denominator of the second term in Eq. /H20849C13 /H20850is /H11011/H9255r4//H92642/H11011r4/rII2. Thus, for r/H11011rII, both terms in the denomi- nator of the sum in the second term are /H11011rII2. The numerator in the sum over nis/H11011/H9255−1 /2forr/H11011rII. Then, the estimate for the second term in Eq. /H20849C13 /H20850is/H92642/rII2/H92551/2, so that the second term is smaller than the first term in parameter /H92642/rII2/H11011/H9255. Next, we notice that for r/H11011rII, both u0and u1are of the same order and are /H11011/H9255−1 /2. On the other hand, as seen fromT. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-24Eq. /H20849C12 /H20850, the parameter wforr/H11011rIIis small, w/H11011/H92551/2. This allows us to disregard wboth in the numerator and denomi- nator in Eq. /H20849C11 /H20850, so that the ratio of integrals reduces to /H20849u0/u1/H208501/2. Using Eq. /H20849C4/H20850, this ratio can be rewritten as /H208511−/H208492i/3/H20850/H20849r/rII/H208502/H20852−1 /2. Substituting it into Eq. /H20849C11 /H20850,w ea r - rive at Eq. /H208495.28 /H20850in Sec. V . APPENDIX D: ANALYSIS OF THE INTEGRALS EQUATION ( 8.6) The dimensionless function I/H20849z/H20850defined by Eq. /H208498.6/H20850can be naturally divided into two parts, I/H20849z/H20850=I++I−, where I−/H20849z/H20850=/H20885 /H92672/H11022/H92671d/H92672d/H92671 /H20849/H92672/H92671/H208503/2/H20885 0z dz/H11032sin/H20851/H20849z−z/H11032/H20850/H20849/H92672+/H92671/H20850/H20852 /H11003/H20881/H92672−/H92671/H20877sin/H20851/H9266/4+ /H20849z+z/H11032/H20850/H20849/H92672−/H92671/H20850/H20852 /H11003/H208811+/H208811+/H926722/H926712/H20849/H92672−/H92671/H208502 1+/H926722/H926712/H20849/H92672−/H92671/H208502+ cos /H20851/H9266/4+ /H20849z+z/H11032/H20850/H20849/H92672 −/H92671/H20850/H20852/H20881/H208811+/H926722/H926712/H20849/H92672−/H92671/H208502−1 1+/H926722/H926712/H20849/H92672−/H92671/H208502/H20878 /H20849D1/H20850 and I+/H20849z/H20850=/H20885 /H92672/H11022/H92671d/H92672d/H92671 /H20849/H92672/H92671/H208503/2/H20885 0z dz/H11032sin/H20851/H20849z−z/H11032/H20850/H20849/H92672+/H92671/H20850/H20852 /H11003/H20881/H92672+/H92671/H20877sin/H20851/H9266/4− /H20849z+z/H11032/H20850/H20849/H92672+/H92671/H20850/H20852 /H11003/H208811+/H208811+/H926722/H926712/H20849/H92672+/H92671/H208502 1+/H926722/H926712/H20849/H92672+/H92671/H208502+ cos /H20851/H9266/4− /H20849z+z/H11032/H20850/H20849/H92672 +/H92671/H20850/H20852/H20881/H208811+/H926722/H926712/H20849/H92672+/H92671/H208502−1 1+/H926722/H926712/H20849/H92672+/H92671/H208502/H20878. /H20849D2/H20850 The complexity in numerical evaluation of I+andI−stems from the fact that, upon integration over z/H11032, both integrals turn into the sums of two contributions, each of which isdivergent in the limit z→0. Therefore, it is necessary to re- write the result of integration over z /H11032inI+and in I−in such a way that cancellation of the divergent contributions is ex-plicit. We start with I −. Integration over z/H11032generates the com- bination of three terms: /H92671+/H92672 /H92671/H92672cos/H20875/H9266 4+2z/H20849/H92671−/H92672/H20850/H20876−1 /H92672cos/H20873/H9266 4+2z/H92671/H20874 −1 /H92671cos/H20873/H9266 4−2z/H92672/H20874. /H20849D3/H20850 In order to treat all these three terms on an equal footing, in the first term of Eq. /H20849D3/H20850, we introduce the following new variables: z˜=z/H20849/H92672−/H92671/H20850,x=/H92672/H92671 z3/H20849/H92672−/H92671/H208502. /H20849D4/H20850 In the second term, we introduce z˜=z/H92671, and finally, in the third term, z˜=z/H92672. After that, the expression for I−assumes the form I−/H20849z/H20850=1 /H208812z3/H20885 0/H11009dx x5/2/H20851F1/H20849x/H20850−F1/H208490/H20850/H20852+1 /H208812z3/H20885 0/H11009dx x5/2F2/H20849x,z/H20850 +1 /H208812z3/H20885 01/4z3dx x5/2/H20851F3/H20849x,z/H20850−F3/H208490,0/H20850/H20852, /H20849D5/H20850 where the functions F1,F2, and F3are defined as F1/H20849x/H20850=/H20885 0/H11009dz˜ z˜5/2/H20877/H20849cos 2 z˜− sin 2 z˜/H20850/H20881/H208811+z˜6x2−1 1+z˜6x2 +/H20849cos 2 z˜+ sin 2 z˜/H20850/H208811+/H208811+z˜6x2 1+z˜6x2/H20878, /H20849D6/H20850 F2/H20849x,z/H20850=/H208811 4+xz3−1 2 2/H208811 4+xz3/H20885 0/H11009dz˜ z˜5/2/H20877/H20849sin 2 z˜− cos 2 z˜/H20850 /H11003/H20881/H208811+z˜6x2+1 1+z˜6x2−/H20849cos 2 z˜ + sin 2 z˜/H20850/H20881/H208811+z˜6x2−1 1+z˜6x2/H20878, /H20849D7/H20850 F3/H20849x,z/H20850=−/H20849/H208811 4−xz3+1 2/H208503+/H20849/H208811 4−xz3−1 2/H208503 2/H208811 4−xz3 /H11003/H20885 0/H11009dz˜ z˜5/2/H20877/H20849cos 2 z˜+ sin 2 z˜/H20850/H20881/H208811+z˜6x2+1 1+z˜6x2 +/H20849cos 2 z˜− sin 2 z˜/H20850/H20881/H208811+z˜6x2−1 1+z˜6x2/H20878. /H20849D8/H20850 Subtraction of x=0 values from F1/H20849x/H20850and F3/H20849x,z/H20850in Eq. /H20849D5/H20850ensures the convergence of integrals over z˜in Eqs. /H20849D6/H20850 and /H20849D8/H20850. On the other hand, this subtraction shifts I−by z-independent constant. It is seen that in the limit z→0, the difference F2/H20849x,z/H20850 −F2/H208490,z/H20850behaves as z3, so that the contribution from F2to I−/H20849z/H20850remains finite in this limit. On the other hand, the con- tributions from F1and F3both behave as 1 /z3. To demon- strate that the two divergent contributions cancel out, wedivide the integration domain in the first term of I −into the intervals /H208530,1 /4z3/H20854and /H208531/4z3,/H11009/H20854. We then combine the two integrals from 0 to 1 /4z3to obtainINTERACTION EFFECTS IN A TWO-DIMENSIONAL … PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-25I−/H20849z/H20850=1 /H208812z3/H20885 01/4z3dx x5/2/H20853/H20851F1/H20849x/H20850−F1/H208490/H20850/H20852+/H20851F3/H20849x,z/H20850 −F3/H208490,z/H20850/H20852/H20854+1 /H208812z3/H20885 1/4z3/H11009dx x5/2/H20851F1/H20849x/H20850−F1/H208490/H20850/H20852 +1 /H208812z3/H20885 0/H11009dx x5/2F2/H20849x,z/H20850. /H20849D9/H20850 The second and third terms in Eq. /H20849D9/H20850are convergent in the limit z→0. The integrand in the first term has the form F1/H20849x/H20850−F1/H208490/H20850+F3/H20849x,z/H20850−F3/H208490,z/H20850 =/H208771−/H20849/H208811 4−xz3+1 2/H208503+/H20849/H208811 4−xz3−1 2/H208503 2/H208811 4−xz3/H20878 /H11003/H20885 0/H11009dz˜ z˜5/2/H20877/H20849cos 2 z˜+ sin 2 z˜/H20850/H20875/H20881/H208811+z˜6x2+1 1+z˜6x2−/H208812/H20876 +/H20849cos 2 z˜− sin 2 z˜/H20850/H20881/H208811+z˜6x2−1 1+z˜6x2/H20878. /H20849D10 /H20850 We see that in the limit z→0, the expression in the curly brackets behaves as /H11008z3, and thus cancels the divergent pref- actor. Now all three terms in Eq. /H20849D9/H20850yield a finite contri- bution at z→0. Our numerical results for I−/H20849z/H20850were ob- tained from Eq. /H20849D9/H20850. We now turn to I+/H20849z/H20850. In order to deal with small- zbe- havior in the integral Eq. /H20849D2/H20850, we introduce, after perform- ing integration over z/H11032, the following new variables: z˜=z/H20849/H92672+/H92671/H20850, x=/H92672/H92671 z3/H20849/H92672+/H92671/H208502. /H20849D11 /H20850 Then one obtains I+=I+1+I+2, where the two contributions are given by I+/H208491/H20850=1 /H208812/H20885 01/4z3dx x3/2/H208811 4−xz3/H20885 0/H11009dz˜ z˜5/2/H20877/H208491 − sin 2 z˜ − cos 2 z˜/H20850/H20881/H208811+z˜6x2−1 1+z˜6x2−/H208491 − cos 2 z˜+ sin 2 z˜/H20850 /H11003/H20875/H20881/H208811+z˜6x2+1 1+z˜6x2−/H208812/H20876/H20878 /H20849D12 /H20850 and I+/H208492/H20850=2 /H208812/H20885 01/4z3dx x3/2/H208811 4−xz3/H20885 0/H11009dz˜ z˜3/2 /H11003/H20877/H20849sin 2 z˜− cos 2 z˜/H20850/H20881/H208811+z˜6x2−1 1+z˜6x2+/H20849cos 2 z˜ + sin 2 z˜/H20850/H20875/H20881/H208811+z˜6x2+1 1+z˜6x2−/H208812/H20876/H20878. /H20849D13 /H20850Both these contributions are finite in the limit z→0. APPENDIX E: ANALYSIS OF THE INTEGRALS EQUATIONS ( 7.23) and ( 7.24) In the integral Eq. /H208497.23 /H20850, we perform the following change of variables: /H92671=z 2/H208731−/H20881v v+4/H20874, /H92672=z 2/H208731+/H20881v v+4/H20874, /H20849E1/H20850 after which it acquires the form P1+/H20849x/H20850=3/H20849213 /6/H20850 /H92663/2/H20885 0/H11009dv v1/2/H20885 0/H11009dz z3/2/H20877cos/H20875xz−/H9266 4−z3 v+4/H20876 − cos/H20875xz−/H9266 4/H20876/H20878. /H20849E2/H20850 In the integral Eq. /H208497.24 /H20850, we perform the following change of variables: /H92671=z 2/H208731+/H20881v+4 v/H20874, /H92672=z 2/H20873/H20881v+4 v−1/H20874, /H20849E3/H20850 after which it acquires the form P1−/H20849x/H20850=3/H20849213 /6/H20850 /H92663/2/H20885 0/H11009dv /H20849v+4/H208501/2/H20885 0/H11009dz z3/2/H20877cos/H20875xz+/H9266 4+z3 v/H20876 − cos/H20875xz+/H9266 4/H20876/H20878. /H20849E4/H20850 It is convenient to present /H208480/H11009dvin Eq. /H20849E4/H20850as the following difference of integrals: P1−/H20849x/H20850=3/H20849213 /6/H20850 /H92663/2/H20885 0/H11009dz z3/2/H20873−/H20885 −40dv /H20881v+4+/H20885 −4/H11009dv /H20881v+4/H20874 /H11003/H20873cos/H20875xz+/H9266 4+z3 v/H20876− cos/H20875xz+/H9266 4/H20876/H20874. /H20849E5/H20850 We now observe that the the second term cancels identically the function P+. Then we readily arrive to Eq. /H208497.25 /H20850. APPENDIX F: ASYMPTOTICS OF THE DENSITY OF STATES The idea of derivation of Eq. /H208498.9/H20850from Eq. /H208498.10 /H20850is that the major contribution to the integral Eq. /H208498.10 /H20850comes from the domain /H20841/H92672−/H92671/H20841/H11270/H92671,/H92672, i.e., from the domain where /H92671 and/H92672are close to each other. To make use of this simplifi- cation, we rewrite the argument of the cosine in Eq. /H208498.10 /H20850asT. A. SEDRAKYAN AND M. E. RAIKH PHYSICAL REVIEW B 77, 115353 /H208492008 /H20850 115353-26/H20849/H92671+/H92672/H208503 4+/H9266 4−27/3/H9275 /H9275h/H20849/H92671+/H92672/H20850−/H20849/H92671+/H92672/H20850/H20849/H92672−/H92671/H208502 4, /H20849F1/H20850 where we had introduced /H9275h=/H92750/H20849h/h0/H208502/3. It is seen from Eq. /H20849F1/H20850that the typical value of /H20849/H92672+/H92671/H20850is/H20849/H9275//H9275h/H208501/2/H112711, while the typical value of /H20849/H92672−/H92671/H20850is/H20849/H92672+/H92671/H20850−1 /2/H11011/H20849/H9275//H9275h/H20850−1 /4, i.e., the relevant difference /H92672−/H92671is small indeed. This allows us to extend the integration over /H92672−/H92671from zero to infinity and perform the integral. This yields /H20883/H9254/H9263/H20849/H9275/H20850 /H92630/H20884=−/H20849/H92630V/H208503/H9275/H9275h1/2 /H92661/2EF3/2/H20885 0/H11009d/H9267 /H92673/H20883cos/H20875/H92673 4−27/3/H9267/H9275 /H9275h/H20876/H20884. /H20849F2/H20850 The argument of the cosine in Eq. /H20849F2/H20850has a sharp minimum at/H9267=/H92670=/H20849213 /6/31/2/H20850/H20881/H9275//H9275h, which allows us to perform the integration over /H9267by introducing /H9254/H9267=/H9267−/H92670and extending the integration over /H9254/H9267from minus to plus infinity. This yields the following asymptote of /H9254/H9263/H20849/H9275/H20850:/H20883/H9254/H9263/H20849/H9275/H20850 /H92630/H20884=−1 64/H2084927/12/H20850/H20881/H9266/H20849/H92630V/H208503/H9275h9/4 EF3/2/H92753/4/H20883sin/H2087532/H208812 3/H208813/H20873/H9275 /H9275h/H208743/2 +/H9266 4/H20876/H20884 h/H20849x,y/H20850, /H20849F3/H20850 in which the random magnetic field enters through /H9275h. The argument of the sine contains the term /H11008/H9275h−3 /2, which can be presented as sh0/h, where the constant sis equal to 16/H208492/H9275/3/H92750/H208503/2. The factor in front of the sine contains /H9275h9/4 /H11008h3/2. Then, the Gaussian averaging over hcan be carried out analytically using the fact that s/H112711. 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PhysRevB.101.155205.pdf

PHYSICAL REVIEW B 101, 155205 (2020) Stable single layer of Janus MoSO: Strong out-of-plane piezoelectricity M. Yagmurcukardes*and F. M. Peeters Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium (Received 11 January 2020; revised manuscript received 22 March 2020; accepted 27 March 2020; published 24 April 2020) Using density functional theory based first-principles calculations, we predict the dynamically stable 1H phase of a Janus single layer composed of S-Mo-O atomic layers. It is an indirect band gap semiconductor exhibitingstrong polarization arising from the charge difference on the two surfaces. In contrast to 1H phases of MoS 2and MoO 2, Janus MoSO is found to possess four Raman active phonon modes and a large out-of-plane piezoelectric coefficient which is absent in fully symmetric single layers of MoS 2and MoO 2. We investigated the electronic and phononic properties under applied biaxial strain and found an electronic phase transition with tensile strainwhile the conduction band edge displays a shift when under compressive strain. Furthermore, single-layer MoSOexhibits phononic stability up to 5% of compressive and 11% of tensile strain with significant phonon shifts. Thephonon instability is shown to arise from the soft in-plane and out-of-plane acoustic modes at finite wave vector.The large strain tolerance of Janus MoSO is important for nanoelastic applications. In view of the dynamicalstability even under moderate strain, we expect that Janus MoSO can be fabricated in the common 1H phasewith a strong out-of-plane piezoelectric coefficient. DOI: 10.1103/PhysRevB.101.155205 I. INTRODUCTION The successful exfoliation of graphene [ 1] has opened an exponentially growing research field of two-dimensional(2D) ultrathin materials. Among the 2D ultrathin materials,transition metal dichalcogenides (TMDs) have gained a lotof attention owing to their potential for applications in, e.g.,optoelectronic devices [ 2–6]. Single-layer MoS 2has been extensively investigated both experimentally and theoretically because of its distinctiveelectronic, optical, and mechanical properties [ 7–12]. The direct band gap nature of 1H-MoS 2has made it a suitable candidate for applications in photonics. Previous experimentshave demonstrated that, to synthesize MoS 2by chemical vapor deposition (CVD), MoO 3could be used as the Mo precursor [ 13–15]. In addition, it has also been demonstrated that incomplete sulfurization of MoO 3leads to the forma- tion of either molybdenum oxides, MoO 3−x, or molybdenum oxysulfides, MoO 3−xSy[16–19]. Therefore, the formation of a fully asymmetric MoSO structure can be feasible in suchexperiments. The question is whether the structural phase ofsuch fully asymmetric MoSO can be controlled or not. On the other hand, recent experimental techniques have allowed successful replacement of one Se layer of MoSe 2 by S atoms and construction of a polar single layer, namelyJanus MoSSe [ 20,21]. The induced internal out-of-plane po- larization opens up a way to tune the properties of 2Dultrathin materials. Following the experimental realizationof Janus MoSSe, out-of-plane asymmetric single layers ofvarious TMDs and other 2D materials have been theoreticallyproposed [ 22–26]. The experimentally realized single-layer *Mehmet.Yagmurcukardes@uantwerpen.beMoSSe was constructed by replacing one surface of singl- layer Mo X2by a different chalcogen atom in a common structural phase, namely 1H phase. Similar to the formationprocess of Janus MoSSe, the synthesis of single-layer MoS 2 from MoO 3powders motivated us to predict a possible new 1H phase of Janus MoSO composed of S-Mo-O atomic layers.Alternatively, since the formation of molybdenum oxysul-fides, MoO 3−xSy, has been observed in previous experiments, it may also be feasible to construct a Janus single layer in suchan experiment as an intermediate state. In this study, we show that the formation of a Janus single layer, composed of S and O layers on different surfaces, leadsto the dynamically stable 1H-MoSO, which is an indirectband gap semiconductor. We show that the formation of JanusMoSO can be distinguished by its Raman spectrum in whichthe broken out-of-plane symmetry creates additional Ramanactive phonon modes. In addition, it is shown that JanusMoSO exhibits a large out-of-plane piezoelectric coefficientdue to the significant charge difference between the twosurfaces. Moreover, we investigate the effect of an externalbiaxial strain on the electronic and phononic properties ofsingle-layer MoSO. Our results indicate that Janus MoSO canwithstand large tensile strains and displays a semiconductor-to-metal transition with tensile strain. II. COMPUTATIONAL METHODOLOGY The plane-wave basis projector augmented wave (PAW) method was employed in the framework of density-functionaltheory (DFT) in our first-principles calculations. The gener-alized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form [ 27,28] was used for the exchange- correlation potential as implemented in the Vienna ab initio Simulation Package ( V ASP )[29,30]. For the electronic-band 2469-9950/2020/101(15)/155205(8) 155205-1 ©2020 American Physical SocietyM. YAGMURCUKARDES AND F. M. PEETERS PHYSICAL REVIEW B 101, 155205 (2020) 0100200300400500600700Ω (cm-1) MK ГГ Mo S OTop Side(a) (b) E' E'' A1OA1SE''E' A1SA1O R. A. (a.u.) 625 x25 FIG. 1. Single layer of Janus MoSO crystal. (a) Top and side views of the structure and increasing charge density is shown by a color scheme from blue to red with linear scaling between zero (blue) and 6.9 e/Å3(red). The unit cell is indicated by the blue dashed parallelogram. In addition, the motions of the atoms for the four Raman active modes are shown (bottom). (b) The phonon band dispersions with corresponding Raman spectrum (right panel). structure calculations spin-orbit coupling (SOC) was included with the GGA functional and Heyd-Scuseria-Ernzerhof(HSE06) screened-nonlocal-exchange functional of the gener-alized Kohn-Sham scheme [ 31]. The van der Waals correction to the GGA functional was included by using the DFT-D2method of Grimme [ 32]. Analysis of the charge transfers in the structure was determined by the Bader technique [ 33]. The kinetic energy cutoff for the plane-wave expansion was set to 500 eV and the energy was minimized until itsvariation became less than 10 −8eV during the electronic and structural optimizations. The Gaussian smearing method wasemployed for the total energy calculations and the width of thesmearing was chosen as 0.05 eV . Total Hellmann-Feynmanforce was reduced to 10 −7eV/Å for the structural optimiza- tion. 24 ×24×1/Gamma1centered k-point samplings were used for the primitive unit cells. To avoid interaction between theneighboring layers, a vacuum space of 25 Å was implementedin the calculations. Phonon band dispersions were calculated by using the small displacement method as implemented in the PHON code [34] by considering a 5 ×5×1 supercell. Each atom in the primitive unit cell was initially distorted by 0.01 Å and thecorresponding dynamical matrix was constructed. Then, thevibrational modes were determined by a direct diagonal-ization of the dynamical matrix. The corresponding Ramanactivity of each phonon mode was obtained from the deriva-tive of the macroscopic dielectric tensor by using the finite-difference method. The relaxed-ion elastic stiffness tensorsare calculated by using the small displacement methodologyas implemented in V ASP . In addition, the piezoelectric stress coefficients are obtained directly using density functionalperturbation theory (DFPT) with a sufficiently large k-pointsampling and kinetic energy cutoff of 700 eV . Moreover, the thermal stability of single-layer Janus MoSO is examinedsuch that for the simulations the NVE ensemble is used with a 4 ×4×1 supercell. The temperature is increased from 0 to 1000 K in 10 ps with 2 fs between consequent steps(see Fig. S2 in the Supplemental Material [ 35]). III. SINGLE-LAYER JANUS MoSO The optimized crystal structure of single-layer Janus MoSO possessing the 1H crystal phase is shown in Fig. 1(a). Note that the 1T phase of Janus MoSO is dynamically un-stable (see Fig. S3 in the Supplemental Material [ 35]). As shown in the figure, the Janus structure is constructed suchthat the atomic layer of molybdenum (Mo) is antisymmetri-cally sandwiched between the atomic layers of sulfur (S) andoxygen (O). The optimized in-plane lattice constant of single-layer MoSO is a=b=3.00 Å. Since the lattice constants of single-layer MoS 2(MoO 2) are larger (smaller) than those of the Janus structure (see Table I), compressive (tensile) strains on S (6.0%) and O (6.4%) surfaces, occurs. The Mo-S andMo-O bond lengths are 2.38 and 2.09 Å, respectively, whichare slightly different from those in MoS 2and MoO 2due to the induced surface strains. Bader charge analysis reveals that thecharge transfer from a Mo atom to the chalcogenide atomsis different: 0.5 and 0.9 echarges are donated to S and O atoms, respectively, which results in a strong out-of-planepolarization. Moreover, the charge difference on the surfacesaffects also the anisotropy of the work function, /Phi1, as listed in Table I. The dynamical stability of single-layer Janus MoSO is ver- ified by calculating its phonon band dispersions through the 155205-2STABLE SINGLE LAYER OF JANUS MoSO: STRONG … PHYSICAL REVIEW B 101, 155205 (2020) TABLE I. For the single-layer crystals of MoS 2,M o O 2, and Janus MoSO we give the optimized lattice constants, aandb; atomic bond lengths in the crystal, dMo-SanddMo-O; the amounts of charge depletion, /Delta1ρ(Mo-S) and/Delta1ρ(Mo-O) ; energy band gaps calculated within SOC on top of GGA, EGGA gap and HSE on top of GGA +SOC, EHSE06 gap ; the work functions calculated for two different surfaces, /Phi1Oand/Phi1S; and locations of VBM and CBM edges in the BZ. ab d Mo-S dMo-O /Delta1ρ(Mo-S) /Delta1ρ(Mo-O) EGGA gap EHSE06 gap /Phi1O /Phi1S (Å) (Å) (Å) (Å) ( e)( e) (eV) (eV) (eV) (eV) VBM /CBM MoS 2 3.19 3.19 2.41 0.6 1.04 1.98 5.70 K/K MoO 2 2.82 2.82 2.05 0.8 0.97 1.56 6.52 /Gamma1/K MoSO 3.00 3.00 2.38 2.09 0.5 0.9 1.07 1.62 6.63 4.67 /Gamma1/K whole BZ, which is presented in Fig. 1(b). As shown, phonon branches are almost free from any imaginary frequencies,except around the /Gamma1point, indicating the dynamical stability of the structure. Small imaginary frequencies in the out-of-plane acoustic mode near the /Gamma1point arise from numerical artifacts caused by the inaccuracy of the fast Fourier transform(FFT) grid. In its three-atom primitive unit cell, single-layerMoSO exhibits six optical phonon branches. As shown in thebottom panel of Fig. 1(a), there are two doubly degenerate in-plane modes, namely E /primeandE/prime/prime, and two nondegenerate out-of-plane vibrational modes, denoted by AO 1andAS 1.T h e calculated Raman spectrum of single-layer MoSO revealsthat all six optical phonon modes are Raman active. In asymmetric 1H-phase single-layer TMD, it is known that thereare two doubly degenerate and one nondegenerate Ramanactive modes. However, in the Janus structure there is anadditional Raman active mode arising from the broken out-of-plane symmetry. Except for the A O 1mode, all three atoms contribute to the vibration of the other optical phonon modes.TheA O 1mode arises from out-of-plane vibration of Mo and O atoms against each other. Its relatively high frequency(631.3 cm −1at the /Gamma1point) originates from strong Mo-O bond stretching. The other out-of-plane mode, AS 1, denotes the opposite vibration of the S and Mo-O pair and the vibration isdominated by the S atom. The frequency of A S 1is calculated to be 464.3 cm−1which is slightly larger than that of A2uin single-layer MoS 2(461.5 cm−1). The in-plane phonon modes can be classified as E/prime, which is dominated by the O vibration, and E/prime/primeis dominated by the S vibration. The E/primemode has a frequency of 471.5 cm−1and it is attributed to the opposite vibrations of Mo-O atoms againsteach other. On the other hand, the E /prime/primemode has a much lower frequency, 323.4 cm−1, and the phonon mode arises from the in-plane opposite vibrations of S and Mo-O pairs. Bothmodes are found to be Raman active and the calculated Ramanactivity of E /primeis found to be larger than that of E/prime/prime. Note that in a symmetric single-layer TMD, the E/prime/primephonon mode arises from the opposite vibration of the chalcogen atoms while thetransition-metal atom has no contribution to the vibration.Moreover, its Raman activity is almost three to four ordersof magnitude smaller than those of the other phonon modes. The electronic properties of single-layer MoSO reveal many more distinctive properties than the fully symmetricsingle layers of MoS 2and MoO 2due to the different bond- ing states of Mo-S and Mo-O atoms. The atomic contribu-tions to the electronic band dispersions of Janus MoSO areinvestigated through the whole BZ and are presented inFigs. 2(a),2(b), and 2(c). As in the case of the single-layers MoS 2and MoO 2, Janus MoSO exhibits semiconducting be- havior with a band gap of 1.62 eV calculated with the HSE06functional. In contrast to the direct band gap semiconductingnature of single-layer MoS 2, whose conduction band mini- mum (CBM) and valence band maximum (VBM) reside at theKpoint, MoO 2is an indirect band gap semiconductor with its CBM and VBM residing at the Kand the /Gamma1points, re- spectively (see Fig. S1 in the Supplemental Material [ 35]). As shown in Fig. 2(c), the VBM of Janus MoSO is composed of orbitals hybridized between d-Mo and pz-O atoms. However, as in the case of single-layer MoS 2and MoO 2, the CBM is dominated by the Mo orbitals only. The domination of VBMby the Mo-O bonding states results in the indirect band gapbehavior of Janus MoSO. The piezoelectric effect is known to generate an electric dipole moment as a result of applied mechanical stress innoncentrosymmetric materials. Theoretical predictions andexperimental observations have demonstrated that in the 2Dlimit the piezoelectric constants of materials can be enhanced[36–39]. The relaxed-ion piezoelectric tensor e ijcan be described as the sum of ionic, eion ij, and electronic, eel ij, contributions. The piezoelectric stress tensor eijthen is related to the piezoelec- tric strain tensor dijthrough the elastic stiffness tensor Cijas follows: eij=dikCkj. (1) As already reported before, for structures exhibiting hexag- onal symmetry, the piezoelectric strain coefficients can befound by using the relations d 11=e11 C11−C12,d13=e13 C11+C12. (2) Note that for the out-of-plane symmetric single layers the coefficient d31does not exist; however, in the presence of sur- face chalcogen replacement, i.e., in Janus single layers, out-of-plane piezoelectricity is created. As listed in Table II,t h e single layers of MoS 2and MoO 2have an in-plane piezoelec- tric property while the replacement of S by O on one surfacecreates a considerably strong out-of-plane piezoelectricity.For comparison, we also calculated the piezoelectric constantsfor experimentally synthesized MoSSe and other predictedsingle-layer Janus TMDs. The e 11component is found to be 3.7×10−10C/m for single-layer MoSO, which is close to that of single-layer MoS 2and MoSSe. The breaking of out- of-plane symmetry adds an additional degree of freedom and 155205-3M. YAGMURCUKARDES AND F. M. PEETERS PHYSICAL REVIEW B 101, 155205 (2020) -3-2-10123Energy (eV) M KГ(eV) 0.00 -1.40-0.35 EF KTop-Valence Band 1.07 -0.70 -1.051.642.212.783.35 Mo S O M KГ K M KГ K(c) (b) (a)Bottom-Conduction Band FIG. 2. The atomic contributions to the electronic band dispersions of Janus MoSO with (a) Mo, (b) S, and (c) O contributions. Energy surfaces of the valence and conduction band edges are shown in the right panel. The insets show the atomic orbital characters of the valence and the conduction band edges. The Fermi level is set to zero energy. d13is no longer zero. As shown in Fig. 3,t h e e31coefficient for MoSO is found to be 1 .4×10−10C/m, which is much larger than for the other predicted Janus TMDs. The relativelylarger charge difference between two surfaces of Janus MoSOresults in a large out-of-plane polarization and thus the e 31 and the corresponding d31coefficients are found to be large in Janus MoSO. IV . BIAXIAL STRAINED SMoO Strain is often present in experiments either naturally or controllably. Many reports indicated that strain can alter theelectronic and the vibrational properties of materials [ 40,41]. In the case of the vibrational spectrum, Raman peak positionsand the corresponding intensities depend significantly on thepresence of strain [ 42,43]. In addition, strain modifies the phonons, with stretching usually resulting in phonon modesoftening, and phonon hardening in the case of compression.The rate of change can be obtained from the Grüneisenparameter. Moreover, by the presence of strain, the relativeRaman intensities can be more distinguishable. TABLE II. For the 1H phases of single-layer MoS 2,M o O 2,a n d MoSO, the relaxed-ion elastic coefficients Cij, piezoelectric stress coefficients eij, and the corresponding piezoelectric strain coeffi- cients dij. Note that eijis multiplied by 10−10. C11 C12 e11 e13 d11 d13 (N/m) (N /m) (C /m) (C /m) (pm /V) (pm /V) MoS 2 131 34 3.7 3.8 MoO 2 229 84 3.5 2.4 MoSO 164 48 3.7 1.4 3.2 0.7 SMoSe 135 30 3.8 0.3 3.7 0.2 SMoTe 116 28 4.5 0.5 5.1 0.4SeMoTe 110 23 4.5 0.2 5.2 0.2 SWSe 147 28 2.6 0.2 2.2 0.1 SWTe 131 22 3.2 0.4 3.0 0.2SeWTe 119 19 3.2 0.2 3.2 0.1A. Electronic structure In our work, we extend the applied biaxial strain to the nonlinear regime in order to investigate also the phononicstability of Janus single-layer MoSO. Before discussing theresponse of the vibrational spectrum to the applied biaxialstrain, we first present our results for the electronic structure.As shown in Fig. 4, the semiconducting nature of Janus MoSO can be tuned via the application of biaxial strain. Mainly, tworesults can be obtained from the strain-dependent electronicband dispersions. One of our findings is that single-layer Janus 2.02.53.03.54.04.55.0 0.000.250.500.751.001.251.50e11 (10-10 C/m) e31 (10-10 C/m) MoSe SMoTe SMoTe SeWSe SWTe SWTe SeMoO SIIIIII IV VVIVIII IIIII IVVVI VII I II III IV V VI VII FIG. 3. The in-plane, e11, and the out-of-plane, e31, components of the piezoelectric stress coefficients of Janus MoSO, which are compared with the other predicted Janus TMDs. Side views of the charge differences on both surfaces are given below. 155205-4STABLE SINGLE LAYER OF JANUS MoSO: STRONG … PHYSICAL REVIEW B 101, 155205 (2020) -3-2-10123 -3-2-10123Energy (eV) MK Г Г-5% -4% -3% -2% -1% +1% +2% +3% +4% +5% +6% +7% +8% +9% FIG. 4. In-plane biaxial strain-dependent electronic band dispersions of single-layer Janus MoSO. The Fermi level is set to zero energy. MoSO undergoes a semiconducting-to-metallic transition for tensile strain over 8% ( ε>8%) which is slightly smaller than that reported for single-layer MoS 2(10%) [ 44]. We find a decreasing band gap via increasing tensile strain, whilethe band gap opens under application of compressive strain.Apparently, the conduction band edges are mostly affectedby the applied in-plane biaxial strain. Therefore, as a result 00.511.522.53Band Gap (eV)Egap(Г-Г) Egap(Г-K)Egap(Г/K-Г) ε (%)- 5 - 4 - 3 - 2 - 1 0123456789 FIG. 5. The evolution of the direct and the indirect band gaps in single-layer MoSO under applied biaxial strain.of this, it is seen that the CBM of single-layer MoSO shifts from the Kpoint to a point between Kand/Gamma1, which is the second main result of strain-driven electronic properties ofMoSO. The CBM point crossover can also be seen in thegraph presented in Fig. 5. Basically, the shift of the CBM can be related to the atomic orbital character of the CBM at the K point. As shown in Fig. 2, due to the out-of-plane character of the orbitals occupying the VBM, it is almost unaffected by theapplied in-plane strain; however, due to the mixed d x2anddz2 orbitals of the Mo atom at the CBM, it displays both a shift in position and an energy shift via applied strain. B. Phononic stability and Raman shifts The stress-strain relation for a material can be used to extract its mechanical properties and also can be used toprobe the elastic instability of the material at a certain appliedstrain. Many 2D materials undergo phononic instability beforereaching the elastic instability. Therefore, in the case of single-layer Janus MoSO, we directly examine the strain-dependentphononic stability. As presented in Fig. 6, small imaginary frequencies around the /Gamma1point increase for compressive strain of 5%. On the other hand, under the application of tensilestrain, the structure retains its dynamical stability up to strainvalues of almost 12%. Notably, our reported critical strainvalue is smaller than those reported for graphene (15% forbiaxial [ 45] and 18–24% for uniaxial strains [ 46]). In addition, the phononic instability strain value of MoS 2(20%) [ 47,48]i s 155205-5M. YAGMURCUKARDES AND F. M. PEETERS PHYSICAL REVIEW B 101, 155205 (2020) 0100200300400500600700Ω (cm-1) MK ГГ MK ГГ+12% -5% FIG. 6. Phonon band dispersions of 5% compressively (on the left) and 12% stretched biaxial strain (right) of single-layer MoSO. Imaginary frequencies are shown as negative frequencies. much larger than our predicted value for single-layer Janus MoSO. At 12% of biaxial tensile strain, single-layer MoSOis found to undergo a phononic instability which is dictatedby both out-of-plane and in-plane soft modes between the M and the Kpoints. This is also known as the finite wave vector instability, which is expected to occur in 2D materials. No-tably, in addition to the out-of-plane flexural acoustic mode,the in-plane acoustic mode also causes phononic instability,which can be attributed to the broken out-of-plane symmetryin the Janus structure. Here the underlying mechanism can beunderstood through the bonds Mo-O and Mo-S. As the appliedtensile strain increases, the Mo-S and Mo-O bonds becomemore flat, which softens the out-of-plane and in-plane acousticmodes. The investigation of phonon shifts and the variation of Raman activities of the Raman active modes can beuseful to probe the strain on the material. As shown inFig.7(a), compressive (tensile) biaxial strain hardens (softens) the frequencies of both in-plane and out-of-plane phononmodes. However, as the applied biaxial strain is in-plane, theresponses of the phonon modes are not expected to be the same due to their different mode Grüneisen parameters. The mode Grüneisen parameter for a phonon mode can be calculated using the relation γ(q)=−a 0 2ω0(q)/bracketleftbiggω+(q)−ω−(q) a+−a−/bracketrightbigg , (3) where a0is the optimized lattice constant, ω0(q) is the un- strained phonon frequency at wave vector q,ω+(q) andω−(q) are the phonon frequencies under tensile and compressivebiaxial strain, respectively, and a +−a−is the difference in the lattice constant when single-layer MoSO is under biax-ial strain. In the present study, the phonon frequencies arecalculated in the q=0 limit, at the /Gamma1point. As presented in Fig. 7(b), the frequency shifts of the phonon modes are different from each other due to their different vibrationalcharacters. The calculated mode Grüneisen parameters for theE /prime/prime,AS 1,E/prime, and AO 1modes are 0.70, 0.26, 1.32, and 0.84, respectively. All of these values are larger as compared to fullysymmetric single-layer TMDs (0.52, 0.68, and 0.23 for E /prime/prime,E/prime, andA1modes of single-layer MoS 2)[49]. All phonon modes are found to display phonon softening (hardening) via appliedtensile (compressive) strains. This is a direct indication of thechanging electron distribution or atomic bonding in the struc-ture with strain. The phonon frequency shifts are importantfor probing the type and strength of the applied strain in thestructure. As the A S 1has the lowest mode Grüneisen parameter, the shift of its frequency is the smallest among all the phononmodes of MoSO. Therefore, it can be seen in Fig. 7(a)that the frequency of E /primebecomes smaller than that of AS 1for tensile biaxial strain. Notably, vibrations of the phonon modes, E/prime andAO 1, are dominated by the O atoms and thus the shifts of the two modes are larger with applied strain due to Mo-Obonds, which become more horizontal to the Mo layer. We present our results up to strain value of 9%, at which semiconducting-metallic transition is driven for MoSO.The transition of single-layer MoSO from semiconductingto metallic significantly affects the Raman activities of theRaman active modes. Therefore, we separated our results forthe tensile strains over 6%, after which the Raman activitiesbecome much larger due to the electronic contributions tothe Raman activity. The Raman activities in the range of±5% strain are multiplied by a factor of 200 in order to be comparable to those for strains over 6%. The Raman activity 250 300 350 400 450 500 550 600 650 700 Ω (cm-1)Raman Activity (a.u.)-1% -2% -3% -4% -5%1%2%3%4%5% 0%x200 6%7%8%9% 250 300 350 400 450 500 550 600 650 700 Ω (cm-1)0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1200300400500600700 ε (a/a0)Ω (cm-1) E''E' A1SA1O(a) (b) E'' E' A1SA1O E''E' A1OA1S FIG. 7. (a) The evolution of the Raman spectrum of single-layer MoSO under compressive and tensile strains. (b) The strain-dependent phonon frequency shifts of the four Raman active modes. 155205-6STABLE SINGLE LAYER OF JANUS MoSO: STRONG … PHYSICAL REVIEW B 101, 155205 (2020) of in-plane vibration modes E/primeandE/prime/prime, displays an increasing trend from compressive to tensile strain. Since the applied strain is in-plane, it rearranges the electron clouds betweenthe Mo-S and Mo-O atoms which significantly enhances thepolarizability and hence the Raman activity. This behavioris the same as that reported for fully symmetric single-layerTMDs under biaxial strain [ 49]. The A O 1mode, which cannot be observed in symmetric single-layer TMDs, exhibits also anincreasing trend in its Raman activity with increasing tensilestrain. In fact this behavior is in contrast to what was reportedfor the A 1mode of single-layer MX 2(M=Mo or W and X=S or Se) [ 49]. The main difference between a TMD’s single layer and the Janus structure arises from the differentvibrational motion of the atoms. In fully symmetric TMDs,theA 1mode represents the out-of-plane vibration of only the chalcogen atoms while in a Janus single layer this modearises mainly from the out-of-plane vibration of a chalcogenand transition metal atoms. The direct bonding between thevibrating atoms in the Janus structure influences strongly theRaman activity response of the phonon mode against the ap-plied strain. In contrast to the A O 1mode, the other out-of-plane vibrational mode, AS 1, possesses a decreasing trend in its Raman activity with increasing tensile strain, which is similarto that of symmetric single-layer TMDs [ 49]. Such behavior arises from the vibration of the chalcogen atoms dominatingthe phonon mode. V . CONCLUSION In this study, the 1H phase of a Janus single layer composed of S-Mo-O atomic layers was predicted to be dynamicallystable and to be an indirect band gap semiconductor. Ourfindings revealed that the single-layer Janus structure of 1H-MoSO exhibits relatively strong polarization arising from thelarge charge difference between S and O surfaces. In contrast to 1H phases of MoS 2and MoO 2, Janus MoSO was shown to possess four Raman active phonon modes in its Ramanspectrum, which is a fingerprint for the observation of theJanus crystal. Among the 1H-phase Janus single-layer TMDs,single-layer MoSO was found to exhibit a much stronger out-of-plane piezoelectric coefficient. Moreover, we investigatedthe electronic and phononic properties of Janus MoSO underapplied biaxial strain. Our results indicated the following:(i) single-layer MoSO undergoes a semiconducting-to-metallic phase transition via applied tensile strain while theconduction band edge displays a shift under compressivestrain. (ii) single-layer MoSO preserves its phononic stabilityunder applied 5% of compressive strain and 11% of tensilestrain, displaying significant phonon shifts. (iii) Raman ac-tivity of the phonon modes was shown to display significantenhancement as the semiconducting-to-metallic transition isreached, which is due to the fact that the electronic transi-tions start to contribute to the Raman activity. In addition,the phonon instability was shown to arise from the soft in-plane and out-of-plane acoustic modes at finite wave vector.The large strain tolerance of Janus MoSO is important forits nanoelastic applications. With its strain-free and strain-dependent stability, we propose that Janus MoSO can befabricated in the common 1H phase with a strong out-of-planepiezoelectric coefficient. ACKNOWLEDGMENTS Computational resources were provided by the Flemish Supercomputer Center (VSC). M.Y . is supported by the Flem-ish Science Foundation (FWO-Vl) through a postdoctoralfellowship. [1] S. K. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, Y . Zhang, S. V . Dubonos, I. Grigorieva, and A. A. Firsov, Science 306,666(2004 ). [2] A. H. Castro Neto and K. Novoselov, Rep. Prog. Phys. 74, 082501 (2011 ). [3] K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett.105,136805 (2010 ). [4] A. Splendiani, L. Sun, Y . Zhang, T. Li, J. Kim, C.-Y . Chim, G. Galli, and F. 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PhysRevB.71.205422.pdf

Electronic structure of straight semiconductor-semiconductor carbon nanotube junctions Young-Woo Son, Sang Bong Lee, Choong-Ki Lee, and Jisoon Ihm * School of Physics, Seoul National University, Seoul 151-747, Korea sReceived 28 October 2004; published 31 May 2005 d We calculate the scanning tunneling microscopy images and the scanning tunneling spectroscopy of straight semiconductor-semiconductor single-wall carbon nanotube junctions with two different diameters. Two kindsof localized states associated with topological defects, one donorlike and the other acceptorlike, arise in theenergy gap of the junctions, whose levels strongly depend on the number of topological defects and theirspatial arrangement.We find that defects on one side of the tube surface influence the electronic distribution onthe other sback dside sufficiently strongly that they may be detected by the scanning of the opposite side as well. DOI: 10.1103/PhysRevB.71.205422 PACS number ssd: 81.07.De, 73.20.Hb, 73.40.Lq, 68.37.Ef Single-wall carbon nanotubes sSWNTs dhave attracted the attention of scientists in many fields due to their unique elec-tronic and mechanical characteristics since their discovery. 1 In the electronic device applications, the SWNT can be usedfor the component of the field effect transistors, 2,3single- electron tunneling transistors,4or rectifiers.5The junction of two SWNTs sa kind of “intramolecular junction” dcan be made with the pentagon or heptagon defects which inducethe positive or negative curvature at the interface. 6There have been many theoretical and experimental researches onthe metal-semiconductor SWNT junctions. 7–11Recently, atomically resolved Scanning Tunneling Microscopy sSTM d experiment on the straight si.e., without bending d semiconductor-semiconductor SWNT junction wasreported. 12Their Scanning Tunneling Spectroscopy sSTSd data clearly show that localized states exist in the energy gapand the extended states of the nanotube on one side penetrateand decay slowly into the other side. Furthermore, their STMimages exhibit a remarkable change from the usual cell-periodic atomic arrangement of the SWNT to the longer-range modulation structure as the tip-sample bias voltagechanges. 12In this paper, we present extensive theoretical simulations of the STM and STS of various straightsemiconductor-semiconductor SWNT junctions and explainthe observed experimental data. We employ the one-electron s p-orbital dtight binding ap- proximation with the hopping integral Vpppof −2.66 eV.13 The model of the junction typically consists of as many as several thousand atoms to reduce the finite-size effect andobtain reasonably converged results that can be comparedwith experiment. 12The tunneling currents between the tip and the sample are evaluated following the approach intro-duced by Meunier and Lambin. 14Based on the Tersoff- Hamann theory,15the tip state is assumed to be a single atom sstate and coupled with the carbon porbital of the junction by the hopping interaction. The standard current expressionis 9,14,16 I=s2pd2e hE −eV0 dEo t,t8o s,s8vtsvt8s8rtt8TsEFT+eV+Ed 3rss8SsEFS+Ed, s1d wheretssddenotes the states belonging to the tip ssample dandvtsgives the interaction between the tip and the junction surface atoms; rtt8sss8dTsSdrepresents the density of states matri- ces of the tip ssample dandEFTsSdis the Fermi energy of the tip ssample d. Adopting the exponentially decaying tunneling process, the matrix element vtsis proportional towse−ds/lcosuswhere the weighting factor ws =e−ads2/os8e−ads82 and ussdsdis the angle sdistance dbetween the tip and the porbital of the sth atom in the sample.14The parameters used here are from Ref. 14, where l=0.85 Å and a=0.6 Å−2. The tip height at each point is determined recur- sively to make the tunneling current between the tip and thesample constant when we calculate the topographic STMimage. For the given tip height, the bias voltage is swept toproduce the I-Vcurves and the spatially resolved LDOS is calculated from dI/dV. We obtain the complete STS map by shifting the tip position along the axial direction of the nano-tube. At first, we consider the straight junction consisting of s14,0dands19,0dzigzag SWNTs and the simplest possible defects sone pentagon and one heptagon d. We then study the s12,1d-s18,2dand s15,2d-s19,3djunctions with complex de- fects. In all the cases above, the chiral vectors of two con- stituent SWNTs of the junction are almost parallel since wewant to study a straight junction with relatively smooth bondconnection at the joint. Figure 1 sadshows the atomic con- figuration of the s14,0d-s19,0djunction. The actual length of the junction we consider is 42 nm while the figure showsonly the central part of it. A pentagon creates a convex sur-face spositive Gaussian curvature dand a heptagon creates a concave surface snegative Gaussian curvature d. 17,18By intro- ducing the angle-dependent interaction between the tip andthe porbital, the simulated images around a convex or con- cave region reflect such a geometrical factor. The simulatedscanning tunneling tip moves along the solid line locatedseveral Å away from the defects as shown in Fig. 1 sadin the constant current mode. After the tip height is determined ateach position on the line, the voltage is swept fromV=−0.5 V to V=+0.5 V to obtain the STS data sdI/dVd. The simulated STS is shown in Fig. 1 sbd. There are two localized states indicated by arrows sidandsiid. These local- ized states have been observed in the STS of the metal-semiconductor junction reported in the literature. 9However,PHYSICAL REVIEW B 71, 205422 s2005 d 1098-0121/2005/71 s20d/205422 s5d/$23.00 ©2005 The American Physical Society 205422-1the localization length in the semiconductor-semiconductor junction sa few nm dis greater than that in the metal- semiconductor junction. The magnitude of the imaginary wave vector in the band gap is proportional to uECsVd−«du1/2 whereECsVdis the energy minimum smaximum dof the con- duction svalence dband and «dis the energy level of the de- fect state. Hence, the states spread more in the smaller bandgap SWNT fs19,0don the right side gthan in the larger one fs14,0don the left side gas shown in sidandsiidof Fig. 1 scd. The characteristic features are the same when we calculatethe STSs following other lines along the junction than theone chosen in Fig. 1 sad. We can observe the influence of defects even from the opposite side of the defects. We avoidthe scanning line which crosses exactly the pentagon andheptagon because the dI/dVpeak grows too much beyond control in that case. From Hückel’s rule that the cyclic p-electron system with 4 n+2sn: nonnegative integer delec- trons is most stable, six-membered carbon rings are morestable than five- or seven-membered rings. 9,19Thus, it can be understood that a heptagon tries to give up an electron to itsneighbors and plays the role of a donor. Similarly, the defectstate originated from a pentagon is an acceptor state in thejunction. Consequently, in Fig. 1, the state sidis the donor level associated with the heptagon and the state siidis the acceptor level associated with the pentagon. Because the en-ergy difference between the donor level and the conduction band edge is larger than that between the acceptor level andthe valence band edge, the decay length of the state sidis shorter than that of the state siid. We also examine the ex- tended states of the nanotube. Conduction band and valenceband states of a SWNT on one side penetrate into the otherside of the junction across the defective region and decayslowly. The decay length of these states for the s14,0d-s19,0d junction is over 20 Å. The arrows siiidandsivdin Fig. 1 sbd indicate decaying states of the valence band edge of thes19,0dands14,0dSWNT, respectively and the corresponding density profiles are presented in Fig. 1 scd. In Fig. 2, we show the calculated STM topographic im- ages in the rectangular region on the opposite sback dside of the defects as a function of the bias voltage. At high biasvoltage sV=+1.0 V d, the image shows, instead of the de- fects, a well-ordered pattern corresponding to the periodic atomic structure. At lower bias voltage, for exampleV=±0.4 V, the deviation from the ordered pattern appears at the interface of the two nanotubes. At V=+0.3 V, the STM detects only the localized donorlike defect state since no ex- FIG. 1. Atomic model and calculated STS image of the s14,0d- s19,0dSWNT junction. Only the central part of the simulated sys- tem of 42 nm in length is shown. sadAtomic model of the junction with one pentagon and one heptagon defect 1.23 nm apart. Thesolid line is the scanning line of the tip to obtain the simulated STSdata below. sbdSTS image of the junction. The abscissa is the position along the tube axis and the ordiante is the sample biasvoltage. Localized states in the gap are indicated by arrows sidand siid. The arrow siiiddenotes the state originating from the s19,0d valence band and decaying into the s14,0dSWNT gap across the defective region. The arrow sivdshows the opposite behavior to siiid.scdDensity profiles along the solid line in sadfor individual states indicated by arrows sid–sivdinsbd. FIG. 2. Calculated topographic images of the s14,0d-s19,0djunc- tion in the solid rectangular region at different tip-sample bias volt-ages. All figures presented here are STM images on the oppositeside of the defects sdark atoms in the atomic model denotes the defects on the back side d.sadAtomic model of the junction. sbd STM images of the junction at different bias voltages. FIG. 3. Calculated STS image of the s12,1d-s18,2djunction. sad Atomic model of the junction. sbdSTS image of the junction.SONet al. PHYSICAL REVIEW B 71, 205422 s2005 d 205422-2tended states exist at this energy. It is interesting that the defect states are detected even when the STM tip is probingthe opposite side of the defects as shown here. Two SWNTson the left and right are connected defectlessly with hexa-gons only on this side of the tube face. However, the defectstates extend onto the opposite side and enable the STM tipto detect the defects residing at the opposite side. Chiralities and band gap differences do not seem to affect significantly the essential charateristics of semiconductor-semiconductor SWNT junctions. Figure 3 shows the modelstructure of the s12,1d-s18,2djunction with one pentagon and one heptagon defects and its STS image. The band gap dif-ference between the s12,1dands18,2dSWNTis 0.4 eVwhich is approximately 50% greater than that between the s14,0d ands19,0dSWNT. The overall characteristics of STS does not differ from that of the zigzag junction in Fig. 1.There aretwo localized states in the gap and also a few decaying statesbetween conduction band edges of two SWNTs, as well asbetween valence band edges of two SWNTs. Figure 4 showsits calculated topographic images which show hexagonal net-works under high bias voltage sV=±1.0 V d, while the im- ages under low bias voltage reflect the topological defects more clearly. We then study the effects of different defect configura- tions at the junction for a given pair of semiconductorSWNTs. We examine the s15,2d-s19,3djunction withthe defects of various configurations. The simplest configuration of defects allowed here has one pentagon andone heptagon. Yet, another configuration is possible by thesingle bond rotation of Stone-Wales type. 20If we rotate a bond neighboring a pentagon spddefect by 90°, a pentagon- heptagon-pentagon sphpddefect complex is produced. In the same manner, a heptagon-pentagon-heptagon shphdis gener- ated from a heptagon shdby a 90° bond rotation. We obtain four different combinations of the defect configurations sh-p,h-php,hph-p, andhph-phpdas shown in Fig. 5. Here, we show the model structures and the STS and STM topo-graphic images of three cases sh-php,hph-php, andhph-pd in Figs. 6–8. The donor state of the h-phpjunction in Fig. 6sbdis closer to the conduction band edge than that of the hph-phpjunction in Fig. 7 sbdwhile there is no appreciable difference in the energy level of the acceptor states. In addi-tion, the shape of the localized states are different as shownin Figs. 6 scdand 7 scd. The STS image in Fig. 8 for the hph-pmodel shows a good agreement with recent experi- mental data. 12Particularly, in Figs. 8 sbdand 8 scd, the state indicated by siidis the mixed state of the acceptorslike local- FIG. 4. Calculated topographic images of the s12,1d-s18,2djunc- tion. sadAtomic model of the junction. sbdTopographic images of the junction at different bias voltages. FIG. 5. Four differenct defect configurations of the s15,2d-s19,3djunction as explained in the text. sadh-p,sbdh-php,scdhph-p, and sdd hph-php. FIG. 6. Calculated STS and topographic images of the s15,2d- s19,3djunction with the h-phpdefect. sadAtomic model of the junction. sbdSTS image of the junction. scdTopographic images on the back side of the defects at different bias voltages.ELECTRONIC STRUCTURE OF STRAIGHT … PHYSICAL REVIEW B 71, 205422 s2005 d 205422-3ized state with the extended valence band edge state of the right s19,3dSWNT. If the energy level of the localized state approaches the band edge very closely, it is hard to distin-guish the localized state from the extended one theoretically,which is also the case in experiment. Therefore, the donor-like state indicated as sidis the only localized state without ambiguity in this system. We note that there are more com-plicated defect configurations proposed in the literature 21 which also give a good agreement with experiment. In summary, we have calculated the STS and STM topo- graphic images of straight semiconductor-semiconductorSWNT junctions. A straight junction consisting of twodifferent-diameter SWNTs has at least two defects. Associ-ated with each defect, there exists an acceptor or a donorlikestate in the gap. The defect states manifest themselves in the STM topographic images as well as in the STS. We find thechange in the electronic distribution by defect states even onthe tube surface opposite to the defects.The extended SWNTstates at each side are shown to penetrate and decay slowlyinto the opposite side across the junctions. Y.-W. S. thanks Hajin Kim for helpful discussions. This work was supported by the CNNC of Sungkyunkwan Uni-versity, and the MOST through the National Science andTechnology Program sGrant No. M1-0213-04-0001 d. Com- putations are performed through the support of the KISTI. *Email address: jihm@snu.ac.kr 1S. Iijima, Nature sLondon d354,5 6 s1991 d. 2Sander J. Tans, Alwin R. M. Verschueren, and Cees Dekker, Na- ture sLondon d393,4 9 s1998 d. 3R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and Ph. Avouris, Appl. Phys. Lett. 73, 2447 s1998 d. 4Marc Bockrath, David H. Cobden, Paul L. McEuen, Nasreen G. Chopra, A. Zettl, Andreas Thess, and R. E. Smalley, Science 275, 1922 s1997 d. 5Philip G. Collins, A. Zettl, Hiroshi Bando, Andreas Thess, and R. E. Smalley, Science 278, 100 s1997 d. 6R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Prop- erties of Carbon Nanotubes sImperial College Press, London, 1998 d.7J. Han, M. P. Anantram, R. L. Jaffe, J. Kong, and H. Dai, Phys. Rev. B57, 14 983 s1998 d. 8Min Ouyang, Jin-Lin Huang, Chin Li Cheung, and Charles M. Lieber, Science 291,9 7 s2001 d. 9V. Meunier, P. Senet, and Ph. Lambin, Phys. Rev. B 60, 7792 s1999 d. 10Zhen Yao, Henk W. Ch. Postma, Leon Balents, and Cees Dekker, Nature sLondon d402, 273 s1999 d. 11M. S. Ferreira, T. Dargam, R. B. Muniz, and A. Latgé, Phys. Rev. B62, 16 040 s2000 d. 12Hajin Kim, J. Lee, S.-J. Kahng, Y.-W. Son, S. B. Lee, C.-K. Lee, J. Ihm, and Young Kuk, Phys. Rev. Lett. 90, 216107 s2003 d. 13X. Blase, L. X. Benedict, E. L. Shirley, and S. G. Louie, Phys. Rev. Lett. 72, 1878 s1994 d. FIG. 7. Calculated STS and topographic images of the s15,2d- s19,3djunction with the hph-phpdefect. sadAtomic model of the junction. sbdSTS image of the junction. scdTopographic images of the junction on the back side of the defects at different biasvoltages. FIG. 8. Calculated STS image of the s15,2d-s19,3djunction with thehph-pdefect. sadAtomic model of the junction. sbdSTS image of the junction. The arrow sidindicates the localized state and the arrows siidand siiidindicate the valence band edge states of the s19,3dSWNT and the s15,2dSWNT, respectively. An acceptorlike localized state is mixed with the state siid.scdDensity profiles along the tube of the individual states indicated by arrows sid–siiidinsbd.SONet al. PHYSICAL REVIEW B 71, 205422 s2005 d 205422-414V. Meunier and Ph. Lambin, Phys. Rev. Lett. 81, 5588 s1998 d. 15J. Tersoff and D. R. Hamann, Phys. Rev. Lett. 50, 1998 s1983 d. 16D. Orlikowski, M. Buongiorno Nardelli, J. Bernholc, and C. Ro- land, Phys. Rev. B 61, 14 194 s2000 d. 17B. I. Dunlap, Phys. Rev. B 49, 5643 s1994 d. 18R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B53, 2044 s1996 d. 19T. W. G. Solomons, Organic Chemistry , 6th ed. sWiley, New York, 1996 d, p. 624. 20A. J. Stone and D. J. Wales, Chem. Phys. Lett. 128, 501 s1986 d. 21W. Fa, J. Chen, and J. Dong, Eur. Phys. J. B 37, 473 s2004 d.ELECTRONIC STRUCTURE OF STRAIGHT … PHYSICAL REVIEW B 71, 205422 s2005 d 205422-5

PhysRevB.90.125113.pdf

PHYSICAL REVIEW B 90, 125113 (2014) Relaxation in Luttinger liquids: Bose-Fermi duality I. V . Protopopov,1,2D. B. Gutman,3and A. D. Mirlin1,4,5 1Institut f ¨ur Theorie der Kondensierten Materie and DFG Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany 2L. D. Landau Institute for Theoretical Physics, RAS, 119334 Moscow, Russia 3Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel 4Institut f ¨ur Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany 5Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia (Received 8 April 2014; revised manuscript received 11 July 2014; published 8 September 2014) We explore the lifetime of excitations in a dispersive Luttinger liquid. We perform a bosonization supplemented by a sequence of unitary transformations that allows us to treat the problem in terms of weakly interactingquasiparticles. The relaxation described by the resulting Hamiltonian is analyzed by bosonic and (after arefermionization) by fermionic perturbation theory. We show that the fermionic and bosonic formulations of theproblem exhibit a remarkable strong-weak coupling duality. Specifically, the fermionic theory is characterizedby a dimensionless coupling constant λ=m ∗l2Tand the bosonic theory by λ−1,w h e r e1 /m∗andlcharacterize the curvature of the fermionic and bosonic spectra, respectively, and Tis the temperature. DOI: 10.1103/PhysRevB.90.125113 PACS number(s): 73 .23.−b,73.63.Nm I. INTRODUCTION Quantum kinetics in interacting one-dimensional (1D) systems is a subject of an active experimental and theoreticalinvestigation. There is a variety of experimental realizationsof 1D fermionic systems which include, in particular, carbon nanotubes, semiconductor and metallic nanowires, as well as edge states of quantum Hall systems and of other 2Dtopological insulator structures. Further, cold-atomic gases inoptical traps can be used to engineer 1D fermionic or bosonicsystems with a tunable interaction. A light or microwaves in waveguides with interaction mediated by two-level systems represent another realization of a correlated 1D bosonicsystem. A common and very powerful theoretical approach to interacting 1D systems is the bosonization [ 1–4]. When the spectral curvature and backscattering processes are neglected,bosonization maps the problem of interacting fermions (knownas Tomonaga-Luttinger model) to the Luttinger-liquid theoryof free bosons (plasmons). A mapping to the Luttingerliquid is also obtained if one starts from the problem ofbosons with repulsion. If the interaction is considered asmomentum-independent (i.e., local in the coordinate space),the spectrum of Luttinger-liquid bosonic excitations is linear,and by virtue of refermionization the problem is equivalent tothat of free fermions. In brief, when the momentum dispersionsof excitations are neglected, the fermionic and bosonic 1Dproblems are equivalent, and the interaction can be completelyeliminated. The problem becomes much more complex when both the spectral curvature of constituent particles and the momentumdependence of the interaction are retained. This leads (apartfrom some special cases) to a violation of integrability of thetheory. While the corresponding corrections to the Luttinger-liquid theory are irrelevant in the renormalization-group (RG)sense, they are very important from the physical point ofview. Specifically, they establish a finite relaxation rate ofexcitations when the system is at nonzero temperature or awayfrom equilibrium.Several recent works addressed various aspects of relax- ation in 1D problems. In Refs. [ 5–8] a perturbative analysis of three-particle scattering in a model of weakly interactingfermions with a spectral curvature (inverse mass) m −1was performed. It was found, in particular, that in the caseof spinless fermions the intrabranch scattering processes(RRL →RRL and RLL →RLL, where R and L denote right and left movers, respectively) induce a scattering rate ofan excitation with momentum kthat scales as ( k−k F)8/m3 at zero temperature and ( k−kF)6T/m2at sufficiently high temperature T, where kFis the Fermi momentum. These results were generalized to the case of Coulomb interactionin Refs. [ 9,10]. The opposite limiting case of a Luttinger liquid with inter- action parameter K/lessmuch1 (describing, in particular, fermions with very strong repulsive interaction, when the system canbe viewed as “almost a Wigner crystal”) was considered inRefs. [ 11,12]. The authors of these works analyzed the decay rate of bosonic excitations in such systems and found the decayrate scaling as T 5. The goal of this work is to study systematically relaxation in dispersive Luttinger liquids in the whole space of parameters.We start from the interacting fermionic problem, then bosonizeit and perform a unitary transformation [ 16] (that extends the one originally introduced in Ref. [ 13]; see also Refs. [ 8,14,15]) which allows one to eliminate a major part of the interaction.In particular, in this way the two chiral branches get decoupledup to the third order in density fluctuations. In Ref. [ 16] this formalism was employed to develop a formalism ofkinetic equation for fermionic quasiparticles. A focus in thatwork was on a not too long time scale where the collisionsbetween quasiparticles can be neglected. Here we use thetheory resulting from the above unitary transformation to findthe relaxation rate of excitations. We perform both fermionicand bosonic analysis of this theory, evaluate the correspondingrelaxation rates, and determine regions of the parameter spacewhere each of the approaches is applicable. This allows usto establish a remarkable picture of Fermi-Bose duality indispersive interacting 1D systems. 1098-0121/2014/90(12)/125113(15) 125113-1 ©2014 American Physical SocietyI. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) The structure of this article is as follows. In Sec. IIwe introduce a model of a generic dispersive Luttinger liquidin terms of fermions with a spectral curvature 1 /mand an interaction with an arbitrary strength and a radius l int.T h e fact that both parameters 1 /mandlintare nonzero makes the problem nonintegrable. We perform a bosonization of thismodel supplemented by a unitary transformation that mapsit onto a problem of weakly interacting bosonic quasiparticles.In Sec. IIIwe calculate the lifetime of bosonic excitations within this theory. In Sec. IVwe refermionize the theory obtained in Sec. IIand explore the relaxation of fermionic excitations. For this purpose, we calculate the contributions tothis relaxation rate from both interbranch and intrabranch scat-tering processes. Finally, in Sec. Vwe collect and analyze the obtained results and determine the behavior of the relaxationrate in the whole parameter space. We show that the parameterspace is subdivided in the “fermionic” and “bosonic” domainswith a dimensionless control parameter m ∗l2T, where the effective mass m∗and the plasmon dispersion length lare expressed in terms of the bare parameters m,lintand the Luttinger-liquid constant K0. The emerging picture has a character of Fermi-Bose weak-strong coupling duality. Weclose the paper by summarizing our results and discussingprospects for future research in Sec. VI. II. DISPERSIVE LUTTINGER LIQUIDS In this section we introduce the Hamiltonian of our model and formulate basic questions to be addressed in the paper. Wethen perform a sequence of unitary transformations bringingthe theory to a form amenable to a perturbative treatmentand highlighting the duality between fermionic and bosonicdescriptions of the dispersive Luttinger liquids. A. The model Our starting point is the Hamiltonian of a generic “dis- persive” Luttinger liquid composed of (spinless) right- andleft-moving fermions [created and annihilated by operatorsψ + η(x),ψη(x) with η=R,L ; occasionally, we also use the notation η=± 1] with curved single-particle spectrum /epsilon1η(k)=ηkvF+k2/2minteracting via a generic finite-range density-density interaction g(x): H=/summationdisplay η/integraldisplay dxψ+ η(x)/parenleftbigg −iηvF∂x−1 2m∂2 x/parenrightbigg ψη(x) +1 2/integraldisplay dxdx/primeg(x−x/prime)ρ(x)ρ(x/prime). (1) Hereρ(x)=ρR(x)+ρL(x). We characterize the interaction g(x) by its strength at zero momentum g0and its radius lint/greaterorsimilar 1/(mvF) so that in momentum space gq=g0/parenleftbig 1−q2l2 int/parenrightbig ,q l int/lessmuch1. (2) We are interested in the properties of our model at low momenta q/lessmuchlint/lessorsimilarpF≡mvFand energies much smaller than the Fermi energy EF∼vFpF. In the subsequent consideration we will neglect the pro- cesses changing the total number of fermions Nη(counted from its value in the ground state) within each chiral branch.They are absent in our model Hamiltonian ( 1) but are ofcourse present in any real 1D system. These processes play a crucial role in the ultimate equilibration between branches inthe Luttinger liquid [ 5,17,18] but show up only at exponentially large time scales ∝exp(E F/T) and are completely irrelevant for the physics discussed in this work. Accordingly, fromnow on we consider our system in the sector characterizedbyN R=NL=0 and set the zero Fourier components of the densities ρη(x) to zero. The standard Tomonaga-Luttinger (TL) is the extreme low-energy limit of the Hamiltonian ( 1) corresponding to linear fermionic spectrum ( m=∞ ) and pointlike interaction g(x)= g0δ(x). From the RG perspective, contributions that are ne- glected within this approximation are irrelevant perturbations.Specifically, when setting m=∞ , one drops an irrelevant perturbation of scaling dimension 3, while discarding themomentum dependence of the interaction is equivalent [fora finite-range g(x)] to the neglect of even weaker perturbation of scaling dimension 4. The bosonization approach [ 1–4] allows one to map the TL Hamiltonian onto free dispersionlessbosons, which in turn are equivalent via refermionization[7,8,13,14] to free fermions. Thus, the TL model can be equally well treated in bosonic and fermionic (after the identificationof the proper fermionic modes) languages. This fact is relatedto the conformal invariance of the TL Hamiltonian. Despite the great success of the TL model in the description of thermodynamic properties of 1D interacting fermions, it isnow known that the irrelevant perturbations it neglects can havestrong impact on the dynamical response of the system. Forexample, the fermionic curvature translates upon bosonizationinto a cubic interaction of the density fluctuations [ 19] −1 2m/integraldisplay dxψ+ η(x)∂2 xψη(x)=2π2 3m/integraldisplay dxρ3 η(x). (3) Although irrelevant in the RG sense, this perturbation acts for the case of a short-range interaction g(x) (or just for free fermions) on a highly degenerate linear bosonic spectrum,so that the corresponding perturbation theory suffers fromstrong divergences. As a consequence, the formally irrelevantperturbation alters dramatically the behavior of, e.g., single-particle spectral weight A(k,/epsilon1) in the immediate vicinity of the single-particle mass shell [ 7,8]. While extremely nontrivial in the bosonic representation, the problem of Luttinger liquid with finite fermionic masscan be elegantly addressed via the introduction of the properfermionic quasiparticles (refermionization) [ 7,8,14]. Thus the fermionic curvature breaks the symmetry between fermionicand bosonic languages present in the TL model in favor offermions. Conversely, for fermions with linear spectrum ( m=∞ ) the Hamiltonian ( 1) describes (after bosonization) free bosons with the dispersion relation ω q=uq|q|,u q=vF/parenleftbigg 1+gq πvF/parenrightbigg1/2 ≡vF Kq. (4) At small momenta the boson velocity uqis given by uq=u0(1−l2q2), (5) l2=1 2/parenleftbig 1−K2 0/parenrightbig l2 int, (6) 125113-2RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) where K0is the zero-momentum limit of the Luttinger-liquid parameter Kqintroduced in Eq. ( 4). Finite interaction radius lintleads thus to the appearance of dispersion in the bosonic spectrum. For long-range interactions the Wigner-crystal-typecorrelations proliferate and bosonic excitations are stable against perturbations caused by the curvature of fermionicspectrum [ 12] in a wide range of energies. Note that upon refermionization, the curvature of bosonic spectrum translatesinto an interaction between fermionic quasiparticles [ 7,16]. The consideration just presented raises the fundamental question, What are the proper degrees of freedom for thedescription of a generic dispersive Luttinger liquid havingboth curved fermionic and bosonic spectra? We observea remarkable duality between the fermionic and bosonicdescription of the problem: curved single-particle spectrumfor the excitations of one type (fermions or bosons) introducesthe interaction between the excitations of the other type. Theimportance of this interaction for the dynamics of the particlesof the second type is determined in turn by the curvature oftheir own spectrum. Accordingly, we expect that in a genericdispersive Luttinger liquid there is a competition betweencurvatures (or, equivalently, between interactions) of fermionsand bosons. The particles with the most curved spectrumare the longest living and most well defined excitations. Fora finite-range interaction g(x), the nonlinear corrections to the bosonic and fermionic excitation spectra scale differentlywith momentum. Specifically, at sufficiently large momentathe bosonic correction δω q∝vFl2 intq3dominates over the fermionic correction δξk∝k2/m, while at small momenta the situation is reversed. Thus, one can expect that if thecharacteristic energy scale of the problem (say, temperatureT) exceeds T 0∝1/ml2 int, the bosonic language gives the proper description of relaxation in the system, while at smallerenergies the fermionic language becomes appropriate [ 20]. In the rest of the paper we explore the lifetime of bosonic and fermionic excitations in the dispersive Luttingerliquid. To be definite, we consider the Luttinger liquid atfinite temperature Tand study the decay rate 1 /τ /epsilon1(T)o fa right-moving excitation (boson or fermion) injected into thesystem at energy /epsilon1/greaterorsimilarT. In agreement with the qualitative consideration presented above, we find that at sufficientlylargeT,/epsilon1the perturbatively obtained lifetime of bosonic excitations is much longer than that of fermionic ones. In thisregime, the bosonic perturbation theory is justified and yieldsthe correct relaxation rate. The situation is reversed at lowT,/epsilon1: in this case the fermionic calculation of the relaxation rate becomes controllable, and the fermionic quasiparticlesare proper excitations. The correspondence between the twoapproaches can be viewed as an example of a strong-weak coupling duality in physics. B. Unitary transformations In this subsection we seek for the representation of the dispersive Luttinger liquid in terms of weakly interactingquasiparticles. The original fermions interact strongly. The RGclassification of various terms in the Hamiltonian ( 1) suggests that in order to reduce the interaction we need first to get rid ofthe density-density interaction between right- and left-movingfermions. The natural way to achieve this goal is bosonization.Thus we bosonize the Hamiltonian ( 1) and arrive at H=/summationdisplay η/integraldisplay dx:/parenleftbigg πvFρ2 η(x)+2π2 3mρ3 η(x)/parenrightbigg :B +1 2/integraldisplay dxdx/primeg(x−x/prime):ρ(x)ρ(x/prime):B. (7) Here :: Bstands for normal ordering with respect to the bosonic modes [Fourier components of ρη(x)]. The density-density coupling between left- and right-chiral sectors can be eliminated by a unitary transformation ofbosonic operators [ 14] ρ η(x)=U+ 2˜ρη(x)U2, (8) U2=exp⎡ ⎣2π L/summationdisplay q/negationslash=0κq q˜ρR,q˜ρL,−q⎤ ⎦. (9) Here the function κqis to be chosen from the requirement that interbranch density-density interaction is absent in thetransformed Hamiltonian and we have assumed that thefermions resides on a circle of circumference L. Taking into account the commutation relations of the density components[ρ η,q,ρη,−q]=ηLq/ 2π, we see that U2generates the Bogoli- ubov transformation: ρR,q=coshκq˜ρR,q−sinhκq˜ρL,q, (10) ρL,q=− sinhκq˜ρR,q+coshκq˜ρL,q. (11) The decoupling of the chiral sectors of the theory to quadratic order in densities fixes now the rotation angle tanh 2κq=gq/(2πvF+gq). (12) At small momenta we obtain κq=κ0−1 2l2q2. (13) The unitary transformation U2fully solves the model of Luttinger liquid with linear fermionic dispersion. In the genericsituation that we are considering this is no longer the case. Interms of the new density operators ˜ ρ ηthe Hamiltonian reads H=(π/L )/summationdisplay quq:(˜ρR,q˜ρR,−q+˜ρL,q˜ρL,−q):B +1 L2/summationdisplay q/bracketleftbig /Gamma1B,RRR q :(˜ρR,q 1˜ρR,q 2˜ρR,q 3+R→L):B +/Gamma1B,RRL q :(˜ρR,q 1˜ρR,q 2˜ρL,q 3+R↔L):B/bracketrightbig . (14) Here qstands for {q1,q2,q3}, the bosonic velocity uqwas defined in Eq. ( 4), and the three-boson interaction vertices are given by /Gamma1B,RRR q =2π2 3m[chκ1chκ2chκ3−shκ1shκ2shκ3],(15) /Gamma1B,RRL q=2π2 m[shκ1shκ2chκ3−chκ1chκ2shκ3],(16) withκi≡κqi.I nE q s .( 15) and ( 16) we have suppressed the Kronecker symbol δq1+q2+q3,0expressing the momentum conservation. 125113-3I. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) The vertices /Gamma1B,RRR q and/Gamma1B,RRL q can be expanded at small momenta qi: /Gamma1B,RRR q =2π2 3m∗/parenleftbigg 1−αl2 2/parenleftbig q2 1+q2 2+q2 3/parenrightbig/parenrightbigg , (17) /Gamma1B,RRL q =−2π2α m∗/parenleftbigg 1+1 2l2/parenleftbig q2 1+q2 2/parenrightbig −l2q2 3 2α/parenrightbigg .(18) Here we have introduced the renormalized mass 1 m∗=3+K2 0 4√K01 m(19) and a dimensionless parameter αcharacterizing the interaction strength, α=1−K2 0 3+K2 0. (20) From the RG prospective the Hamiltonian ( 14)i sa Hamiltonian of free bosons with linear spectrum ω=u0q perturbed by (i) terms of scaling dimension 3 due to cubic interaction of bosons /Gamma1B,RRR q=0and/Gamma1B,RRL q=0, (ii) a perturbation of dimension 4 originating from the curvature of bosonicspectrum, and (iii) various terms of higher scaling dimensions.At low energies it is natural to begin by taking care of mostrelevant perturbations, namely, the cubic bosonic couplingswith vertices approximated by their value at zero momentum. We first include the term /Gamma1 B,RRR q=0that couples the bosons on the same highly degenerate branch. The resulting Hamiltonian H=(π/L )/summationdisplay qu0(: ˜ρR,q˜ρR,−q:B+:˜ρL,q˜ρL,−q:B) +1 L2/Gamma1B,RRR q=0/summationdisplay q(: ˜ρR,q 1˜ρR,q 2˜ρR,q 3:B+R→L) (21) is just a bosonized version of a Hamiltonian of noninteracting fermions with the Fermi velocity u0and the spectral curvature 1/m∗, H=/summationdisplay η/integraldisplay dx˜ψ+ η(x)/parenleftbigg −iηu 0∂x−1 2m∗∂2 x/parenrightbigg ˜ψη(x).(22) The Hamiltonian ( 22) is an effective Hamiltonian of the system at lowest energies. It is worth mentioning that theexpressions ( 4) and ( 19) for the coefficients u 0andm∗ in the effective Hamiltonian ( 22) are not exact because of corrections from the neglected perturbations and originatingat the ultraviolet scale. These corrections are small in the limitof long-range interaction, l intpF/greatermuch1, but generate nontrivial renormalization factors of order unity for lintpF∼1; see also Eq. ( 29) and a discussion following it. The exact values u0and m∗can be related to the thermodynamic characteristics of the system [ 8]. We now reintroduce in the bosonized Hamiltonian Eq. ( 22) the terms describing the curvature of the bosonic spectrum aswell as the interbranch cubic couplings /Gamma1 B,RRL q . Remarkably, it turns out to be possible [ 16] to get rid of the /Gamma1B,RRL q terms. Indeed, it is easy to see that vertex /Gamma1B,RRL q does not describe a real scattering of bosons due to impossibility to fulfill themomentum end energy conservation. It is thus possible todesign a unitary transformation U 3eliminating the /Gamma1B,RRL q coupling [ 21]: ˜ρR(x)=U+ 3R(x)U3,˜ρL(x)=U+ 3L(x)U3. (23) The analogy with the Bogoliubov transformation U2sug- gests the following ansatz for U3: U3≡exp[/Omega13] =exp/braceleftBigg 1 L2/summationdisplay q[fqRq1Rq2Lq3−(L↔R)]/bracerightBigg .(24) Performing a perturbative expansion of U3, we obtain ˜ρR,q=Rq−[/Omega13,Rq]+1 2[/Omega13,[/Omega13,Rq]]+O/parenleftbig ρ4/p3 F/parenrightbig , ˜ρL,q=Lq−[/Omega13,Lq]+1 2[/Omega13,[/Omega13,Lq]]+O/parenleftbig ρ4/p3 F/parenrightbig . (25) In Eq. ( 25) we kept terms up to the third order in densities which are required to compute the Hamiltonian in newvariables up to the fourth order. Neglecting the third-order terms in Eqs. ( 25), substituting the resulting expansions into Hamiltonian ( 14), and demanding that the left and right sectors are decoupled at cubic order, onefinds [ 16] f q=/Gamma1B,RRL q uq1q1+uq2q2−uq3q3. (26) Note that the impossibility of conserving momentum and energy in a scattering event involving two right bosons andone left boson guarantees that the energy denominator in ( 26) is nonzero. The behavior of f qat small momenta can be easily inferred from ( 26) and ( 18): fq=π2α m∗u0q3/bracketleftbigg 1+5 4l2/parenleftbig q2 1+q2 2/parenrightbig +l2 4/parenleftbigg 1−2 α/parenrightbigg q2 3/bracketrightbigg .(27) Retaining now the third-order terms in ( 25), one can recast the Hamiltonian ( 14) into the form H=(π/L )/summationdisplay quq(:RqR−q:B+:LqL−q:B) +1 L2/summationdisplay q/Gamma1B,RRR q/parenleftbig :Rq1Rq2Rq3:B+R→L/parenrightbig +1 L3/summationdisplay q/Gamma1B,RRRR q/bracketleftbig Rq1Rq2Rq3Rq4+R→L/bracketrightbig +1 L3/summationdisplay q/Gamma1B,RRRL q/bracketleftbig Rq1Rq2Rq3Lq4+R↔L/bracketrightbig +1 L3/summationdisplay q/Gamma1B,RRLL q Rq1Rq2Lq3Lq4+O(ρ5). (28) In the last three sums qstands for {q1,q2,q3,q4}. The couplings /Gamma1B,RRRR q ,/Gamma1B,RRRL q , and/Gamma1B,RRLL q are symmetric functions of momenta corresponding to the density components of the samechirality. The full expressions for them are cumbersome andwe do not present them here (see Appendix Afor details). We will discuss their relevant properties when appropriate. 125113-4RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) We concentrate now on the general structure of Eq. ( 28) and notice several points that allow us to simplify the Hamiltonian.First, we neglect the O(ρ 5) corrections in the Hamiltonian (28) that have formal smallness of ρ3/p3 Fas compared to the quadratic part. Indeed, our calculation of the relaxation ratesbelow shows that dominant contributions originate from O(ρ 4) terms, so that there is no need to keep terms of higher orders.When taken into account, such terms would give corrections tothe scattering rates which are small compared to the dominantcontributions [see Eqs. ( 42) and ( 61) below] by positive powers of the small parameter T/E F. Second, we note the absence of normal ordering in the last three terms of the Hamiltonian ( 28). Performing the bosonic normal ordering, one generates various quadratic couplingsof densities. For example, the normal ordering of /Gamma1 B,RRRL q coupling generates the contribution δH(2) RL∝/summationdisplay p,q/Theta1(p)p/parenleftbig /Gamma1RRRL p,−p,q,−q+/Gamma1RRRL p,−p,−q,q/parenrightbig RqL−q.(29) The interaction of right and left movers described by ( 29)i s finite at zero momentum. Its precise value is determined by thebehavior of /Gamma1 RRRL q at large momenta. A quick estimate shows that these corrections are small (in the parameter 1 /pFlint) compared to the density-density interaction in the initialHamiltonian ( 1) as long the interaction radius l intis large. One can get rid of the generated quadratic couplings by a suitablemodification of the unitary transformation U 2. Obviously, ( 29) and similar terms arising from normal ordering are responsiblefor the renormalization of the Luttinger parameter K 0and other parameters of the effective theory coming from the residualinteractions at large energies. This renormalization is smallforl intpF/greatermuch1 and becomes of order unity at lint∼λF.W e assume from now on that the transformation U2was suitably adjusted and omit the terms arising from the bosonic normalordering. The third simplification is as follows. The vertex /Gamma1 B,RRRR q is nonsingular at zero momentum: /Gamma1B,RRRR q =−π3α2 2u0m∗2L3/bracketleftBigg 1−8−23α 24αl24/summationdisplay i=1q2 i/bracketrightBigg .(30) Translated to the fermionic representation it gives rise to (i) a correction to the fermionic spectrum δξk∝k3/mp F, (ii) a small correction to the density-density interaction of fermionsof the same chirality ∝q 2RqR−q/p2 F, and (iii) various terms of higher scaling dimension. Since we are interested inphenomena at energies much less than the Fermi energy, wecan neglect these corrections altogether. The last remark to be made on Eq. ( 28) is that the momentum and energy conservation does not allow scattering processes involving two right and two left bosons. Accord-ingly, the /Gamma1 B,RRLL q coupling does not lead to real bosonic transitions in the first order of perturbation theory and canbe removed by a unitary transformation U 4analogous to U3. Apart from a modification of the O(ρ5) terms (which we neglect anyway), the elimination of /Gamma1B,RRLL q coupling from the Hamiltonian ( 28) is the only effect of transformation U4. We are now ready to summarize our findings on the structure of the Hamiltonian. Once the unitary transformations areperformed and terms that give subdominant contributions tothe relaxation are neglected, the Hamiltonian of a dispersive Luttinger liquid can be presented as H=(π/L )/summationdisplay quq:(RqR−q+LqL−q):B +1 L2/summationdisplay q/Gamma1B,RRR q/parenleftbig :Rq1Rq2Rq3:B+R→L/parenrightbig +1 L3/summationdisplay q/Gamma1B,RRRL q/parenleftbig :Rq1Rq2Rq3Lq4:B+R↔L/parenrightbig .(31) The bosonic vertex /Gamma1B,RRRL q has a complicated singular behavior at small momenta. Specifically, we obtain (seeAppendix A) /Gamma1 B,RRRL q ≈˜/Gamma1B,RRRL q +l2˜˜/Gamma1B,RRRL q ,q2l2/lessmuch1, (32) where ˜/Gamma1B,RRRL q =4π3α 3m∗2u0/bracketleftbigg 1−3α 2−α 4/parenleftbiggq4 q1+q4 q2+q4 q3/parenrightbigg/bracketrightbigg , (33) and ˜˜/Gamma1B,RRRL q =5π3α m∗2u0/bracketleftbiggq1q2q3 q4+(6−13α) 20/parenleftbig q2 1+q2 2+q2 3/parenrightbig −α 30q4/parenleftbiggq2 1+q2 2 q3+q2 1+q2 3 q2+q2 2+q2 3 q1/parenrightbigg +26α−53α2−8 60αq2 4 +1−5α 30q3 4/parenleftbigg1 q1+1 q2+1 q3/parenrightbigg/bracketrightbigg . (34) In the next sections we will use the Hamiltonian ( 31)t o study the the lifetimes of bosons and fermions in 1D interactingsystem. The leading processes contributing to the relaxationof bosonic and fermionic distribution functions are illustratedin Fig. 1. For bosons, this is the two-into-two scattering involving a change of the branch for one of the bosons. Forfermions, these are three-fermion collisions involving in theinitial state two particles from one (say, right) branch andone particle from the other (say, left) branch (see Fig. 1). We notice that both these processes arise already in the first orderof perturbation theory in /Gamma1 B,RRRL q . [This is obvious for the case of the bosonic scattering; for fermions this will becomeclear in Sec. IVwhere we discuss the fermionic form of the Hamiltonian ( 31)]. The momentum q 4in/Gamma1B,RRRL q has the meaning of the momentum transfer between the left- andright-chiral sectors in the collision process. The momentumand energy conservation dictates then the estimates q 4∼l2p3 for bosonic and q4∼p2/mfor fermionic collisions, where p is the typical momentum of the right particles involved. Thisobservation allows one to simplify dramatically the expression for the coupling˜˜/Gamma1 B,RRRL q by dropping all the terms which do not contain q4in the denominator. In this approximation we get /Gamma1B,RRRL q =4π3α 3m∗2u0/bracketleftbigg 1−3α 2+15 4l2q1q2q3 q4/bracketrightbigg . (35) 125113-5I. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) FIG. 1. (Color online) Leading relaxation processes for bosonic (upper panel) and fermionic (lower panel) excitations. The Hamiltonian ( 31) with the coupling /Gamma1B,RRRL q given by (35) and its fermionic version derived in Sec. IVconstitute our starting point for the analysis of the lifetime of bosonicand fermionic excitations in the generic Luttinger-liquidmodel. III. LIFETIME OF BOSONIC EXCITATIONS In this section we exploit the Hamiltonian ( 31) to study the decay of bosonic excitations in a dispersive Luttingerliquid. The Fourier components of the densities R(x) andL(x) can be identified with bosonic creation and annihilation operatorsvia R q=/radicalbigg L|q| 2π[/Theta1(q)bq+/Theta1(−q)b+ −q], (36) Lq=/radicalbigg L|q| 2π[/Theta1(−q)bq+/Theta1(q)b+ −q]. Let us consider a boson at momentum Q/greaterorsimilarT/u 0injected into the otherwise equilibrium Luttinger liquid characterizedby temperature T. We assume for definiteness that Q> 0, so that we are dealing with a decay of a right-moving bosonicexcitation. The dominant collision process limiting the lifetimeof the injected boson is a scattering on a thermal left-movingboson at momentum q< 0 which is transferred to the right branch (see Fig. 1). The lifetime of the injected boson is now given by the out-scattering term of the linearized collisionintegral: 1 τQ(T)=1 2/integraldisplaydqdQ/primedq/prime (2π)3WQ/primeq/prime QqNB(ωq)[NB(ωQ/prime)+1] ×[NB(ωq/prime)+1]/Theta1(−q)/Theta1(Q/prime)/Theta1(q/prime). (37) HereNB(/epsilon1) is the equilibrium bosonic distribution function. The transition probability WQ/primeq/prime Qq can be expressed via the corresponding matrix element of the Tmatrix: WQ/primeq/prime Qq=(2π)2|/angbracketleft0|bq/primebQ/primeTb+ Qb+ q|0/angbracketright|2δ(Ei−Ef) ×δ(Pi−Pf). (38) Here the δfunctions express the energy and momentum conservation in the collision process. It is easy to see that the required matrix element /angbracketleft0|bq/primebQ/primeTb+ Qb+ q|0/angbracketrightarises in the first order of the perturbation theory in the coupling /Gamma1B,RRRL q and can be read off from Eqs. ( 35) and ( 36). The result becomes substantially simplified since, according to the conservation laws, the momentum qof the left particle is given by q=−3 2QQ/primeq/primel2+O(q5l4). (39) In view of this relation, the momentum-dependent and momentum-independent terms in /Gamma1B,RRRL q ,E q .( 35), give contributions of the same form to the matrix element/angbracketleft0|b q/primebQ/primeTb+ Qb+ q|0/angbracketrightwhich finally reads /angbracketleft0|bq/primebQ/primeTb+ Qb+ q|0/angbracketright=πα(1+α) m∗2u0/radicalbig |QQ/primeqq/prime|. (40) Assuming now that the energy of the relaxing boson is much larger than temperature but is not too high ( l2u0Q3/lessmuchT), one can replace the thermal factor NB(ωq)i nE q .( 37)b y1/u0|q| and the other two thermal factors by unity. Calculating theresulting integral, we find 1 τQ(T)=πα2(1+α)2 48m∗4u4 0TQ4. (41) Equation ( 41) gives the lifetime of a hot boson in our system and leads to the following estimate for a typical relaxation timeof thermal bosons (i.e., those with momenta Q∼T/u 0): 1 τQ∼T/u 0(T)∝α2(1+α)2 m∗4u8 0T5, (42) up to a prefactor of order unity. Equation ( 42) for the relaxation time of the bosonic excitations in the dispersive Luttinger liquid constitutes themain result of this section. The T 5scaling of the relaxation rate of bosonic excitation has been earlier obtained [ 11,12]f o r a strongly interacting Luttinger liquid which is characterizedby Luttinger parameter K 0/lessmuch1 and is close to the Wigner crystal. Our derivation is valid for any K0and thus represents a generalization of the result of Ref. [ 11,12]. We will return to the question of the range of validity of Eq. ( 42)i n Sec. V. 125113-6RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) IV . LIFETIME OF FERMIONIC EXCITATIONS A. Refermionization The Hilbert space of a 1D chiral bosonic system is isomorphic to the Hilbert space of a 1D complex chiral fermion(more precisely, to its charge-zero sector). Correspondingly,the Hamiltonian ( 31) can be viewed as a Hamiltonian of fermions c η,k,c+ η,kintroduced via Rq=/summationdisplay kc+ R,kcR,k+q,L q=/summationdisplay kc+ L,kcL,k+q. (43) As was discussed in Sec. II, the fermions are expected to be the proper excitations at momenta satisfying m∗u0l2k/lessmuch1. We are now going to discuss the lifetime of these low-energyfermionic quasiparticles created by the operators c η,k. We need to rephrase the Hamiltonian ( 31) into the fermionic language. This can be done by substituting Eq. ( 43)i n t o Eq. ( 31) and performing the normal ordering of the resulting expression with respect to fermionic modes. It is obvious fromthe structure of the bosonic Hamiltonian ( 31) that in terms of fermions H=/summationdisplay kξR,k:c+ R,kcR,k:F +1 L/summationdisplay k/Gamma1F,RR k :c+ R,k 1c+ R,k 2cR,k/prime 2cR,k/prime 1:F +1 L/summationdisplay k/Gamma1F,RL k :c+ R,k 1c+ L,k 2cL,k/prime 2cR,k/prime 1:F +1 L2/summationdisplay k/Gamma1F,RRR k :c+ R,k 1c+ R,k 2c+ R,k 3cR,k/prime 3cR,k/prime 2cR,k/prime 1:F +1 L2/summationdisplay k/Gamma1F,RRL k :c+ R,k 1c+ R,k 2c+ L,k 3cL,k/prime 3cR,k/prime 2cR,k/prime 1:F +R←→L+··· . (44) Here the dots stand for the four-fermion interaction terms (containing eight fermionic operators). These terms do notcontribute to the three-fermion collision processes which weaim to discuss in this work and we omit them altogether [ 22]. We denote by kin each of vertices /Gamma1 F,... kthe set of momenta of the fermionic operators involved [e.g., k=(k1,k2,k3,k/prime 3,k/prime 2,k/prime 1) in the vertex /Gamma1F,RRL k ; note the order of individual momenta ink]. We refer the reader to Appendix Bfor details of the derivation of the couplings entering the fermionic Hamiltonian(44) and state here the final results only. First, the fermionic single-particle spectrum ξ η,kreceives renormalization from the density-density interaction in ( 31) and is given by ξη,k=k2 2m∗+η/integraldisplayk 0u(k)dk≈ηku 0+k2 2m∗−ηl2u0k3 3. (45) Note that the cubic correction to the fermionic spectrum is small compared to the quadratic one at k< 1/m∗l2u0where the fermions are expected to be proper quasiparticles of thesystem.The two-particle intrabranch interaction in the fermionic Hamiltonian ( 44) arises from the intrabranch density-density interaction in ( 31), /Gamma1 F,RR k=πu 0l2 2(k1−k2)(k/prime 1−k/prime 2). (46) Here we have take into account the expansion of the bosonic velocity uqat small momenta. The vertex /Gamma1RR kalso receives corrections from the cubic-in-density intrabranch bosonic cou-pling/Gamma1 B,RRR q . These corrections are however parametrically small, and we neglect them. The interbranch two- and three-particle couplings, which will be most important for our analysis of the relaxation,both arise from the bosonic vertex /Gamma1 B,RRRL q ,E q s .( 31), (35). Remarkably, a singularity at small momentum transfer k3−k/prime 3 which might be expected in view of the last term in Eq. ( 35) does not show up in /Gamma1RL k. This vertex is mostly determined by the first, momentum-independent term in ( 35) and is given by /Gamma1F,RL k=/Gamma1F,LR k=πα(2−3α) 2m∗2u0k1k/prime 1. (47) On the contrary, the three-fermion coupling /Gamma1F,RRL k emerges solely due to the last term in Eq. ( 35), /Gamma1F,RRL k=5αl2π2(k1−k2)(k/prime 1−k/prime 2) 16m∗2u0(k3−k/prime 3) ×[(k1−k2)2−(k/prime 1−k/prime 2)2], (48) and is singular at k3−k/prime 3=0. Let us finally comment on the three-particle intrabranch interaction vertex /Gamma1F,RRR k . It arises from the cubic intrabranch coupling in the bosonic Hamiltonian ( 31). By construction, /Gamma1F,RRR k is antisymmetric in the three incoming ( k1,k2,k3) and the three outgoing momenta k/prime 1,k/prime 2, andk/prime 3. Since the bosonic vertex /Gamma1B,RRR q is analytic at small momenta, /Gamma1F,RRR k should be of the form /Gamma1F,RRR k ∝l6 m∗/productdisplay i>j(ki−kj)(k/prime i−k/prime j). (49) We thus see that /Gamma1F,RRR k is strongly suppressed by a high power of the momenta. In fact, it is exactly zero within ourapproximation for the bosonic coupling /Gamma1 B,RRR q ,E q .( 17). One needs to retain the sixth-order terms in the expansion of /Gamma1B,RRL q over momentum to generate nonzero /Gamma1F,RRR k .F r o mn o wo nw e will largely ignore /Gamma1F,RRR k apart from a short discussion of the intrabranch fermionic relaxation processes at the end of theSec. IV B . With the fermionic Hamiltonian ( 44) and the couplings (46), (47), (48) at hand, we are now in a position to study the lifetime of the fermionic excitations in our problem. Belowwe employ the perturbation theory in the fermionic interac- tion terms /Gamma1 F,ηη/prime k and/Gamma1F,RRL k to evaluate the corresponding scattering rate. B. Fermionic scattering rate Here we calculate the lifetime of fermionic excitations in a dispersive Luttinger liquid. The relaxation of the fermionic 125113-7I. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) distribution function is governed by the three-particle colli- sions. At zero temperature the intrabranch collision processes(with all three particles in the initial and final states residing onthe right branch) are ruled out by the energy and momentumconservation. At finite temperature, the situation we considerin this work, both intrabranch collisions and the scatteringevents involving the creation of a particle-hole pair in theleft branch (see Fig. 1) contribute to the relaxation of a right-moving fermion injected into the system. However,the intrabranch collision rate 1 /τ RRR(T) turns out to be proportional to a very high power of temperature ( T14) and is small compared to the interbranch collision rate 1 /τRRL(T) in the whole “fermionic” part of the parameter space. We willdiscuss this point in more detail at the end of this section andconcentrate now on the contribution of interbranch collisionsto the decay of fermionic quasiparticles. The decay rate 1 /τ RRL k1(T) of a fermion with momentum k1/greaterorsimilarT/u 0is given by the out-scattering term of the linearized three-particle collision integral, 1/τRRL k1(T)=1 2/integraldisplay (dk)Wk/prime 1,k/prime 2,k/prime 3 k1,k2,k3NF(ξR,k 2)NF(ξL,k 3) ×[1−NF(ξR,k/prime 1)][1−NF(ξR,k/prime 2)] ×[1−NF(ξL,k/prime 3)]. (50) Here ( dk)=dk2dk3dk/prime 1dk/prime 2dk/prime 3/(2π)5andNF(/epsilon1) stands for the Fermi-Dirac distribution at temperature T. The transition probability Wk/prime 1,k/prime 2,k/prime 3 k1,k2,k3entering Eq. ( 50) is given by the modulus squared of the appropriate entry of the Tmatrix, Wk/prime 1,k/prime 2,k/prime 3 k1,k2,k3=(2π)2|/angbracketleft1,2,3|T|1/prime,2/prime,3/prime/angbracketright|2 ×δ(Ei−Ef)δ(Pi−Pf). (51) Here the δfunctions express the conservation of energy and momentum in the collision process and |1,2,3/angbracketright=c+ R,k 1c+ R,k 2c+ L,k 3|0/angbracketright. (52) Examination of the fermionic Hamiltonian ( 44) shows that to the leading order in 1 /mthere are two contributions to the matrix element /angbracketleft1,2,3|T|1/prime,2/prime,3/prime/angbracketright. The first one stems from the three-fermion coupling /Gamma1F,RRL k in the first order of perturbation theory and is given by /angbracketleft1/prime,2/prime,3/prime|T|1,2,3/angbracketright1=4/Gamma1F,RRL k/prime 1,k/prime 2,k/prime 3,k3,k1,k2, (53) where the vertex /Gamma1F,RRL k is given by ( 48). The second contribution arises in the second order of the perturbationtheory in the two-fermion couplings /Gamma1 RR kand/Gamma1RL k, /angbracketleft1/prime,2/prime,3/prime|T|1,2,3/angbracketright2 =8/Gamma1F,RR k/prime 1,k/prime 2,k2,q/parenleftBig /Gamma1F,RL q,k/prime 3,k3,k1+/Gamma1F,LR q,k/prime 3,k3,k1/parenrightBig ξL,k 3−ξL,k/prime 3+ξR,k 1−ξR,q +8/Gamma1F,RR k/prime 1,q,k 2,k1/parenleftBig /Gamma1F,RL k/prime 2,k/prime 3,k3,q+/Gamma1F,LR k/prime 2,k/prime 3,k3,q/parenrightBig ξL,k/prime 3−ξL,k 3+ξR,k/prime 2−ξR,q. (54) In Eq. ( 54) we implicitly assume the antisymmetrization of the right-hand side with respect to permutations of k1andk2as well as k/prime 1andk/prime 2. The momentum qof the intermediate virtual state in Eq. ( 54) is fixed by the momentum conservation inthe vertices /Gamma1F,RL k implying that q=k1+k3−k/prime 3for the first term and q=k/prime 2+k/prime 3−k3for the second one. Let us now consider the energy denominators in ( 54)i n more detail. Using the explicit form ( 45) of the single-particle dispersion relations, one finds for the first energy denominator ξL,k 3−ξL,k/prime 3+ξR,k 1−ξR,q =− 2u0(k3−k/prime 3)[1+O(k/pF)+O(k2l2)],(55) withkbeing the characteristic value of the momenta. The second denominator in ( 54) has exactly the same structure. Working to the leading order in k/pF,k2l2and using explicit expressions for the vertices /Gamma1F,RR k,/Gamma1F,RL k derived earlier, we get (after the proper antisymmetrization over momenta) /angbracketleft1/prime,2/prime,3/prime|T|1,2,3/angbracketright2 =π2α(2−3α)l2(k1−k2)(k/prime 1−k/prime 2) 2m∗2u0(k3−k/prime 3) ×/parenleftbig k2 1+k2 2+4k1k2−k/prime2 1−k/prime2 2−4k/prime 1k/prime 2/parenrightbig .(56) Let us assume from now on that the the temperature of the system is low, m∗l2T/lessmuch1; i.e., we are in the situation when the fermions are expected to be the proper quasiparticlesfor the description of the system. Under this condition we canneglect the cubic term in the fermionic dispersion relation ( 45). The matrix elements ( 53) and ( 56) can be further simplified if one takes into account the energy conservation in the collisionprocess. First, we note that the energy and momentum conser-vation requires that at zero momentum transfer k 3−k/prime 3=0 the right-moving particles can only preserve or exchange theirmomenta. Thus, the singularity in the matrix elements ( 53) and (56) is canceled if we consider them on the mass shell. Second, on the mass shell the momentum transfer can be estimated ask 3−k/prime 3∼k2/m∗u0, where kis the characteristic momentum of the colliding particles. Consequently, we can estimate thematrix elements ( 53) and ( 56) on the mass shell as /angbracketleft1 /prime,2/prime,3/prime|T|1,2,3/angbracketright1(2)∝l2(k1−k2)(k/prime 1−k/prime 2) m∗. (57) The accurate calculation presented in Appendix Cconfirms this estimate and yields [ 23] /angbracketleft1/prime,2/prime,3/prime|T|1,2,3/angbracketright=6π2α(1+α)l2(k1−k2)(k/prime 1−k/prime 2) m∗. (58) Assuming now that the energy of the relaxing particle u0k1is not too high ( u0k1/lessmuch/radicalBig Tu2 0m∗) one can linearize the fermionic spectra in the energy conserving δfunction in Eq. ( 51). The integration over k3andk/prime 3is Eq. ( 50)i st h e n straightforward and leads to 1/slashbig τRRL k1(T)=T 8πu2 0/integraldisplaydk2dk/prime 1dk/prime 2 (2π)3Tk/prime 1,k/prime 2 k1,k2NF(ξR,k/prime 2) ×/bracketleftbig 1−NF(ξR,k/prime 1)/bracketrightbig/bracketleftbig 1−NF/parenleftbig ξR,k/prime 2/parenrightbig/bracketrightbig ×2πδ(k1+k2−k/prime 1−k/prime 2). (59) Atk1/greatermuchT/u 0we can approximate the Fermi distributions entering ( 59) by the zero-temperature ones, which leads to the 125113-8RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) following result for the relaxation rate of a hot fermion in our system: 1/τRRL k1(T)=11πα2(1+α)2l4Tk6 1 80m∗2u2 0. (60) The corresponding estimate for the lifetime of the thermal quasiparticles with k1∼T/u 0constitutes the central result of this section: 1/τRRL k1∼T/u 0(T)∝α2(1+α)2l4T7 m∗2u8 0. (61) Equation ( 61) establishes the relaxation rate of the fermionic quasiparticles in a general dispersive Luttingerliquid. Its scaling with temperature coincides with the oneobtained previously for weakly interacting fermions withina perturbation theory [ 8] and for fermionic quasiparticles in the (perturbed) Lieb-Liniger model [ 24]. The T 7scaling in Eq. ( 61) can be traced back to a product of a T4factor arising due to the quadratic scaling of the matrix element ( 58) with the momentum and a T3factor stemming from the phase volume. Modifying the analysis that led us to Eqs. ( 60) and ( 61)f o r the case of extremely hot electrons with u0k/greatermuch/radicalBig Tu2 0m∗,w e get 1/τRRL k1/parenleftbig T/lessmuchk2 1/m∗/parenrightbig ∝α2(1+α)2l4k8 1 m∗3u2 0. (62) This result matches Eq. ( 60)a tT∼k2 1/m∗.T h ek8 1scaling of Eq. ( 62) agrees with the results of perturbative treatment of weakly interacting fermions [ 6,8] and with the the analysis of the lifetime of fermionic quasiparticles in a Luttinger liquidwith a short-range interaction [ 15]. We return now to the scattering rate 1 /τ RRR k1(T) induced by intrabranch three-particle collisions. The correspondingamplitude /angbracketleft123|T|1 /prime2/prime3/prime/angbracketright(with all the particles belonging now to the right branch) arises in the first order of the perturbation theory over the vertex /Gamma1F,RRR k as well as in the second order in the intrabranch two-particle interaction /Gamma1F,RR k. The matrix element induced by /Gamma1F,RRR k i sg i v e nb y[ s e eE q .( 49)] /angbracketleft1/prime2/prime3/prime|T|123/angbracketright1∝l6 m∗/productdisplay i>j(ki−kj)(k/prime i−k/prime j).(63) A careful examination of the second order of the perturbation theory in /Gamma1F,RR k shows that, despite the presence of energy de- nominators, the corresponding matrix element /angbracketleft123|T|1/prime2/prime3/prime/angbracketright2 is nonsingular at the mass shell and has the same momentum dependence (dictated by indistinguishability of the particles)as Eq. ( 63), /angbracketleft1 /prime2/prime3/prime|T|123/angbracketright2∝m∗u2 0l8/productdisplay i>j(ki−kj)(k/prime i−k/prime j). (64) To obtain Eq. ( 64) one has to go beyond the approximation (4) for the momentum-dependent bosonic velocity and the corresponding approximation ( 46) for the intrabranch two- particle interaction /Gamma1RR k. Specifically, one has to retain the O(k4) terms for both vertices /Gamma1RR kinvolved. This is the reason for the appearance of the factor l8in (64). The factor of mass m∗in Eq. ( 64) comes from the energy denominator of thesecond-order perturbation theory and reflects its degenerate nature for dispersionless fermions. Comparing Eqs. ( 63) and ( 64), we observe that the second contribution dominates due to an additional factor ( m∗u0l)2/greaterorsimilar 1, so that the matrix element for the intrabranch triple collisionsis given by Eq. ( 64). The evaluation of the intrabranch transition rate is now a matter of power counting resulting in 1/τ RRR k1∼T/u 0(T)∝m∗3l16T14 u10 0. (65) The second power of mass in Eq. ( 65) comes from the matrix element ( 64) and an additional factor m∗arises from the δfunction expressing the energy conservation due to the fact that energy and momentum conservation coincide forparticles with the linear spectrum. Comparing ( 65)t o( 61), we see that the interbranch collision processes dominate in theentire range of temperatures m ∗l2T/u 0/lessmuch1 where the above fermionic analysis is justified [ 25]. V . FERMI-BOSE WEAK-STRONG COUPLING DUALITY In the previous sections we have presented a detailed anal- ysis of relaxation times of bosonic and fermionic excitationsin a dispersive Luttinger liquid. For this purpose, we havecarried out a perturbative treatment of the Hamiltonian ( 31) and of its fermionized version ( 44), respectively. Comparing now the two calculations above, we observe that the fermionicand bosonic relaxations are closely related: they both originatefrom the same interaction term /Gamma1 RRRL q in the Hamiltonian ( 31) and are both dominated by the processes with small momentumtransfer between the right- and left-chiral branches. An impor-tant difference between the bosonic and fermionic scatteringprocesses is the scaling of this momentum transfer with thetypical momentum of the right particles, which is cubic forbosons and quadratic for fermions. Comparison of the bosonic and fermionic relaxation times, Eqs. ( 42) and ( 61), reveals the dimensionless parameter λ= m ∗l2Tanticipated in Sec. II. In agreement with the qualitative discussion in Sec. II, at low temperatures, λ/lessmuch1, the result (61) for the decay rate of fermionic excitations in a dispersive Luttinger liquid is much smaller than Eq. ( 42) resulting from a bosonic perturbative treatment of the Hamiltonian ( 31). Thus, fermions are proper excitations in this regime. The situation isreverse at high temperatures where λ/greatermuch1[26]. To support this subdivision of the parameter space in “fermionic” and “bosonic” domains, let us analyze the per-turbation theories used in the previous sections to evaluatethe bosonic and fermionic lifetimes. We consider first thefermionic formalism. As follows from Eq. ( 45), a finite interaction radius l intinduces a cubic correction to the spectrum of the fermionic quasiparticles. This correctionis small in comparison to the original curvature 1 /m ∗at momenta k/lessmuch1/m∗u0l2, yielding the condition λ/lessmuch1f o rt h e fermionic perturbation theory. Furthermore, an estimate forthe scaling of higher-order diagrams (with 4, 5, . . . fermionsinvolved) confirms that the perturbation theory is controlledby the parameter λ/lessmuch1. Conversely, a finite fermionic mass broadens the support of the dynamical structure factor inthe frequency-momentum plane ( ω,q) by an amount of the order of δω∼q 2/m∗. This broadening exceeds the nonlinear 125113-9I. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) FIG. 2. (Color online) “Phase diagram” of a dispersive Luttinger liquid in the parameter plane ( x=Tl/ u 0,y=T/m∗u2 0). Relaxation rates in the bosonic and fermionic parts of the phase diagram are givenby Eqs. ( 42)a n d( 61), respectively. The solid line y=x 2indicates a crossover between the fermionic and bosonic regimes. The dashed line corresponds to the minimal physically sensible interaction lengthl∼1/m ∗u0. bending of the bosonic single-particle spectrum u0l2q3at momenta q/lessorsimilar1/m∗u0l2and makes the perturbative treatment of the bosonic Hamiltonian inadequate for λ/lessorsimilar1. In other words, a small parameter controlling the bosonic perturbationtheory is λ −1/lessmuch1. To summarize, the fermionic and bosonic descriptions of a dispersive Luttinger liquid are characterizedby the coupling constants λandλ −1, respectively, thus showing a remarkable weak-strong coupling duality. The “phase diagram” of a dispersive Luttinger liquid exhibiting a crossover between the bosonic and fermionicregimes is shown in Fig. 2in the coordinates ( Tl/ u 0,T/m∗u2 0). VI. SUMMARY AND OUTLOOK To summarize, we have explored the lifetime of excitations in a dispersive Luttinger liquid in the whole range of param-eters. We employed a bosonization approach supplementedby a sequence of unitary transformations to a quasiparti-cle representation which allowed us to eliminate many ofinteraction-induced contributions from the Hamiltonian. Theresulting bosonic Hamiltonian is given by Eq. ( 31) and its refermionized version by Eq. ( 44). We have performed both bosonic and fermionic analysis of the relaxation rates in this formalism. The central results of this work, Eqs. ( 42) and ( 61), reveal the Bose-Fermi weak-strong coupling duality controlled by the parameter λ=m ∗l2Tand allow us to establish the “Bose-Fermi phase diagram” of ageneric dispersive Luttinger liquid presented in Fig. 2. Since the collision processes leading to the relaxation rates (42) and ( 61) involve intrabranch energy transfer of the order of temperature, they determine the characteristic time scalefor intrabranch equilibration at relatively high (bosonic time)and low (fermionic time) temperatures. Moreover, straightfor-ward extension of our analysis would yield the interbranchequilibration times in the bosonic and fermionic domains,which are dominated by the same collision processes but differ from Eqs. ( 42) and ( 61) due to a small interbranch energy transfer. On the experimental side, there is a growing interestin nonequilibrium phenomena and relaxation in interacting1D conductors [ 28–31]. We hope that further progress in this direction will result in the measurement of the temperaturedependence of equilibration times in such structures, thusproviding an experimental test of our theory. Remarkably, the parameter λcontrolling the Bose-Fermi crossover in the relaxation mechanisms studied in this workis closely related to the parameter λ ρ=u0m∗l2/Delta1ρ which was shown recently[ 16,27] to govern the character of the collisionless evolution of a density perturbation with anamplitude /Delta1ρin a dispersive Luttinger liquid. Specifically, it was found in Ref. [ 16] that for λ ρ/lessmuch1 (“fermionic” regime) the corresponding collisionless kinetic equation predicts aformation of the population inversion in the distributionfunction of fermions while for λ ρ/greatermuch1 (“bosonic” regime) no such phenomenon occurs and the density evolution followsclosely the predictions of a hydrodynamic theory. Comparingthe expressions for λandλ ρ, we observe that they are identical, up to a replacement of the characteristic energy scale T byu0/Delta1ρ. In this context, our present findings open up the possibility to incorporate the relaxation processes into thedescription of the pulse propagation in dispersive Luttingerliquids. In the “fermionic” regime the fermionic collisionsstudied in this work can be directly included in the kineticequation of Ref. [ 16]. On the other hand, a proper account for the relaxation processes in the “bosonic” regime of the pulsepropagation requires a formulation of a bosonic version of thekinetic equation, which remains a prospect for future research. Another direction for future work is the investigation of a broader class of interaction potentials within our formalism.In particular, it would be interesting to study the Fermi-Boseduality in relaxation of excitations in the case of power-law (1 /r α) interactions. Especially important, in view of applications to charged fermions, is the case of Coulomb(1/r) interaction screened (e.g., due to a remote gate) at a large distance d/greatermuchp −1 F. We also envision an extension of our approach to the situation when the initial interacting particlesare bosons, which is relevant in the context of the physics ofcold atoms in one-dimensional traps. ACKNOWLEDGMENTS We acknowledge useful discussions with D. M. Gangardt, I. V . Gornyi, A. Levchenko, and D. G. Polyakov, and financialsupport by the Israeli Science Foundation, German-IsraeliFoundation, and DFG Priority Program 1666. APPENDIX A: UNITARY TRANSFORMATION U3AND BOSONIC VORTEXES In this appendix we derive explicit expressions for the bosonic vertices /Gamma1B,RRRR q ,/Gamma1B,RRRL q , and/Gamma1B,RRLL q . Our start- ing point is the Hamiltonian after the unitary rotation U2, see Eq. ( 14). In terms of the new density operators ˜ ρηthe 125113-10RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) Hamiltonian reads H=(π/L )/summationdisplay quq:(˜ρR,q˜ρR,−q+˜ρL,q˜ρL,−q):B+1 L2/summationdisplay q/bracketleftbig /Gamma1B,RRR q :/parenleftbig ˜ρR,q 1˜ρR,q 2˜ρR,q 3+R→L/parenrightbig :B +/Gamma1B,RRL q :/parenleftbig ˜ρR,q 1˜ρR,q 2˜ρL,q 3+R↔L/parenrightbig :B/bracketrightbig ≡H(2)+H(3) D+H(3) O. (A1) Here we have split the Hamiltonian into the quadratic part H(2), the diagonal-in-chiralities cubic part H(3) D, and the chirality-mixing cubic part H(3) O. Expressing now the densities ˜ ρR(L)via Eq. ( 25), we find H=H(2)+H(3) D+H(3) O−[/Omega13,H(2)]−/bracketleftbig /Omega13,H(3) D+H(3) O/bracketrightbig +1 2[/Omega13,[/Omega13,H(2)]]+O(ρ5). (A2) The operators H(2),H(3) D(O)on the right-hand side of Eq. ( A2) are obtained from that of Eq. ( A1) by a simple replacement ˜ρR(L)→R(L). The decoupling of the chiral sectors in the third order requires that [/Omega13,H(2)]=H(3) O, (A3) which is equivalent to Eq. ( 26). Using Eq. ( A3), one can bring Eq. ( A2)t oas i m p l e rf o r m H=H(2)+H(3) D−/bracketleftbig /Omega13,H(3) D+1 2H(3) O/bracketrightbig +O(ρ5). (A4) Computing the commutator, we obtain the following result for the fourth-order correction to the Hamiltonian: H(4)=1 2πL3/summationdisplay q,p/bracketleftbig −6p/Gamma1B,RRR −p,q 1,q2fp,q 3,q4+p/Gamma1B,RRL −p,q 4,q3fq1,q2,p−p/Gamma1B,RRL q1,q2,−pfp,q 4,q3/bracketrightbig/bracketleftbig/parenleftbig Rq1Rq2Rq3/parenrightbig sLq4+R↔L/bracketrightbig +1 2πL3/summationdisplay q,p/bracketleftbig 3p/Gamma1B,RRR −p,q 1,q2fq3,q4,p+3p/Gamma1B,RRR −p,q 3,q4fq1,q2,p−2p/Gamma1B,RRL −p,q 1,q3fp,q 2,q4−2p/Gamma1B,RRL −p,q 3,q1fp,q 4,q2/bracketrightbig/parenleftbig Rq1Rq2/parenrightbig s/parenleftbig Lq3Lq4/parenrightbig s +1 4πL3/summationdisplay q,pp/Gamma1B,RRL q1,q2,−pfq3,q4,p/bracketleftbig/parenleftbig Rq1Rq2Rq3Rq4/parenrightbig s+R→L/bracketrightbig . (A5) Here the subscript sin the expressions of the type ( ...)sstands for a symmetrization of the expression inside brackets with respect to the momenta qi. Comparing Eq. ( A5)t oE q .( 28), one can read off explicit expressions for fourth-order bosonic vertices ( /Gamma1B,RRRR q , etc.) in terms of /Gamma1B,RRR q and/Gamma1B,RRL q . Exploiting the expansion of third-order vertices at small momenta, we get /Gamma1B,μνρκ q ≈˜/Gamma1B,μνρκ q +l2˜˜/Gamma1B,μνρκ q,q2l2/lessmuch1, (A6) with ˜/Gamma1B,RRRR q =−π3α2 2u0m∗2, (A7) ˜/Gamma1B,RRRL q =4π3α 3m∗2u0/bracketleftbigg 1−3α 2−α 4/parenleftbiggq4 q1+q4 q2+q4 q3/parenrightbigg/bracketrightbigg , (A8) and ˜˜/Gamma1B,RRRL q =5π3α m∗2u0/bracketleftbiggq1q2q3 q4+(6−13α) 20/parenleftbig q2 1+q2 2+q2 3/parenrightbig −α 30q4/parenleftbiggq2 1+q2 2 q3+q2 1+q2 3 q2+q2 2+q2 3 q1/parenrightbigg +26α−53α2−8 60αq2 4+1−5α 30q3 4/parenleftbigg1 q1+1 q2+1 q3/parenrightbigg/bracketrightbigg . (A9) For completeness we present also the amplitude /Gamma1B,RRLL q although we do not need it in the main text: ˜/Gamma1B,RRLL q =π3α m∗2u0/bracketleftbigg 1+α(q1−q2)2 2q1q2/bracketrightbigg +R↔L, (A10) 125113-11I. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) ˜˜/Gamma1B,RRLL q =π3α 8m∗2u0L3/bracketleftbigg (16−47α)/parenleftbig q2 1+q2 2/parenrightbig −(4−3α2) α(q1+q2)2 +17α/parenleftbig q2 1+q2 2/parenrightbig/parenleftbiggq3 q4+q4 q3/parenrightbigg −(5α+2)/parenleftbiggq3 1 q2+q3 2 q1/parenrightbigg/bracketrightbigg +R↔L. (A11) APPENDIX B: FERMIONIC FORM OF THE HAMILTONIAN In this Appendix we present a detailed derivation of the fermionized form of the bosonic Hamiltonian ( 31). It is obvious from the structure of the Hamiltonian ( 31) that in terms of fermions H=/summationdisplay kξR,k:c+ R,kcR,k:F +1 L/summationdisplay k/Gamma1F,RR k :c+ R,k 1c+ R,k 2cR,k/prime 2cR,k/prime 1:F +1 L/summationdisplay k/Gamma1F,RL k :c+ R,k 1c+ L,k 2cL,k/prime 2cR,k/prime 1:F +1 L2/summationdisplay k/Gamma1F,RRR k :c+ R,k 1c+ R,k 2c+ R,k 3cR,k/prime 3cR,k/prime 2cR,k/prime 1:F +1 L2/summationdisplay k/Gamma1F,RRL k :c+ R,k 1c+ R,k 2c+ L,k 3cL,k/prime 3cR,k/prime 2cR,k/prime 1:F +R←→L+··· . (B1) Here···stand for the four-fermion interactions (i.e, those involving eight fermionic operators). In each of vertices /Gamma1F,... k, we denote by kthe vector of momenta of the fermionic operators involved. [As an example, k=(k1,k2,k3,k/prime 3,k/prime 2,k/prime 1) in the vertex /Gamma1F,RRL k .] To derive explicit expressions for the fermionic vertices /Gamma1F,... k, one substitutes the expansions (43)i n t o( 31) and performs the normal ordering of resulting expressions with respect to fermionic operators. In the rest ofthis appendix we analyze these vertices one by one. 1. Single-particle spectrum ξη,k The quadratic part of the fermionic Hamiltonian stems from the quadratic and the cubic terms in the bosonic Hamiltonian(31). For the sake of clarity, we concentrate here on the single- particle spectrum of the right fermions. To compute ξ R,k,w e consider HRR+RRR=(π/L )/summationdisplay quq:RqR−q:B +1 L2/summationdisplay q/Gamma1B,RRR q :Rq1Rq2Rq3:B (B2) with the densities reexpressed in term of the fermionic operators and perform the normal ordering with respectto fermions, retaining only the contributions quadratic infermions. Neglecting first the momentum dependence of/Gamma1 B,RRR q , we get ξR,k=k2 2m∗+/integraldisplayq 0duq≈u0k+k2 2m∗−l2k3 3. (B3)A quick estimate shows that the contribution of the momentum-dependent terms in the expansion of /Gamma1B,RRR q is of the order l2k4/m∗and is always small. 2. Intrabranch two-particle vertex /Gamma1F,RR k Just as the single-particle spectrum, the coupling /Gamma1F,RR k arises from the terms ( B2) of the bosonic Hamiltonian. The contribution of the quadratic part of the bosonic Hamiltonianis easily found to be /Gamma1 F,RR k=πu 0l2 2(k1−k2)(k/prime 1−k/prime 2). (B4) As for the cubic coupling /Gamma1B,RRR q , its zero momentum part does not contribute to /Gamma1F,RR k, while the contribution of its O(q2) terms is of the order k3l2/m∗and can be neglected. 3. Interbranch two-particle vertices /Gamma1F,RL kand/Gamma1F,LR k We turn now to a derivation of the fermionic vertex /Gamma1F,RL k and/Gamma1F,LR k entering the two-particle interaction between left and right sectors of the theory, δH=1 L/summationdisplay k/Gamma1F,RL k :c+ R,k 1c+ L,k 2cL,k/prime 2cR,k/prime 1:F +1 L/summationdisplay k/Gamma1F,LR k :c+ L,k 1c+ R,k 2cR,k/prime 2cL,k/prime 1:F.(B5) Note that the terms with couplings /Gamma1F,RL k and/Gamma1F,LR k in Eq. ( B5) have obviously the same structure with respect to fermionicoperators, and the corresponding splitting of the interaction isdone only for notational convenience. The interbranch two-particle vertex /Gamma1 F,RL k originates from the/Gamma1B,RRRL q coupling in the Hamiltonian ( 31): δHB,RRRL =1 L3/summationdisplay q/Gamma1B,RRRL q :Rq1Rq2Rq3Lq4:B =6 L3/summationdisplay /Gamma1B,RRRL q /Theta1(q1>q 2>q 3)Rq3Rq2Rq1Lq4.(B6) Here we have introduced the shorthand notation /Theta1(q1>q 2> q3)≡/Theta1(q1−q2)/Theta1(q2−q3). We transform now the product of three right densities into a form normal-ordered with respect to fermions. In order to find /Gamma1F,RL k , we have to collect the terms with all but one pairs of fermionic operators replaced by thecorresponding Wick contractions: c + Rk3cR,k 3+q3c+ Rk2cR,k 2+q2c+ Rk1cR,k 1+q1 −→ − :c+ R,k 3cR,k 2+q2:F/angbracketleftcR,k 3+q3c+ R,k 1/angbracketright/angbracketleftc+ R,k 2cR,k 1+q1/angbracketright +:c+ R,k 3cR,k 1+q1:F/angbracketleftcR,k 3+q3c+ R,k 2/angbracketright/angbracketleftcR,k 2+q2c+ R,k 1/angbracketright 125113-12RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) −:c+ R,k 2cR,k 3+q3:F/angbracketleftcR,k 2+q2c+ R,k 1/angbracketright/angbracketleftc+ R,k 3cR,k 1+q1/angbracketright −:c+ R,k 2cR,k 1+q1:F/angbracketleftcR,k 3+q3c+ R,k 1/angbracketright/angbracketleftc+ R,k 3cR,k 2+q2/angbracketright +:c+ R,k 1cR,k 3+q3:F/angbracketleftc+ R,k 2cR,k 1+q1/angbracketright/angbracketleftc+ R,k 3cR,k 2+q2/angbracketright −:c+ R,k 1cR,k 2+q2:F/angbracketleftc+ R,k 3cR,k 1+q1/angbracketright/angbracketleftcR,k 3+q3c+ R,k 2/angbracketright. (B7) Using the contractions of Fermi operators /angbracketleftcR,kc+ R,k/angbracketright=1− /angbracketleftc+ R,kcR,k/angbracketright=/Theta1(k), we find δHB,RRRL→1 L/summationdisplay k/Gamma1F,RL k1,k2,k/prime 2,k/prime 1:c+ Rk1c+ Lk2cL,k/prime 2cR,k/prime 1:F(B8) with /Gamma1F,RL k=6 L2/summationdisplay p1,p2>0/Gamma1B,RRRL −k1+p1+p2,k1+k/prime 1−2p1−p2,−k1+p1,k1−k/prime 1 ×[/Theta1(2k1+k/prime 1−3p1−2p2)−/Theta1(k1−2p1−p2)] +(k1,k/prime 1)→− (k1,k/prime 1). (B9) The behavior of the interbranch two-particle interaction at small momenta can be now inferred from Eqs. ( B9), (32), (33), and ( 34). To present the corresponding expression in a transparent form, it is convenient to classify contributions to /Gamma1F,RL k according to their scaling with the momentum transfer between left and right movers Q=k1−k/prime 1: /Gamma1F,RL k=π m∗2u0(s0+s1Q+s2Q2+s3Q4+s4Q4), (B10) where s0=α/parenleftbigg 1−3α 2/parenrightbigg k1k/prime 1+5(8−13α) 32αl2k2 1k/prime2 1,(B11) s1=α2 8/bracketleftbigg k1+k/prime 1+1 3l2k1k/prime 1/bracketrightbigg lnk2 1 k/prime2 1, (B12) s2=α−3α2 6−259α2−114α+24 48l2k1k/prime 1,(B13) s3=1 48α(17α−3)(k1+k/prime 1)l2lnk2 1 k/prime2 1, (B14) s4=1 576(−883α2+276α−48)l2. (B15) In this work we use the amplitude /Gamma1F,RL k to evaluate the lifetime of the fermionic quasiparticles caused by triple collisions. Aswe discuss in the main text, the momentum transfer Qbetween left and right movers in three-fermion collisions is paramet-rically smaller than the typical momentum pof the colliding particles, Q∼p 2/u0m∗. As a consequence, all but the first term in the expansion of /Gamma1F,RL k are effectively suppressed by additional powers of mass m∗in the denominator and can be neglected. We further note that the vertex /Gamma1F,RL k is nonsingular at small Q.T h e1 /Qsingularity present in the bosonic vertex /Gamma1B,RRRL q [the term q1q2q3/q4in˜/Gamma1q,E q .( 34)] is canceled here. We can thus neglect also the second term in the coefficient s0(cf. Sec. B4). We thus obtain /Gamma1F,RL k=πα(2−3α) 2m∗2u0k1k/prime 1. (B16) The second contribution to the two-particle interaction ( B5) originates from the bosonic term δHRLLLand can be obtained from the first one by applying the R↔Loperation. Obvi- ously, the corresponding vertex /Gamma1LR kis identical to /Gamma1RL k. 4. Interbranch three-particle vertex /Gamma1F,RRL k Here we derive an explicit expression for the three-particle interbranch interaction vertex /Gamma1F,RRL k .J u s tl i k e /Gamma1F,RL k it orig- inates form the bosonic interaction term δHB,RRRL,E q .( B8). The difference is that now we have to collect terms resultingfrom a single Wick contraction in the product of the rightdensities, c + Rk3cR,k 3+q3c+ Rk2cR,k 2+q2c+ Rk1cR,k 1+q1 −→ − /angbracketleft c+ R,k 3cR,k 2+q2/angbracketright:c+ R,k 2c+ R,k 1cR,k 1+q1cR,k 3+q3:F −/angbracketleftc+ R,k 3cR,k 1+q1/angbracketright:c+ R,k 1c+ R,k 2cR,k 2+q2cR,k 3+q3:F +/angbracketleftcR,k 3+q3c+ R,k 2/angbracketright:c+ R,k 3c+ R,k 1cR,k 1+q1cR,k 2+q2:F −/angbracketleftc+ R,k 2cR,k 1+q1/angbracketright:c+ R,k 1c+ R,k 3cR,k 3+q3cR,k 2+q2:F +/angbracketleftcR,k 3+q3c+ R,k 1/angbracketright:c+ R,k 3c+ R,k 3cR,k 2+q2cR,k 1+q1:F +/angbracketleftcR,k 2+q2c+ R,k 1/angbracketright:c+ R,k 2c+ R,k 3cR,k 3+q3cR,k 1+q1:F.(B17) After straightforward algebra, one finds the interbranch three- fermion coupling /Gamma1F,RRL k=− 6s i g n ( k2+k/prime 1) ×|k2+k/prime 1|/2/summationdisplay p=0/Gamma1RRRL k/prime 2−k1,k/prime 1−k2 2+p,k/prime 1−k2 2−p,k/prime 3−k3.(B18) The result ( B18)f o r/Gamma1F,RRL k should be understood as antisym- metrized with respect to incoming ( k1,k2) and outgoing ( k/prime 1, k/prime 2) momenta of the right particles. Substituting now the small-momentum expansion ( 32), (33), and ( 34)o f/Gamma1B,RRRL q , we find /Gamma1F,RRL k=π2(k1−k2)(k/prime 1−k/prime 2) 16m∗2u0/bracketleftbiggs−1 Q+s0+.../bracketrightbigg ,(B19) where Q=k3−k/prime 3is the momentum transfer between right and left movers, and s−1=5αl2[(k1−k2)2−(k/prime 1−k/prime 2)2], (B20) s0=3 2α(13α−6)l2(k1+k/prime 1+k2+k/prime 2). (B21) In Eq. ( B19) we dropped terms containing higher powers of Q; see discussion in Appendix B3. Unlike the two-particle interaction vertex /Gamma1F,RL k, the three- fermion coupling /Gamma1F,RRL k is dominated by its singular behavior at small momentum transfer Qoriginating from the singularity in/Gamma1B,RRRL q at small q4. 125113-13I. V . PROTOPOPOV , D. B. GUTMAN, AND A. D. MIRLIN PHYSICAL REVIEW B 90, 125113 (2014) APPENDIX C: ON-SHELL MATRIX ELEMENTS FOR TRIPLE INTERBRANCH COLLISIONS The aim of this Appendix is to derive the expression ( 58)f o r the matrix element corresponding to interbranch three-particlecollisions. Our starting point is Eqs. ( 53), (56), and ( 48). Let us consider the three incoming particles with momenta k 1,k2, andk3and denote by pandEtheir total momentum end energy, respectively: k1+k2+k3=p, (C1) u0(k1+k2−k3)+1 2m∗/parenleftbig k2 1+k2 2+k2 3/parenrightbig =E. It is convenient to parametrize the momenta satisfying ( C1)b y an angle θvia k1=−2p∗ F 3+p 3+2P0 3(cosθ+√ 3s i nθ), (C2) k2=−2p∗ F 3+p 3+2P0 3(cosθ−√ 3s i nθ), (C3) k3=4p∗ F 3+p 3−4 3P0cosθ. (C4)Here p∗ F=m∗u0, (C5) P0=/radicalbigg p∗2 F+3 4m∗E−p2 8−pp∗ F 4. (C6) The momenta of the three outgoing particles are given by the same expressions with the replacement θ→θ/prime. Note that the requirement that k1,k2are much smaller than p∗ Frestricts the angles θandθ/primeto|θ|,|θ/prime|/lessorsimilarmax(k1,k2)/u0p∗ F. We now substitute the momenta parametrized by the angles θandθ/primeinto Eqs. ( 56) and ( 48) and observe that (k1−k2)2−(k1/prime−k/prime 2)2 k3−k/prime 3=4P0(cosθ+cosθ/prime)≈8p∗ F (C7) and k2 1+k2 2+4k1k2−k/prime2 1−k/prime2 2−4k1/primek/prime 2 k3−k3/prime =− 2p+2p∗ F−4P0(cosθ+cosθ/prime)≈− 4p∗ F.(C8) The result ( 58) then follows immediately. [1] M. Stone, Bosonization (World Scientific, Singapore, 1994). [2] J. von Delft and H. Schoeller, Ann. Phys. 7,225(1998 ). [ 3 ] A .O .G o g o l i n ,A .A .N e r s e s y a n ,a n dA .M .T s v e l i k , Bosoniza- tion in Strongly Correlated Systems (Cambridge University Press, Cambridge, 1998). [4] T. Giamarchi, Quantum Physics in One Dimension , (Claverdon Press, Oxford, 2004). [5] A. M. Lunde, K. Flensberg, and L. I. Glazman, P h y s .R e v .B 75, 245418 (2007 ). [6] M. Khodas, M. Pustilnik, A. Kamenev, and L. I. Glazman, Phys. Rev. B 76,155402 (2007 ). [7] A. Imambekov and L. I. Glazman, Science 323,228 (2009 ); ,Phys. Rev. Lett. 102,126405 (2009 ). [8] A. Imambekov, T. L. Schmidt, and L. I. Glazman, Rev. Mod. Phys 84,1253 (2012 ). [9] T. Micklitz and A. Levchenko, Phys. Rev. Lett. 106,196402 (2011 ). [10] Z. Ristivojevic and K. A. Matveev, Phys. Rev. B 87,165108 (2013 ). [11] S. Apostolov, D. E. Liu, Z. Maizelis, and A. Levchenko, Phys. Rev. B 88,045435 (2013 ). [12] J. Lin, K. A. Matveev, and M. Pustilnik, Phys. Rev. Lett. 110, 016401 (2013 ). [13] D. C. Mattis and E. H. Lieb, J. Math. Phys. 6,304(1965 ). [14] A. V . Rozhkov, Phys. Rev. B 77,125109 (2008 ); ,74,245123 (2006 ); ,Eur. Phys. J. 47,193(2005 ). [15] K. A. Matveev and A. Furusaki, Phys. Rev. Lett. 111,256401 (2013 ). [16] I. V . Protopopov, D. B. Gutman, M. Oldenburg, and A. D. Mirlin, Phys. Rev. B 89,161104 (R) ( 2014 ). [17] T. Micklitz, J. Rech, and K. A. Matveev, P h y s .R e v .B 81,115313 (2010 ); K. A. Matveev, J. Exp. Theor. Phys. 117,508(2013 ).[18] A. P. Dmitriev, I. V . Gornyi, and D. G. Polyakov, P h y s .R e v .B 86,245402 (2012 ). [19] M. Schick, Phys. Rev. 166,404(1968 ). [20] Since our bosonization analysis is justified for temperatures T/lessmuchEF, in the case of a moderate interaction strength, K0∼1, the existence of the bosonic domain requires that lintpF/greatermuch1. [21] Similar approach is often used in the theory of classical weakly non-linear waves, see e.g., V . E. Zakharov and E. A. Kuznetsov,JETP 86,1035 (1998 ). [22] In fact, in the approximation ( 35) (which is the leading order in a small momentum transfer q 4/lessmuchq1,q2,q3), the four-fermion vertex /Gamma1F,RRRL k vanishes due to requirement of antisymmetry with respect to the momenta of right fermions. Retaining higherorder terms in Eq. ( 34), we can estimate /Gamma1 F,RRRL k ∼T3l2/u5 0m∗3. Taking into account the integration measure together with energyand momentum conservation laws [cf. Eqs. ( 50), (59)], we find the contribution of the corresponding collision processes into therelaxation of a fermionic quasiparticle at momentum k 1∼T/u 0 to be 1 /τRRRL k1∼T/u 0(T)∝T11l4/u16 0m∗6. This contribution to the relaxation rate is small in comparison to the dominant term ( 61) by a factor ∼T4/u8 0m∗4∼(T/E F)4/lessmuch1. [23] For weak interaction Eq. ( 58) reduces to /angbracketleft1/prime,2/prime,3/prime|T|1,2,3/angbracketright= (3g2 0/4v2 F)l2 int(k1−k2)(k/prime 1−k/prime 2) and reproduces the result of Ref. [ 6]. [24] M. Arzamasovs, F. Bovo, and D. M. Gangardt, P h y s .R e v .L e t t . 112,170602 (2014 ). [25] While intrabranch collisions yield only a subdominant contribu- tion in the present context, they are the only source of relaxationin a chiral LL describing a single-channel quantum Hall edge.In this case the relaxation rate of fermionic excitations is givenby Eq. ( 65) [I. V . Protopopov, D. B. Gutman, and A. D. Mirlin (unpublished)]. 125113-14RELAXATION IN LUTTINGER LIQUIDS: BOSE-FERMI . . . PHYSICAL REVIEW B 90, 125113 (2014) [26] In the case of very strong interaction (almost-Wigner-crystal limit) with the Luttinger liquid parameter K0/lessmuch1w eh a v e m∗∝√K0andl/similarequallint. Using further u0=vF/K 0, we obtain the following expression for the crossover momentum qc≡ 1/m∗u0l2∼√K0/pFl2 int. Our condition for the bosonic domain q/greatermuchqcagrees then with that found in Ref. [ 12] for this limiting case. [27] I. V . Protopopov, D. B. Gutman, P. Schmitteckert, and A. D. Mirlin, P h y s .R e v .B 87,045112 (2013 ).[28] Y . Fu Chen, T. Dirks, G. Al-Zoubi, N. O. Birge, and N. Mason, Phys. Rev. Lett. 102,036804 (2009 ). [29] G. Barak, H. Steinberg, L. N. Pfeiffer, K. W. West, L. Glazman, F. von Oppen, and A. Yacoby, Nat. Phys. 6,489(2010 ). [30] H. le Sueur, C. Altimiras, U. Gennser, A. Cavanna, D. Mailly, and F. Pierre, Phys. Rev. Lett. 105,056803 (2010 ). [31] M. G. Prokudina, S. Ludwig, V . Pellegrini, L. Sorba, G. Biasiol, and V . S. Khrapai, P h y s .R e v .L e t t . 112,216402 (2014 ). 125113-15

PhysRevB.97.224408.pdf

PHYSICAL REVIEW B 97, 224408 (2018) Chiral surface and edge plasmons in ferromagnetic conductors Steven S.-L. Zhang1,2,*and Giovanni Vignale1,† 1Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA 2Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA (Received 18 April 2018; revised manuscript received 25 May 2018; published 11 June 2018) The recently introduced concept of “surface Berry plasmons” is studied in the concrete instance of a ferromagnetic conductor in which the Berry curvature, generated by spin-orbit (SO) interaction, has oppositesigns for carrier with spins parallel or antiparallel to the magnetization. By using collisionless hydrodynamicequations with appropriate boundary conditions, we study both the surface plasmons of a three-dimensionalferromagnetic conductor and the edge plasmons of a two-dimensional one. The anomalous velocity and the brokeninversion symmetry at the surface or the edge of the conductor create a “handedness” whereby the plasmonfrequency depends not only on the angle between the wave vector and the magnetization, but also on the directionof propagation along a given line. In particular, we find that the frequency of the edge plasmon depends on thedirection of propagation along the edge. These Berry curvature effects are compared and contrasted with similareffects on plasmon dispersions induced by an external magnetic field in the absence of Berry curvature. Weargue that Berry curvature effects may be used to control the direction of propagation of the surface plasmonsvia coupling with the magnetization of ferromagnetic conductors, and thus create a link between plasmonics andspintronics. DOI: 10.1103/PhysRevB.97.224408 I. INTRODUCTION The discovery of collective oscillations of electrons in quan- tum solid-state plasmas in the 1950s was a major milestone inthe evolution of condensed matter physics [ 1]. It exposed the fundamental dichotomy in the character of electronic elemen-tary excitations, which can be either individual quasiparticlesor organized collective oscillations (plasmons), and spawneda variety of theoretical treatments of the electron gas (therandom-phase approximation [ 1,2] being one of the earliest and most successful), effectively igniting the field of many-electronphysics [ 3]. By the end of the 20th century the interest began to shift to the possible technological applications of plasmons,as it was realized that the wavelengths of these oscillations,being much shorter then the wavelength of light at the samefrequency, could be used to compress electromagnetic energyto a nanometric scale—the scale of integrated circuits anddevices. A thriving area of research, known as “plasmonics”[4,5], was born. At about the time that plasmonics was taking off, major advances were made in the band theory of solids [ 6,7]. It was realized that, under quite common conditions, the Bloch wavefunctions of electrons in a periodic solid, regarded as functionsof the Bloch wave vector kin the Brillouin zone, have nontrivial geometric properties. When the nth eigenstate |u n(k)/angbracketrightof the periodic Hamiltonian H(k) is adiabatically transported around a closed loop in the Brillouin zone, the final state differs fromthe initial one by a gauge invariant “Berry phase” /Delta1φ, which equals the flux of “Berry curvature” through the area enclosed *shulei.zhang@anl.gov †vignaleg@missouri.eduby the loop [ 8]. The mathematical expression for the Berry curvature /Omega1n(k)=i/angbracketleft∇kun(k)|×|∇kun(k)/angbracketright (1) is one of the most important properties of a solid-state system, its integral over the Brillouin zone being connected to topo-logical quantum numbers and quantized conductivities [ 9]. A question that naturally arises at this point is, how does the Berry curvature of a band affect, if at all, the properties of theplasmons of the carriers in that band? One of the simplest waysto address the question is to set up the collisionless hydrody-namic equations for the collective motion of the electron fluid[2]. These will in turn be based on the quasiclassical equations of motion for wave packets in the band [ 9,10]: ˙r=¯h −1∇kEn(k)+/Omega1n(k)×˙k, (2a) ˙k=− ¯h−1∇rV(r), (2b) where rand ¯hkare the position and the momentum of the wave packets, En(k) is the energy of the Bloch state, and V(r)i st h e potential energy arising from the self-consistent electric field.The Berry curvature enters the equations of motion throughthe second term in the expression for ˙r. This is often referred to as the “anomalous velocity”: v a(k)≡/Omega1n(k)×˙k=− ¯h−1/Omega1n(k)×∇rV(r). (3) Physically, the anomalous velocity reflects the nonconser- vation of the Bloch momentum. As the Bloch momentumchanges under the action of a force according to Eq. ( 2b), the quantum state of the electron no longer coincides with theinstantaneous eigenstate |u n(k(t))/angbracketright. The difference between the actual state |un(t)/angbracketrightand the instantaneous eigenstate |un(k(t))/angbracketright is reflected in the expectation value of velocity operator 2469-9950/2018/97(22)/224408(11) 224408-1 ©2018 American Physical SocietySTEVEN S.-L. ZHANG AND GIOV ANNI VIGNALE PHYSICAL REVIEW B 97, 224408 (2018) ˆv(k)≡¯h−1∇kˆH(k), where ˆH(k) is the Hamiltonian at wave vector k. It is easily seen that the expectation value of ˆv(k)i nt h e instantaneous eigenstate equals ¯ h−1∇kEn(k). The anomalous velocity is the correction to that result, arising from the factthat, in a dynamical situation, |u n(t)/angbracketright/negationslash=|un(k(t))/angbracketright. The study of the effect of the anomalous velocity on the dynamics of the plasmons was pioneered in a recent paper bySong and Rudner (SR) [ 11]. Working within the framework of collisionless hydrodynamics (see Sec. II) they first showed that the anomalous velocity has no effect on the bulk modes ofa homogeneous electron liquid. This is because the anomalousvelocity enters the bulk hydrodynamics only in the continuityequation, and only through its divergence, which is zero dueto∇ r·[/Omega1n(k)×∇rV(r)]=0. The situation changes when one considers surface or edge plasmons [ 12]. These collective oscillations are exponentially localized near the surface or the edge of the system, witha localization length of the order of v F/ωp, where vFis the Fermi velocity and ωpis the plasmon frequency. These are also the modes that are of greatest interest in plasmonicapplications, because they hybridize with electromagneticwaves to produces surface plasmon polaritons [ 13–16]. In the hydrodynamic approach, surface plasmons are derived byimposing a boundary condition on the current density at thesurface or edge of the system. The boundary condition statesthat there is no electron flux through the boundary of thesystem, i.e., j z(z=0)=0, where zis the direction perpendicular to the boundary and jzis thezcomponent of the particle current. Notice that the impo- sition of the boundary condition is the way hydrodynamics—along-wavelength theory—handles the sharp variation of theelectronic density across the boundary. It is precisely throughthe boundary condition that the anomalous velocity enters thesolution for the plasmon. This point was clearly demonstratedby SR [ 11] for the edge plasmon of a two-dimensional (2D) system. Taking a rather abstract approach in which a Berry cur-vature of unspecified origin was assumed to exist, SR showedthat the frequency of right-propagating modes (along the edge)can be significantly different from that of left-propagatingmodes. At finite wave vector one of the two modes can be welldefined while its time-reversed partner may be severely Landaudamped. Under this scenario an essentially unidirectionalpropagation of edge plasmons is achieved, which is of greattechnological interest. The scenario is similar, but not identicalto that of surface and edge plasmons in a magnetic field (theso-called magnetoplasmons), which were studied, for example,in Refs. [ 17–20]. Both scenarios require broken time-reversal symmetry to produce chiral plasmons, but the magnetoplasmonarises from the classical Lorenz force exerted by the magneticfield, whereas the Berry plasmon arises from the anomalousvelocity. In this paper, we study a concrete realization of the abstract SR scenario, namely the Berry plasmons at the surface of aferromagnetic conductor, with the magnetization lying in theplane of the boundary surface in 3D or perpendicular to theplane of the system in 2D. Spontaneous magnetization breakstime-reversal symmetry, but we assume that the magnetic fieldassociated with the magnetization has negligible effect on the electrons: in particular, there is no sizable Lorenz force. Onthe other hand, the Berry curvature of electrons in (say) theconduction band is assumed to be different from zero andspin dependent, having opposite signs for electrons of oppositespins. For example, in the conduction band of 3D GaAs, asimple calculation based on the 8-band model [ 21] predicts the Berry curvature [ 22] /Omega1 c(k)/similarequalλ2σ (4) at the bottom of the band. The effective Compton wavelength λis related to band parameters by the well-known formula λ2=2¯h2|P|2 3m2e/bracketleftBigg 1 E2g−1 (Eg+/Delta1)2/bracketrightBigg , (5) where Egis the fundamental band gap, /Delta1is the gap separating the light/heavy hole bands from the so-called SO split band,andPis the matrix element of the momentum operator between atomic sandpstates. Thus, the /Delta1gap is a direct measure of the SO-induced splitting of atomic energy levels with J=1/2 and 3/2, and its nonzero value is essential to the emergence of a finite Berry curvature in the conduction band. Becauseelectrons of opposite spins have opposite Berry curvatures, andhence opposite anomalous velocities, no effect is expected onthe surface plasmons of a spin-unpolarized system. But, if thesystem is magnetic, then the opposite anomalous velocities of majority and minority spin electrons give a nonvanishing contribution to the net particle current, which affects thecollective motion of the electron liquid, and maximally sowhen the electron liquid is fully spin polarized. We refer tothese collective motions as ferromagnetic surface (or edge) plasmons . The results of our study, presented below, pertain to long- wavelength plasmons (wave vector q/lessmuchk F, where kFis the Fermi wave vector), but the wavelength is not so large thatretardation effects must be taken into account: namely, weassume q/greatermuchω p/c, where cis the speed of light. Even in this limit, we find that the frequency of the ferromagnetic surfaceplasmon depends on the angle between the wave vector andthe magnetization. Similarly, the frequency of the ferromag-netic edge plasmon depends on the direction of propagationalong the edge. The fact that charge oscillations, such asthe plasmons, “sense” the magnetization is a consequence ofthe spin-orbit-induced Berry curvature. It has nothing to dowith the well known anisotropy of plasmons in a magneticfield. The relation between chiral magnetoplasmons and chiralferromagnetic plasmons is reminiscent of the relation betweenthe regular Hall effect and the anomalous Hall effect, wherethe former arises from the Lorentz force, while the latterarises from the concerted action of spin-orbit coupling andmagnetization. Another significant difference between ferro-magnetic surface plasmons and ordinary magnetoplasmons isthat we find a single surface mode, as opposed to two. Thetwo magnetoplasmons arise from the interplay of two classicalforces, the electrostatic force and the Lorentz force, which caneither work together or against each other. We have no Lorentzforce, and therefore find a single ferromagnetic plasmon mode. Chiral edge plasmons were recently predicted [ 23]i n two-dimensional gapped Dirac systems under pumping with 224408-2CHIRAL SURFACE AND EDGE PLASMONS IN … PHYSICAL REVIEW B 97, 224408 (2018) circularly polarized light, which produces a population im- balance between two valleys. In contrast, the chirality of theplasmons in the present study arises from a spontaneous spinpolarization under equilibrium conditions. A recent study oftopological edge magnetoplasmons [ 24] is not directly relevant to the present scenario, since it relies on a magnetic field ratherthan a spontaneous magnetization. The angular dependence and chirality of the plasmon frequency is a potentially important issue in plasmonics, sinceit can be used to control the direction of propagation ofplasmon waves. Even more interesting, in our view, is theunusual coupling between magnetism and charge oscillationsthat this work foreshadows. The coupling should persist in fullydynamical situations, when both the magnetization and thecharge density are time dependent. This suggests the intriguingpossibility of coupling plasmons and spin waves, thus bringingtogether the fields of spintronics [ 25–29] and plasmonics. The remaining of the paper is organized as follows. In Sec. II, we introduce a hydrodynamic model of electron fluid to investigate the dispersion of surface (or edge) plamson inferromagnetic conductors. Our treatment is based on a setof linearized hydrodynamic equations and the correspondingboundary conditions for the surface or edge plasmon modes inwhich the anomalous velocity comes into play. We then presentthe exact solution of the 3D ferromagnetic surface plasmondispersion in Sec. III. An approximate solution of the 2D problem will be discussed in Sec. IVwith a simplified treatment of the electrostatics. Following the thorough investigation of both the ferromagnetic surface and edge plamsons, weshow, in Sec. VA , that they are distinguishable from the classical surface and edge magnetoplasmons (arising from theLorentz force) by providing a comparison between the twokinds of plasmons in the long wavelength limit. And finally,in Sec. VB, we discuss possible experimental observation of the ferromagnetic surface and edge plasmons in variousferromagnetic systems. The conclusion is given in Sec. VI. II. COLLISIONLESS HYDRODYNAMICS Unlike proper hydrodynamics, which presupposes slow col- lective motion on the scale of the particle-particle collision fre-quency, collisionless hydrodynamics applies to high-frequencycollective motion, such as plasmons, in which collisions be-tween quasiparticle can be disregarded [ 1]. It is well known that the full-fledged collisionless hydrodynamic treatment mustinclude a viscoelastic stress tensor, which produces not onlythe “hydrostatic force” (gradient of pressure), but also anelastic shear force and a viscous friction force [ 2]. These two additional forces are essential to obtain, respectively, thecorrect dispersion of plasmons at finite wave vector and thenon-Landau damping [ 1]. In this paper, however, we limit ourselves to a more crude model, in which we neglect shear andviscous forces. This approach is expected to become essentiallyexact in the long wavelength limit, due to the dominanceof electrostatic forces, but will become inaccurate at shorterwavelength. These inaccuracies are of secondary importancehere, since our primary interest is in the qualitatively newfeatures of the solution, which appear already in the longwavelength limit. With the above discussion in mind, the bulkhydrodynamic equations are ∂ tδn+∇r·j=0( 6 ) (continuity equation) and ∂tjp+s2∇rδn−en0 me∇rϕ=0, (7) the Euler equation, where n0is the uniform equilibrium density of electrons, meis the effective mass, jp=p me(8) is the canonical current density, proportional to the canonical momentum density passociated with the momentum variable ¯hk,sis the velocity of the hydrodynamic sound (this is of the order of the Fermi velocity, and is related to the bulk modulus K by the well-known relation [ 30]s2=K n0me),δnis the deviation of the electron density from the equilibrium density n0, and j is the physical current density given by j=jp+Pn0e ¯hλ2ˆm×∇rϕ (9) with ˆmbeing the unit vector along the magnetization, and P=n0↑−n0↓ n0the spin polarization of the electron density. Note that the total physical current density is a superposition of thecurrent densities arising from the canonical momentum andthe anomalous velocity in conjunction with the Berry curvatureg i v e nb yE q .( 4). For the latter contribution, notice that majority and minority electrons in ferromagnetic conductors have op-posite spin projections on the magnetization direction ˆmand hence opposite anomalous velocities. The net contribution ofthe anomalous velocity to the current is therefore proportionalto the spin asymmetry of the electron density characterizedby the spin polarization P. The hydrodynamic equations contain the electrostatic potential ϕ, which is assumed to be instantaneously created by the charge density according to thePoisson equation ∇ 2 rϕ=4πeδn. (10) The solution of the Poisson equation is straightforward in 3D, but not at all in 2D, unless some simplifying approximationsare made, which we will discuss later. In the absence of boundary conditions it is easy to see that the solution to the hydrodynamic equations (togetherwith the Poisson equation) is not affected by the anomalousvelocity. This is because, as remarked in the introduction,∇ r·(ˆm×∇rϕ)=0 allows us to replace jwith jpin the continuity equation. Then the coupled equations for δnand pdo not contain the anomalous velocity term and therefore the eigenfrequencies do not depend on λorˆm. On the other hand, for surface or edge modes the boundary condition ofvanishing charge current density, i.e., j z|z=0−=0, creates a coupling between charge and magnetization; this boundarycondition can be explicitly written as /bracketleftbigg j p,z+ePn0 ¯hλ2(ˆm×∇rϕ)·ˆz/bracketrightbigg z=0−=0. (11) In addition, the electric potential and its gradient must be continuous at z=0, i.e., ϕ|z=0−=ϕ|z=0+and∂zϕ|z=0−=∂zϕ|z=0+. (12) 224408-3STEVEN S.-L. ZHANG AND GIOV ANNI VIGNALE PHYSICAL REVIEW B 97, 224408 (2018) III. SOLUTION IN THREE DIMENSIONS Assuming translational invariance in the plane of the sur- face, we seek solutions in the form of plane waves of wavevector qdecaying exponentially in the bulk ( z<0) as e κz withκ> 0. We let all the physical quantities (e.g., δn,p, and ϕ) take the form /Psi1(r,z;t)∼eκzei(q·r−ωt)up to some constant coefficients to be determined by the boundary conditions.Notice that boldface symbols are used to indicate vectors in theplane of the surface. Inserting this ansatz in the hydrodynamicequations, we find a set of homogeneous algebraic equations,i.e., −iωδn+κj p,z+iq·jp,/bardbl=0, (13a) −iωjp,z+s2κδn−en0 meκϕ=0, (13b) −iωjp,/bardbl+iqs2δn−iqen0 meϕ=0, (13c) where jp,/bardbl=(jp,x,jp,y) are the in-plane components of the particle current density, and ϕandδnare related by ϕ=4πeδn κ2−q2(14) through the Poisson equation. The set of equations has non- trivial solutions only if κ2=q2+s−2/parenleftbig ω2 B−ω2/parenrightbig , (15) where ω2 B=4πe2n0 meis the bulk plasmon frequency. Now we can write down the general solution for the density oscillation as δn=δn1eκzeiq·r,z < 0, (16) and that for the electric potential as ϕ=ϕ1eκzeiq·r+ϕ2eqzeiq·r,z < 0, ϕ=ϕ0e−qzeiq·r,z > 0, (17) withδn1,ϕ0,ϕ1, and ϕ2being integration constants to be determined by the boundary conditions. Putting the general solutions for the electron density fluc- tuation ( 16) and the electric potential ( 17) into the boundary conditions given by Eqs. ( 11) and ( 12), we obtain a set of three linear homogeneous equations, the solution of which gives thedispersion relation (κ+q)ω 2 B−2κω2+P(κ−q)ω2 Bωτsosinφq=0,(18) where φq, with −π<φ q/lessorequalslantπ, is the angle between the wave vector qand the in-plane magnetization, and the quantity τso(≡meλ2 ¯h) has the dimension of time which characterizes the strength of the SO interaction. Making use of Eq. ( 15), one can rewrite Eq. ( 18) as follows: (κ−q)/bracketleftbig ω2−s2(κ+q)2−Pω2 Bωτsosinφq/bracketrightbig =0.(19) While the solution κ=qgives the bulk frequency of ω=ωB as can be easily seen from Eq. ( 15), the general surface plasmon frequency is given by the equation ω2−s2(κ+q)2−Pω2 Bωτsosinφq=0. (20)It is instructive to first examine the solutions in the long wavelength limit ( q→0) for which Eq. ( 20) reduces to 2ω2−ω2 B−Pω2 Bωτsosinφq=0. (21) Equation ( 21) has two possible solutions: ω+=ωS 2/bracketleftbig PωSτsosinφq+/radicalBig 4+(PωSτsosinφq)2/bracketrightbig (22a) ω−=ωS 2/bracketleftbig PωSτsosinφq−/radicalBig 4+(PωSτsosinφq)2/bracketrightbig ,(22b) where ωS=ωB√ 2is the 3D surface plasmon frequency at q=0. We observe that, for any angle φq, the solution ω+is positive whereas the ω−solution is negative. Moreover, the solutions satisfy the relation ω+(−φq)=−ω−(φq).The existence of two solutions connected in this manner is a necessary conditionfor being able to construct real solutions of the hydrodynamicequations. Physically, the two solutions together describe a single chiral wave whose frequency is determined, for eachwave vector q(specified by its magnitude qand angle φ q), by the positive branch ω+(q). A real wave that propagates in the direction of qis described by the superposition eiq·re−iω+(q)t+ e−iq·re−iω−(−q)t, where we have used the fact that changing the sign of φqamounts to reversing the direction of q. Similarly, a real wave that propagates in the direction of −qis described by the superposition e−iq·re−iω+(−q)t+eiq·re−iω−(q)t. Crucially, the two waves, with wave vectors qand−qrespectively, exhibit different phase velocities, and different dependences onmaterial parameters such as the strength of the SO interactioncharacterized τ so, spin polarization P,e t c . Another interesting feature of the 3D ferromagnetic surface plasmon is that the decay length κ−1behaves quite differently for waves that propagate in opposite directions. More specif-ically, if the decay length κ −1evaluated from Eq. ( 15) with ω=ω+(q)increases with increasing value of |PωSτso|for surface plasmons propagating along qdirection, then it must decrease with increasing value of |PωSτso|for those propagat- ing along −qdirection. This can be easily observed in the long wavelength limit for which ω+is explicitly given by Eq. ( 22a). Furthermore, we note that when the product |PωSτso|becomes sufficiently large, the surface plasmon mode is forbidden in arange of directions where κremains imaginary (physically this means that the surface mode merges with the bulk mode in thesedirections). For the long-wavelength limit, one can show thatwhen|Pω Sτso|>1√ 2, there exist two intervals for the plasmon propagation angle, φq∈[φcrit q−π,−φcrit q]∪[φcrit q,π−φcrit q] withφcrit q=arcsin (1√ 2|PωSτso|), in which the surface mode is absent, as shown schematically in Fig. 1(d). In Fig. 1, we show the ferromagnetic surface plasmon frequency as a function of the direction and the magnitudeof the wave vector q. For a given q(=0.1k F), the surface plasmon frequency exhibits a sinusoidal-like dependence onφ qas shown in Fig. 1(b): It reaches a maximum at φq=+π 2and a minimum at φq=−π 2(i.e., when the surface plasmon propagates in directions perpendicular to the magnetization),and coincides with the normal surface plasmon frequency in theabsence of SO interaction (indicated by the black dotted line)when the surface plasmon propagates parallel or antiparallelto the magnetization (i.e., φ q=0o rπ). 224408-4CHIRAL SURFACE AND EDGE PLASMONS IN … PHYSICAL REVIEW B 97, 224408 (2018) FIG. 1. Variation of the 3D ferromagnetic surface plasmon frequency ω+(scaled by ωS) with the in-plane wave vector q. The setup of the system is shown schematically in panel (a) with φqdefined as the angle between the direction of propagation of the surface plasmon and the direction of magnetization fixed on the xaxis. Panel (b) shows ω+as a function of φqat a fixed magnitude of the wave vector of q=0.1kF for several different values of τso(scaled by ω−1 B), panel (c) shows ω+as a function of the magnitude of the wave vector qfor several different angles φqwith a given SO interaction strength ωBτso=0.5, and panel (d) is a schematic picture showing the range of φq(plasmon propagation directions) for which the surface mode is forbidden when the quantity |PωSτso|is greater than1√ 2. Note that the Fermi wave vector kFis related with the 3D equilibrium electron density via n0=k3 F 3π2. Figure 1(c) shows the surface plasmon frequency as a function of the magnitude of the plasmon wave vector forthree different directions of propagation: φ q=−π 2,0,π 2.W e note that while ω+grows monotonically with qfor all three directions [ 31], their frequencies remain non-degenerate for anyqdue to the presence of the SO interaction. One interesting consequence of the anisotropic dispersion is that the phasevelocities, given by ω+(q) q, for surface plasmons propagating in opposite directions ( φq/negationslash=0,π) are always different: this implies that as long as the direction of propagation deviatesfrom the direction of the magnetization, no standing wave canbe formed for surface plasmons. In Fig. 2, we show the dependences of the frequency ω + and decaying length κ−1of the surface plasmons on the strength of the SO interaction characterized by ωBτso.A s the effect of the SO interaction is most prominent for sur-face plasmons propagating in the directions perpendicular tothe magnetization, we shall focus on the cases of φ q=−π 2and π 2for a finite magnitude of the wave vector q. Consistent with the qualitative analysis we performed for the long wave lengthlimit, we find that for surface plasmons propagating at an angleφ q=−π 2with respect to the magnetization, both ω+andκ−1 decrease monotonically with increasing ωBτso, whereas for those propagating in the opposite direction (i.e., φq=π 2) both ω+andκ−1increase monotonically with increasing ωBτsoand are terminated when τsoreaches a certain threshold (indicated by the vertical dashed line in the Fig. 2) where the decay length κ−1diverges as ω2approaches ω2 B+s2q2[see Eq. ( 15)] and the surface mode merges into the bulk mode [ 32]. IV . SOLUTION IN TWO DIMENSIONS An exact treatment of the electrostatics in a two- dimensional plane is quite more complicated than in 3D,due to the fact that the electric field exists in the wholethree-dimensional space, while the electron density is confined to a plane. Fortunately, the treatment can be greatly simpli-fied by making the approximation adopted by Fetter [ 19]i n his treatment of the two-dimensional edge magnetoplasmon,namely replacing the exact electrostatic Green’s function (thenonlocal kernel that connects the density to the potential inthe plane) by an approximate Green’s function that has thesame integrated area and second moment. What is lost inthe approximation is a weak logarithmic dependence of theedge magnetoplasmon frequency on the magnitude of the wavevector, ω∼q|lnq|, which is confirmed by a more accurate FIG. 2. Dependence of 3D ferromagnetic surface plasmon on the strength of the SO interaction (characterized by ωBτso): (a) the ferromagnetic surface plasmon frequency ω(scaled by ωS)a sa function of ωBτsoand (b) the decay length of the surface mode given byκ−1as a function of ωBτsoat a given magnitude of wave vector q=0.1kF, including two opposite propagation directions: φq=π 2 and−π 2respectively. 224408-5STEVEN S.-L. ZHANG AND GIOV ANNI VIGNALE PHYSICAL REVIEW B 97, 224408 (2018) treatment making use of the Wiener-Hopf technique [ 18,33]. This is not a very serious drawback in our case, since thelong-wavelength dispersion continues to be largely controlledby classical electrostatics, which mandates a√ qdependence. With this approximation, the equations remain essentially thesame as in 3D, except that Eq. ( 14), connecting the potential to the density, takes the slightly different form ϕ=4πe|q| κ2−2q2δn. (23) We let the electron liquid be confined to the y-zplane, and consider the edge plasmon localized in the zdirection and propagating in the ydirection, while the magnetization is along the xaxis, perpendicular to the electron liquid (since the plasmon only propagates in the ydirection, we have suppressed the subscript for the wave vector q). Combining Eq. ( 23) with the set of equations ( 13a)–(13c) (with qreplaced by qˆy), we arrive at the equation relating the edge plasmon frequency ω and the decaying constant κ, i.e., κ4−κ2/parenleftbig k2 0+k2 ω+3q2/parenrightbig +2q2/parenleftbig k2 ω+q2/parenrightbig =0, (24) where we have defined k2 ω≡ω2 q−ω2 s2andk2 0≡ω2 q s2(25) withωq=/radicalBig 2πn0e2|q| methe bulk 2D plasmon frequency. Also note that the sound velocity in the 2D case shares the same expression as that in the 3D case, i.e., s2=K n0mbut with n0 being the equilibrium density of the 2D electron fluid. The equation has two solutions: κ2 1,2=1 2/bracketleftbigg k2 0+k2 ω+3q2 ±/radicalBig/parenleftbig k2 0+k2ω/parenrightbig2+q2/parenleftbig 6k2 0−2k2ω+q2/parenrightbig/bracketrightbigg .(26) Similar to the 3D case, we write the general solutions for the electrostatic potential and the electron density fluctuation asfollows: ϕ=ϕ 1eκ1zeiqy+ϕ2eκ2zeiqy,z < 0, ϕ=ϕ0e−√ 2qzeiqy,z > 0, (27) δn=1 4πe|q|/bracketleftbig/parenleftbig κ2 1−2q2/parenrightbig ϕ1eκ1z+/parenleftbig κ2 2−2q2/parenrightbig ϕ2eκ2z/bracketrightbig eiqy, (28) where we have suppressed the common time-dependent com- ponents. The general solution for the density fluctuation δn was derived by invoking the approximate Poisson relation ( 23). Putting these equations in the boundary conditions, we derivethe following equation for ω, the solution of which gives the edge plasmon frequency: κ 1/parenleftbig κ2 1−2q2−2k2 0/parenrightbig +2Pk2 0ωτsoqmx κ2/parenleftbig κ2 2−2q2−2k2 0/parenrightbig +2Pk2 0ωτsoqmx=κ1+√ 2|q| κ2+√ 2|q|.(29) Note that the magnetization enters the dispersion relation only through the xcomponent of the magnetization (i.e., the one perpendicular to the plane of the 2D electron liquid). This canbe understood as follows: The magnetization enters the formula for the anomalous velocity as va∼ˆm×∇rϕ; as the surface plasmon propagates along the edge in the ydirection, only thexcomponent of the magnetization generates an anomalous velocity in the zdirection (perpendicular to the edge), which affects the boundary condition and hence the edge plasmonfrequency. In the long-wavelength limit, we obtain simpler forms for κ 1andκ2by keeping only leading-order terms in q, i.e., κ1/similarequalCω|q|+O(q3) (30) withC2 ω=ω2 q−ω2 ω2q−1 2ω2and κ2/similarequal1 s/radicalBig 2ω2q−ω2+O(q2). (31) Making use of these expressions, the general equation ( 29) can be reduced to ω2−2 3ω2 q+2√ 2 3Pω2 qωτsomx=0, (32) where we have made the approximation of Cω/similarequal√ 2 2by taking ωto be the unperturbed edge plasmon frequency ω(0)=/radicalBig 2 3ωq [19]. This equation has two solutions, ω+=√ 2 3ωq/bracketleftbigg/radicalBig 3+(Pωqτsomx)2−Pωqτsomxsgn(q)/bracketrightbigg (33a) and ω−=−√ 2 3ωq/bracketleftbigg/radicalBig 3+(Pωqτsomx)2+Pωqτsomxsgn(q)/bracketrightbigg . (33b) Notice that in the absence of the SO interaction, we recover the unperturbed edge plasmon frequency. As in the 3D case,the reality of the classical wave fields requires ω +(−q)= −ω−(q),which shows the solutions of opposite frequencies and momenta to be parts of the same wave . Also, reversing the direction of propagation of the edge plasmon is equivalentto reversing the direction of the magnetization direction:therefore we find ω +(−m)=−ω−(m) as expected. For this reason, we shall concentrate on the positive solution ω+of the 2D ferromagnetic edge plasmons in the following discussions. Properties of the 2D ferromagnetic edge plasmon at fi- nite wavelength are readily obtained by numerically solvingEq. ( 29). In Fig. 3(a),w ep l o t ω +as a function of wave vector qwith the magnetization direction fixed along the x axis. In the absence of the SO interaction (i.e., ωkFτso=0), ω+is symmetric in qas indicated by the black dotted lines. The frequency of the left-propagating mode increases withincreasing strength of the SO interaction, whereas that of theright-propagating mode decreases. In addition, we observethat the left- and right-propagating modes remain gapless atq=0, in contrast to the 2D edge magnetoplasmon [ 19], which develops a gap in one direction. We will come back to this pointin the next section when we compare the ferromagnetic surfaceplasmon with the more familiar surface magnetoplasmon. 224408-6CHIRAL SURFACE AND EDGE PLASMONS IN … PHYSICAL REVIEW B 97, 224408 (2018) FIG. 3. Dependence of the 2D ferromagnetic edge plasmon frequency (scaled with bulk plasmon frequency with q=kF)o n (a) the wave vector along the ydirection with magnetization direction (denoted by m)fi x e di nt h e xdirection and (b) angle θMbetween the magnetization mand the xdirection (i.e., the normal direction of 2D electron liquid plane), including three different strengths of the SO interaction. Note that the Fermi wave vector kFis related to the 2D equilibrium electron density via n0=k2 F 2π. A similar chiral plasmon dispersion was also found in massive Dirac systems [ 23]. In Fig. 3(b),w ep l o t ω+as a function of the polar angle θMbetween the magnetization and the xaxis for a right- propagating wave with a given wave vector of q=0.1kF, where kFis the Fermi wavelength. The extremes in ω+occur when the magnetization is perpendicular to the plane of the2D electron fluid with a maximum at θ M=πand a minimum atθM=0. The SO interaction has no effect on the edge ferromagnetic plasmon when the magnetization lies in theplane of the 2D electron fluid, as shown by the crossing pointatθ M=π 2. Lastly in Fig. 4, we show the variations of the frequency ω+and decaying lengths κ−1 1andκ−1 2[given by Eq. ( 26)] as functions of the strength of the SO interaction. For the right-propagating mode (i.e., q=0.1k F), both ω+andκ−1 i(i=1,2) decrease with increasing ωkFτso, whereas for left-propagating mode (i.e., q=−0.1kF) both ω+andκ−1 i(i=1,2) increase monotonically with increasing ωkFτsoand terminate, similarly to the 3D case, when ωkFτsoreaches a threshold (as indicated by the vertical dashed line) beyond which the edge mode mergeswith the bulk mode. FIG. 4. (a) Variation of the ferromagnetic edge plasmon fre- quency ωvs SO interaction strength parameter ωkFτsoand (b) variation of the decay lengths of the edge modes κ−1 1,2vsωkFτso(as indicated by the black arrows, the dashed lines refer to the variation ofκ−1 1and the solid lines refer to the variation of κ−1 2). Two opposite wave vectors q=0.1kFandq=−0.1kFare considered. The vertical grey dashed lines indicate the critical magnitude of the SO interaction beyond which the edge mode with q=−0.1kFno longer exists. V . DISCUSSION A. Comparison with surface and edge magnetoplasmon Now that we have thoroughly investigated both the surface and edge plasmons in 3D and 2D ferromagnetic conductors,it is worthwhile discussing the features that distinguish themfrom the classical surface or edge magnetoplasmons. In theclassic magnetoplasmon, the direction dependence of theplasmon dispersion arises from the Lorentz force exerted bythe applied magnetic field, while in the present case thereis no magnetic field, but an anomalous velocity connectingthe collective charge oscillation with the bulk magnetization.In addition, we note that the anomalous velocity term playsa role in altering surface ferromagnetic plasmon frequencyonly through the boundary conditions, whereas the Lorentzforce contributes to the time rate of change of the canonicalcurrent (or momentum) density and hence enters the bulk Eulerequation [ 19]. These essential differences are reflected in the dispersion relations. Qualitative differences emerge already inthe long-wavelength limit, as we show below. For a surface magnetoplasmon that propagates in the ( x,y) plane, with magnetic field lying in the same plane, the q→0 limit of the dispersion is given by ω 3D mp(φq)=ωcsinφq 2+/radicalBigg ω2csin2φq 4+ω2 B+ω2ccos2φq 2, (34) where ωc(>0) is the cyclotron frequency and φqis the angle between qand the applied magnetic field. A detailed derivation of surface magnetoplasmon dispersion with arbitrary propa-gation direction is presented in the Appendix. For plasmonspropagating perpendicularly to the in-plane magnetic field, i.e.,φ q=±π 2,E q .( 34) reduces to ω3D mp,⊥=1 2/parenleftbig/radicalBig ω2c+2ω2 B±ωc/parenrightbig , (35) 224408-7STEVEN S.-L. ZHANG AND GIOV ANNI VIGNALE PHYSICAL REVIEW B 97, 224408 (2018) which is exactly the result for the special case discussed by Fetter [ 19]. The general dispersion ( 34) also shows that even when the plasmon wave vector is collinear with the magneticfield (i.e., when φ q=0), the magnetic field still gives rise to a correction to the surface magnetoplasmon frequency of secondorder in ω c, i.e., ω3D mp,/bardbl=/radicalBigg ω2 B+ω2c 2, (36) This may seem a little counterintuitive at first glance, as one may think the Lorentz force, given bye cj×Hwith Handj the magnetic field and the current density respectively, wouldvanish in this geometry; however, this is in fact not the casesince the in-plane current density is in general notparallel to the wave vector in the presence of the magnetic field (seethe Appendix for the general relation between jandq;t h e component of the electric field perpendicular to the surfacecombines with the magnetic field to produce a drift velocityperpendicular to the magnetic field). In addition, due to mirrorsymmetry about the plane perpendicular to the magnetic field,the frequencies of surface plasmons propagating parallel orantiparallel to the applied magnetic field must be identical,giving rise to an effect of the order of O(ω 2 c). Although the frequency of the ferromagnetic surface plasmons alsoremains finite at q=0, it reduces to the normal surface plasmon frequency of ω S=ωB√ 2when the propagation direction of the plasmon waves become collinear with the in-plane magnetization (i.e., φq=0o rπ). The reason for this different behavior is that the anomalous velocity ceases to be operativein the boundary condition for the current density when qis parallel to ˆm[see Eq. ( 11)]. Another difference is that surface magnetoplasmons in the long-wavelength limit remain welldefined for all directions of qand all values of the magnetic field, at variance with ferromagnetic plasmons which in certaindirections may merge with the bulk plasmons when the SOinteraction is strong enough. More significant differences arise in two dimensions. Quot- ing from Ref. [ 19], the dispersion of the low-frequency edge magnetoplasmon for q→0i s ω 2D mp=√ 2 3/bracketleftbig/radicalBig 3ω2q+ω2c+ωcsgn(q)/bracketrightbig , (37) where ωc>0 is the cyclotron frequency, and ωq(=/radicalBig 2πn0e2|q| me) is the bulk 2D plasmon frequency. We see that in this case the right-propagating mode with q> 0 is gapped and approaches a frequency2√ 2 3ωcin the long-wavelength limit, whereas the left-propagating mode with q< 0 has a frequency that goes to zero linearly for q→0, as shown in Fig. 5. This is quite different from the ferromagnetic edge plasmons, where bothright- and left-propagating waves have a dispersion ω∝ω q∝√qforq→0, only with different proportionality constants in the two directions, as shown by Fig. 3(a). B. Material considerations In transition metal ferromagnets, contributions to the anomalous Hall effect due to the intrinsic or side jumpmechanism can be attributed to an anomalous velocity [ 34] FIG. 5. 2D edge magnetoplasmon dispersion relation with the external magnetic field (denoted by H)fi x e di nt h e xdirection, for three different magnitudes of the cyclotron frequency ωc. which, in the present paper, has been shown to play an essential role in imparting chiral properties to the ferromagnetic surfaceplasmons. Consequently, one would expect the ferromagneticsurface plasmons to be observable in transition metal ferromag-nets with large anomalous Hall effect. To obtain some order ofmagnitude estimation for the anisotropy of the ferromagneticsurface plasmon frequency in the long-wavelength limit, as parametrized by η S≡ω(φq=π 2)−ω(φq=0) ω(φq=0), let us consider the side- jump contribution to the anomalous Hall conductivity, which is given by σsj yx∼n0e2 ¯hλ2[35–37]. The effective Compton wavelength λcan thus be estimated from the experimentally accessible anomalous Hall angle θah(≡σyx σxx, with σxx=e2n0τ me the longitudinal conductivity) by τso=meλ2 ¯h=θahτ, where τis the momentum relaxation time. Taking the parameters τ∼10−14s,θah=0.01,P=0.5, and ωB∼50 THz [ 38], we findτso=0.1 fs and ωBτso=0.005, which lead to ηS∼0.1%. Another promising class of materials to observe the 3D ferromagnetic surface plasmons are the diluted magnetic semi-conductors [ 39]. For example, using the material parameters for GaMnAs ω B∼100 THz, λ2/similarequal4.4˚A2, andme∼10−31kg [40,41] we find τso=0.01 fs and ωBτso=0.001, for which the parameter characterizing the anisotropy of the ferromagneticsurface plasmon frequency is evaluated to be η S∼0.01%. In addition to the surfaces of ferromagnetic single layers, it has been shown that the interface between a heavy metaland a ferromagnetic insulator may be host to both strongSO interaction and magnetism [ 42–44]. Therefore, bilayer structures such as Pt/YIG, Au/YIG, etc. may be exploredas another platform for probing the surface ferromagneticplasmons. Recently, spin current generated by surface plasmonresonance was observed [ 45] in a bilayer of Pt /BiY 2Fe5O12 with Au nanoparticles embedded in the Pt layer, indicating the existence of a coupling between surface plasmons andmagnetic ordering in such heterostructures. Ferromagnetic edge plasmons can also be hosted in various systems. One possibility is the generation of edge plasmonsin the conducting surface of magnetic topological insulators.For instance, the quantum anomalous Hall effect was observedin magnetically doped topological insulator (Bi ,Sb) 2Te3[46] 224408-8CHIRAL SURFACE AND EDGE PLASMONS IN … PHYSICAL REVIEW B 97, 224408 (2018) with quantized anomalous Hall conductivity of σQAH yx/similarequale2 ¯h. Given this value, one can estimate the effective “Comp- ton wavelength” to be λ2/similarequal5˚A2[47]. Similar to the 3D ferromagnetic surface plasmons, we can define a quantityγ E=ω(q)−ω(−q) ω(q)|q→0to characterize the chirality of the 2D ferromagnetic edge plasmons. If we use typical values of the parameters for topological insulators [ 48](ωq∼1T H z , me∼10−31kg), we find γE∼0.02%. Another system that may provide interesting results for the 2D ferromagnetic edge plasmons is the 2D electron gasformed at the interface of two dielectric perovskites, suchas LaAlO 3/SrTiO 3[49]o rL a T i O 3/SrTiO 3[50]. The recent observation of ferromagnetism at LaAlO 3/SrTiO 3interfaces [51] is of great relevance for the present study, providing a realization of a high-mobility magnetic 2D electron gas. VI. CONCLUSION A new type of surface plasmon, which depends not on the Lorenz force but on the spin-orbit coupling to the mag-netization, has been identified. We call it a “ferromagneticsurface plasmon.” The frequency and the angular dependenceof the ferromagnetic surface plasmon can be controlled byvarying the direction and the magnitude of the bulk magne-tization. Because the magnetization of a ferromagnetic systemis a dynamical variable with its own intrinsic oscillations(spin waves or magnons) our results foreshadow the excitingpossibility of a coupling between spin waves and surfaceor edge plasmons. Such a coupling could be exploited tocontrol the plasmon dispersion by acting on the magnetizationvia a magnetic field or a current, or, reciprocally, to inducechanges in the magnetization by pumping surface plasmons ina ferromagnetic material. Such possibilities, if realized, couldcreate an unexpected link between the two apparently distantfields of plasmonics and spintronics. ACKNOWLEDGMENTS We thank Alessandro Principi and Olle G. Heinonen for helpful discussions. G.V . and S.S.-L.Z gratefully acknowledgesupport for this work from NSF Grant No. DMR-1406568. Partof the work done by S.S.-L.Z at Argonne National Laboratorywas supported by the Department of Energy, Office of Science,Basic Energy Sciences, Materials Sciences and EngineeringDivision. APPENDIX: 3D SURFACE MAGNETOPLASMON WITH ARBITRARY PROPAGATION DIRECTION In an earlier paper [ 19], Fetter studied the dispersion of the surface magnetoplasmons for the special case in whichboth the propagation direction of the plasmon waves and theapplied magnetic field are lying in the plane of the surfaceand perpendicular to each other. In this Appendix, we examinethe more general case where the plasmons propagate in anyarbitrary direction. To be more specific, we consider surfaceplasmons in a semi-infinite metal layer occupying the spacez<0 and the magnetic field is applied in the ˆxdirection parallel to the surface of the metal layer.Let us start with the following set of bulk hydrodynamic equations, which include the current continuity equation ∂δn ∂t+∇r·j=0( A 1 ) with jthe particle current density, the Euler equation involving the Lorentz force term associated with the cyclotron frequencyω cgiven by ∂j ∂t=−s2∇rδn+en0 me∇rϕ+ωcˆx×j, (A2) and the Poisson equation for the electrostatic potential ϕgiven by ∇2 rϕ=4πeδn. (A3) Note that in the exterior of the metal layer ( z>0) the electrons are absent and hence the Poisson equation reduces to aLaplacian equation, ∇ 2 rϕ=0. (A4) Assuming translational invariance in the x-yplane, we seek solutions in the form of plane waves of wave vector q, but ones that decay exponentially along the zdirection towards the bulk (z<0), i.e., /Psi1(r,z;t)∼eκzei(q·r−ωt), where /Psi1stands for the various hydrodynamic variables under consideration (e.g., δn, j,ϕ, etc.), κ(>0) is the decaying constant, and both qandrlie in thex-yplane. Inserting this ansatz in the bulk hydrodynamic equations, we find a set of linear algebraic equations for thehydrodynamic variables, i.e., −iωδn+κj z+iq·j/bardbl=0,(A5a) −iωjx+iqx/parenleftbigg s2δn−en0 meϕ/parenrightbigg =0,(A5b) −iωjy+iqy/parenleftbigg s2δn−en0 meϕ/parenrightbigg +ωcjz=0,(A5c) −iωjz+κ/parenleftbigg s2δn−en0 meϕ/parenrightbigg −ωcjy=0,(A5d) where j/bardbl=(jx,jy) are the in-plane components of the current density, and the Poisson equation ( A3) establishes a relation between δnandϕas ϕ=4πe κ2−q2δn. (A6) Note that the above equations are invariant under q→− q, ω→−ω, and a complex conjugation, as required by the reality of the electromagnetic waves. It is straightforward to show thatthe set of equations ( A5) have nontrivial solutions only if the following equation is satisfied: (κ 2−q2)2−/bracketleftbigg k2 ω−/parenleftbiggqxωc ω/parenrightbigg2/bracketrightbigg (κ2−q2)−/parenleftbiggqxωBωc sω/parenrightbigg2 =0, (A7) where ωB≡/radicalBig 4πn0e2 meis the 3D bulk plasma frequency, and we have defined k2 ω≡ω2 B+ω2 c−ω2 s2. 224408-9STEVEN S.-L. ZHANG AND GIOV ANNI VIGNALE PHYSICAL REVIEW B 97, 224408 (2018) Note that κmay have two positive solutions, as given by κ2 1,2=q2+1 2/bracketleftbigg k2 ω−q2 xω2 c ω2 ±/radicalBigg/parenleftbigg k2ω−q2xω2c ω2/parenrightbigg2 +/parenleftbigg2qxωBωc sω/parenrightbigg2/bracketrightbigg ,(A8) where κ1andκ2correspond to the solutions with the +sign and the−sign respectively. Having found the decaying constants for the hydrodynamic variables, we can now write down thegeneral solutions for electrostatic potential ϕand the electron density fluctuation δnas follows: ϕ=e iq·r×/braceleftbiggϕ1eκ1z+ϕ2eκ2z,z < 0, ϕ0e−qz,z > 0,(A9) and δn=eiq·r 4πe/bracketleftbig ϕ1/parenleftbig κ2 1−q2/parenrightbig eκ1z+ϕ2/parenleftbig κ2 2−q2/parenrightbig eκ2z/bracketrightbig ,(A10) where we have invoked Eq. ( A6) in deriving the expression for δn, and note that we have dropped the common time-dependent multiplier e−iωtfor ease of notation. Similarly, one can write down the general solution for the normal component of thecurrent density as 4πe j z=/parenleftbig ϕ1J1eκ1z+ϕ2J2eκ2z/parenrightbig eiq·r, (A11) where we have defined the quantities Jα=i/bracketleftbig ω2 B−s2/parenleftbig κ2 α−q2/parenrightbig/bracketrightbig/parenleftbiggωκα+ωcqy ω2−ω2c/parenrightbigg (A12) withα=1,2. Now by imposing the boundary conditions, i.e., the con- tinuity of ϕand∂zϕas well as the condition jz=0a tt h e surface z=0, we arrive at a general equation that determines the dispersion of the surface magnetoplasmon: (κ1+q)J2−(κ2+q)J1=0. (A13) Despite the complicated appearance of the general dispersion equation, its solutions of interests in the long-wavelengthlimit in fact can be solved analytically. Up to O(q 1), the two decaying constants given by Eq. ( A8) reduce to κ2 1→ω2 B+ω2 c−ω2 s2(A14) and κ2 2→q2/bracketleftBigg 1−/parenleftbig ωBωccosφq/parenrightbig2 ω2/parenleftbig ω2 B+ω2c−ω2/parenrightbig/bracketrightBigg , (A15) where we have let qx=qcosφqandqy=qsinφq. By plug- ging Eqs. ( A14) and ( A15)i nE q .( A13) and making somerearrangements, we arrive at a simpler equation for the disper- sion: /parenleftbig ω2 c−ω2/parenrightbig/bracketleftBig 2ω2−2ωcωsinφq−/parenleftbig ωccosφq/parenrightbig2−ω2 B/bracketrightBig =0. (A16) Discarding the unphysical solution of ω=ωc, the surface magnetoplasmon frequency in the q→0 limit is given by ω3D mp/parenleftbig φq/parenrightbig =ωcsinφq 2+/radicalBigg ω2csin2φq 4+ω2 B+ω2ccos2φq 2. (A17) Interestingly, the magnetic field gives rise to a correction to the surface magnetoplasmon frequency even when the plasmonspropagate along the direction of the magnetic field; this can beseen by setting φ q=0o rπ, for which the dispersion relation reduces to ω3D mp,/bardbl=/radicalBigg ω2 B+ω2c 2. (A18) This can be understood since it is the velocity density v(which is parallel to the current density j=n0v) that enters the bulk hydrodynamic equation via the Lorentz force of the form Fl= e cv×Hwith Hthe applied magnetic field; the velocity density, however, is notproportional to the the wave vector qin the presence of the magnetic field. 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PhysRevB.84.054449.pdf

PHYSICAL REVIEW B 84, 054449 (2011) Magnetic dichroism in angle-resolved hard x-ray photoemission from buried layers Xeniya Kozina, Gerhard H. Fecher,*Gregory Stryganyuk, Siham Ouardi, Benjamin Balke, and Claudia Felser Institut f ¨ur Anorganische und Analytische Chemie, Johannes Gutenberg Universit ¨at, D-55099 Mainz, Germany Gerd Sch ¨onhense Institut f ¨ur Physik, Johannes Gutenberg Universit ¨at, D-55099 Mainz, Germany Eiji Ikenaga, Takeharu Sugiyama, Naomi Kawamura, and Motohiro Suzuki Japan Synchrotron Radiation Research Institute, SPring-8, Hyogo 679-5198, Japan Tomoyuki Taira, Tetsuya Uemura, and Masafumi Yamamoto Division of Electronics for Informatics, Hokkaido University, Sapporo 060-0814, Japan Hiroaki Sukegawa, Wenhong Wang, and Koichiro Inomata National Institute for Materials Science, Tsukuba 305-0047, Japan Keisuke Kobayashi National Institute for Materials Science, SPring-8, Hyogo 679-5148, Japan (Received 20 June 2011; published 15 August 2011) This work reports the measurement of magnetic dichroism in angular-resolved photoemission from in-plane magnetized buried thin films. The high bulk sensitivity of hard x-ray photoelectron spectroscopy (HAXPES) incombination with circularly polarized radiation enables the investigation of the magnetic properties of buriedlayers. HAXPES experiments with an excitation energy of 8 keV were performed on exchange-biased magneticlayers covered by thin oxide films. Two types of structures were investigated with the IrMn exchange-biasinglayer either above or below the ferromagnetic layer: one with a CoFe layer on top and another with a Co 2FeAl layer buried beneath the IrMn layer. A pronounced magnetic dichroism is found in the Co and Fe 2 pstates of both materials. The localization of the magnetic moments at the Fe site conditioning the peculiar characteristicsof the Co 2FeAl Heusler compound, predicted to be a half-metallic ferromagnet, is revealed from the magnetic dichroism detected in the Fe 2 pstates. DOI: 10.1103/PhysRevB.84.054449 PACS number(s): 75 .25.−j, 79.60.−i, 85.75.−d, 71.20.Lp I. INTRODUCTION Rapid breakthroughs in the area of spintronics have led to the development of electronic devices with improved perfor- mance. Being a principal constituent part of such devices, com- plex multilayer structures have caused considerable interest in exploring their unique properties and at the same time have made this task rather sophisticated. Along with investigationof micromagnetic properties, an improved understanding of magnetoelectronic properties of deeply buried layers and interfaces in magnetic multilayer structures is of the most importance in the viewpoint of their potential applications in the field of magnetic recording, as data storage devices andsensors. Magnetic circular dichroism (MCD) in photoabsorption and photoemission has become a very powerful tool for theelement-specific investigation of the magnetic properties ofalloys and compounds. Thus far, such studies have beenmainly carried out using soft x-rays, resulting in a rathersurface sensitive technique due to the low-electron mean freepath of the resulting low-energy electrons. The applicationof hard x-rays 1results in the emission of electrons with high kinetic energies and thus, it increases the probingdepth. 2The bulk sensitivity of this technique was recently proved and, for hν > 8 keV , the bulk spectral weight wasfound to reach more than 95%.3Hard X-ray photoelectron spectroscopy (HAXPES) has been found to be a well-adaptablenon-destructive technique for the analysis of chemical andelectronic states. 4,5It was recently shown that HAXPES can be combined easily with variable photon polarization when usingphase retarders. 6Linear dichroism in the angular distribution of the photoelectrons is achieved using linearly polarized hardx-rays and is successfully applied to identify the symmetry ofvalence band states in Heusler compounds. 7In combination with excitation by circularly polarized x-rays,6this method will serve as a unique tool for the investigation of theelectronic and magnetic structure of deeply buried layers andinterfaces. Baumgarten et al. 8carried out a pioneering study on magnetic dichroism in photoemission and observed thisphenomenon in the core-level spectra of transition metals. Theeffect, however, was rather small (a few percentage points)because of the limited resolution of the experiment. It was later shown that dichroic effects are also obtained using linearly or even unpolarized photons. 9,10The observed intensity differ- ences in photoemission are essentially a phenomenon specificto angular-resolved measurements, and therefore, these havebeen termed as magnetic circular dichroism in the angulardistribution (MCDAD). 11,12 054449-1 1098-0121/2011/84(5)/054449(8) ©2011 American Physical SocietyXENIYA KOZINA et al. PHYSICAL REVIEW B 84, 054449 (2011) II. MAGNETIC DICHROISM IN THE ANGULAR DISTRIBUTION OF PHOTOELECTRONS MDAD Theoretical atomic single-particle models were quite suc- cessful in describing, explaining, and predicting many aspectsof magnetic dichroism. Cherepkov et al. elaborated the general formalism for the dichroism in photoemission excited bycircularly, linearly, and unpolarized radiation. 11They showed that MCDAD is very sensitive to the geometry of theexperiment and depends strongly on the relative orientationbetween the magnetization, helicity, and momentum of theexcited electrons. The maximum effect is obtained whenthe magnetization and helicity vectors are parallel; the ef-fect decreases with an increase in the angle between thesevectors. The electronic states in solids usually do not carry a spherical or axial symmetry as in free atoms but have to followthe symmetry of the crystal. 13The angular distribution Ij(k,n) of the photoemitted electrons—as derived, for example, inRef.11for the case of axially symmetric polarized atoms—has to account for the nondiagonal density matrix ρ n NM/prime N.14This leads to the following equation for the case of a nonaxial symmetry: Ij(k,n)=cσ [l]/radicalbigg 3[j] 4π/summationdisplay κ,L/summationdisplay N[N]1/2Cj κLN/summationdisplay x,M/summationdisplay MN,M/prime N ×ργ κxρn NM/prime N(j)Y∗ LM(k)DN MNM/prime N(/Omega1)/parenleftbiggκL N xMM N/parenrightbigg , (1) where landjare the orbital and the total angular momentum of an electron in the initial state. Cj κLNare the dynamic parameters derived from the radial matrix elements and ργ κxare photon state multipoles.14Dj mmj(/Omega1) is the Wigner rotation matrix with /Omega1being the set of Euler angles describing the rotation from the laboratory to the atomic coordinate frame. The direction of the electron momentum→ k=k(θ,φ) is defined by the angles θandφ(see Fig. 1). Finally, cσis a photon-energy-( hν) dependent constant, cσ=4π2αh ν 3, where αis the fine structure constant. FIG. 1. (a) The coordinate system used for the investigation of photoemission. k(θ,φ) is the electron momentum, qis the photon beam and nis the principal axis of alignment. θandφare the angles defining the direction of the outgoing photoelectrons. αis the angle of photon incidence (in the x-zplane) as defined in optics. It is seen that the angle describing the photon propagation in spherical coordinatesis given by /Theta1 q=α+π. The direction of the zaxis corresponds to the quantization axis n. (b) The direction of the in-plane axes xand yis illustrated for an object with C2vsymmetry.TABLE I. State multipoles of |L,J/angbracketright=| 0,1/2/angbracketright,|1,1/2/angbracketright,|1,3/2/angbracketright, and|2,3/2/angbracketrightstates. J1 23 2 MJ +1 2−1 2+3 2+1 2−1 2−3 2 ρ001√ 21√ 21 21 21 21 2 ρ101√ 2−1√ 23 2√ 51 2√ 5−1 2√ 5−3 2√ 5 ρ20 ––1 2−1 2−1 21 2 ρ30 ––1 2√ 5−3 2√ 53 2√ 5−1 2√ 5 This formalism can also be used to consider open-shell atoms and the multiplets resulting from the interaction betweenthe core states and the open-shell valence states. In that case, the dynamic parameters Cj JκLN have to be calculated for the appropriate coupling scheme ( jj, LSJ, or intermediate) with the single particle quantum numbers j,m being replaced by those ( J,M ) describing the complete atomic state.11In that case, the dynamic parameter will redistribute the single-electron results in a particular way over the states of a multiplet(see Refs. 15and16). The state multipoles of the s,p, anddstates that define the intensity and the sign and magnitude of the dichroism aresummarized in Tables IandII. Note that the state multipoles are independent of the orbital angular momentum L, and they depend only on the total angular momentum Jand its projection M J. A. MDAD equations for the grazing incidence geometry In the following, let us consider the special case of geometry with the photons impinging in the x-zplane with unit vector of the photon momentum ˆq=(−cos(α),−sin(α),0). At such a grazing incidence with α=π/2 it becomes ˆq=(−1,0,0). The electrons are observed in the direction perpendicu-lar to the photon beam ( θ= π 2−α) with the momentum ˆk=(−sin(θ),0,cos(θ)). At a photon incidence of α=π/2 it becomes ˆk=(0,0,1). (Compare also Figs. 1and3.) Now examine the case:→n→−→nwhere the magnetic dichroism emerges from a switching of the direction of magnetization with the initial direction→n=(1,0,0) that is along the xaxis. Applying Eq. ( 1) and the state multipoles TABLE II. State multipoles of |L,J/angbracketright=| 2,5/2/angbracketrightstates. J5 2 MJ −5 2−3 2−1 2+1 2+3 2+5 2 ρ001√ 61√ 61√ 61√ 61√ 61√ 6 ρ10 −5√ 70−3√ 70−1√ 701√ 703√ 705√ 70 ρ205 2√ 21−1 2√ 21−2√ 21−2√ 21−1 2√ 215 2√ 21 ρ30 −5 6√ 57 6√ 52 3√ 5−2 3√ 5−7 6√ 55 6√ 5 ρ401 2√ 7−3 2√ 71√ 71√ 7−3 2√ 71 2√ 7 ρ50 −1 6√ 75 6√ 7−5 3√ 75 3√ 7−5 6√ 71 6√ 7 054449-2MAGNETIC DICHROISM IN ANGLE-RESOLVED HARD X- ... PHYSICAL REVIEW B 84, 054449 (2011) of Table I, the circular magnetic dichroism in the angular distribution for pstates is given by the equations: CMDADσ+(p) =−ρ10sin(α)/parenleftbig/radicalBig 2 3C(1,0,1) JkLN+/radicalBig 1 15C(1,2,1) JkLN (1−6 cos2(α))/parenrightbig CMDADσ−(p) =+ρ10sin(α)/parenleftbig/radicalBig 2 3C(1,0,1) JkLN+/radicalBig 1 15C(1,2,1) JkLN (1−6 cos2(α))/parenrightbig (2) The circular magnetic dichroism in the angular distribution (CMDAD) for opposite helicity of the photons has an oppositesign. The equations for the p 1/2andp3/2states are the same. The magnitude differs, however, because of the differences inthe state multipoles ρ 10and dynamical parameters CJkLN . Forα=π/2 the CMDAD of the pstates ( J=1/2,3/2) becomes simply: CMDADσ±(pJ)=∓ρ10/parenleftbigg/radicalBig 2 3C(1,0,1) JkLN+/radicalBig 1 15C(1,2,1) JkLN/parenrightbigg . (3) The linear counterpart LMDAD vanishes in that geometry, independent whether the photons are sorppolarized. At α=π/2 the magnetic dichroism in the angular distribution vanishes for all pstates independent of the polarization of the photons if the magnetization is perpendicular to the planespanned by the photon incidence and the electron momentum [here for the x-zplane with→n=(0,±1,0)]. III. EXPERIMENTAL DETAILS The present study reports on the MCDAD experiment in the HAXPES range on different types of exchange-biased structures with epitaxially grown ferromagnetic layersof CoFe and Co 2FeAl, these being typical materials used in tunnel magnetoresistive devices (see Fig. 2). The on- top approach multilayers were deposited in the sequenceMgO(100) substrate/MgO buffer layer (10 nm) /Ir 78Mn 22 (10 nm) /CoFe (3 nm) /MgO barrier (2 nm) /AlOx(1 nm)17 that corresponds to the lower exchange-biased electrode of a magnetic tunnel junction (MTJ). After growth the stacks wereannealed at 350 ◦C for 1 h in vacuum of 5 ×10−2Pa in a magnetic field of 0.4 MAm−1to provide exchange biasing of the CoFe layer film through the IrMn/CoFe interface (seealso Ref. 18). The on-bottom configuration was realized in the multilayer sequence MgO(100) substrate/Cr buffer layer(40 nm) /Co 2FeAl (30 nm) /Ir78Mn 22(10 nm) /AlOx(1 nm).19 The sample stacks were annealed at 400◦C for 1 h in vacuum under a magnetic field of 0.4 MAm−1to provide exchange biasing to the Co 2FeAl thin film through the Co 2FeAl/IrMn interface (see also Ref. 19). In both cases, the topmost AlOxlayers served as a protective coating. All metal layers were deposited by magnetron sputtering and electron beamevaporation was used to epitaxially grow the MgO barrier.IrMn serves as an exchange-biasing layer that keeps CoFe orCo 2FeAl magnetized in preset directions. The magnetized samples were mounted pairwise with opposite magnetization on the same sampleholder and canbe selected via sample shift. Care was taken that the mag-netization directions were antiparallel and that surfaces were MgO (001) substrateMgO buffer (10 nm)Ir78Mn22(10 nm)CoFe (3 nm)MgO (2 nm) FM AFM MgO (001) substrateCrbuffer (40 nm)Ir78Mn22(10 nm) Co2FeAl (30 nm)AlO x(1 nm) AFM FMAlO x(1 nm)(b) (a) FIG. 2. (Color online) Sketch of the exchange-biased films used in the dichroism experiments. The multilayer structure in (a) corresponds to the lower part of the electrode and is realized inon-top configuration with CoFe ferromagnetic layer. The structure shown in (b) presents on-bottom configuration with Co 2FeAl ferro- magnetic layer. In both films a 1-nm-thick AlO xlayer is used as a protective cap. parallel to avoid different detection angles. The mounting of the samples at the fixed sample manipulator was chosen to haveup/down as well as left/right pairs as it is shown in Fig. 3). This allowed to probe the dichroism by varying both the directionof magnetization and the direction of helicity. The HAXPES experiments with an excitation energy of 7.940 keV were performed using beamline BL47XU atSPring-8. 20The energy distribution of the photoemitted electrons was analyzed using a hemispherical analyzer (VG-Scienta R4000-12 kV) with an overall energy resolution of150 or 250 meV . The angle between the electron spectrometerand the photon propagation was fixed at 90 ◦. The detection angle was set to θ=2◦in order to reach the near-normal emission geometry and to ensure that the polarization vectorof the circularly polarized photons is nearly parallel ( σ −) or antiparallel ( σ+) to the in-plane magnetization M+.T h e MM e(a) MM e(b) FIG. 3. (Color online) Scheme of the experimental geometry. The incidence angle θ(with respect to the surface plane) of the circularly polarized photons was fixed to 2◦. X-rays of opposite helicity (σ+andσ−) were provided by a phase retarder. Further, samples with opposite directions of magnetization are used. In (a) the in-planemagnetization Mis nearly parallel to the beam axis and in (b) the in-plane magnetization is perpendicular to the beam axis. The electron detection is fixed and perpendicular to the photon beam. 054449-3XENIYA KOZINA et al. PHYSICAL REVIEW B 84, 054449 (2011) sign of the magnetization was varied by mounting samples with opposite directions of magnetization ( M+,M−). The polarization of the incident photons was varied using anin-vacuum phase retarder based on a 600- μm-thick diamond crystal with (220) orientation. 21The direct beam is linearly polarized with Pp=0.99. Using the phase retarder, the degree of circular polarization is set such that Pc>0.9. The circular dichroism is characterized by an asymmetry that is defined asthe ratio of the difference between the intensities I +andI−and their sum, A=(I+−I−)/(I++I−), where I+corresponds toσ+- andI−toσ−- type helicity. Magnetic dichroism may be defined in a similar manner using the differences in theintensities if the direction of the magnetization is changedkeeping the polarization of the photons fixed. The photon flux on the sample was about 10 11photons per second in a bandwidth of 10−5during the measurements at the given excitation energy. The vertical spot size on the sampleis 30μm, while in horizontal direction, along the entrance slit of the analyzer, the spot was stretched to approximately 7 mm.The measurements were performed using grazing incidencegeometry. The resulting count rates (taken from the equivalentgray scale values provided by the spectrometer software) werein the order of 0.6 to 6 MHz for the core level spectra, includingshallow core levels and about 0.25 MHz for the valence band. IV . RESULTS AND DISCUSSION Figure 4shows the 2 pcore-level spectra of Co that were taken from an exchange-biased CoFe film that wascovered by oxide films. A pronounced difference was observedin the spectra taken with photons having opposite helicityfor a fixed direction of magnetization. The pure difference/Delta1I=I +−I−presented in the figure is already free of the influence of the background and gives the correct shapeof the magnetic dichroism. This means that it containsall characteristic features of the magnetic dichroism. For FIG. 4. (Color online) Polarization-dependent photoelectron spectra of the Co 2 pcore-level emission from CoFe on top of an IrMn exchange-biasing layer and the difference of two spectra. Asymmetry values are marked at selected energies.quantification and comparison of the dichroic effects, the MCDAD asymmetry was determined from A=(I+−I−) (I++I−)=/Delta1I 2I(4) after subtracting a Shirley-type background from the spectra to find the asymmetry caused only by the direct transition.The background subtraction leads, however, to a very lowintensity in the beginning, in the end of the spectral energyrange as well as in the range between the spin-orbit split peaksin both spectra (that is in the ranges of the spectra where nosignal from the transition itself is expected). This, in turn, leadsto very high and rather nonphysical values of the calculatedasymmetry in these energy ranges. From the above remark on/Delta1Iit is, therefore, advantageous to show the differences of the intensities and to mark the asymmetry for characteristicenergies only. Here the largest obtained asymmetry value is−42% at Co 2 p 3/2. As one can see, the spin-orbit splitting of the Co 2 pstates is clearly resolved, as expected. When going from p3/2to p1/2, the dichroism changes its sign across the 2 pspectra in the sequence: −+ + − ; as appears characteristic of a Zeemann-type mjsublevel ordering. This sequence of signs is directly expected from Eq. ( 3) and the state multipoles ρ10 given in Table Iwhen identifying the states of the magnetically split 2 pdoublet as |j,mj/angbracketrightin the single-particle description. The details of the MCDAD reveal, however, that the situationis more complicated. In particular, the dichroism in the Fe 2 p spectra does not vanish in the region between the spin-orbitdoublet. The multiplet formalism to describe the spectra inmore detail will be given below. MCDAD has previously been used to investigate the itinerant magnetism of ferromagnetic elements such as Co,Fe, and Ni, where it was explained in terms of single-particlemodels. 12,22–24As demonstrated in the case of Ni, however, the single-particle approach poorly describes all the peculiaritiesof the complex spectra. van der Laan and Thole considered theMCDAD phenomenon by taking into account the influenceof electron correlation effects in the frame of atomic many-particle models that were successfully used to describe bothlocalized and itinerant magnetism phenomena. 11,15,24Many- body effects play an important role when using polarizedincident photons. The correlation between spin and orbitalmoments, 2 pcore-hole, and spin-polarized valence band results in a rich multiplet structure that spreads out over a wideenergy range of a spectrum. 25In strongly correlated systems, the bulk magnetic and electronic properties differ markedlyfrom the surface ones. However, as observed previously,MCDAD with radiation in the soft x-ray range is highlysensitive to the surface where the dichroism is influencedby symmetry breaking. 26Because of the strong inelastic electron scattering in this energy range, the escape depthof the photoemitted electrons of a few angstroms becomescomparable to the thickness of a monolayer. The tuning ofthe excitation energy also affects the photoionization crosssections. At high energies, the intensities from the dstates of transition metals are reduced as compared to the partial crosssections of the sandpstates. 2,27,28The shape and magnitude of the asymmetry depend on the partial bulk to surface spectral 054449-4MAGNETIC DICHROISM IN ANGLE-RESOLVED HARD X- ... PHYSICAL REVIEW B 84, 054449 (2011) FIG. 5. (Color online) Illustration of the vanishing dichroism in photoemission when the photon polarization vector is perpendicularto the in-plane magnetization vector demonstrated for the Co 2 p state of CoFe. Shown are the photoelectron spectra I +,I−and their difference I+−I−obtained with different helicity at fixed magnetization perpendicular to the photon beam. weights; hence, only at high energies, the dichroism effects appear to be related to the bulk properties. It was carefully proven that the dichroism vanished in the geometry in which the projection of the photon vector isperpendicular to the magnetization, independently of whetherthe photon helicity or the magnetization was reversed. Thisindicates that the films are perfectly magnetized in the directionforced by the exchange-biasing layer magnetization. As anexample, Fig. 5confirms the absence of the dichroic signal at t h eC o2 pstates of the CoFe film in agreement to the theoretical description given above. Figure 6shows the polarization dependence of the CoFe valence band spectra together with the resulting magneticdichroism. The MCDAD observed for the valence band ismuch smaller as compared to the core-level photoemission.The largest asymmetry is approximately −2% at−1 eV below the Fermi energy. Such low asymmetry values were also observed when using low photon and kinetic energies. 29Only for excitation close to threshold, higher asymmetries arise inthe case of one- 30and two-photon photoemission.31In the range of the valence states, the detection is further complicatedby the signal from the underlying IrMn layer that does notcontribute to the dichroism. Because of the thin layer of CoFeand the large escape depth of the nearly 8 keV fast electrons,the two layers cannot be distinguished in the valence band.It is worthwhile to note that the dichroic signal itself arisesexclusively from the buried, ferromagnetic CoFe layer. For studies aimed toward the development of novel devices, it is necessary to also detect the magnetic signal from deeplyburied layers. To prove the reliability of the proposed method,experiments were also performed on samples in which theIrMn exchange-biasing layer was on top of the layer structure. Figure 7compares the MCDAD results for the shallow core levels of CoFe in the on-top configuration [Fig. 7(a)] and the deeply buried Co 2FeAl in the on-bottom configurationFIG. 6. (Color online) MCDAD in valence band of CoFe on top of IrMn. The asymmetry is given at −1 eV below Fermi level. beneath a 10-nm-thick IrMn film [Fig. 7(b)]. For such complex multilayer structures, the situation becomes complicated inthat the signals from all the elements contained in thesystem are detected. In both cases the shallow core levelsof all elements of the multilayers are detected. The intensitydifferences between Fe and Co 3 pemission or Ir 4 fand Mn 3 p in the different configurations are obvious and arise from thedamping of the intensity when the electrons pass through thelayers above the emitting layer. Strong signals are still detectedfrom the buried elements even though the ferromagneticCo 2FeAl layer lies 10 nm beneath the antiferromagnetic IrMn layer, as is clearly seen in the inset of Fig. 7(b).Al a r g e asymmetry is clearly observed at the Co and Fe signals, andthese are the ones responsible for the ferromagnetic properties FIG. 7. (Color online) MCDAD for the shallow core level spectra obtained from the buried CoFe on top and Co 2FeAl beneath a 10-nm- thick IrMn film. The insets show an enlarged view of the Fe 3 pstates. 054449-5XENIYA KOZINA et al. PHYSICAL REVIEW B 84, 054449 (2011) FIG. 8. (Color online) Polarization-dependent photoelectron s p e c t r ao ft h eF e2 pcore-level emission from CoFe on top of an IrMn exchange-biasing layer, Co 2FeAl beneath IrMn and the corresponding differences of the spectra taken with the opposite helicity of light. Asymmetry values are marked at selected energies. The insets show an enlarged view of I+at the Fe 2 p3/2states in both cases. of the system. The asymmetries of −56% (CoFe) and −45% (Co 2FeAl) in the Fe 3 psignal are quite evident. In Co 3 p,i ti s well detected even though the direct spectra overlap with theIr 4fstates. Figure 8shows the polarization dependent HAXPES spectra and the MCDAD at the Fe 2 pstates of the buried CoFe (a) and Co 2FeAl (b) layers. The multiplet splitting at the Fe 2 p3/2is very well resolved and the MCDAD is well detected in both materials. The emission from the Co 2FeAl has a lower intensity and the resolution was therefore reducedto 250 meV in order to keep the counting rates comparableto those of the CoFe measurements. (Note that this doesnot influence the spectra much as they are governed by alifetime broadening that is in the same order of magnitude.)It was shown [32] that linear magnetic dichroism (LMDAD)along with the circular one can be successfully applied toinvestigate the electronic and magnetic properties of surfacesand interfaces. The LMDAD asymmetry observed at Fe 2 p 3/2, however, was only at most −9% for a low excitation energy. In our studies the maximum asymmetries are −59% for CoFe and−41% for Co 2F e A la tF e2 p3/2, and this is ideal for the analysis of the magnetic properties. Closer inspection of the MCD spectra [see insets of Figs. 8(a) and 8(b)] reveals a striking distinction between t h eF e2 pspectra of the two layer systems. Even though taken with a slightly lower resolution, the multiplet splitting of theFe 2p 3/2emission from Co 2FeAl appears more pronounced as compared to the corresponding spectrum from CoFe. Themean splitting /Delta1E of the Fe 2 p 3/2states is 0.8 and 1.0 eV for CoFe and Co 2FeAl, respectively. Co 2FeAl is supposed to be a half-metallic ferromagnet with a magnetic moment of5μ Bin the primitive cell and about 2.8 μBper Fe atom,33 whereas CoFe is a regular band ferromagnet with a very high magnetic moment (about 2.5 μBat Fe).34In both cases the Fe moment is clearly above that of pure Fe (2.1 μB). One of the major differences is the localized magnetic moment of Fein Co 2FeAl that is caused by a strong localization of the t2gFIG. 9. (Color online) Calculated polarization-dependent photo- electron spectra of the Fe 2 pcore-level emission obtained by means of atomic multiplet calculations and their difference for CoFe (a) and Co2FeAl (b). The insets show the enlarged views of the difference curve in the region between spin-orbitally split components of Fe 2 p states. The bars mark the multiplet states. bands. In the ordered case of both compounds, the Fe atoms are in a cubic environment and are surrounded by 8 Co atoms.Co 2FeAl forms a perfect 23CsCl supercell with every second Fe atom of CoFe replaced by Al. This causes additional Co-Albonds that reduce the Co-Fe d-state overlap. The result is a localized moment at the Fe sites. From this viewpoint, Fe inCo 2FeAl is in closer to an covalent than a metallic state. For the Fe atoms, this causes a more pronounced interaction of thecore hole at the ionized 2 pshell with the partially filled 3 d valence shell. As mentioned above, the single-particle theory cannot explain the details of the spectra and their dichroism. Itis necessary to respect the coupling between the ionizedcore and open valence shells. In the present case, this isthe interaction between the 2 p 5core hole and the open 3 d valence shell of Fe. Therefore, multiplet calculations werecarried out to explain the experimentally obtained resultsfor the two different materials. They were performed bymeans of the charge transfer multiplet calculations for x-rayabsorption spectroscopy ( CTM4XAS )5.2 program,35using its x-ray photoelectron spectroscopy (XPS) option. The resultsare shown in Fig. 9. The simulations were made for a Fe 3+ ionic ground state with 4 s03d5configuration that describes well the emission from the Fe-2 pstates of both systems, CoFe and Co 2FeAl. The Slater integrals ( Fdd,Fpd, andGpd)w e r e reduced to 0.65; 0.55; 0.65 and 0.7; 0.5; and 0.5 of the freeatom values to describe the spectra of CoFe and Co 2FeAl, respectively. As exchange interaction plays an important rolein ferromagnetic materials, the effect of exchange splitting wastaken into account by setting the magnetic splitting parameterMto 50 meV for CoFe and 450 meV for Co 2FeAl. The obtained values for the splitting /Delta1E o ft h eF e2 p3/2states are 0.9 and 1.1 eV for CoFe and Co 2FeAl, respectively. The applied parameters resulted in a quite good agreementbetween calculated and experimental spectra and dichroism.Possible, slight disagreements may be attributed to the factthat the observed spectra depend on the degree of localization 054449-6MAGNETIC DICHROISM IN ANGLE-RESOLVED HARD X- ... PHYSICAL REVIEW B 84, 054449 (2011) or itineracy of the magnetic moment at the Fe site through the coupling of the 2 p5core hole with the d-valence bands. Fractional d-state occupancies (for example, d5+x,0<x< 1) that might better describe the partial delocalization of d electrons of Fe in metallic systems, however, are not availablein the atomic model. The insets in Fig. 9present a enlarged view of the region of the dichroism between the main linesof the multiplet. In those insets one clearly recognizes theappearance of multiplet states over the entire energy range.These states form the characteristic structure of the dichroismthat is in a good agreement with the experiment. It is worthwhile to note that such differences in the multiplett structure of two very similar alloys are not resolvedby X-ray circular dichroism (XMCD) in soft x-ray photoabsorption. 36This is found easily if comparing the here shown photoelectron spectra and dichroism to previously reportedXMCD spectra of Fe containing Heusler compounds 37–39 where the XMCD spectra and dichroism appear rather without any resolved splitting of the L2;3lines. V . SUMMARY AND CONCLUSIONS In summary, MCDAD in hard x-ray photoelectron spec- troscopy was used to study the magnetic response of thecore level of buried, remanently magnetized layers. Usingbulk-sensitive HAXPES-MCDAD, it was shown that IrMnexchange-biasing layers keep thin films of CoFe or Co 2FeAl remanently magnetized in a well-defined direction. Dichroismin the valence band spectroscopy is complicated in metal/metallayers; however, the situation will improve in metal/insulatorstructures in which the insulator does not contribute to thestates at the Fermi energy. 4The magnetic dichroism from core levels, including shallow core levels, of CoFe and buriedCo 2FeAl multilayer has asymmetries up to above 58% when it is excited by circularly polarized hard x-rays and is thusmuch larger compared to that in the case of excitation by softx-rays. As a noteworthy result, the differences in the Fe 2 p emission from a regular ferromagnet (CoFe) and a suggestedhalf-metallic ferromagnet (Co 2FeAl) were demonstrated. The splitting observed in Co 2FeAl points to the covalent character of the compound. Overall, the high bulk sensitivity of HAXPES combined with circularly polarized photons will have a major impacton the study of the magnetic phenomena of deeply buriedmagnetic materials. The combination with recently proposedstanding wave methods 40,41will allow an element-specific study of the magnetism of buried layers and make feasiblethe investigation of the properties of magnetic layers not onlyat the surface but also at buried interfaces. ACKNOWLEDGMENTS Financial support by Deutsche Forschungsgemeinschaft and the Strategic International Cooperative Program of JST(DFG-JST: FE633/6-1) is gratefully acknowledged. We arethankful to the Japan Synchrotron Radiation Research Institute(JASRI) for the support of experiments within the approved2009B0017 proposal. The team of Hokkaido Universityacknowledges the support of MEXT, Japan (Grants-in-Aid20246054, 21360140, and 19048001) X.K. acknowledges thesupport of MAINZ. *fecher@uni-mainz.de 1I. Lindau, P. Pianetta, S. Doniach, and W. E. Spicer, Nature 250, 214 (1974). 2K. Kobayashi et al. ,Appl. Phys. Lett. 83, 1005 (2003). 3S. Suga and A. Sekiyama, Eur. Phys. J. Special Topics 169, 227 (2009). 4G. H. Fecher et al. ,Appl. 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PhysRevB.97.035101.pdf

PHYSICAL REVIEW B 97, 035101 (2018) Interplay between bandwidth-controlled and filling-controlled pressure-induced Mott insulator to metal transition in the molecular compound [Au(Et-thiazdt) 2] B. Brière,1J. Caillaux,1Y. L e G a l ,2D. Lorcy,2S. Lupi,3A. Perucchi,4M. Zaghrioui,1J. C. Soret,1 R. Sopracase,1and V . Ta Phuoc1 1GREMAN, CNRS UMR 7347-CEA, Université François Rabelais, UFR Sciences, Parc de Grandmont, F-37200 Tours, France 2Univ Rennes, CNRS, ISCR (Institut des Sciences Chimiques de Rennes) – UMR 6226, F-35000 Rennes, France 3CNR-IOM and Dipartimento di Fisica, Università di Roma Sapienza, P .le Aldo Moro 2, I-00185 Roma, Italy 4Elettra – Sincrotrone Trieste S.C.p.A., AREA Science Park, I-34149 Basovizza, Trieste, Italy (Received 4 July 2017; published 3 January 2018) Optical properties of the quasi-two-dimensional single-component molecular Mott insulator [Au(Et-thiazdt) 2] (Et-thiazdt =N-ethyl-1,3-thiazoline-2-thione-4,5-dithiolate) have been investigated under pressure at room temperature. At 1.5 GPa, [Au(Et-thiazdt) 2] undergoes an insulator to metal transition (IMT). Optical conductivity spectra exhibit a clear Drude peak at high pressure. In addition, we observed a clear anisotropy of pressure-inducedmodifications of the electronic structure. With increasing pressure, along the molecule stacks, a strong increaseof the spectral weight below 1 eV is observed, while in the transverse direction, it remains barely constant witha redistribution from midinfrared to low energy. Besides the increase of the singly occupied molecular orbital(SOMO) bandwidth, calculations show that the SOMO-1 bands cross the Fermi level at the transition. Moreover,we have calculated the optical conductivity as a function of pressure to provide a picture of the compound physicsunder 1 eV . Our results indicate that the pressure-induced IMT is simultaneously due to a bandwidth and aband-filling phenomenon that imply both Mott physics and uncorrelated charge carriers. DOI: 10.1103/PhysRevB.97.035101 One fascinating phenomenon in strongly correlated fermion systems is the transition from an insulator to a metallic statedriven by electronic correlations. An electron in a crystalgoes from localized to itinerant (or vice versa) when a con-trol parameter such as doping, pressure, or temperature ismodified. For a half-filled system, this problem is knownas the Mott transition [ 1]. Understanding the properties of Mott insulators has become of primary importance since thediscovery of molecular and high- T csuperconductors [ 2–5] which have in common the fact that from a half-filled bandinsulator they become metallic and superconducting withdoping (a so-called “doped Mott insulator”). On this basis,extensive efforts have been devoted to developing new typesof molecular metals and superconductors by enhancing theinterstack interactions [ 6]. One solution to raise conductivity and to overstep the charge repulsion issue is to increase theelectronic bandwidth Wwith respect to the on-site Coulomb repulsion U, which can easily be done by applying pressure [7,8], especially in organic materials due to their soft nature. It has been extensively studied in Mott organic salts suchas the one-dimensional (1D) tetramethyl-tetrathiafulvalene(TMTTF) [ 9–11] or the two-dimensional (2D) κ-(Bis(ethylene dithio)tetrathiafulvalene) 2X(BEDT-TTF) [ 12,13]. Moreover, crystals of molecular conductors present numerous advantagesfor organic electronics compared to traditional inorganic ma-terials: low fabrication cost, high mechanical flexibility, lightweight, and ease of fabrication [ 14]. For this purpose, the use of neutral organic radicals as building blocks is a promisingtrack to design materials with unpaired electrons which serveas charge carriers without any doping process [ 15].Among single-component organic conductors based on bis(1,2-dithiolene) ligands [ 6,16–21], [Au(Et-thiazdt) 2]( E t - thiazdt =N-ethyl-1,3-thiazoline-2-thione-4,5-dithiolate) is the first well characterized molecular metal without TTF dithiolateligands. It also is the first molecular member of a new class ofresistive random access memory (RRAM) called “Mott mem-ories.” RRAM applications are made possible by the resistiveswitching phenomenon (resistive transition induced by electricpulses). Whereas the resistive switching is usually obtained byelectrochemical effects or charge trapping, it is based on a pureelectronic intrinsic effect in inorganic Mott GaTa 4Se8[22] and in [Au(Et-thiazdt) 2][23]. This phenomenon could constitute a generic feature of Mott insulators. Indeed, with a 2D structuremade of molecules stacked along the baxis and interstacked along the aaxis [see Supplemental Material (SM) [ 24] part I], [Au(Et-thiazdt) 2] is a half-filled electron system with signifi- cant electronic correlations that place it close to an insulator tometal transition (IMT) controlled by U/W . Pressure reduces U/W sufficiently to induce the IMT at 1.3 GPa [ 19]. In this paper, we report a pressure-dependent optical study of [Au(Et-thiazdt) 2] single crystals made possible by the use of a home-made high-vacuum microspectrometer and asynchrotron radiation source (see SM [ 24] and Ref. [ 25]). In order to understand the mechanisms of the IMT, we performedfirst-principles density functional theory (DFT) calculations.The study highlights the “Mottness” of the compound anddescribes a different type of Mott transition which involvestwo types of charge carriers. In order to elucidate the ambient pressure electronic struc- ture at room temperature, we measured the polarized optical 2469-9950/2018/97(3)/035101(7) 035101-1 ©2018 American Physical SocietyB. BRIÈRE et al. PHYSICAL REVIEW B 97, 035101 (2018) FIG. 1. Optical conductivity spectra between 43 meV and 1.2 eV for (a) E||aand (b) E||bat ambient pressure and room temperature. Dotted lines provide the gap. Curves with red circles represent the sum of all the contributions. conductivity (see Fig. 1)[26,27]. The spectra exhibit an insu- lating behavior in both E||aand E||bdirections. Although both spectra can be decomposed in the midinfrared (MIR) with atleast two absorption bands roughly located at 0.43 and 0.74 eV ,they differ in shape and level. Along the aaxis, the amplitude of the 0.74 eV band is higher than the 0.43 eV one. Alongthebaxis, it is to the contrary and the spectrum also exhibits an overall lower absolute level. Despite these differences, theoptical gap is roughly the same in both directions (250 meV),in good agreement with transport experiments and previousoptical measurements [ 23]( s e eS M[ 24] part I for the gap extraction procedure). In order to assign the absorption features to electronic transi- tions, we performed ab initio calculations [ 28–31]. Calculation details are provided in the SM [ 24]. Although DFT does not fully include electronic correlations, it provides valuable infor-mation, as evidenced in various studies [ 15,32]. The band struc- ture with the “fatband” representation shown in Fig. 2allows one to identify the singly occupied molecular orbital bands(SOMO) (green), the SOMO-1 (cyan), the SOMO-2 (blue),and the lowest unoccupied molecular orbital bands (LUMO)(orange). Remarkably, two SOMO bands cross the Fermilevel/epsilon1 F, driving the state metallic, contrary to experiments. This is a typical signature of strong electronic correlationsinsofar as DFT underestimates on-site interactions. Moreover,the SOMO-1 bands remain below /epsilon1 Fand the bandwidth of SOMO bands W=0.37 eV . Note that the band dispersion along the /Gamma1-Ydirection is directly related to electron hopping FIG. 2. (a) Band structure with the “fatband” representation calculated at 0 GPa, showing the molecular orbital character. (b)Density of states (DOS).FIG. 3. Calculated optical conductivity tensor at ambient pressure. along the baxis. The /Gamma1-Xdirection is close to the interstack direction asince ˆ ac=91.83(5)◦[19]. Nevertheless, due to the monoclinic symmetry, the optical conductivity tensor isnondiagonal. The nonzero elements of the tensor are providedin Fig. 3. The experimental conductivity along the aaxis [Fig. 2(a)] involves both σ xxandσxzcomponents (see SM [24] part II and Ref. [ 33]).σyyis directly compared to the experimental conductivity along b[Fig. 1(b)]. Remarkably, σxxandσxzare dominated by an interband contribution SOMO-1 →SOMO at 0.4 eV (cyan curves, Fig.3). Note that a small intraband contribution (green curve) appears in σxxdue to the presence of SOMO bands at /epsilon1F. Moreover, two interband transitions SOMO-2 →SOMO and SOMO →LUMO are predicted around 1 eV . SOMO-1 → LUMO is predicted around 1.5 eV (above the experimentalrange). The calculated conductivity for the E||bcase (σ yy)i s dominated by the SOMO intraband contribution. In addition,σ zzindicates an insulating character, in agreement with the 2D character of the system. It is crucial to notice that the experiment exhibits a first contribution around 0.43 eV (green contribution, Fig. 1) and a second one around 0.74 eV (cyan contribution) whereasthe DFT predicts an intraband (green curve, Fig. 3) and an interband contribution SOMO-1 →SOMO around 0.4 eV (cyan curve). This discrepancy can be understood bytaking into account the on-site correlations U. For a large enough U, SOMO bands are split into a filled lower Hubbard band (LHB) and an empty upper Hubbard band (UHB),resulting in an Mott insulator [Fig. 11(a) ]. By considering the correlations, the DFT intraband SOMO contribution shouldshift by U, towards higher energies [ 34]. Consequently, the contribution at 0.43 eV , which has to correspond to the DFTintraband contribution, is assigned to the electronic transitionsLHB→UHB. Moreover, the DFT interband contribution SOMO-1 →SOMO should shift by U/2 towards higher energies in the presence of correlations. Therefore, theexperimental contribution at 0.74 eV could be assigned toSOMO-1 →UHB (empty states of SOMO) excitations. Hence, with an overall good agreement with the experimentaldata, we deduce U=0.43 eV . Such a value of Uis commonly found in this class of Mott organic compounds [ 35,36]. Note that U−W=0.43–0.37≈0.1 eV , which is the same order of the electronic gap. The higher-energy excitations 035101-2INTERPLAY BETWEEN BANDWIDTH-CONTROLLED AND … PHYSICAL REVIEW B 97, 035101 (2018) FIG. 4. Optical conductivity pressure dependence between 43 meV and 1 eV for (a) E||aand (b) E||bat room temperature. Circles between 210 and 330 meV represent the diamond absorption. (c) shows the low-frequency extrapolated behavior of the conductivity along b. This extrapolation was obtained by fitting the low-frequency reflectivity Rsd(measured through the diamond) with a Drude model. (d) provides the gap pressure dependence and (e) shows SW calculated forωc=1 eV along both directions. SOMO →LUMO and SOMO-2 →SOMO above the experimental spectral range will not be discussed. The experimental polarized conductivity spectra presented in Fig. 4from 0 to 5 GPa (at room temperature) exhibit an obvious rising Drude-like response at low frequency above1.5 GPa, a decrease, and a closing of the gap between 1 and1.5 GPa [Fig. 4(d)]. Hence, by applying pressure, [Au(Et- thiazdt) 2] undergoes an IMT. Remarkably, the low-frequency data extrapolation repre- sented in Fig. 4(c) are qualitatively consistent with room- temperature transport measurements realized up to 2.1 GPa byTenn et al. [19]. Indeed, as shown in Fig. 5, they both exhibit the FIG. 5. Comparison between room-temperature σdcvalues (up to 2.1 GPa) obtained by transport measurements by Tenn et al. [19]a n d by extrapolation of the optical conductivity data at ω=0. They are qualitatively consistent according to both the order of magnitude andthe pressure dependence. FIG. 6. Pressure dependence of the band structure calculated by DFT. Gray zones underline the bandwidth of the SOMO bands. same pressure dependence and order of magnitude. Note that this extrapolation was obtained by fitting the low-frequencyreflectivity R sd(measured through the diamond) with a Drude model. Also, error bars were extracted by evaluating both thehighest and the lowest dc values obtained within the Drudemodel framework, which reasonably fit the experimental data. The IMT exhibits several features consistent with the dynamical mean field theory (DMFT) picture of the Mott transitions [ 37,38]: the rise of a Drude peak due to a coherent quasiparticle response and the persistence of the main MIR ab-sorption band at 0.43 eV assigned to LHB →UHB transitions at 0 GPa. In order to rule out possible structural changes (from which the IMT could originate), we performed vibrational proper-ties using infrared and Raman spectroscopies. The phononfrequencies as a function of pressure obtained by fitting theinfrared and Raman spectra with Drude-Lorentz oscillators andGaussian-Lorentzian profiles, respectively, are reported in theSupplemental Material [ 24]. Within experimental resolution, apart from the usual blueshift of phonon modes (due tovolume lattice shrinking), no phonon anomaly (disappearance,appearance, splitting, kink in the phonon frequency pressuredependence) has been noticed at 1.5 GPa and beyond, in bothinfrared and Raman spectra. This indicates that no structuralphase transition has been detected under pressure, in agreementwith the Mott transition picture. Note that the anisotropy still persists at high pressure as it is clearly evidenced by the higher absolute level of the low-frequency conductivity along bthan along aand the spectral weight (SW) values [see Fig. 4(e)]. In the bdirection, SW increases between 0 and 3 GPa and is constant above 3 GPa,whereas along a, it slightly decreases between 0 and 1.5 GPa and is constant above 1.5 GPa. In order to understand the pressure effects on the elec- tronic structure and to disentangle the IMT mechanisms, weperformed a full optimization of cell parameters and internalcoordinates at different pressures between 0 and 4 GPa withinDFT [ 39], as successfully done for other organic molecular compounds [ 15,40]. Pressure effects on the band structure are depicted in Fig. 6. Note that the lack of dispersion along /Gamma1-Zindicates that the system mainly remains 2D at high pressure. In addition, theband shape is almost unchanged between 0 and 4 GPa butthe bandwidth increases from W=0.37 to 1.05 eV for the SOMO and from 0.70 to 1.05 eV for the SOMO-1 bands.As a consequence, U/W goes from 1.16 at 0 GPa to 0.41 035101-3B. BRIÈRE et al. PHYSICAL REVIEW B 97, 035101 (2018) FIG. 7. Pressure dependence of the SOMO bandwidth Wcalcu- lated by DFT. Pressure dependence of the experimental LHB →UHB contribution damping γ. They both increase with pressure and they are in agreement with a usual bandwidth enlargement observed in correlated organic 2D conductors through the Mott transition [ 12,43]. at 4 GPa, well below the critical value of U/W ≈1f o rw h i c h IMT is theoretically expected to occur [ 41,42]. Consequently, the role of correlations is expected to be weaker when pressureincreases and charge carriers are supposed to be less correlated.In addition, as shown in Fig. 7, the evolution of the bandwidth Wand the pressure dependence of the damping γof the optical conductivity contribution assigned to LHB →UHB transitions are compatible with a usual bandwidth enlargementobserved in correlated 2D organic conductors through the Motttransition [ 12,43]. The damping was obtained by fitting the Hubbard midinfrared absorption band from 0 to 5 GPa with aDrude-Lorentz model. Remarkably, the increase of dispersion in both /Gamma1-Xand/Gamma1-Y directions raises the energy level of the SOMO-1 bands goingat/Gamma1from 0 −to 0.10 eV between 0 and 2 GPa. As a result, the SOMO-1 bands cross /epsilon1Fand become hole filled. In contrast, the SOMO bands (previously assigned to UHB and LHB whencorrelations are included) go slightly deeper under /epsilon1 Fwith pressure, getting electron filled. Hence, part of the IMT isachieved by tuning the electronic filling of the UHB, suggestinga pressure-induced doping of the Mott insulator. Thus, twomechanisms are driving the IMT: the bandwidth enlargement FIG. 8. Calculated optical conductivity tensor at 4 GPa.FIG. 9. Comparison between calculated and experimental con- ductivities, for (a) 0 and (b) 4 GPa. Inset: DFT SW calculated at 1 eV . and the band filling. Pressure induces a bandwidth enlargement and an electronic doping of the SOMO bands (UHB) thatimplies physics related to correlated electrons (“Mottness”).On the contrary, since the SOMO-1 bands are far from halffilling at 0 and 4 GPa, the charge carriers arising from the holedoping of SOMO-1 bands are expected to be less correlated oruncorrelated. Comparing the conductivity tensors calculated at 4 GPa (Fig. 8) and at 0 GPa (Fig. 3), we notice an enhancement of the low-frequency conductivity for both σ xxandσyydriving the IMT. Moreover, the SOMO-1 →SOMO contribution is shifted by ≈0.2 eV towards low frequency and its amplitude is also clearly lower than at ambient pressure. In contrast,forσ zz, apart from the decrease of the magnitude of the SOMO-1 →SOMO contribution between 0 and 4 GPa, there is no significant changes at low frequency. Hence the systemremains 2D, as previously observed. Remarkably, the Drudecontribution present in σ xxandσyyis composed of two different intraband contributions: one due to SOMO bands and anotherone produced by the hole filling of the SOMO-1 bands (cf.green curves of Fig. 8). Note that the SW of the total Drude is lower in σ xxthan in σyy, underlining the remaining anisotropy at high pressure. In order to get a global picture of the IMT, in Fig. 9we plot both the calculated (deduced from the nondiagonal opticalconductivity tensor) and experimental optical conductivity at0 and 4 GPa along the aand baxis. Calculations at 4 GPa in both directions are in good agreement with the experiment inmany respects. Along the aaxis, the DFT SW [see the inset, Fig.9(b)] decreases slightly from 0 to 4 GPa, and along b,i t increases between 0 and 4 GPa. Moreover, the low-frequencyDFT conductivity level is consistent with the anisotropyevidenced experimentally since σ DFT b>σDFT aat high pressure. Noticeably, the agreement between DFT and experiments isclearly better at high pressure since the role of electroniccorrelations is expected to be weaker than at ambient pressure. FIG. 10. Contributions of optical conductivity spectra for (a) E||a and (b) E||bat 4 GPa and room temperature. 035101-4INTERPLAY BETWEEN BANDWIDTH-CONTROLLED AND … PHYSICAL REVIEW B 97, 035101 (2018) FIG. 11. Schematic representation of the electronic structure. (a) Effects of correlations. (b) Effects of pressure. The vertical arrows represent the electronic transitions. Naively, in correlated systems, we could consider that the electronically active bands at /epsilon1Fare ruled/dominated by electronic correlations. However, for [Au(Et-thiazdt) 2], other uncorrelated interactions seem to occur. The high-pressure experimental conductivity can be fitted under 1 eV with a minimal model consisting of three Lorentzoscillators and one Drude peak (Fig. 10). By comparing the experimental and theoretical results (done at 0 GPa), aschematic view of the electronic structure in the high-pressurephase is depicted in Fig. 11(b) . Note that at 4 GPa, DFT reproduces quite well the experimental data, except for thecontribution centered at 0.40 eV (green contribution of Fig. 10), which is missing in DFT (Fig. 8). Accurately, the two Drude contributions predicted by DFT (dark and dashed green curves,Fig.8) are included in the experimental Drude (red contribution of Fig. 10). They are assigned to two types of intraband transitions since the SOMO-1 bands and the quasiparticle peak(QP) cross /epsilon1 F. Note that QP originates from the bandwidth enlargement of the Hubbard bands, as predicted by DMFT [ 38]. The lower-frequency interband contribution at ≈0.2 eV (cyan contribution, Fig. 10) is assigned to both SOMO-1 →SOMO and LHB →QP transitions. Note that the LHB →QP transi- tions usually cannot be identified unambiguously in Mott or-ganic compounds because the coherent response is not clearlyseparated from the LHB →UHB excitations [ 43,44]. At 4 GPa the empty states of the SOMO bands are composed of the UHBand the empty states of the QP peak. They are “the unoccupiedHubbard states.” Therefore, at 4 GPa, SOMO-1 →SOMO transitions are equivalent to SOMO-1 →unoccupied Hubbard state transitions. The SOMO-1 →SOMO contribution shifts from 0.74 eV (cyan contribution, Fig. 1)t o≈0.2 eV between 0 and 4 GPa because the SOMO-1 bands and UHB are gettingcloser together with pressure. Hence, the energy requiredfor the transition between SOMO-1 bands and unoccupied Hubbard states is reduced. Similarly, as predicted by DFT, theenergy required to induce SOMO-2 →SOMO and SOMO → LUMO transitions (blue and orange curves, Fig. 8) is reduced going from ≈1t o≈0.75 eV between 0 and 4 GPa due to bandwidth enlargement. Therefore, these contributions arefound at ≈0.75 eV (blue/orange dashed contribution, Fig. 10). They are not clearly detected at 0 GPa because they are locatedbeyond 1 eV , but with a rough estimation of the correlationeffect, we find 1 eV +U/2≈1.2 eV , which is consistent with the onset of the contribution at 1.2 eV (blue/orange dashedcontribution, Fig. 1). Furthermore, the experimental contribution, located at 0.4 eV (green contribution, Fig. 10) and unpredicted by DFT, should correspond to transitions reminiscent of LHB →UHB, as predicted by DMFT. In conclusion, a Mott IMT induced by pressure has been evidenced in the organic compound [Au(Et-thiazdt) 2]. The originality of the transition lies in the existence of two co-operative mechanisms (bandwidth and band filling) that drivethe state metallic. On the one hand, SOMO-1 bands are filledwith holes, underlining the existence of uncorrelated chargecarriers. On the other hand, the bandwidth enlargement ofHubbard bands and the filling (doping) of the UHB implythe presence of correlated carriers. Therefore, two types ofcharge carriers are expected to coexist in the metallic phase,as it appears in other intriguing systems [ 45–47]. 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PhysRevB.100.195132.pdf

PHYSICAL REVIEW B 100, 195132 (2019) Rashba splitting of Dirac points and symmetry breaking in strained artificial graphene Vram Mughnetsyan ,1Aram Manaselyan,1Manuk Barseghyan ,1Albert Kirakosyan,1and David Laroze2,* 1Department of Solid State Physics, Yerevan State University, Alex Manoogian 1, 0025 Yerevan, Armenia 2Instituto de Alta Investigación, CEDENNA, Universidad de Tarapacá, Casilla 7D, Arica, Chile (Received 1 August 2019; revised manuscript received 30 October 2019; published 20 November 2019) The effect of Rashba spin-orbit interaction and anisotropic elastic strain on the electronic, optical, and thermodynamic properties of an artificial graphenelike superlattice composed of InAs /GaAs quantum dots has been theoretically considered. The electronic energy dispersions have been obtained using Green’s functionformalism in combination with the Fourier transformation to the reciprocal space and an exact diagonalizationtechnique. We have observed a splitting of Dirac points and the appearance of additional Dirac-like points due tothe Rashba spin-orbit interaction. Furthermore, a breaking of the hexagonal symmetry of the dispersion surfacescaused by the strain anisotropy is observed as well. It is shown that both the spin-orbit interaction and strainanisotropy have a qualitative impact on the measurable characteristics of the considered structure and can beused as effective tools to control the performance of devices based on artificial graphene. DOI: 10.1103/PhysRevB.100.195132 I. INTRODUCTION Dirac materials are foreseen to be of paramount importance because of their universal behavior and the robustness of theirproperties which are linked to symmetry [ 1,2]. Their band structure is similar to one of relativistic massless particleswhere the energy dependence on the momentum is linearin the vicinity of touching (Dirac) points of the electronicbands. Graphene is an innate example of a one-atom-thick 2Delectron system composed by carbon atoms on a honeycomblattice with two inequivalent sites in the unit cell. Due toits unique electronic spectrum, graphene makes possible theobservation and test of table-top quantum relativistic phenom-ena in experiments, which are unobservable in high-energyphysics [ 3]. The preference of graphene for spin logic appli- cations instead of other metals and semiconductors has beenrecently experimentally tested [ 4,5]. The main features which make graphene an advantageous option for these applicationsare the large spin signal, the long spin diffusion length, andthe relatively long spin lifetime [ 6,7]. In principle, Dirac-type singularities may exist in any 2D lattice with the similar underlying symmetry as graphene.Advanced methods such as atom-by-atom assembling [ 8], nanopatterning of 2D electron gas in semiconductors [ 9], and optical trapping of ultracold atoms in crystals of light [ 10] make it possible to design and fabricate artificial honeycomblattices or artificial graphene (AG), which is a unique structurefor the investigation and manipulation of several systemsdisplaying massless Dirac quasiparticles, topological phases,and strong correlations. One of the reasons for pursuing thestudy of AG is the opportunity for regimes difficult to achievein these systems, such as high magnetic fluxes, tunable latticeconstants, and precise manipulation of defects, edges, andstrain [ 11]. These studies enable tests of several predictions *dlarozen@uta.clfor massless Dirac fermions. For future experiments based on AG, the availability of semiconductors and metals with largespin-orbit coupling opens new exciting potential features toinvestigate topological phases of artificial matter. It has been shown that two-dimensional electron gas in a periodic potential of the honeycomb array of GaAs /GaAlAs quantum dots (QDs) can result in isolated massless Diracpoints with controlable Fermi velocity (FV) [ 12]. The control- lable FV , in turn, can lead to bound states of Dirac fermions[13], which is crucial for building practical digital devices with well-defined on /off logical states [ 14]. The realization of massless Dirac fermions in standard semiconductors opensinteresting possibilities regarding the impact of the spin-orbitinteraction (SOI), especially when using InAs-based materialssuch as honeycomb lattice of InAs /GaAs QDs [ 15]. Although the growth of homogeneous and spatially or- dered arrays of InAs /GaAs QDs is a technological challenge [15,16], recent studies point to the possibility of controlling the size and shape, as well as the electron concentration inthem, using strain engineering and selective area epitaxy.In this regard, there is a good prospect to achieve uniform,position-controlled InAs QDs in the near future [ 17–22]. It is known that the elastic strains at the InAs /GaAs heterojunction due to the lattice mismatch dramatically altersthe electronic band structure [ 23–26]. It has been shown that the strain anisotropy in InAs /GaAs AG leads to the shift of the Dirac points from the Kand the K /primepoints of the first Brillouin zone (FBZ) resulting in anisotropy in the FV and qualitativechanges in the density of states (DOS) [ 27]. The optical properties of transistors [ 28], optical switches [29–31], midinfrared photodetectors [ 32,33], photovoltaic de- vices [ 34], ultrafast lasers [ 35], etc., significantly rest on the light-matter interaction, limited in graphene (optical absorp-tion is less than 2 .5%). One of the advantages of AG is the possibility to overcome this limitation and tune the absorptioncoefficient (AC) by means of external factors such as RashbaSOI. The possibility to study the collective optical response 2469-9950/2019/100(19)/195132(8) 195132-1 ©2019 American Physical SocietyVRAM MUGHNETSY AN et al. PHYSICAL REVIEW B 100, 195132 (2019) of modulated nearly 2D electrons [ 36,37] and holes [ 38]i n semiconductors is another advantage of AGs based on QDs. The heat capacity (HC) is a measurable thermodynamic quantity that can be considered as a sensitive tool to bringout the modifications in the electron energy spectrum inQD, as well as in graphene structures due to internal andexternal factors [ 36,39–41]. The study of HC in AG is of great interest due to the possibility to observe the combinedeffects originated by the quantum confinement in QDs and theunderlying honeycomb symmetry. In this regard, the consideration of the Rashba SOI and the elastic strain field in AG opens perspectives for the control ofthe optical and thermal properties of Dirac fermions. In the present paper, the effect of Rashba SOI on the electronic band structure and DOS, as well as optical andthermodynamic properties of AG composed of highly strainedInAs/GaAs QDs has been considered. The paper is organized as follows: In Sec. II, the model and the method are presented. In Sec. III, the results are displayed and the corresponding discussion is given. The conclusions are presented in Sec. IV. II. THEORETICAL MODEL Our theoretical model is based on the following assump- tions. In the view of strong quantization in the directionperpendicular to the plane of the superlattice (SL), we willassume that electron makes a two-dimensional motion in theplane of the SL. Further, due to very weak dependence of thehydrostatic strain on the coordinate in the transverse direction,only the in-plane variations of the strain will influence themotion of the electron [ 42]. The method developed in Ref. [ 43] allows one to derive an analytic expression for the Fourier components of thestrain tensor for a single QD of arbitrary shape in a materialwith a lattice of cubic symmetry (see the Appendix). In theframework of the mentioned approach, the hydrostatic strainin a two-dimensional SL of honeycomb symmetry, composedof cylindrical QDs of the height h dand the radius rdis as follows [ 27]: ˜εh(ξ1,ξ2)=3/summationdisplay i=1/integraldisplay dξ3˜εii(ξ1,ξ2,ξ3)=ε0˜χQD(n1,n2) ×/parenleftbigg 3−C11+2C12 π/integraldisplay∞ −∞ξ−1 3sin(ξ3hd/2)dξ3 C12+C44+/Lambda1−1 ξ/parenrightbigg , (1) where /Lambda1ξ=3/summationdisplay p=1ξ2 p C44ξ2+Canξ2p(2) and ˜χQD(n1,n2)=2πrdJ1(rd|/vectorG|) s0|/vectorG|A(n1,n2)(1−A(n1,n2)) (3) is the Fourier component of the SL’s shape function [ 42,44], J1(t) is the first kind of Bessel function of the first order, /vectorG=n1/vectorg1+n2/vectorg2is the 2D lattice vector in reciprocal space, /vectorg1=(2π/3a)(1;√ 3) and /vectorg2=(2π/3a)(1;−√ 3) are theelementary vectors of the reciprocal lattice, ais the smallest distance between the centers of QDs in the SL, A(n1,n2)= exp (−i2π(n1+n2)/3),n1,2are integers, s0is the area of the SL’s unite cell, ˜ εii(/vectorξ) is the 3D Fourier transform of the diagonal element of the strain tensor in SL, ξ1=Gx,ξ2=Gy, ξ2=/summationtext3 i=1ξ2 i,C11,C12, and C44are the elastic moduli of the matrix material (GaAs), Can=C11−C12−2C44is the parameter of anisotropy, ε0=(a1−a2)/a2is the initial strain [45], and a1anda2are the lattice constants of the GaAs and InAs lattices, respectively. It should be noted that when the condition hd/lessmuchrdis satis- fied, the dependence of hydrostatic strain on the zcoordinate is weak [ 42] and Eq. ( 1) can be used for the calculation of the hydrostatic strain in 2D space: εh(/vectorr)=/summationdisplay /vectorG˜εh(/vectorG)ei/vectorG/vectorr.(4) The Hamiltonian of the considered system is H=1 2ˆp1 m(/vectorr)ˆp+HSO+V(/vectorr), (5) where HSO=α ¯h(/vectorσ×/vectorp)z (6) is the Rashba SOI Hamiltonian which arises in 2D electron systems due to an inversion of asymmetry of the confinementpotential perpendicular to the 2D plain direction. The SOIconstant αcan be tuned by an external electric field in that direction [ 46,47]. In Eq. ( 5),V(/vectorr)=v 0(/vectorr)+acεh(/vectorr)i st h e periodic potential of QD SL, v0(/vectorr)=Q(Eg,GaAs−Eg,InAs) is the potential of unstrained structure, Eg,GaAs(InAs) is the band gap of GaAs(InAs) material, Qis the conduction band offset, acis the hydrostatic potential constant, and m(/vectorr)i s the electron effective mass. Due to the periodicity of theHamiltonian Eq. ( 5), one can make a Fourier transformation to the momentum space [ 44,48]: ψ ↑(↓)(/vectorr)=1 Sei/vectork/vectorru/vectork↑(↓)(/vectorr)=1 S/summationdisplay /vectorGu/vectork,/vectorG↑(↓)ei(/vectork+/vectorG)/vectorr, (7) V(/vectorr)=/summationdisplay /vectorGV/vectorGei/vectorG/vectorr, (8) 1 m(/vectorr)=/summationdisplay /vectorGm−1 /vectorGei/vectorG/vectorr. (9) Note that in Eq. ( 7),u/vectork↑(↓)(/vectorr) and u/vectork,/vectorG↑(↓)are the Bloch amplitude and its Fourier transform for the spin-up (spin-down) component of the spinor /hatwideψ, respectively. Also, V /vectorGand m−1 /vectorGare the Fourier transforms of the SL potential and inverse effective mass, respectively. Finally, /vectorkis quasimomentum and Sis the effective area of the AG. Substituting the expressions Eq. ( 7)–(9) to the Ben Daniel-Duke’s equation H/hatwideψ=E/hatwideψ, one can arrive at the following set of linear equations in 195132-2RASHBA SPLITTING OF DIRAC POINTS AND SYMMETRY … PHYSICAL REVIEW B 100, 195132 (2019) reciprocal space: /summationdisplay /vectorG/prime/parenleftbigg/bracketleftbigg¯h2 2m−1 /vectorG−/vectorG/prime(/vectork+/vectorG)(/vectork+/vectorG/prime)+V/vectorG−/vectorG/prime−Eδ/vectorG,/vectorG/prime/bracketrightbigg u/vectork,/vectorG/prime↑ +αδ/vectorG,/vectorG/prime[i(kx+G/prime x)+(ky+G/prime y)]u/vectork,/vectorG/prime↓/parenrightbigg =0, (10) /summationdisplay /vectorG/prime/parenleftbigg/bracketleftbigg¯h2 2m−1 /vectorG−/vectorG/prime(/vectork+/vectorG)(/vectork+/vectorG/prime)+V/vectorG−/vectorG/prime−Eδ/vectorG,/vectorG/prime/bracketrightbigg u/vectork,/vectorG/prime↓ −αδ/vectorG,/vectorG/prime[i(kx+G/prime x)−(ky+G/prime y)]u/vectork,/vectorG/prime↑/parenrightbigg =0, (11) where m−1 /vectorG=δ/vectorG,0m−1 GaAs+(m−1 InAs−m−1 GaAs)˜χQD(/vectorG) and V/vectorG= (v0/s0)˜χQD(/vectorG) are the Fourier transforms of the electron’s inverse mass and the SL potential, respectively. The electronicdispersions are obtained by means of diagonalization of the set of Eqs. ( 10) and ( 11) for each value of the quasimomentum /vectork. The DOS of the considered structure can be expressed as follows: ρ(E)=1 (2π)2/summationdisplay j/integraldisplay FBZδ(E−Ej(/vectork))d2k, (12) where the integration is carried out over the FBZ and jdenotes the number of the miniband. Assuming that the Fermi energy EFis on the touching point between two couples of splitted minibands, the AC caused bythe allowed direct transitions is α(ω)=α 02/summationdisplay i=14/summationdisplay j=3/integraldisplay FBZd2k|Mi,j(/vectork)|2 ×δ(¯hω−(Ej(/vectork)−Ei(/vectork)), (13) where Mi,j(/vectork)=¯h/summationdisplay /vectorG/parenleftbig u(i) /vectork,/vectorG↑u(j) /vectork,/vectorG↑+u(i) /vectork,/vectorG↓u(j) /vectork,/vectorG↓/parenrightbig (/vectorG/vectorη)(14) is the dipole matrix element of the transitions from the ith to the jth miniband, α0=e2(m2 0chdω√/epsilon1)−1,ωand/vectorηare the frequency and the polarization vector of the incident photon,/epsilon1is the dielectric constant, m 0andeare the free mass and the electron charge, respectively, and cis speed of light.We have also calculated the electronic HC of the system using the following expression [ 41,49]: cV=/integraldisplay Eρ(E)∂f(E,T) ∂TdE, (15) where integration is carried out over all the conduction bands, f(E,T)=(eβ(E−μ(T))+1)−1is the Fermi-Dirac distribution function, and β=1/kBTandμ(T) is the chemical potential. One can obtain the dependence of the chemical potential onthe temperature by solving the following equation: n=/integraldisplay ρ(E)f(E,T)dE, (16) where it is assumed that the electron 2D concentration nin the conduction band is constant and the Fermi energy E F= μ(T=0) is on the touching point between two couples of split minibands. III. DISCUSSION The numerical calculations are carried out for the fol- lowing values of the parameters: a=22 nm, rd=10 nm, hd=2n m , mInAs=0.023m0,mGaAs=0.067m0,Eg,GaAs= 1518 meV, Eg,InAs=413 meV, and Q=0.6[50]. Taking into account that the electron is mostly localized in the QD regions,we use the value of the dielectric constant in InAs material(/epsilon1=12.3) for the AC Eq. ( 13). The energy level broadening is taken into account, replacing the Dirac δfunction in Eq. ( 13) by the Lorentzian function with the value of the broadeningparameter /Gamma1=0.2m e V[ 51]. Figure 1represents the electronic dispersion surfaces with- out (a) and with [(b) and (c)] Rashba SOI for isotropicallystrained AG. The vicinity of the K /primepoint is mentioned by a dashed rectangle in Fig. 1(b), while the zoom of the cor- responding region is shown in Fig. 1(c). It is obvious from the comparison of Figs. 1(a) and1(b) that each surface splits in two due to SOI. Moreover, in the zoom of the vicinity oftheK /primepoint one can observe an obvious multiplication of Dirac points [see Fig. 1(c)]. Namely, around each Dirac point which is in the corner of the FBZ [the red line in Fig. 1(c)], three extra Dirac-like points appear where the minibands areattached. These points are shifted from the corner of the FBZalong the diagonals of the three hexagons with the same FIG. 1. Dispersion surfaces for the splitted by SOI electronic minibands of isotropically strained AG. (a) The entire picture, (b) the first four minibands in the vicinity of the K/primepoint, and (c) two touching minibands in the vicinity of K/primepoint (the position of the K/primepoint is indicated by a red line). 195132-3VRAM MUGHNETSY AN et al. PHYSICAL REVIEW B 100, 195132 (2019) FIG. 2. Dispersion surfaces for the splitted by SOI electronic minibands of anisotropically strained AG. (a) The entire picture, (b) two touching minibands in the vicinity of K/primepoint (the position of the K/primepoint is indicated by a red line), and (c) the top view of the first miniband dispersion surface (the red dashed lines indicate the diagonals of three hexagons with the same corner in reciprocal space and their crossing point coincides with the K/primepoint). corner and compose an equilateral triangle K1K2K3[Fig. 1(c)]. There are also three Dirac-like points of touching for eachcouple of the splitted surfaces which we refer as points S 1,S2, andS3. These points compose two equilateral triangles which are rotated by 180owith respect to the triangle K1K2K3around the energy axis passing through the K/primepoint [the red line in Fig. 1(c)]. Figure 2represents the electronic dispersion surfaces in the presence of Rashba SOI for anisotropically strained AG.Figure 2(b) shows the zoom of the region mentioned by the dashed rectangle in Fig. 2(a), while Fig. 2(c) represents the top view of the dispersion surface of the first miniband. FromFig. 2(b), the effect of the strain anisotropy on the symmetry of the dispersion surfaces is obvious. One can observe thatthe dispersion surfaces in the vicinity of Kpoints coincide with those in the vicinity of K /primepoints when rotated by 180o [Fig. 2(a)]. Importantly, both the Dirac points and the Dirac- like points are shifted from the corners of the FBZ [Figs. 2(b) and2(c)]. A more detailed examination shows that the shift ofK1is along one of the axes of the FBZ, while the shifts ofK2andK3are no longer along corresponding diagonals. In addition, the shift of K1is significantly larger than the shifts of the two other Dirac-like points [Fig. 2(c)]. As a result, the dispersion surfaces are neither of hexagonal nor of square FIG. 3. Dependence of the FV on the angle between the x axis and the /vectork−/vectorkDin the vicinity of 1: K/primepoint for isotropically strained AG, 2: K1point for isotropically strained AG, 3: Dirac point near the K/primepoint for the anisotropically strained AG, 4: K1point for anisotropically strained AG, 5: K3point for anisotropically strained AG.symmetry and they keep only the symmetry of reflection with respect to kxandkyaxes. The accurate analysis of the solutions of the set of Eqs. ( 10) and ( 11) in the vicinity of Dirac and Dirac-like points indicate the linear dependence of the energy on |/vectork−/vectorkD|, where /vectorkDstands for the position of the Dirac or the Dirac-like point in the FBZ. Furthermore,the proportionality coefficient, which is the analog of the FV , depends on the orientation of the /vectork−/vectork Din contrast to the case of the conventional honeycomb lattice. Based on theabove-mentioned regularities, one can introduce an effectivelow-energy Hamiltonian in the vicinity of each Dirac and Dirac-like point in the following way: H eff=¯hvF(ϕ)|/vectork−/vectorkD|, where vF(ϕ) is the projection of the energy gradient at point /vectorkDon the direction of /vectork−/vectorkDandϕis the angle between the /vectork−/vectorkDand the xaxis. (a) (b) FIG. 4. Density of states for isotropically (a) and anisotropically (b) strained AG with (red lines) and without (black lines) Rashba SOI. Arrows indicate the touching points of the first and the second minibands. 195132-4RASHBA SPLITTING OF DIRAC POINTS AND SYMMETRY … PHYSICAL REVIEW B 100, 195132 (2019) (a) (b) FIG. 5. Absorption coefficient for isotropically (a) and anisotrop- ically (b) strained AG with Rashba SOI for different directions of polarization vector of incident photon. The insets represent thecorresponding graphs in the absence of SOI. Figure 3illustrates the dependence of the FV on the angle ϕ in the vicinity of the touching points (Dirac and Dirac-like) ofthe second and the third minibands. For isotropically strainedAG, there is a third-order rotational symmetry around the K /prime point; that is why the results for only the K/primeandK1points are illustrated. At the K/primepoint, the FV is almost constant leading to a dispersion like in graphene. However, the FVat the Dirac-like point K 1has an oscillatory dependence on ϕ. For anisotropically strained AG, K1andK2(3)represent physically different points in the FBZ, which leads to differentdependencies of FV on ϕ. One can observe that the curve, which corresponds to K 3point, is not symmetric regarding the line ϕ=π/2 in contrast to all other curves. This fact is connected with the shift of the point K3from the diagonal of corresponding hexagon and the breaking of the structurehexagonal symmetry due to the strain anisotropy. The DOS in the presence (red lines) and the absence (black lines) of Rashba SOI is plotted in Fig. 4for isotropically [Fig. 4(a)] and anisotropically [Fig. 4(b)] strained AG (in the figure, a B≈3.57 nm is the effective Bohr radius in InAs). The results for AG without SOI are taken from the Ref. [ 27] for comparison. An obvious multiplication of the maxima ofDOS is observed. Namely, for an isotropically strained AG[Fig. 4(a)], two peaks are replaced by eight. Each of these peaks corresponds to the energy when the gradient of one of(a) (b) FIG. 6. Heat capacity of isotropically (a) and anisotropically (b) strained AG with (red solid lines) and without (black dashed lines) Rashba SOI. The insets show the corresponding chemical potential dependence on the temperature. the splitted surfaces is zero. The comparison of Figs. 4(a) and 4(b) shows that each peak of the DOS is duplicated because of the strain anisotropy. The splittings of the left and theright peaks are very weak (less then 0.05 meV) because theycorrespond to the zero gradient regions of dispersion surfaceswhich are very close to the center of FBZ. The effect of the miniband splitting and the symmetry change of the dispersion surfaces on the AC of AG is pre-sented in Fig. 5. The dependencies of the AC on incident pho- ton energy for four different values of the angle ϕbetween the light polarization vector and the xaxis are shown for isotrop- ically [Fig. 5(a)] and anisotropically [Fig. 5(b)] strained AG. The insets correspond to the absorption spectrum due to thetransitions between the first and the second minibands in theabsence of the Rashba SOI. It is noteworthy that when SOIis absent, the absorption curves corresponding to differentlight polarizations almost coincide for isotropically strainedAG [see the inset of Fig. 5(a)]. This effect is connected with the hexagonal symmetry (symmetry of the SL) of thesection of dispersion surfaces by the plane of constant energycorresponding to the allowed optical transitions in momentumspace. However, the strain anisotropy removes the above-mentioned symmetry, leading to the significant splitting ofthe curves which intersect at a fixed value of the incidentphoton energy [see the inset of Fig. 5(b)]. Furthermore, one can observe a pronounced maxima of AC in the presence of 195132-5VRAM MUGHNETSY AN et al. PHYSICAL REVIEW B 100, 195132 (2019) SOI [in both Figs. 5(a) and5(b)], which is associated with the corresponding peaks of the DOS. An obvious splitting of the curves corresponding to dif- ferent polarizations of incident photon caused by SOI is alsoobserved for both isotropically and anisotropically strainedAG. Like the anisotropically strained SL without SOI [theinset of Fig. 5(b)], there is a certain value of the incident photon energy [indicated by arrows in Figs. 5(a) and5(b)]a t which the values of the AC for different light polarizationscoincide. In Fig. 6, the temperature dependence of the HC on the AG is presented with and without Rashba SOI for two con-sidered different cases of strain. The insets of Fig. 6show the temperature dependence of the chemical potential, assumingthat the 2D concentration of electrons in the conduction bandis constant and defined by the Fermi level, which is on thetouching point between two couples of splitted minibands. Anobvious increase can be observed for the chemical potentialwhich has larger values when there is no SOI. As we cansee from the figures, HC has a nonmonotonic behavior. It iszero at T=0, because of the vanishing DOS at the Fermi energy. With the increase of the temperature, the HC initiallyincreases as well due to the occupation of the states in theenergy regions where the DOS is maxima. However, withfurther increase of the temperature, the states with higherdensity in the energy scale become saturated, leading to asmaller increase of the system’s mean energy. As a result, theHC starts to decrease at some value of T. A comparison of Figs. 6(a) and6(b) shows that the strain anisotropy results in the shift of the maximum of the HC to the region of lowertemperatures. On the other hand, the Rashba SOI alwaysincreases the value of the HC because it removes the twofoldspin degeneracy of minibands, leading to a necessity of extraenergy for occupation of the splitted minibands. IV . CONCLUSION In summary, we have considered the effect of Rashba SOI on the energy dispersion, DOS, AC, and the HC of theAG composed by InAs /GaAs QDs, taking into account the anisotropic elastic strain field due to the lattice mismatchbetween the materials of QDs and the matrix. Splitting ofDirac points due to the SOI have been observed. The Dirac-like points K 1,K2, and K3are shifted from the Kpoint along the diagonals of adjacent hexagons in k-space when an isotropic strain is considered. However, in the case ofanisotropic strain, only the K 1point is shifted along a diagonal of the Brillouin zone. The detailed analysis of the dispersionsurfaces in the vicinity of touching points of minibands indi-cate the oscillatory behavior of the FV as a function of therotation angle around the touching point. The oscillations arenot only observed for the rotation around the K /primepoint in the isotropically strained structure. The DOS in the presence ofSOI has eight characteristic peaks. Moreover, each of thesepeaks is duplicated due to the strain anisotropy. Additionally,it is shown that both the Rashba SOI and the strain anisotropyhave a qualitative effect on the measurable quantities of AG,like AC and HC. In particular, the splitting of the absorptionspectra for different polarizations of incident photons, as wellas the significant change in the HC make the Rashba couplingan effective tool for controlling the optical and thermal char- acteristics of AG. ACKNOWLEDGMENTS This work was supported by the State Committee of Sci- ence of RA (Research Project No. 18T-1C223). V .M. ac-knowledges partial financial support from EU H2020 RISEProject CoExAN (Grant No. H2020-644076). D.L. acknowl-edges partial financial support from Centers of Excellencewith BASAL /CONICYT financing, Grant No. FB0807, CE- DENNA. APPENDIX: DERIV ATION OF THE FOURIER TRANSFORM OF HYDROSTATIC STRAIN IN TWO-DIMENSIONAL SUPERLATTICE It is well known that in an elastic media, the displacement at position /vectorrin the ldirection due to the united point force applied at the origin of coordinates in the ndirection can be expressed by Green’s tensor Gln(/vectorr), which satisfies the following equation [ 52]: /summationdisplay klmλiklm∂Gln(/vectorr) ∂xk∂xm=−δ(/vectorr)δi,n, (A1) where λiklmis the tensor of elastic moduli. Making the follow- ing Fourier transformations in Eq. ( A1): Gln(/vectorr)=/integraldisplay ˜Gln(/vectorξ)e x p ( i/vectorξ/vectorr)d3ξ, δ(/vectorr)=(2π)−3/integraldisplay exp(i/vectorξ/vectorr)d3ξ, (A2) one arrives at the following equation for the Green’s function Fourier transform: /summationdisplay klmλiklmξkξm˜Gln(/vectorξ)=(2π)−3δin. (A3) In particular, for materials with cubic crystal structure λiklm= C12δikδlm+C44(δilδmk+δimδkl)+Can/summationtext3 p=1δipδkpδlpδmp, where Can=C11−C12−2C44is the parameter of anisotropy. For this case, after simple mathematical manipulations, onegets from Eq. ( A3) the following expression: (/vectorξ˜G) n≡3/summationdisplay l=1ξl˜Gln(/vectorξ)=1 (2π)3ξn C44ξ2+Canξ2n ×⎛ ⎝1+(C12+C44)3/summationdisplay p=1ξ2 p C44ξ2+Canξ2p⎞ ⎠−1 .(A4) In the framework of the method of inclusions [ 53], the i component of the displacement caused by the existence of asingle QD is as follows: D s i(/vectorr)=Ds iχQD(/vectorr)+/summationdisplay n,k/integraldisplay Gi,n(/vectorr−/vectorr/prime)σs nkdS/prime k, (A5) where σs nk=/summationtext prλnkprεs pris the initial stress tensor, εs prand Ds iare the initial strain tensor component and the initial displacement due to the lattice mismatch between the QD 195132-6RASHBA SPLITTING OF DIRAC POINTS AND SYMMETRY … PHYSICAL REVIEW B 100, 195132 (2019) and the surrounded material, χQD(/vectorr) is the so-called QD shape function which is 1 inside the QD and is 0 outside it. The superscript sindicates that the expression refers to a single QD. The integration in Eq. ( A5) is carried out over the surface of the QD. Inserting Eq. ( A5) in the definition of the strain tensor, εs ij=1 2/parenleftbigg∂Ds i(/vectorr) ∂xj+∂Ds j(/vectorr) ∂xi/parenrightbigg , (A6) and implying the Gauss’s theorem, one obtains εs ij(/vectorr)=εs ijχQD(/vectorr)+1 2/summationdisplay nkpr/integraldisplay/bracketleftbigg∂2Gin(/vectorr−/vectorr/prime) ∂xj∂xk +∂2Gjn(/vectorr−/vectorr/prime) ∂xi∂xk/bracketrightbigg λnkprεs prχQD(/vectorr/prime)d3r/prime,(A7) where integration is carried out over the whole 3D space. Applying the operator Fof the inverse Fourier transformation to both sides of Eq. ( A7) and taking into account the convo- lution theorem according to which F(/integraltext P(/vectorr−/vectorr/prime)Q(/vectorr/prime)d/vectorr/prime)= (2π)3F(P(/vectorr))F(Q(/vectorr)) for the functions P(/vectorr) and Q(/vectorr), we arrive at the following expression of the strain tensor Fourier transform εs ij(/vectorξ) for a single QD in an elastic media: ˜εs ij(/vectorξ)=εs ij˜χQD(/vectorξ)−(2π)3 2 ×/summationdisplay nkpr(ξi˜Gjn(/vectorξ)+ξj˜Gin(/vectorξ)) ˜χQD(/vectorξ)λnkprξkεs pr,(A8) where ˜ χQD(/vectorξ) is the Fourier transform of the shape function and/vectorξis the position vector in the inverse space. Taking into account that for cubic crystals, the initial strain tensorε s ij=ε0δij, it is not hard to obtain from Eq. ( A8) the following expression: ˜εs ij(/vectorξ)=ε0˜χQD(/vectorξ)/parenleftbigg δij−(2π)3 2(C11+2C12) ×[ξi(/vectorξ˜G)j+ξj(/vectorξ˜G)i]/parenrightbigg . (A9)Substituting the dot products ( /vectorξ˜G)i nE q .( A9) by their cor- responding expressions presented in Eq. ( A4), we arrive at an analytic expression for the Fourier transform of the straintensor: ˜ε s ij=ε0˜χQD(/vectorξ)⎛ ⎝δij−1 2(C11+2C12)ξiξj/ξ2 1+(C12+C44)/summationtext3 p=1ξ2p C44ξ2+Canξ2p ×/bracketleftBigg 1 C44+Canξ2 i/ξ2+1 C44+Canξ2 j/ξ2/bracketrightBigg⎞ ⎠. (A10) Because of the linearity of the elasticity problem, the strain tensor component for a one-layer QD SL is as follows: εij(/vectorr)=/summationdisplay /vectorRεs ij(/vectorr−/vectorR)=/summationdisplay /vectorR/integraldisplay ˜εs ijexp (/vectorξ(/vectorr−/vectorR))d/vectorξ, (A11) where /vectorRruns over the in-plane cite vectors of QDs. On the other hand, the Fourier expansion of εij(/vectorr)f o rt h e2 DQ D lattice has the following form: εij(/vectorr)=/summationdisplay /vectorGexp(i/vectorG/vectorρ)/integraldisplay∞ −∞˜ε(/vectorG,ξ3)e x p ( iξ3z)dξ3,(A12) where /vectorGruns over the vectors of the 2D reciprocal lattice and /vectorρis the in-plane position vector. Comparison of Eqs. ( A11) and ( A12) leads to the expression for the strain tensor Fourier transform for a one-layer QD SL: ˜εij(/vectorG,ξ3)=(2π)2 s0˜εs ij(/vectorG,ξ3). (A13) The hydrostatic strain is defined as the trace of the strain tensor: εh=3/summationdisplay i=1εii. (A14) Finally, acting on the both sides of Eq. ( A14) by the operator Fand applying Eqs. ( A10) and ( A13), we arrive at Eq. ( 1)o f Sec. II. [1] T. O. Wehling, A. M. Black-Schaffer, and A. V . Balatsky, Adv. Phys. 63,1(2014 ). [2] M. Polini, F. Guinea, M. Lewenstein, H. C. Manoharan, and V . Pellegrini, Nat. Nanotech. 8,625(2013 ). [3] A. K. Geim and K. S. Novoselov, Nat. 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PhysRevB.95.121110.pdf

RAPID COMMUNICATIONS PHYSICAL REVIEW B 95, 121110(R) (2017) Inelastic Kondo-Andreev tunneling in a vibrating quantum dot Zhan Cao,1Tie-Feng Fang,1,*Qing-Feng Sun,2,3and Hong-Gang Luo1,4,† 1Center for Interdisciplinary Studies and Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, China 2International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 3Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 4Beijing Computational Science Research Center, Beijing 100084, China (Received 25 September 2016; revised manuscript received 21 February 2017; published 27 March 2017) Phonon-assisted electronic tunneling through a vibrating quantum dot embedded between normal and superconducting leads is studied in the Kondo regime. In such a hybrid device, with the bias applied to the normallead, we find a series of Kondo sidebands separated by half a phonon energy in the differential conductance,which are distinct from the phonon-assisted sidebands previously observed in conventional Andreev tunnelingand in systems with only normal leads. These Kondo sidebands originate from the Kondo-Andreev cooperativecotunneling mediated by phonons, which exhibit an interesting Kondo transport behavior due to the interplay ofthe Kondo effect, the Andreev tunneling, and the mechanical vibrations. Our result could be observed in a recentexperiment setup [J. Gramich et al. ,Phys. Rev. Lett. 115,216801 (2015 )], provided that their carbon nanotube device reaches the Kondo regime at low temperatures. DOI: 10.1103/PhysRevB.95.121110 Introduction. Hybrid quantum systems have a potential to exhibit new emergent phenomena by merging the strength ofdifferent media [ 1]. A quantum dot (QD) embedded between normal (N) and s-wave superconducting (S) leads (N-QD-S) is one such device, which has received considerable attentionfrom both the theoretical [ 2–12] and experimental [ 13–15] communities in the past two decades. In such a hybridsystem, two important phenomena may arise: One is theAndreev tunneling (AT) [ 16] and the other is the screening of the localized spin in the QD by conduction electrons inthe leads. While the former induces Andreev bound states(ABSs) located in the superconducting gap, the latter is thefamous Kondo effect [ 17]. The competition between these two processes results in a profound influence on the ground-stateproperties [ 6–12] as well as the transport behavior of the devices [ 2–9,11,12]. For a molecular QD, it was found that vibrational degrees of freedom are easily excited when electronic tunneling takesplace [ 18–20], which has a dramatic influence on the transport of the system due to the presence of inelastic tunnelingprocesses mediated by the emission or absorption of phonons[21–24]. In recent years, phonon-assisted inelastic AT in an N-QD-S system has also led to interesting physics on, forexample, the electronic transport [ 25–27], the heat generation [28], the ground-state cooling [ 29], the steady-state shot noise [30], as well as the transient dynamics under a step bias [ 31]. More interestingly, a phonon-assisted AT can lead to resonantpeaks every time the bias voltage changes by one phononenergy or the gate voltage changes by half a phonon energy[32,33], which has been unambiguously observed in a recent experiment [ 34]. This is somewhat reminiscent of normal systems where phonon sidebands of Kondo cotunneling[35–40] and single-electron tunneling [ 41–43] are also sep- arated by one phonon energy in the bias voltage. Since the *fangtiefeng@lzu.edu.cn †luohg@lzu.edu.cnN-QD-S setup fabricated in the experiment [ 34] is indeed an ideal platform to explore the Kondo physics, it is our aim inthis Rapid Communication to provide a theoretical study ofthe Kondo transport in such a device. Our investigation reveals that the interplay of the Kondo correlations, the superconductivity, and the mechanical vibra-tions of the QD gives rise to distinct transport characteristics,as compared with those arising from conventional phonon-assisted AT [ 32–34]. The main physical scenario is illustrated in Fig. 1, where elastic and inelastic AT with and without the Kondo effect are schematically shown. We set the chemicalpotentials of the N ( μ N) and S ( μS) leads as μN=V,μS=0, and the superconducting gap /Delta1is taken as the largest energy scale in the problem. We consider the parameter regime wherethe QD-S tunnel coupling is much larger than the N-QDcoupling and both are several times smaller than the on-siteCoulomb repulsion, such that the Kondo effect and the on-sitepairing coexist [ 15]. In this case, dot electrons would undergo frequent Andreev reflections at the QD-S interface, which formtwo Andreev bound states (ABSs) with energies ±E Ain the spectrum of the QD. The ABSs are separated roughly by theCoulomb energy and their widths are determined by the N-QDcoupling [ 8]. At zero bias V=0, a spin- ↑localized electron and a spin- ↓lead electron at μ Ncan convert to a Cooper pair in S, while another spin- ↓lead electron at μNtransits into the QD simultaneously [Fig. 1(a)]. This spin-flip cotunneling process, which we refer to as the Kondo-Andreev tunneling,is elastic and accounts for the zero-bias conductance peakpreviously observed in this system [ 13,15]. When the bias increases to V=ε ph/2(εphis the phonon energy), besides the elastic Kondo-Andreev tunneling process, additional inelasticKondo-Andreev tunneling emitting one phonon can also takeplace [Fig. 1(b)]. Here, the emission of a phonon fulfills the energy conservation of the transition that two N-lead electronseach with energy ε ph/2 in the initial state are annihilated and a Cooper pair with zero energy is created in the final state,while the QD energy under spin flipping remains the same. Atnegative bias V=−ε ph/2, similar inelastic Kondo-Andreev 2469-9950/2017/95(12)/121110(5) 121110-1 ©2017 American Physical SocietyRAPID COMMUNICATIONS CAO, FANG, SUN, AND LUO PHYSICAL REVIEW B 95, 121110(R) (2017) (d) (e) (f) N SEA EA N S N S(a) (b) (c) N SEA EA N S N S1 32 21 32 23 12 2ΓN ΓS Δ ΔμS V = εph / 2 V = 0 V = εph / 2 V = EA+εph V = EA V = EA+εph εph εph Δ ΔμSΓN ΓS FIG. 1. Schematics of elastic [(a), (d)] and inelastic [(b), (c), (e), (f)] electronic tunneling in an N-QD-S system. (a)–(c) rep- resent Kondo-Andreev tunneling through an interacting QD with the numbers 1 ,2,3 denoting the tunneling sequences. (d)–(f) show conventional AT through a noninteracting QD. The solid lines in the QD indicate the two ABSs with energies ±EA, while the wavy arrows represent the emission of phonons during the inelastic tunneling. tunneling can occur from the S lead to the N lead [Fig. 1(c)]. The opening of these additional tunneling channels wouldgive rise to additional conductance peaks at V=±ε ph/2. When multiple-phonon processes are involved, a series ofsidebands separated by half a phonon energy are thus expectedatV=nε ph/2 with n=0,±1,±2,... . For comparison, we also give a general scenario of con- ventional phonon-assisted AT for a noninteracting N-QD-S,where the QD without the on-site Coulomb interaction favorseven electron occupation and the distance between the twoABSs is determined roughly by the QD-S coupling [ 8]. In this system, there are two interleaved sets of phonon sidebands,each separated by ε ph, in the differential conductance, since additional phonon-emitted inelastic AT can be triggered atV=±E A+nεph.F o rn=0, the AT is elastic [see Fig. 1(d) forV=EA]. Forn> 0, an N-lead electron at μNcan transfer to the S lead through the lower [Fig. 1(e),V=−EA+nεph]o r upper [Fig. 1(f),V=EA+nεph] ABS by emitting nphonons, while another electron passes directly through the other ABS.Similar inelastic AT takes place from the S lead to the N leadforn< 0. When the two ABSs are indistinguishable (e.g., their widths being larger than their interval) or separated bymultiples of ε ph, the two sets of phonon sidebands merge into a single set of sidebands separated by one phonon energy. This isexactly the special case discussed in Ref. [ 32]. In the following, we perform a model calculation to demonstrate these transportscenarios. Model and formalism. Our N-QD-S system is mod- eled by the Hamiltonian H=H leads+Hph+HQD+Htunnel . The first term represents the normal ( β=N) and su- perconducting ( β=S) leads, Hleads=/summationtext k,σ,βεkc† kσβckσβ− /Delta1/summationtext k(c† k↑Sc† −k↓S+c−k↓Sck↑S).Hph=εpha†amodels the lo- cal phonon mode. HQD=/summationtext σεdd† σdσ+Und↑nd↓+λ(a+ a†)/summationtext σndσdescribes an interacting single-level QD, withCoulomb repulsion energy U, coupled with the local phonon byλthe Holstein-type electron-phonon interaction (EPI). The last term Htunnel=/summationtext k,σ,β (Vβc† kσβdσ+H.c.) describes the electronic tunneling between the dot and the leads. Fromthe tunneling matrix elements V β, the dot level εdacquires an intrinsic broadening /Gamma1β≡2πρ0|Vβ|2withρ0the density of states of lead N and lead S in the normal state. By the standardKeldysh nonequilibrium Green’s function (GF) theory [ 44], the electronic current flowing from the N lead into the QD canbe expressed as I=2ie h/integraldisplay dω/Gamma1 N[(1−fN)G< 11(ω)+fNG> 11(ω)], (1) where fN(ω) is the Fermi distribution function of lead N. The boldfaced GF matrices are defined in the well-known2×2 Nambu representation [ 45], from which the local density of states (LDOS) per spin can be calculated by ρ(ω)= −(1/π)ImG r 11(ω). Due to the presence of EPI, calculating the GFs needed in the current and the LDOS is nontrivial [ 42,46], even if the QD itself is noninteracting. Various approximationstreating the EPI from the weak to strong coupling regimeand from equilibrium to nonequilibrium have been established[47–49]. In this work, we focus on the strong EPI regime. It is thus appropriate to make the nonperturbative Lang-Firsovtransformation [ 50]˜H=e SHe−SwithS=(λ/ε ph)(a†−a)/summationtext σndσto eliminate the linear EPI. This gives us ˜H=Hleads+ Hph+˜HQD+˜Htunnel , where ˜HQD=/summationtext σ˜εdndσ+˜Und↑nd↓ and ˜Htunnel=/summationtext k,σ,β (˜Vβc† kσβdσ+H.c.), with ˜ εd=εd−gεph, ˜U=U−2gεph,˜Vβ=VβX, and X=exp[−(λ/ε ph)(a†− a)]. Here, a dimensionless measure of EPI g≡λ2/ε2 phis introduced. As in dealing with the localized polarons, we adoptthe approximation replacing the operator Xwith its expec- tation value /angbracketleftX/angbracketright=exp[−g(N ph+1/2)], where the average is taken over the independent phonon bath Hph, andNphis the Bose distribution. Hence, the renormalized ˜/Gamma1β=/angbracketleftX/angbracketright2/Gamma1β. This zero-order approximation which ignores the backactionof electrons on the phonons is valid when V β/lessmuchλand has been widely employed in the literature [ 25–28,30,33,36,41,43,51]. A previous study [ 42] which compares a full self-consistent calculation and the zero-order approximation shows that thelatter can predict accurate positions of the phonon sidebands,even though their exact line shapes are missed to some extent.This is sufficient for the purpose of our work. Applying theabove decoupling scheme and the Feynman disentanglingtechnique [ 52], one obtains G r 11(ω)=/summationtext∞ n=−∞Ln[˜Gr 11(ω− nεph)+1 2˜G< 11(ω−nεph)−1 2˜G< 11(ω+nεph)] and G<(>) 11(ω)=/summationtext∞ n=−∞Ln˜G<(>) 11(ω±nεph), where Ln=exp[−g(2Nph+ 1)]exp[ nβε ph/2]In(x), with x=2g/radicalbigNph(Nph+1) and In(x) being the modified Bessel function of the first kind. Notethat the new GF ˜Gis defined according to the Hamilto- nian ˜Hin which the Bose degrees of freedom are totally decoupled. We solve the retarded GF ˜G rusing the equation-of- motion method [ 53–55]. This method forms probably one of the simplest bases for qualitatively capturing the Kondophysics and thus has been widely used in the literature[2,3,5,8,11,56–67]. Here, ˜G ris solved under the truncation 121110-2RAPID COMMUNICATIONS INELASTIC KONDO-ANDREEV TUNNELING IN A . . . PHYSICAL REVIEW B 95, 121110(R) (2017) FIG. 2. (a) Differential conductance in the Kondo regime with different EPI strengths for ˜ εd=− 2.5,˜/Gamma1S=4,˜U=10, and εph= 0.1. The curve for g=1.5 is offset by 0.025. (b) LDOS at different bias voltages V=0 (top), −εph/4,−εph/2,−3εph/4, and −εph (bottom). The curves are offset for clarity. The dashed lines guide the shift of the Kondo satellites, while the circles mark their mergence.(c) Illustrations of the Kondo cotunneling processes corresponding to the Kondo peaks in (b). (d) Currents vs ˜ ε dat different bias voltages ( V decreases from −εph/4t o−2εphby a step εph/4, from top to bottom). The inset shows the corresponding differential conductance. scheme previously adopted by Sun et al. [3] (see details in the Supplemental Material [ 68]). The lesser and greater GFs ˜G<(>)are then obtained through the Keldysh equation ˜G<(>)=˜Gr˜/Sigma1<(>)˜Ga, with ˜Ga=(˜Gr)†. Having these GFs self-consistently determined [ 68], the current I, differential conductance G≡dI/dV , and LDOS ρ(ω) can be directly calculated. In the Supplemental Material [ 68], we also use the modified second-order perturbation theory in the Coulombinteraction [ 7] to calculate ρ(ω) andG, which agree with and complement the equation-of-motion results here. Results and discussions. In the numerical results presented below, we take all the renormalized parameters to be freelytunable. ˜/Gamma1 Nis taken as the energy unit and the temperature is always set at zero. We consider first the phonon-assistedinelastic AT in the Kondo regime. To this end, we adopt theparameters ˜ ε d=− 2.5,˜/Gamma1S=4, and ˜U=10 such that the Kondo effect and the on-dot pairing coexist. In Fig. 2(a),i t is shown that remarkable differential conductance peaks, inaddition to the zero-bias Kondo peak, develop whenever thebias voltage varies by half a phonon energy. These Kondosidebands, with their typical temperature dependence givenin the Supplemental Material [ 68], are consistent with the scenarios previously discussed in Figs. 1(a)–1(c), and are very different from those occurring in N-QD-N systems that areseparated by one phonon energy [ 35–40]. Note also that the Kondo sidebands at positive bias are much weaker than those atnegative bias, which can be ascribed to the Kondo effect beingsuppressed (enhanced) at positive (negative) bias since the dot energy level gets away from (closer to) the Fermi level of leadN. Furthermore, as compared with the Kondo resonance atzero EPI [see the red dashed curve in Fig. 2(a)], the zero-bias peak at finite EPI is significantly reduced and narrowed. The underlying physics about why the conductance peaks are separated by ε ph/2 can be acquired by examining the LDOS, since the conductance from the Kondo-Andreevtunneling processes is roughly proportional to the convolutionof the electron and hole density of states [ 15,69]. Figure 2(b) presents the LDOS for several bias voltages decreasing in stepsofε ph/4. In equilibrium, multiple Kondo satellites ( ω=nεph) are exhibited on each side of the main Kondo resonance(ω=0) due to the EPI. In the following, we will focus on the two nearest satellites around the main resonance. In nonequi-librium, the main resonance and the two Kondo satellites allsplit into two subpeaks, resulting in a total of six Kondo peaksin the LDOS, as indicated by L 1(ω=V), L 2(ω=V−εph), L3(ω=V+εph), R 1(ω=−V), R 2(ω=−V−εph), and R3(ω=−V+εph)i nF i g . 2(b). When the bias is tuned to V=−εph/2, the two peaks L 1and R 2,a sw e l la sL 3and R1, merge into a single pronounced resonance (marked by red circles), respectively. Clearly, the convolution of these twomerged Kondo resonances is larger than the convolution of L 1 and R 1atV=−εph/4, thereby cooperatively giving rise to a conductance peak at V=−εph/2. Similarly, at V=−εph, the two Kondo satellites L 3and R 2merge at ω=0 and thus result in a conductance peak. In other words, the Kondosidebands always appear in the conductance at the bias voltageVunder which the LDOS exhibits Kondo-peak cooperative enhancement within the bias window ω∈[−V,V ]. The cotunneling processes associated with some Kondo peaks in the LDOS are illustrated in Fig. 2(c).I ti ss h o w n that the cotunneling processes of the L i(i=1,2,3) and R i Kondo peaks are of the second and fourth order, respectively. This explains why the L iKondo resonances are stronger than the R iresonances. Specifically, in the Kondo process of L 1, a localized spin- ↑electron tunnels out to lead N, followed closely by a spin- ↓electron at μNtunneling into the QD. At low temperatures, a coherent superposition of suchsecond-order spin-flip cotunneling events yields a many-bodyspin singlet comprising localized and N-lead electrons, whichmanifests itself as the sharp Kondo resonance L 1in the LDOS. When ˜/Gamma1S>˜/Gamma1Nas in our case, the AT can also take part in the Kondo cotunneling process. For example, in theKondo-Andreev process of R 1, the localized spin- ↑electron first tunnels out to μNand a Cooper pair in the S lead splits into two electrons with opposite spins. The split spin- ↓electron then tunnels into the QD while the other electron transfersthrough the QD to the empty state with energy −Vin lead N. The coherent superposition of such fourth-order spin-flipcotunneling events leads to the weak Kondo resonance R 1in the LDOS [ 2,3,7]. Other Kondo peaks such as L 2,L 3,R 2, and R 3are produced by similar Kondo and Kondo-Andreev cotunneling processes but with one phonon being emitted. The current and conductance as a function of the dot level ˜εdare further investigated [Fig. 2(d)]. As we can see, both quantities change monotonously with ˜ εd. This is different from those in a conventional AT regime where characteristicpeaks show up whenever the the dot level ˜ ε dchanges by εph/2 121110-3RAPID COMMUNICATIONS CAO, FANG, SUN, AND LUO PHYSICAL REVIEW B 95, 121110(R) (2017) 0.00.20.40.6 0.00.20.40.6 -4 -3 -2 -1 0 1 2 3 40.00.20.40.60.00.20.40.60.8 0.00.20.40.60.8 -3 -2 -1 0 1 2 30.00.20.40.60.8V+,0 (f)(e)(d) (c)(b)G(4e2/h)ΓS=εph/3 (a) V-,0 V-,1V+,1 V-,2V+,2G(4e2/h)ΓS=εph/2 V-,0V+,0 V-,1V+,1 V-,2V+,2 V-,3V+,2V+,0G(4e2/h) V/εphΓS=εph V-,2V+,1V-,0V-,1 ρ(ω) ρ(ω) ρ(ω) ω/εph˜ ˜ ˜ FIG. 3. Left panel: The conductance of a noninteracting ( ˜U=0) N-QD-S system with finite EPI g=0.8 for different ˜/Gamma1S, as indicated. Right panel: The corresponding LDOS at four different bias voltages V=0,εph/2,εph,a n d3 εph/2 (from top to bottom in each figure). The curves are offset for clarity. The horizontal distance between the two dashed lines represents the bias window ω∈[−V,V ]. Other parameters are ˜ εd=0a n dεph=30. [32,34]. The featureless nature of our Ivs ˜εdandGvs ˜εdcurves can be readily understood. As long as ˜ εdis always restricted in the Kondo regime, the resulting Kondo resonances are robustand no additional phonon-assisted channel could be opened orclosed when ˜ ε dis varied. For comparison, we now investigate the conventional inelastic AT in a noninteracting ( ˜U=0) N-QD-S with the QD level ˜ εd=0 fixed at the Fermi energy. In this parameter regime, the two ABSs appear at ±EAwithEA=˜/Gamma1S/2. In Figs. 3(a)–3(c), the conductance is displayed for three valuesof˜/Gamma1S. Different from the conductance behavior in the Kondo regime, there are indeed two sets of phonon sidebands atV ±,n≡±EA+nεph, each separated by one phonon energy, in agreement with our previous discussions of Figs. 1(d)–1(f). Generally, the two sets of sidebands are interleaved [Figs. 3(a) and3(b)]. ForEA=εph/2 [Fig. 3(c)], the two sets of sidebands merge. This corresponds to the I-Vstaircases addressed previously [ 32]. These conductance behaviors displayed can also be traced back to the LDOS at different bias voltages,as shown in Figs. 3(d)–3(f). At zero bias, only hole-type (electron-type) sidebands of the upper (lower) ABS appear atE A+nεph(−EA−nεph), with n> 0, which can be attributed to the fact that the upper (lower) ABS is fully empty (occupied)and the phonon absorption is unavailable at zero temperature[41]. For a finite bias larger than E A, the upper ABS becomes occupied, therefore phonon sidebands develop on both sidesof each ABS. Upon adjusting ˜/Gamma1 Ssuch that EA=nεph/2, the sidebands associated with the two ABSs merge [see Fig. 3(f)]. For a weak but nonzero ˜U, the ground state is still a BCS singlet as ˜U=0. Only the height and distance between the two ABSs are slightly affected, while the general scenario ofthe phonon sidebands remains unchanged [ 33,68]. Conclusions. We have predicted in N-QD-S systems a series of differential conductance subpeaks developed at V= nε ph/2 and resulting from phonon-assisted inelastic Kondo- Andreev cotunneling. These structures are truly remarkablewhen compared with the transport characteristics of (i) aconventional inelastic AT in N-QD-S systems [ 32–34] and (ii) inelastic Kondo cotunneling in N-QD-N systems [ 35–40]. Our prediction might be observed in a carbon nanotube devicefabricated by Gramich et al. [34] as long as the Kondo regime is achieved at low temperatures. 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PhysRevB.102.201105.pdf

PHYSICAL REVIEW B 102, 201105(R) (2020) Rapid Communications Photoinduced Floquet mixed-Weyl semimetallic phase in a carbon allotrope Tingwei Deng,1Baobing Zheng ,2,1Fangyang Zhan,1Jing Fan ,3Xiaozhi Wu,1and Rui Wang1,4,* 1Institute for Structure and Function and Department of Physics, Chongqing University, Chongqing 400044, People’s Republic of China 2College of Physics and Optoelectronic Technology, Nonlinear Research Institute, Baoji University of Arts and Sciences, Baoji 721016, People’s Republic of China 3Center for Computational Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, People’s Republic of China 4Center for Quantum Materials and Devices, Chongqing University, Chongqing 400044, People’s Republic of China (Received 15 September 2020; accepted 22 October 2020; published 5 November 2020) The interplay between light and matter attracts tremendous interest for exploring novel topological quantum states and their phase transitions. Here we show by first-principles calculations and the Floquet theoremthat a carbon allotrope body-centered tetragonal C 16(bct-C 16), a typical nodal-line semimetal, exhibits exotic photoinduced Floquet mixed-Weyl semimetallic features. Under the irradiation of a linearly polarized light,bct-C 16undergoes a topological phase transition from a nodal-line semimetal to a Weyl semimetal with two pairs of tunable Weyl points. With increasing the light intensity, left-handed Weyl points evolve from type I intotype II while right-handed ones are always preserved to be type I, giving rise to light-induced unconventionalWeyl pairs composed of distinct types of Weyl points. Importantly, a special Weyl pair formed by type-I andtype-III Weyl points is present at the critical transition point. The photon-dressed Fermi arcs connecting theprojections of two different types of Weyl points are clearly visible, further revealing their unique topologicalfeatures. Our work not only realizes promising unconventional Weyl pairs but also paves a reliable avenue forinvestigating light-induced topological phase transitions. DOI: 10.1103/PhysRevB.102.201105 Topological semimetals possess topologically protected fermionic quasiparticles, extending the topological classifi-cation of condensed mater beyond insulators due to theirnontrivial electronic wave functions [ 1–3]. For these ma- terials, the valence and conduction bands cross near theFermi level in the momentum space, forming the pointlikeor linelike Fermi surfaces [ 3]. Accordingly, various topolog- ical fermions, such as Dirac fermions [ 4,5], Weyl fermions [6–9], nodal-line fermions [ 10–13], triple fermions [ 14,15], as well as beyond [ 16–18], have been proposed. Most of them have been verified in experiments [ 5,15,19–23]. Among these nontrivial fermionic quasiparticles, Weyl fermions in Weylsemimetals (WSMs) are of particular importance. A Weylpoint (WP) in WSMs is characterized by specific chirality(right- or left-handed), acting as a topological monopole inthe field of Berry curvature. According to the manifold ofthe Fermi surface, two types of Weyl fermions have beenidentified [ 9,24,25]. The first type (i.e., type-I Weyl fermions) corresponds to a standard Weyl cone, which is character-ized by a closed isoenergetic contour eventually evolvinginto a pointlike Fermi surface. The second type (i.e., type-IIWeyl fermions) corresponds to a tilted Weyl cone, which ischaracterized by an open Fermi surface with two crossingisoenergetic contours. Furthermore, a particularly interestingsituation occurs at the critical state between type-I and type-II Weyl cones. This critical transition point is accompaniedby a flatband along one direction, termed as a type-III WP *rcwang@cqu.edu.cn[24,26–28]. For the type-III Weyl fermions, the Fermi surface is a single line, inducing highly anisotropic effective masses[28]. More importantly, the type-III WPs offer the possibility to study the event horizon of a black hole in crystalline solids[29]. A significant hallmark of WSMs is the nontrivial Fermi arc [ 6], which connects the projections of two WPs with opposite chirality. These two WPs form a Weyl pair. Usually,a conventional Weyl pair contains two same types of WPs.The presence of paralleled electric and magnetic fields canswitch the number of paired Weyl fermions with opposite chi-rality, inducing that the classical conservation of topologicalcharge is broken in Weyl systems. This effect is known as thechiral anomaly [ 3]. One can guess that if a special Weyl pair composed of two different types of WPs is present, the chiralanomaly will switch the number of distinct types of Weylfermions [ 30]. Since the Fermi surfaces of distinct types of WPs are different, it is expected that the unconventional Weylpair composed of two different types of WPs would give riseto exotic transport phenomena. However, unfortunately, thisunconventional Weyl pair has not been reported in a realisticmaterial. As is well known, due to twofold-degenerate features, WPs always appear in a system with either the parity ( P)o rt i m e - reversal ( T) symmetry broken. Therefore, besides the intrinsic WSMs, the Weyl fermions can also be obtained from othertopological phases [e.g., Dirac semimetals (DSMs) and nodal-line semimetals (NLSMs)] by artificially breaking the relatedsymmetries, such as strain [ 31], external fields [ 3], light ir- radiation [ 32], etc. Among these approaches, the application 2469-9950/2020/102(20)/201105(6) 201105-1 ©2020 American Physical SocietyDENG, ZHENG, ZHAN, FAN, WU, AND W ANG PHYSICAL REVIEW B 102, 201105(R) (2020) FIG. 1. Light-modulated topological states in bct-C 16driven by linearly polarized layer (LPL) A(τ)=[0,0,Azsin(ωτ)]. (a) Schematic figure of bct-C 16irradiated by incident laser A(τ). (b) An ideal Dirac nodal line located at the kx-kyplane with kz=0 is transitioned to two pairs of WPs once the laser is applied. (c)–(e) The types of WPs evolve with increasing the light intensity. (f)–(h) The evolution of Fermi surface corresponds to (c)–(e), respectively. The band gap is present when the light intensity exceeds a criticalvalue. of light irradiation is highly effective due to the absence of disorder [ 26,32–40]. On the one hand, the breaking of specific symmetries can be conveniently controlled by the propagationor polarization direction of an incident light. On the otherhand, the light irradiation produces fascinating Floquet-Blochstates [ 36]. Therefore, the light irradiation provides a reliable pathway for exploring promising topological features withwide applications [ 32,41,42]. By using low-energy effective models, Weyl fermions have been investigated in several light-driven NLSMs [ 32,38–40]. However, this fascinating phenomenon has rarely been re-alized in realistic materials. Here, based on first-principlescalculations and the Floquet theorem, we demonstrate thatthe light-induced topological phase transition in a three-dimensional (3D) carbon phase can generate the unconven-tional Weyl pair. 3D carbon allotropes with topological-protected fermionic quasiparticles have been intensely investi-gated [ 43–48]. Due to the extremely tiny spin-orbital coupling (SOC) of the carbon element, the interplay between the SOCand light irradiation can be ignored. Therefore, a carbon al-lotrope can be considered as an ideal platform to study thephoton-dressed topological states. In this work, we focus onthe light-modulated topological states in bct-C 16, a typical NLSM protected by the parity-time ( PT) reversal symmetry [48,49]. In ambient conditions, the carbon allotrope bct-C 16 crystallizes in a body-centered tetragonal (bct) structure with space group I41/amd [see Fig. 1(a)], which can be obtained from the famous T carbon through a temperature-driven struc-tural transition [ 49]. Here, the Cartesian coordinates are fixed with crystal axes of bct-C 16. As shown in Fig. 1(b), ourcalculations indicate that bct-C 16hosts an ideal Dirac nodal line located at the mirror reflection invariant kx-kyplane with kz=0, which agrees well with the previous results [ 48,49]. Under a periodic field of a linearly polarized laser (LPL),we show that the NLSM phase is transitioned to a mixed-WSM phase with two pairs of tunable WPs [see Fig. 1(b)]. With increasing the light intensity, right-handed WPs W − 1and W− 2are always preserved to type I, while left-handed WPs W+ 1andW+ 2undergo an evolution from type I to type II [see Figs. 1(c)–1(h)]. During the transition process, the left- handed WPs can go through a critical type-III state, i.e., theWeyl pairs formed by type-I and type-III WPs can be present[see Figs. 1(d) and1(g)]. To reveal the light-induced topological phase transition in bct-C 16, we carried out first-principles calculations to ob- tain the basis of plane waves as implemented in the Viennaab initio simulation package [ 50] [see the details in the Sup- plemental Material (SM) [ 51]]. By projecting plane waves of Bloch states onto localized Wannier basis of C atoms usingthe W ANNIER 90 package [ 56,57], we constructed the Wannier tight-binding (TB) Hamiltonian as HW=/summationdisplay m,n,R,R/primetmn(R−R/prime)C† m(R)Cn(R/prime)+H.c., (1) where RandR/primeare lattice vectors, ( m,n) is the index of Wannier orbitals, tmn(R−R/prime) are the hopping integrals be- tween Wannier orbital mat site Rand Wannier orbital nat siteR/prime, and C† m(R)o rCm(R) creates or annihilates an electron of Wannier orbital mon site R. When a time-periodic and space-homogeneous monochromatic laser field is applied tobct-C 16[see Fig. 1(a)], the time-dependent hopping integrals are obtained by using the Peierls substitution [ 58,59] tmn(R−R/prime,τ)=tmn(R−R/prime)ei(e/¯h)A(τ)·dmn, (2) where A(τ) is the time-dependent vector potential of an ap- plied laser field, and dmnis the related position vector between Wannier orbital mat site Rand Wannier orbital nat site R/prime. The corresponding light-driven operator is Cm(R,τ)=/summationtext∞ α=−∞ Cαm(R)eiαωτwith the Floquet operator Cαm(R)[59]. In this case, the time-dependent HW(τ) hosts both lattice and time translational symmetries, so we can map it onto atime-independent Hamiltonian according to the Floquet the-ory [ 51,58,59]. By carrying out a dual Fourier transformation, the static Floquet Hamiltonian can be expressed as H F(k,ω)=/summationdisplay m,n/summationdisplay α,β/bracketleftbig Hα−β mn(k,ω) +(α−β)¯hωδmnδαβ/bracketrightbig C† αm(k)Cβn(k)+H.c.,(3) where ωis the frequency of an incident layer and thus ¯ hω represents the energy of photon, and the matrix Hα−β mn(k,ω) can be obtained by the Wannier Hamiltonian as Hα−β mn(k,ω)=/summationdisplay R/summationdisplay R/primeeik·(R−R/prime)/parenleftbigg1 T/integraldisplayT 0tmn(R−R/prime) ×ei(e/¯h)A(τ)·dmnei(α−β)ωτdτ/parenrightbigg . (4) Generally, the incident layer spans the Hilbert space of HF(k,ω) to infinite dimensions, but the matrix Hα−β mn(k,ω) 201105-2PHOTOINDUCED FLOQUET MIXED-WEYL SEMIMETALLIC … PHYSICAL REVIEW B 102, 201105(R) (2020) FIG. 2. Floquet band structure evolution of bct-C 16under the irradiation of a LPL. The black dashed lines, blue dot-dashed lines, and red solid lines represent a light intensity eAz/¯h=0.0, 0.015, 0.03 Å−1, respectively. The inset shows enlarged views around the original nodal point along /Gamma1-M. decays rapidly with an increase of its order |α−β|. Here, we truncate HF(k,ω) to the second order ( |α−β|=0,1,2) and thus there are 120 bands in the static energy spectrum, whichcan accurately describe the photon-dressed band structures ofbct-C 16[51]. In the main text, we focus on the results of a LPL. The results of a circularly polarized light (CPL) shownin the SM [ 51] indicate that the CPL also leads to a topological phase transition but cannot induce a mixed-WSM phase. Next, we illustrate the topological phase transition of bct-C 16under the light irradiation by diagonalization of the Floquet TB Hamiltonian equation ( 3). To reveal the transition process, we employ a LPL with a time-periodic vector po-tential A(τ)=[0,0,A zsin(ωτ)], where Azis its amplitude. The incident direction parallels the x-yplane, and the po- larization is along the zaxis. In order to avoid the Floquet subbands crossing each other, we set the photon energy to ¯hω=5 eV , which is larger than the bandwidth of bct-C 16. Once the light irradiation A(τ) is applied, the wave vector will be coupled with vector potential by kz→kz+eAzsin(ωτ)/¯h, which may destroy the TP symmetry-protected nodal line in bct-C 16. As shown in Fig. 2, we compare band structures without light irradiation (black dashed lines) with those oflight intensities eA z/¯h=0.015 Å−1(blue dot-dashed lines) and 0.03 Å−1(red solid lines). One can see that the light irradiation obviously influences the electronic band struc-tures of bct-C 16. As expected, the previous band crossings in the /Gamma1-Xand/Gamma1-Mdirections are both gapped, indicating that the nodal-line fermions in bct-C 16disappear. With in- creasing the light intensity, the band gaps are further enlarged(see the inset of Fig. 2). However, Fig. 2also exhibits that the band inversion at the /Gamma1point is preserved. Hence, bct-C 16un- der the light irradiation of Azsin(ωτ) still keeps the nontrivial band topology. Through carefully checking energy differ-ences between valence and conduction bands, we find thatthere are four nodal points in the whole Brillouin zone (BZ)[see Fig. 1(b)]. The nodal points are below the Fermi level, making electron doping in bct-C 16. Each nodal point pos- sesses specific chirality, forming two pairs of WPs: ( W+ 1, W− 1) and ( W+ 2,W− 2). As a result, the light-induced WSM phase of bct-C 16exhibits excellent topological features with a minimum number of WPs in a T-preserved system, in which the WPs with same chirality are symmetrically distributedwith respect to the /Gamma1point, i.e., k W+ 1=−kW+ 2andkW− 1= −kW− 2. Since the light coupling in the BZ is momentum dependent, the positions of WPs will evolve with the lightamplitude A z. The coordinates of WPs in momentum space at several typical light intensities are listed in the SM [ 51]. In addition, it is worth noting that there is a light intensitythreshold of eA z/¯h=0.069 Å−1(corresponding to the electric field strength of 3 .454×109V/m or peak intensity of 1 .6× 1012W/cm2; this light intensity is experimental feasibility [41,42]). When a light intensity exceeds it, all WPs annihilate with each other and bct-C 16becomes a trivial insulator (see Fig. 1). In experiments, Floquet states can be realized by irradiating a laser pulse [ 34,41,42]. Usually, a laser pulse with a trapezoidal envelope of ten or so cycles is long enough toobserve Floquet states [ 41], whose fluence is far less than the energy threshold to break bct-C 16[48]. To better understand the photon-dressed Weyl fermions dependent on the wave vector kin bct-C 16, we present the band profiles around one pair of WPs (i.e., W+ 1andW− 1) evolving with increasing the amplitude of a LPL as shown in Fig. 3. The other pair of WPs (i.e., W+ 2andW− 2)s h o w s the same behaviors with respect to the Tsymmetry. The band dispersions around the W− 1with a light intensity eAz/¯h of 0.03, 0.059, and 0 .066 Å−1are illustrated in Figs. 3(a), 3(b), and 3(c), respectively. It is found that the right-handed WPW− 1remains type I though it becomes more tilted with increasing the light intensity. On the contrary, light-dependent change around W+ 1is more remarkable. When the light inten- sityeAz/¯hincreases from 0.03 to 0 .066 Å−1, the left-handed WPW+ 1undergoes a transition from type I [see Fig. 3(d)]t o type II [see Fig. 3(f)]. In this transition process, the critical type-III WP is present between type-I and type-II states with eAz/¯h=0.059 Å−1[see Fig. 3(e)]. The 3D plots of band profiles around W+ 1with eAz/¯h=0.03, 0.059, and 0 .066 Å−1 are respectively shown in Figs. 3(g),3(h), and 3(i), which are consistent with topologically nontrivial features of type-I, type-II, and type-III Weyl fermions. In addition, we calculatethe evolution of Wannier charge centers around the nodal points by employing the Wilson loop method [ 60][ s e et h e insets of Figs. 3(a)–3(f)]. The results demonstrate that the unconventional Weyl pairs composed of distinct types of WPs are realized in bct-C 16. One significant consequence of unconventional Weyl pairs in bct-C 16is the existence of topologically protected surface states. To reveal this nontrivial properties, we calculate surfacestates using the iterative Green’s method [ 61,62] based on the Floquet TB Hamiltonian equation ( 3). The calculated photon- dressed Fermi surfaces and local density of states (LDOS)projected on the semi-infinite (001) surface of bct-C 16with a light intensity eAz/¯hof 0.03, 0.059, and 0 .066 Å−1are respec- tively shown in Figs. 4(a)–4(f). The LDOS show that there is a visible gap along ˜/Gamma1-˜Xand a projected Weyl cone with 201105-3DENG, ZHENG, ZHAN, FAN, WU, AND W ANG PHYSICAL REVIEW B 102, 201105(R) (2020) FIG. 3. Band profiles around one pair of WPs W+ 1andW− 1under the irradiation of a LPL with a light intensity [(a), (d), and (g)]eA z/¯h=0.03 Å−1, [(b), (e), and (h)] eAz/¯h=0.059 Å−1, and [(c), (f), and (i)] eAz/¯h=0.066 Å−1. Panels (a), (b), and (c) indicate that the right-handed WP W− 1is always kept to be type I. Panels (c), (d), and (f) indicate that the left-handed WP W+ 1undergoes a transition from type I to type II, and a critical type-III WP is present between type-I and type-II states. The insets show the evolution ofthe Wannier charge centers around the WPs W − 1andW+ 1, respec- tively. Panels (g), (h), and (i) show the 3D plots of band dispersion around W+ 1, exhibiting standard type-I Weyl cone, type-III Weyl cone combining with a flatband, and tilted type-II Weyl cone, respectively. The conduction and valence bands are respectively marked as blue and green. The isoenergetic contours corresponding to the Fermisurfaces are colored by yellow. linear dispersion along ˜/Gamma1-˜W. Under different intensities, the projected band profiles around ˜W+ 1respectively exhibit type-I [Fig. 4(a)], type-III [Fig. 4(c)], and type-II [Fig. 4(e)]W e y l features. We can see that there are always two Floquet Fermi arcs connecting two projected WPs with opposite chirality(i.e., ˜W + 1and ˜W− 1or˜W+ 2and ˜W− 2). In particular, the exotic Fermi arcs connecting two distinct types of WPs are presentin Figs. 4(d) and4(f). In addition, it is worth noting that the separation between the paired WPs decreases with increasinglight intensities. In conclusion, we theoretically propose that the mixed- WSM features with tunable WPs are present in bct-C 16under a periodic field of a LPL. This exotic WSM phase is derivedfrom a nodal line since the light irradiation of LPL breaksthePsymmetry, resulting in a minimum number of WPs in a T-preserved system. With increasing the light inten- sity, left-handed WPs evolve from type I into type II whileright-handed ones are always preserved to be type I, real-izing unconventional Weyl pairs composed of distinct typesof WPs. A very interesting issue is that a special Weyl pair FIG. 4. The calculated photon-dressed LDOS and Fermi surfaces projected on the semi-infinite (001) surface of bct-C 16with a light intensity eAz/¯hof (a), (b) 0.03, (c), (d) 0.059, and (e), (f) 0 .066 Å−1. The green and blue dots denote the projected WPs with left-handed and right-handed chirality, respectively. In panels (b), (d), and (f), Floquet Fermi arcs connecting two projected WPs with oppositechirality are clearly visible. The paths for LDOS in panels (a), (c), and (e) are marked in white-dashed lines in panels (b), (d), and (f). with type-I and type-III WPs also appear in bct-C 16. These unconventional Weyl pairs can be expected to exhibit an un-known effect related to the chiral anomaly. Photon-dressedFermi arcs connecting the projections of two different types ofWPs give rise to a unique one-way dissipationless electronicpropagation channel. It is expected that the transient changesin topology and Weyl fermions may be detected by ultrafasttransport [ 42,63,64] or time-resolved angle-resolved photoe- mission spectroscopy [ 65]. In addition, because of heating and decoherence effects, one needs to employ effective methodsto reduce dissipation from environments, such as multicolorlaser fields [ 66]. Considering the extremely tiny SOC effect in carbon materials, bct-C 16offers an ideal candidate to investi- gate light-induced mixed WSMs. Our work not only realizesthe exotic unconventional Weyl pairs constructed by differenttypes of WPs but also demonstrates that the light irradiation isa fascinating avenue for controlling topological states. This work was supported by the National Natural Science Foundation of China (NSFC; Grants No. 11974062, No.11704177, and No. 11947406), the Chongqing Natural Sci-ence Foundation (Grant No. cstc2019jcyj-msxmX0563), andthe Fundamental Research Funds for the Central Univer-sities of China (Grants No. 2019CDXYWL0029 and No.2020CDJQY-A057). 201105-4PHOTOINDUCED FLOQUET MIXED-WEYL SEMIMETALLIC … PHYSICAL REVIEW B 102, 201105(R) (2020) [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) . [2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011) . [3] N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev. Mod. Phys. 90, 015001 (2018) . [4] Z. Wang, Y . Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, P h y s .R e v .B 85, 195320 (2012) . [5] Z. K. Liu, B. Zhou, Y . Zhang, Z. J. Wang, H. M. Weng, D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai, Z. Hussain,and Y . L. Chen, Science 343, 864 (2014) . [6] X. Wan, A. M. Turner, A. Vishwanath, and S. Y . 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PhysRevB.87.075403.pdf

PHYSICAL REVIEW B 87, 075403 (2013) Shaping a time-dependent excitation to minimize the shot noise in a tunnel junction Julien Gabelli1and Bertrand Reulet1,2 1Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, F-91405 Orsay Cedex, France 2Universit ´e de Sherbrooke, Sherbrooke, Qu ´ebec J1K 2R1, Canada (Received 16 May 2012; published 4 February 2013) We report measurements of shot noise in a tunnel junction under biharmonic illumination, Vac(t)= Vac1cos(2πνt)+Vac2cos(4πνt+ϕ). The experiment is performed in the quantum regime, hν/greatermuchkBTat low temperature T=70 mK and high frequency ν=10 GHz. From the measurement of noise at low frequency, we show that we can infer and control the nonequilibrium electronic distribution function by adjusting the amplitudesand phase of the excitation, thus modeling its shape. In particular, we observe that the noise depends not onlyon the amplitude of the two sine waves but also on their relative phase, due to coherent emission/absorption ofphotons at different frequencies. By shaping the excitation we can minimize the noise of the junction, which nolonger reaches its minimum at zero dc bias. We show that adding an excitation at frequency 2 νwith the proper amplitude and phase can reduce the noise of the junction excited at frequency νonly. DOI: 10.1103/PhysRevB.87.075403 PACS number(s): 72 .70.+m, 05.40.−a, 42.50.Lc, 73 .23.−b I. INTRODUCTION In recent years the dynamical control of mesoscopic conductors has gained increasing interest, mainly motivatedby the realization of phase-coherent electronics for quan-tum computation. One major challenge is the experimentalachievement of a single electron excitation above the Fermisea. The way to drive the ground state of a metallic conductorto reach the single electron excitation should be optimized tominimize the creation of electron-hole excitations; this canbe probed by noise measurements. 1–7For a conductor with energy-independent transmission, the variance of the currentfluctuations due to the discrete nature of electrons, the so-calledshot noise, reaches a minimum when the excitation V L(t)i sa T=1/νperiodic sequence of Lorentzian peaks of quantized area/integraltextT 0eVL(t)dt=Nheach, with Ninteger. In a tunnel junction this leads to a noise spectral density S2=Ne2ν, i.e., the same as the shot noise of a purely dc current I=Neν. Thus, this ac excitation creates a nonequilibrium electrondistribution function with the remarkable property that it yieldsto a charge transfer of Nelectrons per cycle in average with a variance /Delta1N 2=N. It is experimentally difficult to generate Lorentzian pulses with precise shape and high repetition rate, a condition neces-sary to observe the predictions. 1,2The simplest ac excitation consists of a pure sine wave Vaccos(2πνt). Unfortunately, the presence of the ac voltage always increases the noise,8–11i.e., S2(Vdc,Vac)>S 2(Vdc,Vac=0), where S2is the noise spectral density measured at low frequency. A much richer waveform,which we have used in the present work, is the biharmonicexcitation: V ac(t)=Vac1cos(2πνt)+Vac2cos(4πνt+ϕ). (1) By controlling the three parameters Vac1,Vac2,ϕ, one can modify the shape of the ac excitation, which modifies theout-of-equilibrium electron distribution function and thusthe noise. As we show below, adding the excitation atfrequency 2 νmay lower the noise, i.e., S 2(Vdc,Vac1,Vac2)< S2(Vdc,Vac1,Vac2=0), thus partially erasing the extra noise created by the excitation at frequency ν. This occurs because the absorption/emission of two photons of frequency νmayinterfere destructively with that of one photon of frequency 2ν. The paper is organized as follows. In Sec. II, we describe the experimental setup. In Sec. III, we calculate the nonequi- librium stationary distribution function generated by any time-dependent, periodic excitation. From the noise measurementswe deduce in Sec. IVthe experimental electron distribution function in the presence of the biharmonic excitation. Weshow in Sec. Vthat a biharmonic excitation with two spectral components νand 2νcan reduce the monoharmonic photon- assisted noise at frequency ν. In Sec. VI, we summarize our discussion. For the sake of clarity, some technical details aremoved from the main body of the paper to the Appendices. II. EXPERIMENTAL SETUP We have measured the shot noise of an Al/Al oxide/Al tunnel junction similar to that used for noise thermometry12 cooled to 70 mK. We apply a 0.1 T perpendicular magneticfield to turn the Al normal. We measure the noise at lowfrequency while the junction is excited by the biharmonic acvoltage (1), as depicted in Fig. 1. To generate the biharmonic signal, a microwave source of frequency ν=10 GHz is split in two arms. A frequency doubler in the upper arm generates theoscillating voltage at 2 ν=20 GHz. Its phase ϕcan be tuned by a mechanical phase shifter while its amplitude V ac2is set by the tunable output power of the source. In the lower arm, a variableattenuator allows us to modify V ac1. The signals from the two arms are recombined at point A and sent to the sample througha directional coupler placed at liquid helium temperature. Abias tee, sketched by an inductor and a capacitor in Fig. 1, allows us to add the dc voltage V dcto the ac one coming from the coupler. An example of an achievable waveform isshown in Fig. 1(b), together with a Lorentzian. The ac voltages experienced by the sample are measured by fitting the data ofthe photo-assisted noise with a single frequency, as in Ref. 11. The resistance of the sample 1 /G=48/Omega1is close enough to 50/Omega1to provide a good impedance matching to the coaxial cable and avoid reflection of the ac excitation. Thus, onlythe fluctuating current due to the tunneling process is ampli-fied by a low noise cryogenic amplifier (noise temperature 075403-1 1098-0121/2013/87(7)/075403(7) ©2013 American Physical SocietyJULIEN GABELLI AND BERTRAND REULET PHYSICAL REVIEW B 87, 075403 (2013) tunnel junction0.02 K 4.2 K 300 K square law detector linear amplifierbandpass filter Bdirectionnal coupler bias Tdephaser attenuatordoubler A C2f 1 0 -1 -10 0 10 3 2 1 0 2 1 0BC FIG. 1. (Color online) Experimental setup for the measurement of the photon-assisted noise in a tunnel junction under biharmonic excitation. Inset B: T-periodic sequence of biharmonic excitation (blue line) with eVdc=eVac1=2eVac2=hνandϕ=0, approximat- ing Lorentzian pulses of width τ=ln 2/(2πT) and height N=1 (red dashed line). Inset C: normalized differential noise spectral density with (blue line) and without (black dotted line) microwave excitation vs normalized dc bias. The power of the generator, the variable attenuator, and the phase shifter are tuned to obtaineV ac1=2eVac2=5.4hνandϕ=0. TN/similarequal7 K). The noise is filtered to keep frequencies in the range 0 .5–1.8 GHz before impinging on a power detector. The dc voltage at point C is proportional to the noise power densityS 2integrated over the bandwidth of the filter, see Appendix A. The derivative of the noise ∂S2/∂eV dcis measured with an additional 77 Hz, small voltage modulation, and a usual lock-in detection. To calibrate the measurement, we use the noisespectral density at high voltage being S 2(eV/greatermuchkBT)=eIas in shot noise thermometry.12Figure 1(c)shows measurements of the differential noise ∂S2/∂eV dcwith and without ac excitation. From the data without ac excitation and taking intoaccount the finite bandwidth of the detection, we determinethe electrons temperature: T el=70 mK =0.14hν/k B. When the ac excitation is switched on, the differential noise exhibitsan intermediate step, a consequence of the electron energydistribution function differing from that of Fermi-Dirac. III. MEASUREMENTS OF DISTRIBUTION FUNCTIONS Distribution functions in samples driven out of equilib- rium by a dc voltage have been obtained by the mea-surement of the differential conductance in systems wherethe density of states depends on energy. This occurs withsuperconducting electrodes, 13,14in the presence of dynamical Coulomb blockade15,16or with a quantum dot.17In our case, the differential conductance of the junction is totallyvoltage-independent and its measurement does not provideany spectroscopic information. However, as we show below,the differential noise does. As with differential conductancemeasurements, we cannot access the distribution functionsof the two contacts separately. We measure the distributionfunctions that are involved in the transport, which depends onlyon the voltage difference between the contacts. This can bedescribed by taking one of the contacts at equilibrium while theother one experiences the full time-dependent voltage. For nottoo small energy ξ,ξ/greatermuchh/Delta1ν f,kBTelwhere /Delta1νf=1.3 GHz is the bandwidth of the noise detection, we show in Appendix A that the distribution function ˜fis related to the differential noise∂S2/∂eV dcby ˜f(/epsilon1F+ξ)/similarequal1 2/parenleftbigg 1−1 G∂S2 ∂eVdc/parenrightbigg eVdc=ξ. (2) An energy resolution better than h/Delta1ν,k BTelcan be achieved by numerical deconvolution of the noise data, as explained inAppendix A. IV . DISTRIBUTION FUNCTION FOR A TIME-DEPENDENT EXCITATION In the presence of a periodic voltage Vac(t) of frequency ν, the electron wave functions acquire an extra phase factor:19 /Xi1(t)=exp/parenleftbigg−i ¯h/integraldisplayt 0eVac(t/prime)dt/prime/parenrightbigg . (3) Electronic states with energy /epsilon1are split into subbands with energies /epsilon1±nhν and spectral weight given by the modulus squared of the Fourier coefficients cnof/Xi1(t)=/summationtext+∞ n=−∞cnei2πνnt. The corresponding nonequilibrium distri- bution function is20 ˜f(/epsilon1)=+∞/summationdisplay n=−∞|cn|2f(/epsilon1+nhν), (4) where fis the equilibrium Fermi-Dirac distribution. For har- monic excitation ( Vac2=0),cn=Jn(eVac1/hν) with Jnthe Bessel functions of the first kind. For biharmonic excitation: cn=+∞/summationdisplay m=−∞Jn−2m/parenleftbiggeVac1 hν/parenrightbigg Jm/parenleftbiggeVac2 2hν/parenrightbigg e−imϕ. (5) The sum in Eq. (5)expresses the interference involving several absorption/emission processes of photons of frequencies νand 2ν. This interference depends on the relative phase ϕ. Figure 2 shows measured nonequilibrium distribution functions ˜ffor different ac excitations. They are obtained from numericaldeconvolution of the noise data (see Appendix A). Although biharmonic excitation depends on only three parameters V ac1, Vac2, and ϕ, a large class of distribution functions can be realized, which allows us to control related physical propertiessuch as the shot noise. For example, taking V ac1=2Vac2 creates a distribution function with two steps. The height and width of the steps can be controlled by tuning the phaseshift [see Fig. 2(a)] or the amplitude of the ac excitation [see Figs. 2(b)–2(c)]. We show in the following that this 075403-2SHAPING A TIME-DEPENDENT EXCITATION TO ... PHYSICAL REVIEW B 87, 075403 (2013) FIG. 2. (Color online) Nonequilibrium distribution functions obtained from numerical deconvolution of the measured differential noise. (a) Noise is measured for eVac1=2eVac2=5.4hν, for phase shiftsϕ=0 (blue), ϕ=π/2 (green), and ϕ=π(red). (b) [resp. (c)] Noise is measured for various amplitudes of excitation Vac1andVac2 keeping Vac1=2Vac2,f o rϕ=0 (resp. ϕ=π/2). distribution minimizes the shot noise for a given amplitude Vac1. Making the spectroscopy of a system with discrete levels has been performed in solid state qubits with harmonic21and biharmonic22excitation. In such systems, one can directly measure the population of the levels in the presence of theexcitation. In our case, the noise, i.e., the variance of thefluctuations of the populations, provides the spectroscopicinformation. It should be noted that the energy distributionfunction ˜f(/epsilon1) refers to a single-particle distribution function and takes, by definition, no account of potential correlationsbetween electrons and holes. V . NOISE MINIMIZATION In the following we show how controlling the distribution function via the shape of the exciting waveform allows us toreduce the shot noise in the tunnel junction. The current noiseof a coherent conductor biased by a time-dependent, periodicvoltage has been calculated for a pure sine wave excitation. 8,23 For a tunnel junction and an arbitrary periodic excitation, we obtain S2,ac(eVdc)=+∞/summationdisplay n=−∞|cn|2S(0) 2(eVdc+nhν), (6) where S(0) 2(hν)=Ghν coth(hν/2kBTel) is the Johnson- Nyquist equilibrium noise, and cnare given by Eq. (5).I n the case of a harmonic excitation, one observes features on (a) (b) -15 -10 -5 0 5 10 15 FIG. 3. (Color online) Calculated (a) and measured (b) second derivative of the biharmonic photon-assisted noise ∂2S2,ac/∂V2 dcas a function of normalized dc bias and phase shift. In both cases eVac1=2eVac2=5.4hνwithν=10 GHz, and the temperature isTel=0.14hν/k B=70 mK. Red curves correspond to the cal- culated (a) and measured (b) minimum of the photon-assisted noise, ∂S2,ac/∂Vdc=0. Dashed lines correspond to phase shifts that are used in Fig. 4. S2,ac(eVdc)a tb i a s eVdc=nhν withninteger (discontinuities ofdS2/dV rounded by the finite temperature and detection bandwidth).9,11For biharmonic excitation the interferences between multi-photon-assisted processes at frequency νand 2νinduce interference fringes on a larger scale. We show this additional complexity in the interference pattern forV ac1=2Vac2in Fig. 3(b), where the second derivative of the noise ∂2S2,ac/∂eV2 dcis plotted. The choice Vac1=2Vac2 has been motivated by a numerical calculation described in Appendix B. The interference pattern in the ( eVdc,ϕ) space exhibits fringes with a fringe spacing /similarequal5hν[see Fig. 3(b)], in agreement with numerical calculations using Eqs. (5)and(6), see Fig. 3(a), whereas the substructure at hνis almost washed out by thermal broadening. Red curves in Fig. 3correspond to the calculated (a) and measured (b) eVdcvalue at which the photon-assisted noise is minimal ( ∂S2,ac/∂eV dc=0). It exhibits steps at eVdc=±hνandeVdc=±3hν. The appearance of fringes at a scale larger than hνis similar to what is observed when systems with discretespectrum are driven by a large amplitude signal: 21,24The fringes due to individual photon resonances, characterizedby the energy scale hν, are superimposed fringes with larger characteristic scale corresponding to St ¨uckelberg oscillations. The latter may persist even if the hνpattern is lost and are a direct consequence of quantum coherence. In our case,two contacts with time-dependent chemical potentials arecoupled by tunneling. The phase acquired by the electron-hole 075403-3JULIEN GABELLI AND BERTRAND REULET PHYSICAL REVIEW B 87, 075403 (2013) 1.0 0.5 0.0 -0.5 -10 -5 0 5 1010 8 6 4 2 0(b) (c)(a) -505 10 5 0 FIG. 4. (Color online) (a) Shape of the ac excitation with the different phase shifts: ϕ=0 (blue), ϕ=π/2 (green) and ϕ=π (red). Dashed line: monoharmonic signal. (b) Normalized biharmonic photon-assisted noise S2,ac/Ghν vs normalized dc bias for eVac1= 5.4hν. Blue square, green circle, red triangle symbols: data for eVac2=2.7hνand phase shifts ϕ=0,π/2,π. Black circles: data forVac2=0, i.e., pure sine wave excitation. Cross symbols ( ×): data forVac1=Vac2=0, i.e., shot noise without any ac excitation. Solid lines: theoretical predictions, Eqs. (5)and(6). (c) Difference between biharmonic and monoharmonic photon-assisted noise /Delta1S 2,ac(Vdc)= S2,ac(Vdc,Vac1,Vac2)−S2,ac(Vdc,Vac1,Vac2=0). pairs involved in the transport mechanism depends on the time dependence of the voltage. The probability to crossthe barrier involves interferences between several processes,which results in the St ¨uckelberg-like oscillations we observe. 25 This behavior is generic for driven quantum systems and is a part of the more general effect of Ramsey multiple-time-slitinterferences. 26 We have calculated numerically the set of parameters (eVdc,eVac2,ϕ) minimizing the photon-assisted noise S2,ac for a given eVac1and temperature Tel, see Appendix B. The optimal dc voltage is zero only for ϕ=π/2, which corresponds to the existence of a symmetry in the waveform:For each positive value of V ac(t) there is a symmetric, negativevalue [green curve on Fig. 4(a)]. When this symmetry is lost there is no reason for the noise to reach its minimum at Vdc=0. For experimental parameters T=0.14hν/k BandeVac1= 5.4hν, we obtain that optimal values are eVac2=eVdc= 2.4hνandϕ=0. For ϕ=π, the waveform is reversed [see Fig. 4(a)] and the minimum occurs at the opposite value of Vdc. Figure 4(b) shows noise measured for eVac2=2.7hν (i.e., close to optimal) for ϕ=0 (blue), π/2 (green), and π (red). All the data (symbols) are very well fitted by the theory(solid lines). One observes that the minima for ϕ=0 and π occur at opposite values of eV dc=±2.3hνin agreement with the numerical result. The black curve on Fig. 4(b) (black circles) shows the noise for Vac2=0. There is a clear region of Vdcwhere it is above the red or blue curve, which correspond to Vac2/negationslash=0. We have emphasized this result by plotting in Fig.4(c) the difference /Delta1S 2,ac(Vdc)=S2,ac(Vdc,Vac1,Vac2)− S2,ac(Vdc,Vac1,Vac2=0) between the noise under biharmonic and monoharmonic excitations, which can be negative. Thisproves that the addition of the excitation at frequency 2 νmay reduce the noise. It is also noticeable that the noise under biharmonic excitations drops below the absolute minimum of the noise with monoharmonic excitation, which occurs atzero bias, in agreement with our numerical simulations (seeAppendix B). Noise has been predicted to be minimal when the excitation is a sequence of Lorentzian peaks of a quantized area/integraltext T 0eV(t)dt=Nh.1,2Such an excitation does not add more noise than its dc voltage alone. In other words, the noise as afunction of the dc voltage has minima for quantized values ofV dc. This property seems to be valid for many ac waveforms at zero temperature,6including the biharmonic excitation (see Appendix C). We observe that this is no longer the case at finite temperature for the biharmonic excitation (data not shown), inagreement with numerical calculations (see Appendix C). Let us now consider the difference in the noise for two excitationsat the same frequency. Obviously, it should have extrema forthe same quantized values of V dcat zero temperature. As shown in Fig. 4(c), this property seems to survive at finite temperature if we consider the difference between the monoharmonic andbiharmonic excitations /Delta1S 2,ac, which has minima at ±3hν. VI. CONCLUSION We have observed the effect of biharmonic illumination on the nonequilibrium current noise in a tunnel junction. We havemeasured the low frequency shot noise of the junction whilevarying the shape of the ac excitation and showed that fromthese measurements we can determine the out-of-equilibriumdistribution function induced by the excitation. This opens theway of engineering the waveform of an ac signal to control theout-of-equilibrium distribution function of the electrons in amesoscopic conductor, thus modifying its physical properties.We have demonstrated this ability by reducing the shot noisein a tunnel junction irradiated at frequency νby adding another coherent irradiation at frequency 2 νof controlled amplitude and phase. Such a procedure may be used in many situations.For example, it may be used to dynamically control theamplitude of the critical current of a superconductor/normalmetal/superconductor tunnel junction, 27or even reverse it as 075403-4SHAPING A TIME-DEPENDENT EXCITATION TO ... PHYSICAL REVIEW B 87, 075403 (2013) with a dc current.28,29This would realize a Josephson junction that can be switched from 0 state to πstate dynamically, an interesting device in the context of quantum computation. ACKNOWLEDGMENTS We are very grateful to Lafe Spietz for providing us with the sample and to Leonid Levitov for many stimulatingdiscussions. We thank Marco Aprili, Wolfgang Belzig, SophieGu´eron, and Mihajlo Vanevic for fruitful discussions. This work was supported by ANR-11-JS04-006-01 and the CanadaExcellence Research Chair program. APPENDIX A: TUNNELING SPECTROSCOPY OF DISTRIBUTION FUNCTIONS The quantity we measure is /Delta1I2(eVdc)=/integraldisplay+∞ −∞S2(eVdc,hν/prime)|H(ν/prime)|2dν/prime,(A1) where S2(eVdc,hν/prime) is the spectral density of current fluctua- tions at frequency ν/primeandH(ν/prime) the frequency response of the bandpass filter, of width /Delta1νf, and central frequency νf(in our experiment, νf=1.15 GHz, /Delta1νf=1.3 GHz). S2(eVdc,hν/prime) depends on the ac excitation Vac(t). For a tunnel junction with energy-independent transmissions, S2(eVdc,hν/prime)i ss i m p l y S2(eVdc,hν/prime)=1 2/bracketleftbig S(0) 2(eVdc+hν/prime)+S(0) 2(eVdc−hν/prime)/bracketrightbig , (A2) withS(0) 2(eVdc)=S2(eVdc,0) the zero-frequency noise spec- tral density, given by S(0) 2(eVdc)=G/integraldisplay+∞ −∞[fL(/epsilon1)(1−fR(/epsilon1)) +fR(/epsilon1)(1−fL(/epsilon1))]d/epsilon1. (A3) HerefL(respectively fR) is the energy distribution function of electrons in the left (resp. right) contact. In the presence ofboth dc and ac bias, we can without loss of generality considerthat the dc bias is applied to the left reservoir while the acvoltage is applied to the right one. Thus f Lis the Fermi- Dirac distribution fwith a shifted electrochemical potential, fL(/epsilon1)=f(/epsilon1−eVdc), whereas fR=˜fis the nonequilibrium distribution function we wish to measure. Defining the differ-ence in the noise with and without ac excitation, M(eV dc)= /Delta1I2(eVdc,eVac/negationslash=0)−/Delta1I2(eVdc,eVac=0), we obtain the following convolution: ∂M ∂eVdc(eVdc)=/integraldisplay+∞ −∞K(eVdc−/epsilon1)[˜f(/epsilon1)−f(/epsilon1)]d/epsilon1 (A4) with a kernel: K(/epsilon1)=−G h/integraldisplay+∞ −∞|H(ν/prime)|2∂f ∂ν/prime(hν/prime−/epsilon1)dν/prime.(A5) Thus, the nonequilibrium function ˜fcan be calculated using the Fourier transform FT: ˜f(/epsilon1)=f(/epsilon1)+FT−1/braceleftBigg FT/bracketleftbig∂M ∂eVdc/bracketrightbig FT[K]/bracerightBigg (/epsilon1). (A6)In the limit ξ/greatermuchhνF,h/Delta1ν f,kBTel,u s i n g K(/epsilon1)= −4G/Delta1ν fδ(/epsilon1−/epsilon1F), Eq. (A6) reduces to ˜f(/epsilon1F+ξ)/similarequal1 2/parenleftbigg 1−1 G∂S2 ∂eVdc/parenrightbigg eVdc=ξ. (A7) In our experiment, hνf/kB=50 mK, h/Delta1ν f/kB=56 mK, andTel=70 mK. APPENDIX B: OPTIMIZATION OF THE BIHARMONIC PHOTON-ASSISTED NOISE AT FINITE TEMPERATURE The photon-assisted noise in the tunnel junction depends on the shape of the ac excitation. In the case of biharmonicexcitation, we determine numerically the set of parameters(eV ⋆ dc,eV⋆ ac1,eV⋆ ac2,ϕ⋆) which minimize the noise at temper- ature Tel. At each temperature, the noise spectral density is calculated for 100 ×100×100×100 different values of(eVdc/hν,eV ac1/hν,eV ac2/hν,ϕ )in the range [ −5,5]× [0,10]×[0,5]×[0,π]. Let us suppose we excite at frequency νwith an amplitude Vac1and we want to minimize the low frequency shot noise. Figures 5(a)and5(b) show respectively how to choose VdcandVac2to reach this goal, as a function ofVac1and for various Tel. The obtained noise reduction, R= S2(Vdc=0,Vac1,Vac2=0)−S2(V⋆ dc,Vac1,V⋆ ac2), is plotted in Fig. 5(c). It appears that one always has V⋆ dc/similarequalV⋆ ac2.F o r example, for our experimental parameters kBTel=0.14hν andeVac1=5.4hν, the optimum is eVdc=2.38hν,eVac2= 2.4hν, andϕ=0 (or the opposite Vdcforϕ=π). Adding dc voltage and ac voltage at frequency 2 νallows us to reduce the noise below that with no dc bias and the same excitation atfrequency ν, by an amount R=0.04Ghν (or in terms of noise temperature, by 20 mK). We observe this effect in Fig. 4(b) of the paper: The minimum of S 2(Vdc,Vac1,V⋆ ac2) drops below S2(Vdc=0,Vac1,Vac2=0) for Vdc∼V⋆ dc. For a given Vac1, there is a temperature Tmax(eVac1) above which the optimal point does not exist anymore: V⋆ dc=V⋆ ac2= 0, see inset of Fig. 5(c). Above that temperature, adding a second harmonic will never reduce the noise. In particular, foreV ac1<2hνit is never possible to reduce the noise with a biharmonic excitation. In our experiment, Tmax/similarequal250 mK > Tel, so the addition of the sine wave at frequency 2 νmay lead to a reduction of the noise, as we observe. It is interesting to remark that the waveform we found that minimizes the noise for a given Vac1at finite temperature is not close to Lorentzian, but corresponds almost to the first twoharmonics of a Lorentzian with a dc offset [see Fig. 1(b)]. The Lorentzian pulses are optimal only if we consider thenoise at zero frequency, zero temperature, and integer valuesofeV dc/hν. They are no longer optimal if we work at finite detection frequency, finite temperature, or a noninteger valueofeV dc/hν, see Appendix C. APPENDIX C: PHOTON-ASSISTED NOISE AT FINITE TEMPERATURE FOR VARIOUS WA VEFORMS We consider the three waveforms shown in Fig. 6(a):VL(t) is the Lorentzian shape of width τ=ln 2/2πT (this value is chosen to have the same first two harmonics as the one we havechosen in our experiment), V 1(t)=Vdc[1+cos(2πνt)] is the same Lorentzian waveform truncated to the same dc and first 075403-5JULIEN GABELLI AND BERTRAND REULET PHYSICAL REVIEW B 87, 075403 (2013) (a) (b) (c)5 4 3 2 1 0 5 4 3 2 1 0 0.4 0.3 0.2 0.1 0.0 8 6 4 2 01 01.0 0.5 0.08 6 4 2 0 FIG. 5. (a) Optimal value of the reduced dc voltage eVdc/hν. Optimal value of the reduced amplitude eVac2/hνat frequency 2 ν.( c ) : Noise reduction R=S2(Vdc=0,Vac1,Vac2=0)−S2(V⋆ dc,Vac1,V⋆ ac2) between the monoharmonic photon-assisted noise at zero dc bias and the optimal biharmonic one with the same ac amplitude Vac1at frequency ν. (a), (b), and (c) are plotted as a function of the reduced amplitude eVac1/hν at frequency ν, for various reduced temperature kBTel/hν ranging from 0 (blue line) to 0.65 by steps of 0.05. Inset: Temperature above which the noise cannot be reduced by biharmonic excitation. harmonic, and V2(t)=Vdc[1+cos(2πνt)+0.5 cos(4 πνt)] is again the same Lorentzian waveform but truncated to dcand first two harmonics. We show in Fig. 6(b) the numerical (b)0.15 0.10 0.05 0.00 5 4 3 2 1 03.0 2.0 1.0 0.0 2.0 1.0 0.0(a) FIG. 6. (Color online) (a) T-periodic sequence of Lorentzian pulses of width τ=ln 2/2πT [red line, VL(t)] and its harmonic [green line, V1(t)] and biharmonic [blue line, V2(t)] approximations. (b) Noise difference S2−S2,dcfor the different waveforms. Dashed lines correspond to zero temperature whereas solid lines correspond to our experimental temperature kBT=0.14hν. difference in the noise, S2−S2,dcbetween ac+dc excitation and dc-only excitation for these three waveforms. At zerotemperature (dashed lines), there is a sharp minimum foreach integer value of eV dc/(hν) for the three waveforms. The Lorentzian reaches zero and the biharmonic is better than themonoharmonic. For eV dc<h ν the Lorentzian is the worst. At finite temperature (solid lines, kBT=0.14hνas in the experiment), none of the waveforms minimize the noise atquantized values of eV dc/hν. 1L. S. Levitov, H. W. Lee, and G. B. Lesovik, J. Math. Phys. 37, 4845 (1996). 2D. A. Ivanov, H. W. Lee, and L. S. Levitov, Phys. Rev. B 56, 6839 (1997). 3J. Keeling, I. Klich, and L. S. Levitov, Phys. Rev. Lett. 97, 116403 (2006). 4C. Grenier, R. Herv ´e, E. Bocquillon, F. D. Parmentier, B. Plac ¸ais, J.-M. Berroir, G. F `eve, and P. Degiovanni, New J. Phys. 13, 093007 (2011). 5N. d’Ambrumenil and B. Muzykantskii, Phys. Rev. B 71, 045326 (2005).6M. Vanevi ´c, Y . V . Nazarov, and W. Belzig, Phys. Rev. B 78, 245308 (2008). 7M. Vanevic and W. Belzig, P h y s .R e v .B 86, 241306(R) (2012). 8G. B. Lesovik and L. S. Levitov, Phys. Rev. Lett. 72, 538 (1994). 9R. J. Schoelkopf, A. A. Kozhevnikov, D. E. Prober, and M. J. Rooks,Phys. Rev. Lett. 80, 2437 (1998). 10L.-H. Reydellet, P. Roche, D. C. Glattli, B. Etienne, and Y . Jin, Phys. Rev. Lett. 90, 176803 (2003). 11J. Gabelli and B. Reulet, P h y s .R e v .L e t t . 100, 026601 (2008). 075403-6SHAPING A TIME-DEPENDENT EXCITATION TO ... PHYSICAL REVIEW B 87, 075403 (2013) 12L. Spietz, K. W. Lehnert, I. Siddiqi, and R. J. Schoelkopf, Science 300, 1929 (2003). 13H. Pothier, S. Gu ´eron, Norman O. Birge, D. Esteve, and M. H. Devoret, P h y s .R e v .L e t t . 79, 3490 (1997). 14F. Pierre, A. Anthore, H. Pothier, C. Urbina, and D. Esteve, Phys. Rev. Lett. 86, 1078 (2001). 15F. Pierre, H. Pothier, P. Joyez, Norman O. Birge, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. 86, 1590 (2001). 16A. Anthore, F. Pierre, H. Pothier, and D. Esteve, Phys. Rev. Lett. 90, 076806 (2003). 17C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly, and F. Pierre, Nat. Phys. 6, 34 (2010). 18Y . M. Blanter and M. B ¨uttiker, Phys. Rep. 336, 1 (2000). 19P. K. Tien and J. P. Gordon, Phys. Rev. 129, 647 (1963). 20The ac excitation also generates electron-hole correlations. Those are irrelevant for a conductor with small transmission such as ourtunnel junction (Ref. 18).21D. M. Berns, M. S. Rudner, S. O. Valenzuela, K. K. Berggren, W. D. Oliver, L. S. Levitov, and T. P. Orlando, Nature (London) 455, 51 (2008). 22J. Bylander, M. S. Rudner, A. V . Shytov, S. O. Valenzuela, D. M. Berns, K. K. Berggren, L. S. Levitov, and W. D. Oliver, Phys. Rev. B80, 220506(R) (2009). 23M. H. Pedersen and M. B ¨uttiker, P h y s .R e v .B 58, 12993 (1998). 24D. M. Berns, W. D. Oliver, S. O. Valenzuela, A. V . Shytov, K. K. Berggren, L. S. Levitov, and T. P. Orlando, P h y s .R e v .L e t t . 97, 150502 (2006). 25L. S. Levitov (private communication). 26E. Akkermans and G. V . Dunne, P h y s .R e v .L e t t . 108, 030401 (2012). 27F. Chiodi, M. Aprili, and B. Reulet, Phys. Rev. Lett. 103, 177002 (2009). 28J. J. A. Baselmans, A. F. Morpurgo, B. J. Van Wees, and T. M.Klapwijk, Nature (London) 397, 43 (1999). 29D. Prober (private communication). 075403-7

PhysRevB.70.113301.pdf

Dissipationless spin current in anisotropic p-doped semiconductors Bogdan A. Bernevig Department of Physics, Stanford University, Stanford, California 94305, USA JiangPing Hu Department of Astronomy and Physics, University of California at Los Angeles, Los Angeles, California 90095, USA Eran Mukamel and Shou-Cheng Zhang Department of Physics, Stanford University, Stanford, California 94305, USA (Received 13 April 2004; published 8 September 2004 ) Recently, dissipationless spin current has been predicted for p-doped semiconductors with spin-orbit cou- pling. Here we investigate the effect of the breaking of spherical symmetry on the dissipationless spin current,and obtain values of the intrinsic spin Hall conducitivity for realistic semiconductor band structures with cubicsymmetry. DOI: 10.1103/PhysRevB.70.113301 PACS number (s): 72.10. 2d, 72.15.Gd, 73.50.Jt Spintronics is a new field of science and technology aimed at manipulating the spin of electrons to build func-tional logic and storage devices. 1The creation, manipulation and transport of spin currents is a central challenge in thisfield. Recently, Murakami et al. 2found an important law of spintronics, which relates the spin current and the electricfield by the response equation, j ji=sseijkEk, s1d wherejjiis the current of the ith component of the spin along the direction jandeijkis the totally antisymmetric tensor in three dimensions (3D). This effect arises because of spin- orbit coupling in the valence band of conventional semicon-ductors such as GaAs and Ge. Sinova et al. 3also found a similar effect in the electron doped conduction band. Trans-port equation (1)is similar to Ohm’s law in electronics. However, unlike Ohm’s law, this new law describes a purelydissipationless spin current, in the sense that Eq. (1)is in- variant under time reversal and the intrinsic part of ssdoes not depend on impurity scattering. These effects have beenfurther discussed in recent literature. 4–9 Fundamental to the proposal of Murakami et al.2is the spin-orbit coupling that exists in the Lüttinger effective-massmodel in degenerate valence bands: H=1 2mXSg1+5 2g2Dk2−2g2sk·Sd2C, s2d wherekis the momentum operator of the valence holes, and Sis the four-by-four spin-3/2 operator that describes the four hole states at a given value of k. In this “isotropic,” or spherically symmetric model, the helicity l=kˆ·SWis a good quantum number of the isotropic Lüttinger Hamiltonianabove, and it labels the two doubly degenerate Kramers’bands that correspond to the heavy holes l=±3/2and light holes l=±1/2. The spin current effect can be intuitively understood as a consequence of the conservation of total an-gular momentum: J="x3k+S, wherexis the holes’ posi- tion operator. The spin current flows in such a way that thechange of the orbital angular momentum L="x3kexactly cancels the change of the spin angular momentum S. When an electric field is applied on the arbitrary zaxis, thezcom- ponent of Jis conserved. The topological nature of the spin current is manifested in the gauge-field formulation of Ref.5, where the spin conductance is defined in terms of a linearcombination of the components of a gauge field, G ij=lsl2 −13/4 deijlkl/k3, clearly reflecting a monopole structure in k space. The singularity at k!0 exemplifies the confluence of the Kramers’ doublets at the Gpoint where the band be- comes fourfold degenerate, but the flux of the gauge fieldthrough a two-dimensional surface in kspace is constant and set by the helicity eigenvalue. The picture presented above is valid as long as the Hamil- tonian is isotropic, that is to say, it has spherical symmetry.In the real materials in which the dissipationless spin currentis predicted, 2all of which are characterized by large aniso- tropy (see Table I ), the angular momentum Jand the helicity l=kˆ·SWare no longer good quantum numbers. It is therefore vital to ask whether the topological spin current is preservedin materials which are not rotationally invariant. In this briefreport, we investigate the effect of the breaking of sphericalsymmetry on the dissipationless spin current, and calculatethe values of the intrinsic spin Hall conductivity for aniso-tropic band structure parameters. TABLE I. Valence-band parameters for some common materials (after Ref. 10 ). Following Ref. 11 we define d=sg3−g2d/g1as a measure of the anisotropy. g1 g2 g3 d Si 4.22 0.39 1.44 0.248 Ge 13.35 4.25 5.69 0.108GaAs 7.65 2.41 3.28 0.114InSb 35.08 15.64 16.91 0.036InAs 19.67 8.37 9.29 0.047GaP 4.20 0.98 1.66 0.162PHYSICAL REVIEW B 70, 113301 (2004 ) 1098-0121/2004/70 (11)/113301 (4)/$22.50 ©2004 The American Physical Society 70113301-1The most general Hamiltonian which respects time- reversal and cubic symmetries was derived by Lüttinger:12 H0=1 2mSg1+5 2g2Dk2−g2 mskx2Sx2+ky2Sy2+kz2Sz2d−2g3 mfhkx,kyj 3hSx,Syj+hky,kzjhSy,Szj+hkz,kxjhSz,Sxjg, s3d where we define hA,Bj=1 2sAB+BAdandk2=kx2+ky2+kz2. The parameters, g1,g2, and g3, are material dependent. In the special case of g2=g3(which we call isotropic ), the last two terms simply combine to yield − g2/mskW·SWd2. In real materials, however, the values of g2and g3are very different. Table I lists the values of these parameters insome important materials. The anisotropy, characterized bythe parameter d;sg3−g2d/g1, is relevant and substantial for all the materials, and especially relevant for Si. In order to understand the dissipationless spin current generated in thesereal materials, including its dependence on the orientation ofthe field and current with respect to the crystal axes, we mustconsider the full anisotropic Hamiltonian Eq. (3). When g2Þg3, the Hamiltonian is no longer isotropic and the helicity is not a good quantum number. However, theenergy spectrum of the Hamiltonian retains the same struc-ture as in the isotropic case, albeit with a different dispersionrelation. After diagonalizing the Hamiltonian, we obtain twodoubly degenerate energy levels, which we call light andheavy holes in analogy within the isotropic case: Eskd=1 2mg1k2±g3 mdskd, d2skd=Sg2 g3D2 skx4+ky4+kz4d +S3−Sg2 g3D2Dsky2kx2+kx2kz2+ky2kz2d. s4d Following Ref. 5 we can expand the spin-dependent terms in the anisotropic Lüttinger Hamiltonian in terms of Cliffordalgebra of dirac Gmatrices hG a,Gbj=2dabI434: H0=eskd+g3 mdaGa, s5d eskd=g1 2mk2, d1=−˛3kzky,d2=−˛3kxkz,d3=−˛3kxky, d4=−˛3 2g2 g3skx2−ky2d,d5=−1 2g2 g3s2kz2−kx2−ky2ds6d withdada=d2. Whereas in the isotropic Lüttinger model the matrix used to diagonalize the Hamiltonian belongs to theSOs3dgroup of rotations in kspace, 2in the anisotropic ma- terials the matrix that diagonalizes the anisotropic Hamil- tonian belongs to the SOs5drotations in daspace. The SOs5d Clifford algebra representation of the Hamiltonian (5)natu- rally unifies both the isotropic and the anisotropic Lüttingermodel on the same footing. Since this form of the Hamil- tonian depends on konly through the five-dimensional (5D) vectorda, a large part of the results in Ref. 5 is directly applicable to the anisotropic case. In this sense, the SOs5d Clifford algebra formalism shows its full power in the aniso- tropic case studied here. The projection operators onto thetwo-dimensional subspace of states of the heavy-hole (HH) and light-hole (LH)bands read: P L=1 2s1+dˆaGad,PH=1 2s1−dˆaGad. s7d For finite k, the Hamiltonian maintains the SOs4dsymmetry observed in Ref. 5. This symmetry reflects the degeneracy of the two Kramers’doublets at each value of k, corresponding to the doubly degenerate HH and the LH bands. Each of thebands has a SUs2dsymmetry, which we denote by SUs2d HH andSUs2dLH. Therefore, the total symmetry is SUs2dHH 3SUs2dLH=SOs4d.At the Gpoint,k=0, there is a enhanced SOs5dsymmetry. The symmetry generators read: rab=Gab+dbdcGca−dadcGcb=PLGabPL+PHGabPH, s8d where Gab=−i/2fGa,Gbgandfrab,H0g=0 trivially since the Hamiltonian is diagonal in the HH and LH bands. The spin operators Siare related to the Gabmatrices through the tensor habi, whose entries were given in Ref. 5: Si=habiGab. The concept of a conserved spin current is still valid in an-isotropic materials, since the projected spin is a constantof motion in virtue of its being a linear combination of the symmetry generators, S scdl=hablrab=PLSlPL+PHSlPH. We can therefore define the conserved spin current as Jil =1 2h]H/]ki,Sscdlj. Note that the richer anisotropic Lüttinger Hamiltonian yields a very similar structure to the isotropic one when cast in SOs4dlanguage. Although the concept of helicity l=kiSiis not valid in anisotropic materials, we can define a corresponding con-served helicity, l new,a s lnew=kiSscdi=l+2kihabidbdcGca=PLlPL+PHlPH.s9d Since it is a linear combination of the symmetry generators ofH0slnew=kiSscdi=kihabirabd, it is clear that fH,lnewg=0.In the isotropic limit, lnew=l, as can be seen using the identi- tiesfl,PLg=fl,PHg=0, valid in the isotropic case. The recent work reported in Ref. 5 shows that the Kubo formula for the conserved spin current response can be ex-pressed purely in terms of a geometric quantity, G ij=GijabGab=1 4d3eabcdedc]dd ]ki]de ]kjGab, s10d which describes mapping from the 3D kvector space to the 5Ddvector space. This results also include a quantum cor- rection to the semiclassical result in Ref. 2. We shall applythis formula to the anisotropic case here. However, there isone essential difference. Whereas in the isotropic case, thefield strength can be brought, through proper choice ofgauge, to the diagonal form G ij=lsl2−13/4 deijlkl/k3,i nt h eBRIEF REPORTS PHYSICAL REVIEW B 70, 113301 (2004 ) 113301-2anisotropic case this is impossible. Non-Abelian field strengths are, in general, gauge-variant. However, there is afundamental difference between fields that can be diagonal-ized through gauge transformation and fields for which thisis not possible. The former are ultimately Abelian in nature,whereas the latter are truly non-Abelian. The nondiagonalgauge field which describes evolution in anisotropic materi-als reflects the richer structure of the anisotropic LüttingerHamiltonian. We can express the field strength in terms of the (un- projected )spin degrees of freedom if we first note that the tenSOs5dgenerators G abdecompose into the three spin ma- tricesSiand the seven cubic, symmetric and traceless com- binations of the spin operators of the form SiSjSk, namely, A1=sSxd3,A2=sSyd3,A3=sSzd3, A4=hSx,sSyd2−sSzd2j, A5=hSy,sSzd2−sSxd2j, A6=hSz,sSxd2−sSyd2j, A7=SxSySz+SzSySx. s11d Then we can write Gij=1 4d3eijlklfVmAm+UlSlg,l=1,...,3, m=1,...,7, s12d where Ul=1 2g2 g3FS13+28g2 g3Dkl3+S13−28g2 g3Dk2klG, Vl=−2g2 g3FS1+4g2 g3Dkl3+S1−4g2 g3Dk2klG, V4=−3g2 g3kxsky2−kz2d, V5=−3g2 g3kyskx2−kz2d, V6=−3g2 g3kzskx2−ky2d, V7=−12kxkykz, s13d l=1,...,3. When cast in SOs4dlanguage, the expression for the spin conductance in anisotropic materials has the same form as that in the spherical model:sijl=8e2 V"o knLskd−nHskd1 3hablGijab, s14d wherenL=nFseLdandnH=nFseHdare the Fermi functions of the LH and HH bands. This expression can be put into the following elegant form: 1 3hablGijab=1 8d3g2 g3eijmkmklFS1−g2 g3Dkl2+S1+g2 g3Dk2G, s15d where we see that the first term in square brackets vanishes in the isotropic case.The lindex specifies the direction of the spin orientation, and it is not summed on the right-hand sideof Eq. (15). It is now obvious that the only components of sijlthat survive after summing the contributions from the whole Fermi surface are those for which iÞjÞl. Indeed, upon integration over k,sijlbecomes proportional to eijk, just as it should for crystals with cubic symmetry.13 Our result for the spin current can thus be put in the form ss=e2 "n1/3Ssg1,g2,g3d, s16d where the material-specific coefficient, S, is independent of the Fermi energy, and is of the order ,0.05 for most mate- rials (see Table II ). The ss,n1/3scaling is the hallmark of the dissipationless spin current, and has been proposed asa means by which to distinguish from other extrinsiceffects. 2,4To compare the spin conductance in different ma- FIG. 1. Spin conductivity plotted as a function of the anisotropy, parameterized by d;sg2−g3d/g1, with m=s6g3+4g2d/5 andn =1019cm−3held fixed at values corresponding to Si (bottom curve ), GaAs and InSb (top curve ). The circles indicate the real values of the parameters in Si, GaAs and InSb.TABLE II. Material-dependent coefficients of spin conductivity for values of g1,g2,g3that corresponding to common semiconduc- tors. Also given are the actual spin conductivities at n=1019cm−3 for both real anisotropic materials, and their spherical approxima- tions sd=0d. Ssg1,g2,g3d sssV−1cm−1d ssud=0sV−1cm−1d Si 0.028 14.60 21.10 Ge 0.063 32.79 34.31GaAs 0.062 32.64 34.33InSb 0.083 43.63 44.67InAs 0.079 41.61 42.55GaP 0.051 26.50 29.17BRIEF REPORTS PHYSICAL REVIEW B 70, 113301 (2004 ) 113301-3terials, we separate the dependence on the total carrier den- sity, which for the anisotropic Lüttinger model depends on the band parameters: n=s2meFd3/22 3es1/fsg1−2g3d/k2d3/2g +1/fsg1+2g3d/k2d3/2gdd2kˆ/s2pd3. Using this relation, we can find eFas a function of n, and use it to define the aniso- tropic Fermi distribution functions, nL,Hskd=QskFL,H−kd.W e have calculated ssfor band parameters that correspond to a selection of real materials, as well as for band parametersthat correspond to isotropic materials with the same values of g1and m;s6g3+4g2d/5g1. The results, listed in Table II, show that the nonzero anisotropy leads to a decrease in spin conductivity of as much as 30% (for Si ), although the reduc-tion in materials with smaller anisotropy is typically only ,5%. To illustrate the systematic dependence of spin conduc- tance on anisotropy, we plot ssas a function of d=sg3 −g2d/g1with g1andm=s6g3+4g2d/5 held fixed at values corresponding to Si, GaAs and InSb (Fig. 1 ). The spin con- ductance at fixed carrier concentration is maximum at d=0, whereas all real materials have d.0. This observation should guide in the selection of materials with relatively lowanisotropy for spin-injection devices and other applicationswhere strong spin current is desired. Finally, the variation of sswith the carrier concentration, n, is shown in Fig. 2. The authors would like to thank L. Balents, S. Murakami, N. Nagaosa and J. Sinova for useful conversations. Thiswork was supported by the NSF under Grant No. DMR-9814289 and by the U.S. Department of Energy, Office ofBasic Energy Sciences, under Contract No. DE-AC03-76SF00515. One of the authors (B.A.B. )acknowledges sup- port of a Stanford graduate fellowship and a second author(E.M. )acknowledges support of a NSF graduate fellowship. Another author (J.P.H. )was supported by funds from the David Saxon chair at UCLA. 1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M.Treger, Science 294, 1488 (2001 ). 2S. Murakami, N. Nagaosa, and S. C. Zhang, Science 301, 1348 (2003 ). 3J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. MacDonald, Phys. Rev. Lett. 92, 126603 (2004 ). 4D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, and Q. N. A. H. MacDonald, cond-mat/0309475. 5S. Murakami, N. Nagaosa, and S.-C. Zhang, cond-mat/0310005.6J. Hu, B. A. Bernevig, and C. Wu, cond-mat/0310093. 7J. Schliemann and D. Loss, Phys. Rev. B 69, 165315 (2004 ). 8N. A. Sinitsyn, E. M. Hankiewicz, W. Teizer, and J. Sinova, cond- mat/0310315. 9S.-Q. Shen, cond-mat/0310368. 10P. Lawaetz, Phys. Rev. B 4, 3460 (1971 ). 11A. Balderes and N. O. Lipari, Phys. Rev. B 8, 2697 (1973 ). 12J. M. Lüttinger, Phys. Rev. 102, 1030 (1956 ). 13M. Lax,Symmetry Principles in Solid State and Molecular Phys- ics(Dover, New York, 2001 ). FIG. 2. Dependence of the spin conductivity on the carrier den- sity,n, using the band parameters of Si (bottom curve ), GaAs and InSb (top curve ).BRIEF REPORTS PHYSICAL REVIEW B 70, 113301 (2004 ) 113301-4

PhysRevB.103.115132.pdf

PHYSICAL REVIEW B 103, 115132 (2021) Integrable nonunitary open quantum circuits Lucas Sá ,1,*Pedro Ribeiro ,1,2,†and Tomaž Prosen3,‡ 1CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal 2Beijing Computational Science Research Center, Beijing 100193, China 3Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia (Received 20 November 2020; revised 20 February 2021; accepted 25 February 2021; published 18 March 2021) We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneoustransfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit’s dynamicalgenerator. After showing the trace preservation and complete positivity of local maps, we reinterpret them asthe Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrabilityof the map is broken by adding interactions to the local coherent dynamics or by removing the dephasing.In particular, even circuits built from convex combinations of local free-fermion unitaries are nonintegrable.Moreover, the construction allows us to explicitly build circuits belonging to different non-Hermitian symmetryclasses, which are characterized by the behavior under transposition instead of complex conjugation. We confirmall our analytical results by using complex spacing ratios to examine the spectral statistics of the dissipativecircuits. DOI: 10.1103/PhysRevB.103.115132 I. INTRODUCTION Integrability is a fascinating field of mathematical physics. It provides exact solutions to dynamics and equilibrium invery diverse contexts, ranging from deterministic (i) classical[1,2] and (ii) quantum [ 3,4] many-body Hamiltonian dynam- ics to classical stochastic systems, (iii) in[5], and (iv) out[6] of equilibrium. Although the Liouville-Arnold [ 7] (i), Bethe- ansatz [ 8,9] (ii), and Onsager [ 10] (iii) threads of integrability were initially developed independently, they were beautifullyunited within the techniques of (quantum) inverse scattering[3,4,11] and the celebrated Yang-Baxter equation [ 5,12]. Later, quantum inverse scattering methods (a.k.a. algebraic Bethe ansatz) found their way to the exact solution (diagonal-ization) of classical stochastic systems—many-body Markovchains, such as simple exclusion processes [ 13]. More re- cently, related new techniques have been developed for theexact solution of open integrable quantum many-body sys-tems, specifically, by extending the algebraic Bethe ansatzto noncompact (nonunitary) auxiliary spaces [ 14] and by providing an exact mapping between Liouvillians of openmany-body systems and Bethe-ansatz integrable systems on(thermofield) doubled Hilbert spaces [ 15,16]. Very recently, (local) quantum circuits have become an important paradigm of nonequilibrium many-body physics,in particular, due to their simulability by emerging quantumcomputing facilities, where they provide a natural platformfor the demonstration of quantum supremacy [ 17]. Moreover, *lucas.seara.sa@tecnico.ulisboa.pt †ribeiro.pedro@tecnico.ulisboa.pt ‡tomaz.prosen@fmf.uni-lj.si(open) quantum circuits with local projective measurements have been shown to host an exciting new physics paradigm ofmeasurement-induced phase transitions [ 18–22]. The natural and significant question arises if integrability methods can be extended to such a paradigm. The results onintegrable trotterizations of integrable quantum spin chains[23] and classical stochastic parallel update exclusion pro- cesses [ 24] give very encouraging hints. In this paper, we make a key step in this direction by constructing an integrable open (nonunitary) local quantumcircuit. We show that Shastry’s ˇRmatrix [ 25–29], the essential integrability concept of the one-dimensional Fermi-Hubbardmodel, can be interpreted as a completely positive (CP)trace-preserving (TP) map over a pair of qubits (spins 1 /2) after a suitable analytic continuation of the interaction andspectral parameters. Our CPTP map represents a convexcombination of two coherent (unitary) symmetric nearest-neighbor-hopping (XX) processes, one of them composedwith local dephasing. By virtue of the Yang-Baxter equation,we then show the existence of a commuting transfer matrix forthe brickwork quantum circuit built from such CPTP maps,generating a family of local superoperators commuting withthe dynamical map. Integrability of the Floquet dynamics isalso demonstrated empirically by studying spectral statistics(complex spacing ratios (CSRs) [ 30]), whose sensitivity to integrability breaking is shown by studying two alternativefamilies of local open quantum circuits. The rest of the paper is organized as follows. In Sec. II,w e define the dissipative Hubbard circuit and describe in detailits elementary local gates. Next, we prove that the circuit isindeed integrable and CPTP in Secs. IIIandIV, respectively. The subsequent three sections focus on the physical contentof our circuit: We address its Kraus representation in Sec. V, 2469-9950/2021/103(11)/115132(9) 115132-1 ©2021 American Physical SocietySÁ, RIBEIRO, AND PROSEN PHYSICAL REVIEW B 103, 115132 (2021) integrability-breaking regimes in Sec. VI, and its symmetries in Sec. VII. Finally, we present numerical evidence corrobo- rating all our results in Sec. VIII before drawing conclusions and summarizing our findings in Sec. IX. Three Appendices present some additional details. II. THE DISSIPATIVE HUBBARD CIRCUIT We consider a spin-1 /2 chain of even size Lwith pe- riodic boundary conditions. The density matrix ρof the system evolves under the action of the discrete-time quan-tum channel /Psi1,ρ(t+1)=/Psi1[ρ(t)]—a linear map over the 4 L-dimensional state vector ρ—that we choose to be of the brickwork circuit form: /Psi1=/parenleftBiggL/2/productdisplay j=1ˇR2j,2j+1/parenrightBigg/parenleftBiggL/2/productdisplay j=1ˇR2j−1,2j/parenrightBigg (1) Here, ˇRklis the Hubbard ˇRmatrix nontrivially acting on sites kandl. Each wire in Eq. ( 1) carries a four-dimensional operator Hilbert space and ˇRacts as a two-site (16 ×16) elementary gate (grey box). One time step consists of tworows of the circuit—in the second of which the elementarygates are shifted by one site. Accordingly, Eq. ( 1) can also be written as /Psi1=T †/Phi1T/Phi1, (2) where /Phi1=ˇR⊗L/2corresponds to a single row of the cir- cuit. We introduced the one-site translation operator T, defined by its action on the computational operator basis,T|e 1,e2,..., eL/angbracketright=| eL,e1,..., eL−1/angbracketright, where indices ej∈ {0,1,2,3}label four possible spin-1 /2 operators at site j. While each row /Phi1of the circuit is factorizable into two-site elementary gates, the checkerboard pattern renders the fullcircuit /Psi1interacting. Because the same gate ˇRis applied throughout space and time, the repeated action of /Psi1leads to, in general, nonunitary translationally invariant Floquet dy-namics. After a Jordan-Wigner transformation, the Hubbard model can be understood as a spin ladder formed of a pair of XXmodels (corresponding to up- and down-spin fermions or tothebraand the ketof the density matrix [ 15] in our nonuni- tary formulation) coupled by the Hubbard interaction alongthe rungs. Thus, we start with the (two-site) spin-1 /2X X ˇR matrix, ˇR=1 a⎛ ⎜⎜⎜⎝a00 0 0c−ib 0 0ib c 0 00 0 a⎞ ⎟⎟⎟⎠, (3)which admits a simple trigonometric parametrization: a=cosλ, b=sinλ, c=1. (4) ˇR=ˇR(λ) is real orthogonal for imaginary spectral parameter λ∈iR.1We introduce a basis {eβ α}of 2×2 matrices such that the only nonzero entry (equal to 1) of eβ αis in row α and column β. We then consider the action of ˇRon two copies of the system (corresponding to ket ( ↑) and bra ( ↓)o f the vectorized density matrix ρ=/summationtext mnρmn|m/angbracketright/angbracketleftn|/mapsto→|ρ/angbracketright=/summationtext mnρmn|m/angbracketright⊗| n/angbracketright∗), ˇr↑(λ)=ˇRαγ βδ(λ)eβ α⊗12⊗eδ γ⊗12, ˇr↓(λ)=ˇRαγ βδ(λ)12⊗eβ α⊗12⊗eδ γ,(5) where 1dis the d×didentity matrix. Summation over re- peated indices is assumed throughout. In terms of the XX ˇRmatrices, the Hubbard ˇRmatrix reads (choosing the appropriate gauge) [ 29] ˇR(λ,μ)=βˇr(λ−μ)+αˇr(λ+μ)(σz⊗σz⊗14),(6) where ˇ r(λ)=ˇr↑(λ)ˇr↓(λ) and σx,y,zdenote the standard Pauli matrices. The two prefactors α≡α(λ,μ, u) and β≡ β(λ,μ, u) depend on two independent spectral parameters λ andμand on the Hubbard interaction strength u. The Hub- bard ˇRmatrix—which is not of difference form—satisfies the Yang-Baxter equation, (14⊗ˇR(λ,μ))(ˇR(λ,ν)⊗14)(14⊗ˇR(μ,ν)) =(ˇR(μ,ν)⊗14)(14⊗ˇR(λ,ν))(ˇR(λ,μ)⊗14),(7) if the ratio α/β is fixed as α β=cos(λ+μ)s i n h( h−/lscript) cos(λ−μ) cosh( h−/lscript), (8) where hand/lscriptare implicitly defined in terms of λ,μ, and u through sinh(2 h)/sin(2λ)=sinh(2 /lscript)/sin(2μ)=u. Finally, by choosing β=cos(λ−μ) cosh( h−/lscript) cos(λ−μ) cosh( h−/lscript)+cos(λ+μ)s i n h( h−/lscript), (9) we have α+β=1. Furthermore, with this choice of β,ˇR satisfies the unitarity condition: ˇR(λ,μ)ˇR(μ,λ)=116. (10) III. PROOF OF THE INTEGRABILITY OF THE HUBBARD CIRCUIT By construction, the Hubbard circuit /Psi1, defined by Eq. ( 1), is integrable. Indeed, since ˇRsatisfies the (braid) Yang-Baxter equation, Eq. ( 7), there exists a one-parameter family t(ω)o f transfer matrices in involution, i.e., [ t(ω1),t(ω2)]=0 for all 1We have introduced the factors of ±imultiplying b(corresponding to the choice x=−iin Eq. (12.93) of Ref. [ 29]) to compensate for the imaginary spectral parameter (since sin iλ=isinhλ). While these factors could be removed by a trivial similarity transformation,this choice will prove convenient below. 115132-2INTEGRABLE NONUNITARY OPEN QUANTUM CIRCUITS PHYSICAL REVIEW B 103, 115132 (2021) ω1,ω2. After introducing an auxiliary space, labeled a, iden- tical to the (local) four-dimensional physical Hilbert space,the transfer matrix is expressed as the partial trace of themonodromy matrix, t(ω)=Tr aTa(ω), with Ta(ω)=←/productdisplay 1/lessorequalslantj/lessorequalslantLRaj/parenleftbigg ω,λ+μ 2−(−1)jλ−μ 2/parenrightbigg , (11) where R=PˇR,Pis a 16 ×16 permutation matrix de- fined by P(|ρ1/angbracketright⊗|ρ2/angbracketright)=|ρ2/angbracketright⊗|ρ1/angbracketright, and the symbol←/productdisplay j indicates an ordered product with decreasing index j.T h e monodromy matrix Tais inhomogeneous (staggered) to ac- count for the checkerboard pattern of the quantum circuit.Evaluating the monodromy matrix in Eq. ( 11)a tt h et w o special (a.k.a. shift) points ω=λandω=μ, the Floquet propagator /Psi1, defined by Eq. ( 1), can be written as /Psi1=t(μ) −1t(λ). (12) To verify this claim, we start by computing the monodromy matrix at ω=λ. It reads as Ta(λ)=←/productdisplay 1/lessorequalslantj/lessorequalslantL/2Ra,2j(λ,μ)Pa,2j−1 =←/productdisplay 1/lessorequalslantj/lessorequalslantL/2Pa,2jˇRa,2j(λ,μ)Pa,2j−1 =←/productdisplay 1/lessorequalslantj/lessorequalslantL/2Pa,2jPa,2j−1ˇR2j−1,2j(λ,μ),(13) where we have used the identities R(λ,λ)=R(μ,μ )=P and ˇRalPak=PakˇRkl. Because all Pand ˇRoperators com- mute with each other when acting on different Hilbert spaces(i.e., when they have no subscript indices in common), takingthe trace over the auxiliary space yields t(λ)=Tr a/bracketleftBigg←/productdisplay 1/lessorequalslantj/lessorequalslantLPaj/bracketrightBiggL/2/productdisplay j=1ˇR2j−1,2j(λ,μ). (14) To evaluate the remaining trace, we use PalPak=PakPklto permute Pa1over all the other Pajand then use Tr aPa1=14. The transfer matrix finally reads as t(λ)=/parenleftBigg←/productdisplay 2/lessorequalslantj/lessorequalslantLP1j/parenrightBiggL/2/productdisplay j=1ˇR2j−1,2j(λ,μ). (15) The computation for ω=μproceeds similarly and results in the expression Ta(μ)=/parenleftBigg←/productdisplay 1/lessorequalslantj/lessorequalslantLPaj/parenrightBigg ˇRa1(μ,λ)L/2−1/productdisplay j=1ˇR2j,2j+1(μ,λ).(16) To evaluate the trace over the auxiliary space, we first cycle ˇRa1to the left of the product of permutations, permute it over PaLto obtain ˇRL1, take it out of the trace, and, at last, evaluate the resulting trace of permutations as above. Finally, imposingperiodic boundary conditions (i.e., identifying j=L+1 withj=1) and using the unitarity condition of Eq. ( 10), the trans- fer matrix at ω=μis given by t(μ)=/parenleftBigg ←/productdisplay 2/lessorequalslantj/lessorequalslantLP1j/parenrightBiggL/2/productdisplay j=1(ˇR2j,2j+1(λ,μ))−1. (17) It is now evident that the dynamical generator /Psi1can be written as in Eq. ( 12): /Psi1=t(μ)−1t(λ) =/parenleftBiggL/2/productdisplay j=1ˇR2j,2j+1(λ,μ)/parenrightBigg/parenleftBiggL/2/productdisplay j=1ˇR2j−1,2j(λ,μ)/parenrightBigg .(18) The involution property of the transfer matrix implies the integrability of the circuit since /Psi1commutes with t(ω)f o r allωand, in particular, with the two infinite sets of local superoperator charges generated from t(ω) by logarithmic differentiation: Q(1) n=dn dωnlogt(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=λ,Q(2) n=dn dωnlogt(ω)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=μ.(19) IV . THE HUBBARD ˇRMATRIX AS A LOCAL CPTP MAP Having proved that the circuit is integrable, it remains to be shown that it describes proper open quantum dynamics,i.e., that it is a CPTP map. It suffices to show this for theelementary gates ˇR. Indeed, choosing λ,μ, u∈iR(purely imaginary interaction), then α,β∈Rand ˇRbecomes a bis- tochastic quantum map [ 31,32] (i.e., a unital CPTP map). To check this result, we first reshuffle the indices of ˇRto obtain the dynamical Choi matrix D[31], such that D αγεη βδζθ=ˇRαβεζ γδηθ. Due to the channel-state duality [ 33,34], the map ˇRis CP if D is non-negative; it is TP if the partial trace of Dover the first copy of the system is the identity and it is unital if the partialtrace over the second copy of the system is the identity. TheTP and unitary conditions can be written as D αγεη αδεθ=ˇRααεε γδηθ=δγ δδη θ, (21a) Dαγεη βγζη=ˇRαβεζ γγηη=δα βδε ζ, (21b) respectively. To see that Eq. ( 21a) holds, we write out the components of the Choi matrix using Eq. ( 6), Dαγεη βδζθ(λ,μ)=βˇRαε γη(λ−μ)ˇRβζ δθ(λ−μ) +αˇRαε ιη(λ+μ)ˇRβζ κθ(λ+μ)(σz)ι γ(σz)κ δ,(21) compute the trace of Eq. ( 21a), Dαγεη αδεθ=(ˇR†ˇR)γη δθ×/braceleftbigg β+αifγ=δ β−αifγ/negationslash=δ =(β+α)δγ δδη θ=δγ δδη θ,(22) and find that the map is indeed TP. In Eq. ( 22), the three equalities hold because (i) ˇRadmits a real representation, (ii) it is unitary, and (iii) we have fixed α+β=1, respectively. Similarly, a computation starting from Eq. ( 21b) leads to a term proportional to ˇRˇR†, which again is nonvanishing only when the prefactor is β+α=1 and the map is therefore 115132-3SÁ, RIBEIRO, AND PROSEN PHYSICAL REVIEW B 103, 115132 (2021) unital. Finally, since we have a rank-two map, of the 16 eigen- values of D, 14 are zero and the remaining 2 are explicitly found to be 4 α> 0 and 4 β> 0.2The Choi matrix is therefore non-negative and the map is CP. We have thus shown that asuitable analytic continuation of ˇRis a unital CPTP map. V . KRAUS REPRESENTATION The previous result implies that ˇRcan be written in the Kraus form [ 31,35,36]. Abandoning the formal identification with the Fermi-Hubbard model, we identify, by reorderingtensor factors, φαγεη βδζθ(q+,q−,p)≡ˇRαεγη βζδθ(λ,μ, u) (23) with a vectorized quantum map parametrized by three in- dependent real parameters: the coherent hopping strengthsq ±≡−i(λ±μ)∈Rand the relative weight of the channels p≡α∈[0,1]. Swapping the second and third tensor-product factors in Eq. ( 6),φcan be written in the vectorized Kraus representation, φ(q+,q−,p)=K−⊗K∗ −+K+⊗K∗ +, (24) acting on (local two-site) states as φ[ρ]=K−ρK† −+K+ρK† +, with Kraus operators K−=/radicalbig 1−pˇR(iq−),K+=√pˇR(iq+)(σz⊗12).(25) We see that the Kraus map of Eqs. ( 24) and ( 25), which we dub the Hubbard-Kraus map, describes the discrete-timedynamics of free fermions (after undoing the Jordan-Wignertransformation) subjected to local dephasing. 3Indeed, after a suitable change of basis, the ˇRmatrix given in Eq. ( 3) with the parametrization of Eq. ( 4) can be written as ˇR(iq±)= exp{igd(q±)HXX}, where HXX=(σx⊗σx+σy⊗σy)/2i s the XX-chain Hamiltonian and gd( q)=/integraltextq 0dx/cosh xis the Gudermannian function. In the Trotter limit, q±→0, gd(q±)→q±, and the quantum map of Eq. ( 24) describes the quantum stochastic process in which, at each (discrete)half-time-step, with probability 1 −p, a fermion hops from the first to the second site (or vice versa) with amplitude q −; or, with probability p, it hops with amplitude q+; in the latter case, it is also subject to dephasing when at the first site. Weagain emphasize that only the local Kraus maps describe freedynamics (with dephasing), as the checkerboard pattern of thecircuit makes the full circuit strongly interacting. VI. BREAKING INTEGRABILITY A natural way of breaking the integrability of the circuit is by adding interactions to the local coherent processes. This can be achieved by replacing, in Eq. ( 25), the XX ˇRmatrices by more general XXZ (six-vertex) ˇRmatrices [ 5,38], which 2The analytic continuation of λ,μ,a n d uto the imaginary axes can always be chosen to render αandβpositive. 3Note that the circuit /Psi1does notdescribe a dissipative Hubbard model. ˇRis used as a mathematical device to build an integrable circuit which, ap r i o r i , is unrelated to the original model. For a recent study of an exactly solvable dissipative Hubbard model, seeRef. [ 37].have the same form of Eq. ( 3) but admit a two-parameter trigonometric parametrization, a=sin(λ+γ),b=sinλ, c=sinγ, (26) where γ∈(−π,π ] is related to the anisotropy parameter of the XXZ chain and, as before, λ∈iR.T h eX X ˇRmatrix given by Eq. ( 4) follows from Eq. ( 26) upon setting γ=π/2. The Kraus operators, defined in Eq. ( 25), have five independent real parameters p,q±,γ±and the extensive dephasing-XXZ circuit is built from them exactly as before. Note that integra-bility is broken because the ˇRmatrix obtained this way from Eqs. ( 5) and ( 6) no longer satisfies the Yang-Baxter equation, Eq. ( 7). Furthermore, while one might be tempted to conjecture the integrability of the quantum map of Eq. ( 25) for general ˇRmatrices at a free-fermion point [ 39] (i.e., satisfying a 2= c2−b2), this turns out not to be correct. One such model is obtained from the Hubbard-Kraus map by removing dephas-ing from the second channel. The resulting quantum map, stillof the form of Eq. ( 24), is a convex combination of two unitary free Kraus channels, K −=/radicalbig 1−pˇR(iq−),K+=√pˇR(iq+), (27) which no longer satisfies the Yang-Baxter equation, Eq. ( 7). We thus arrive at the strong conclusion that even the sim-plest local dynamics (i.e., the convex combination of freeunitaries) can lead to nonintegrable quantum circuits. Thisresult highlights the special, and rather nontrivial, nature ofthe construction of the Hubbard-Kraus circuit above. Below,we will give numerical evidence for the breaking of integra-bility in the preceding two examples (dubbed XXZ circuit andtwo-free-channel circuit). VII. SYMMETRIES AND SPECTRAL STATISTICS According to the quantum chaos conjectures [ 40,41]o f dissipative systems [ 30,42], the statistics of the complex eigenvalues of an integrable circuit are the same as those ofuncorrelated random variables (henceforth, Poisson statistics),while nonintegrable models follow the predictions of randommatrix theory (RMT), in the corresponding symmetry class.The comparison can only be done once all the (unitary andmutually commuting) symmetries of the model have beenresolved, i.e., separating sectors with a fixed set of eigenvaluesof the unitary symmetries. 4 Let us describe the unitary symmetries of our circuits (see Appendix Afor details). Because of the structure of the quan- tum circuit defined in Eq. ( 1), all the considered models are invariant under translation by two sites ([ /Psi1,T2]=0) and, therefore, have L/2 sectors of conserved quasimomentum k∈{0,1,..., L/2−1}. Furthermore, the circuits are invariant under simultaneous space translation by one site (half a unit cell) and temporaltranslation by one circuit layer (half a time step), which canbe encoded in the commutation relation [ T/Phi1,T †/Phi1]=0. For 4Otherwise, levels from different symmetry sectors overlap without interacting and one obtains apparent Poisson statistics regardless ofthe actual statistics. 115132-4INTEGRABLE NONUNITARY OPEN QUANTUM CIRCUITS PHYSICAL REVIEW B 103, 115132 (2021) a sector of fixed quasimomentum k,(/Psi1)k=e−4πik/L(T/Phi1)2 k, where ( A)k≡PkAPkandPkare orthogonal momentum- projection operators: Pk=2 LL/2−1/summationdisplay n=0T2nexp−2πikn L/2. (28) Therefore, resolving the space-time symmetry of /Psi1amounts to examining the spectral statistics of ( T/Phi1)k. Besides the kinematic symmetries of the circuit, the XX and XXZ ˇRmatrices display conservation of (total) magneti- zation in each (bra and ket) copy of the system independently.5 Accordingly, the quantum map /Phi1splits into ( L+1)2sectors, each of dimension N=/parenleftbigL M↑/parenrightbig/parenleftbigL M↓/parenrightbig , where M↑andM↓denote the total magnetization in the two copies. Once inside a fixed sector of the unitary symmetries, the symmetry class to which each circuit belongs is determinedby its behavior under transposition (which can be understoodas non-Hermitian time reversal) instead of complex conjuga-tion [ 43,44]. Transposition symmetry imposes local correla- tions and completely determines the short-distance spectral statistics [ 43,45]. We argue in Appendix Bthat both the Hubbard-Kraus and two-free-channel circuits admit a trans-position symmetry; the latter—being nonintegrable—has,therefore, the same spectral statistics as matrices from classAI †[30,43,46] (complex symmetric random matrices with Gaussian entries). In contrast, the XXZ circuit breaks transpo-sition symmetry and has, therefore, the same spectral statisticsas matrices from the Ginibre orthogonal ensemble (GinOE,real asymmetric random matrices with Gaussian entries). 6 VIII. NUMERICAL RESULTS To probe the statistics of the dissipative quantum circuits, we consider the complex spacing ratios (CSRs) [ 30]o ft h e eigenvalues of the operator ( T/Phi1)k. We denote the set of eigenvalues by {/Lambda1j}and, for each /Lambda1j, we find its nearest neighbor, /Lambda1NN j, and its next-to-nearest neighbor, /Lambda1NNN j.T h e CSRs are defined by zj=(/Lambda1NN j−/Lambda1j)/(/Lambda1NNN j−/Lambda1j). In the thermodynamic limit, the probability distribution of zjis flat on the unit disk for Poisson statistics; for non-Hermitianrandom matrices, it has a characteristic C shape (an ana-lytic surmise is given in Ref. [ 30]), see Figs. 1(a)–1(c) .I n Figs. 1(d)–1(f) , we plot the CSR distribution of ( T/Phi1) kfor, respectively, the Hubbard-Kraus, XXZ, and two-free-channelcircuits (eigenvalues obtained from exact diagonalization).The flat distribution of the integrable Hubbard-Kraus circuit(d) and the C-shaped distribution of the chaotic XXZ (e) andtwo-unitary-channel (f) circuits are clearly visible. In the lattertwo cases, the CSR distribution also allows us to distinguishthe different symmetry classes to which the circuits belong. 5This also restricts the allowed incoherent processes to dephasing, which is the case in all our models. 6Because matrices from the Ginibre orthogonal ensemble (GinOE) and Ginibre unitary ensemble (GinUE) differ by their behavior un- der complex conjugation and not transposition, they share the same spectral correlations. For this reason, the large- Nresults presented for the GinUE in Ref. [ 30] carry over to the GinOE. FIG. 1. (a)–(c) CSR distributions obtained from sampling 105 uncorrelated random variables (Poisson spectrum) (a) or exact di- agonalization of 104×104random matrices from the GinOE (b) or class AI†(c) [104 realizations superimposed in (b) and (c)]. (d)–(f) CSR distributions of (a single realization of) the operator ( T/Phi1)kfor the Hubbard-Kraus (d), XXZ (e), and two-free-channel (f) circuits.The eigenvalues were obtained by exact diagonalization in sectors of fixed L=12,M ↑=4,M↓=3, and k=0,1,2—leading to three sectors of fixed quasimomentum kwith 18 150 eigenvalues each, the ratios of which were then superimposed. The numerical parameters were chosen as follows: q+=−0.2,q−=0.6,p=0.55 (for all three circuits) and γ+=γ−=1 (for the XXZ circuit). (g)–(i) /angbracketleftr/angbracketright versus /angbracketleft−cosθ/angbracketrightplots for the (g) Hubbard-Kraus, (h) XXZ, and (i) two-free-channel circuits, obtained by randomly sampling (40 samples) p,q±,a n dγ±, for different L,M↑,a n d M↓and fixed k=1. Larger, black-rimmed dots mark the average (center of mass) for each symmetry sector. The points accumulate around the Poisson (2 /3,0), AI†(0.722,0.188), and GinOE (0.738,0.241) points or spread over the line between them. To provide a more quantitative measure of spectral chaotic- ity, we express the CSR in polar coordinates, z=rexpiθ, and characterize its distribution by two numbers, namely, the meanratio/angbracketleftr/angbracketright, measuring the degree of radial level repulsion, and the angular correlation /angbracketleft−cosθ/angbracketright. A Monte Carlo sampling over the quantum circuits with varying model parameters thenyields an /angbracketleftr/angbracketrightversus /angbracketleft−cosθ/angbracketrightscatter plot, which can again be compared with the results for Poisson random variablesand GinOE and AI †matrices. Details on the numerical pro- cedure can be found in Appendix C. Figures 1(g)–1(i) show the/angbracketleftr/angbracketrightversus /angbracketleft−cosθ/angbracketrightplots for the Hubbard-Kraus, XXZ, and two-free-channel circuits, respectively. For the Hubbard-Kraus circuit (g), which is integrable by construction, weobtain a high concentration of points around the Poisson point,even for modest system sizes. For the chaotic XXZ circuit(h) of the same system sizes, while points spread over theline connecting the Poisson and GinOE points, there is now ahigh accumulation of data around the GinOE point, signalingintegrability breaking. Note that the center-of-mass values areflowing to the GinOE point as sector dimension increases. 115132-5SÁ, RIBEIRO, AND PROSEN PHYSICAL REVIEW B 103, 115132 (2021) Finally, the two-free-channel circuit (i) displays the same qualitative integrability-breaking behavior as the XXZ circuitand, for the largest system sizes, has reached the AI †point. IX. CONCLUSIONS Let us summarize the two key findings of this paper, ad- dressing the critical question we posed at the start, namely,if integrability methods can be extended to the realm ofdissipative quantum circuits. First, we answered it in the pos-itive, by showing that Shastry’s celebrated ˇRmatrix of the Fermi-Hubbard model can be interpreted as a unital CPTPmap of a pair of qubits (spin 1 /2’s) for imaginary values of interaction and spectral parameters. By consequence of theYang-Baxter equation, this implies integrability of the brick-work open circuit built from such nonunitary two-qubit maps.This result opens an avenue for studying general integrableopen (driven /dissipative) quantum Floquet circuits. For ex- ample, our result straightforwardly generalizes to SU( d) open qudit circuits using Maassarani’s ˇRmatrix [ 28]. Second, our construction shows that building integrable dissipative circuitsis highly nontrivial, in the sense that deformations of theHubbard-Kraus circuit (even convex combinations of localfree-fermion unitaries) are nonintegrable. Finally, we note that for a real interaction parameter u∈R the staggered transfer matrix, defined in Eq. ( 11), generates a unitary Floquet circuit, via Eq. ( 12), for a 2 ×Lspin ladder which represents an integrable trotterization of the HermitianFermi-Hubbard model by taking λ=iτ,μ=0, where τ∈R is the time step. This remarkable side result parallels the result[23] for the Heisenberg chain. ACKNOWLEDGMENTS T.P. thanks D. Bernard and F. Essler for inspiring dis- cussions and Institut Henri Poincaré for hospitality inthe last stage of this work. L.S. acknowledges supportby FCT through Grant No. SFRH /BD/147477 /2019. L.S. and P.R. acknowledge support by FCT through Grant No.UID/CTM/04540 /2019. T.P. acknowledges ERC Advanced Grant No. 694544-OMNES and ARRS Research Program No.P1-0402. APPENDIX A: KRAUS REPRESENTATION AND UNITARY SYMMETRIES OF THE EXTENSIVE BRICKWORK CIRCUIT In this Appendix, we recast the extensive circuit into the Kraus representation and use it to give a more detailed expo-sition of the unitary symmetries (kinematic and dynamical) ofthe various Kraus circuits discussed in the paper. 1. Kraus representation of the extensive circuit To build the extensive quantum circuit of length Lout of the elementary two-site building blocks in the Kraus repre-sentation, we define a row Kraus operator F νby tensoring L/2 copies of the elementary Kraus operators K±,Fν=/circlemultiplytextL/2 j=1Kν2j. Here, ν=(ν2,ν4,...,ν L) is a multi-index with all two-site indices, ν2j=±, and Kν2jis a Kraus operator cou- pling sites 2 j−1 and 2 j. The quantum map correspondingto the entire row is then /Phi1=ˇR⊗L/2=/summationtext νFν⊗F∗ ν(where tensor-product factors are reordered such that all second-copydegrees of freedom come after the first copy). The secondrow of the circuit is again obtained by translation by one site,T †/Phi1T. Then, one complete time step is given by /Psi1=T†/Phi1T/Phi1 (A1) (A2) where the superimposed layers represent the two copies of the system—red gates act on the ket of the density matrix whilecomplex conjugate blue gates act on the bra. 2. Translational invariance The two-site translation invariance of the quantum circuit defined in Eq. ( A2), [/Psi1,T2]=0, leads to the conserva- tion of quasimomentum. Since TL=1, the eigenvalues ofTare exp 2 πi(k/L), with k=0,1,..., L−1. Then, the translational-invariant Kraus circuit with Lsites ( L even) has L/2 sectors of conserved quasimomentum k∈ {0,1,..., L/2−1}. Note that each row Kraus operator Fν is not translationally invariant, only their sum is, and hence there is no conservation of momentum in each bra /ket copy individually. To project states into sectors of fixed k,w eu s e the orthogonal projection operators given in Eq. ( 28). 3. Magnetization conservation The XX and XXZ ˇRmatrices display conservation of (to- tal) magnetization (or particle-number in a fermion picture) ineach copy of the system independently. In these conditions,for a given copy of the system, each row Kraus operator F ν splits into L+1 sectors of total magnetization Sz=M, where Szacts on the computational-basis states as Sz|s1,..., sL/angbracketright=/summationtextL j=1sj|s1,..., sL/angbracketright. Each sector Mhas dimension/parenleftbigL M/parenrightbig .A c - cordingly, the quantum map /Phi1splits into ( L+1)2sectors, each of dimension N=/parenleftbigL M↑/parenrightbig/parenleftbigL M↓/parenrightbig (here M↑andM↓denote the magnetization in the two copies). We restrict ourselves to sectors with M↑,↓/negationslash=L/2 to avoid an additional Z2spin-flip (particle-hole) symmetry. Finally, there is another Z2symme- try connecting the two copies of the system; we also avoid thissymmetry by considering only sectors with M ↑/negationslash=M↓. APPENDIX B: TRANSPOSITION-SYMMETRY CLASSES In this Appendix, we elaborate on the symmetry classifica- tion of general non-Hermitian matrices and CPTP generators,in particular in terms of the transposition symmetry. We alsoargue for the presence (absence) of transposition symmetry inthe Hubbard-Kraus and two-free-channel (XXZ) circuits. 115132-6INTEGRABLE NONUNITARY OPEN QUANTUM CIRCUITS PHYSICAL REVIEW B 103, 115132 (2021) FIG. 2. Schematic representation of the steps involved in determining whether there exists a transposition symmetry of the circuit. Gray gates represent local quantum maps φ, while orange and magenta filled circles depict local unitaries v−,v+∈SU(4), respectively, and empty circles their inverses. Transposition is signaled by the flip of the wedge in the corner of the local maps. To respect the kinematical symmetries of the circuit, the allowed unitary transformations are one-site translations and the local unitaries v±. 1. Symmetry classification of non-Hermitian matrices and CPTP generators The symmetry class to which each circuit belongs is determined by its antiunitary (and anticommuting unitary)symmetries. While there are 38 symmetry classes of non-Hermitian matrices [ 46–48]—dictated by the behavior under sign inversion, complex conjugation, transposition, and Her-mitian conjugation—considering only the generators of CPTPdynamics restricts the allowed symmetry classes back to ten[44,49], which are in one-to-one correspondence with the Altland-Zirnbauer [ 50,51] classes of closed quantum systems. Indeed, for a genuine quantum channel /Psi1, its Hermiticity- preserving property guarantees the existence of a symmetry/Psi1=S/Psi1 ∗S†for some unitary S, while complete positivity forbids the existence of a symmetry /Psi1=−S/Psi1/latticetopS†or/Psi1= −S/Psi1S†. Of the remaining three types of symmetries, only transposition symmetry (i.e., the existence of a unitary Tsuch that/Psi1=T/Psi1/latticetopT†with TT∗=±1) imposes local correlations and, hence, completely determines the short-distance spectralstatistics [ 43,45]. So, while there are ten remaining symmetry classes, only three different universal statistics exist, differingby the amount of level repulsion. In the absence of transposi-tion symmetry, the generator is represented by a general realasymmetric matrix and shares spectral statistics with randommatrices from the GinOE. If there is a unitary Tsatisfying TT ∗=+1, then the generator shares the spectral statistics with the complex symmetric matrices from class AI†. 2. Transposition symmetry of the Kraus circuits We now analyze the behavior of the three circuits consid- ered in the paper under transposition. We start by showing thatit suffices to consider the properties of elementary two-sitequantum maps. Taking the transpose of the circuit, we find/Psi1 /latticetop=/Phi1/latticetopT/latticetop/Phi1/latticetopT∗=/Phi1/latticetopT†/Phi1/latticetopT. We want to bring it back to/Psi1by a unitary transformation that satisfies the symmetries of the model. These include translations (necessary to bringthe circuit back to the correct order of applying first gatesas odd-even bonds followed by even-odd bonds) and, pos-sibly, local 4 ×4 unitaries in each local Hilbert space (local gauge transformations). The procedure is depicted pictoriallyin Fig. 2. We conclude that the circuit satisfies the transposition sym- metry /Psi1=T/Psi1 /latticetopT†if the local quantum maps φsatisfy φ= =(v−⊗v+)φ/latticetop(v† +⊗v† −), (B1) for some v−,v+∈SU(4) respecting the dynamical sym- metries of the circuits (i.e., magnetization conservation).Moreover, from Eq. ( 24), we see that the behavior under trans- position of φis fully determined by the behavior of the Kraus operators K±of each circuit (since tensoring and transposing commute). It thus suffices to analyze the behavior of the Krausoperators under transposition. a. Two-free-channel circuit The local quantum map φis a convex combination of a pair of unitary channels whose Kraus operators are givenin Eq. ( 27). The transposition symmetry is evident from Eq. ( 3) after a unitary change of basis by conjugation with V=diag(1 ,exp{iπ/4},exp{−iπ/4},1)—which corresponds to the choice of gauge v −=v∗ +=Vin Eq. ( B1). Writing out the Kraus operators explicitly (we omit all zero entries) in thenew basis, K ±(λ±)=⎛ ⎜⎝cosλ± 1s i n λ± sinλ± 1 cosλ±⎞ ⎟⎠,(B2) we see that they are complex symmetric, where we defined the spectral parameters λ±≡λ±μ∈iR(related to the hopping amplitudes q±by multiplication by i). (The normalization of theˇRmatrix is irrelevant for the purpose of this Appendix and will be dropped throughout.) It follows that, in this basis,φ(λ −,λ+)=φ/latticetop(λ−,λ+), and the circuit enjoys a transposi- tion symmetry. b. Hubbard-Kraus circuit Although the Hubbard-Kraus circuit is integrable and, therefore, exhibits Poisson spectral statistics, it is instructiveto determine its symmetry class to provide contrast to theXXZ circuit case discussed below. The Kraus operators de-fined in Eq. ( 25) read as K −(λ−)=⎛ ⎜⎜⎜⎝cosλ− 1 −isinλ− isinλ− 1 cosλ−⎞ ⎟⎟⎟⎠and K+(λ+)=⎛ ⎜⎜⎜⎝cosλ + 1 isinλ+ isinλ+ −1 −cosλ+⎞ ⎟⎟⎟⎠. (B3) 115132-7SÁ, RIBEIRO, AND PROSEN PHYSICAL REVIEW B 103, 115132 (2021) While both Kraus operators cannot be symmetrized simultaneously by a change of basis as before, they still satisfy K/latticetop −(λ−)=K−(−λ−)=K−(λ∗ −) and K/latticetop +(λ+)=K+(λ+). (B4) It follows that φ(λ−,λ+)=φ/latticetop(λ∗ −,λ+). For nonintegrable cases, generalizing the local Hubbard-Kraus map to more than two Kraus operators of the form of Eq. ( 25), we have made the empirical observation that the equality of the Kraus operators and their transposes up to complex conjugation of the imaginary spectral parameters is enough to guarantee the convergence of their spectral statistics to those of the AI†class. This condition is fulfilled by the Hubbard-Kraus map. c. XXZ circuit By introducing interactions into the coherent dynamics, i.e., by considering Kraus operators K−(λ−)=⎛ ⎜⎝sin(γ−+λ−) sinγ−−isinλ− isinλ− sinγ− sin(γ−+λ−)⎞ ⎟⎠and K+(λ+)=⎛ ⎜⎝sin(γ++λ+) sinγ+ isinλ+ isinλ+−sinγ+ −sin(γ++λ+)⎞ ⎟⎠,(B5) it follows that neither can both Kraus operators be simultane- ously symmetrized nor do they satisfy Eq. ( B4). Hence, the XXZ circuit does not enjoy a transposition symmetry. APPENDIX C: DETAILS ON THE NUMERICAL ANALYSIS In this Appendix, we discuss in more detail the random sampling of the circuits and the numerical analysis of CSRs. To obtain the scatter plots of Figs. 1(g)–1(i) , we express the CSR in polar coordinates and characterize its distributionswith two numbers (the mean ratio /angbracketleftr/angbracketrightand the angular correla- tion/angbracketleft−cosθ/angbracketright). For each model, we randomly sample the two independent hopping parameters q ±from a standard normal distribution (i.e., zero mean, unit variance), the channels’ relative weight pfrom a uniform distribution on [0,1], and, in the case of the XXZ circuit, the anisotropy parameters γ±from a uniform distribution on ( −π,π ]. For a fixed random realiza- tion of the parameters, we exactly diagonalize the quantumcircuit for four different system sizes and conserved magne-tization sectors—( L,M ↑,M↓)=(12,3,2), (10,4,3), (14,4,2), and (12,4,2), corresponding to sector sizes N=2420, 5040, 13 013, and 18 150, respectively—and compute the pairs(/angbracketleftr/angbracketright,/angbracketleft−cosθ/angbracketright). A Monte Carlo sampling of the quantum cir- cuits then yields a /angbracketleftr/angbracketrightversus /angbracketleft−cosθ/angbracketrightscatter plot.There are three special points: For a set of un- correlated (Poisson) random variables, we have exactly (/angbracketleftr/angbracketright,/angbracketleft−cosθ/angbracketright)=(2/3,0), while for random matrices from the GinOE and class AI †we have ( /angbracketleftr/angbracketright,/angbracketleft−cosθ/angbracketright)≈ (0.738,0.241) and ( /angbracketleftr/angbracketright,/angbracketleft−cosθ/angbracketright)≈(0.722,0.188), respec- tively [ 30]. When approaching the thermodynamic limit, we expect that the points obtained by sampling an integrablecircuit concentrate around the Poisson point, while they ac-cumulate around the RMT point of the respective symmetryclass when sampling from a nonintegrable model. For finitesystem sizes L, the points spread over the line connecting the two fixed points and one tries to determine to which oneof them the scatter points are flowing as Lincreases. This last step may be difficult to perform if only small systemsizes are available and /or finite-size effects are pronounced. Moreover, the flow may be nonmonotonic; for instance, itmay depend more strongly on the system length Lthan on the magnetization sectors M, or sectors with different parity ofMmay flow differently. Finally, there are deviations from the pattern described above if the spectrum is close to one-dimensional, for in-stance, if the quantum map is very close to being unitary(which happens when pis very close to either 0 or 1). 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PhysRevB.92.045125.pdf

PHYSICAL REVIEW B 92, 045125 (2015) Protection of a non-Fermi liquid by spin-orbit interaction T. K. T. Nguyen1and M. N. Kiselev2 1Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan, Hanoi, Vietnam 2The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151 Trieste, Italy (Received 1 April 2015; published 27 July 2015) We show that a thermoelectric transport through a quantum dot–single-mode quantum point contact nanodevice demonstrating pronounced fingerprints of nonFermi liquid (NFL) behavior in the absence of external magneticfield is protected from magnetic field NFL destruction by strong spin-orbit interaction (SOI). The mechanism ofprotection is associated with the appearance of additional scattering processes due to lack of spin conservationin the presence of both SOI and small Zeeman field. The interplay between in-plane magnetic field /vectorBand SOI is controlled by the angle between /vectorBand/vectorB SOI. We predict strong dependence of the thermoelectric coefficients on the orientation of the magnetic field and discuss a window of parameters for experimental observation of NFLeffects. DOI: 10.1103/PhysRevB.92.045125 PACS number(s): 73 .23.Hk,73.50.Lw,72.15.Qm,73.21.La I. INTRODUCTION The paradigm of Landau Fermi liquid (FL) [ 1] is one of the cornerstones of modern condensed matter theory. Basedon the concepts of quasiparticles—well-defined excitationswhose energy in the long-wave limit is greater than theirdecay rate—the FL theory successfully explains the behaviorof normal and superconducting metals giving universal pre-dictions for thermodynamic and transport properties [ 2]. The FL phenomenology is justified in many microscopic modelsdescribing interacting fermions in and out of equilibrium.However, there are several cases where a violation of the FLpicture is observed experimentally (e.g., in strongly correlatedelectron systems such as heavy fermion compounds [ 3], unconventional superconductors [ 2], and quantum transport through nanostructures [ 4,5]). The pronounced non-Fermi liquid (NFL) behavior of these systems is attributed to abreakdown the quasiparticle concept: The decay rate of low-energy excitations becomes greater than the energy of theexcitations itself. While the fingerprints of NFL physics in thermodynamics of strongly correlated systems and quantum transport had beenseen experimentally [ 3,5], it is also generally accepted that the NFL picture is extremely sensitive to variation of externalparameters being unstable against the FL ground state. Thus,the stability of the NFL domain and the possibility to observestrong deviations from the Landau FL paradigm poses majorchallenges including the development of theoretical modelsand predictions for the stabilization of NFL states. Importantquestions are as follows: Is it possible at all to protect unstableNFLs? What are the physical observables which demonstratethe most pronounced manifestation of the NFL physics? In this paper we present an example based on a window of parameters within which the observation of strong deviation from the FL picture can be protected and extended by effects ofstrong spin-orbit interaction (SOI). The physical observableswe consider in this work are thermoelectric coefficients ofa nanodevice (Fig. 1). Our theoretical model justifying NFL behavior is a two-channel Kondo (2CK) model [ 6–8]. While the scattering of single orbital channel electrons on a resonance quantum impurity itself leads to strong modification of the thermoelectric transport properties within the Landau FLparadigm through strong renormalization of the FL energy scale [ 9–11], the detour from the FL picture is predicted to change completely both electric [ 12–15] and thermoelectric transports [ 16,17]. For example, one of the manifestations of the NFL behavior in quantum transport is associated with the logarithmic enhancement of the thermoelectric power [ 16]i n the situation when the 2CK model originates from the chargeKondo effect in a single mode quantum point contact (QPC)–quantum dot (QD) setup tuned by gate voltages to the Coulombblockade (CB) peak regime [ 12–16]. In that case two channels are the electron spin degrees of freedom while the almost transparent QPC (weak back-scatterer) works as a quantum impurity. The 2CK physics is known to be unstable withrespect to any effects which can potentially break (staticallyor dynamically) the symmetry between the channels [ 5,17]. In particular, it has been shown that the effects associated withtime-reversal symmetry breaking (TRS) due to an externalmagnetic field restore the FL properties at temperatures below T efftunable by the field [ 17]. The universality class of the unstable 2CK model than changes to a single channel Kondoproblem (1CK). Fully screened 1CK is characterized by stable local FL properties. Therefore, while being very attractive from the theoretical point of view, the 2CK physics suffersfrom serious experimental obstacles [ 4,5,18] impeding a direct observation of NFL behavior. The paper is organized as follows: We describe possible experimental setup for observing the NFL transport in Sec. II. A theoretical model accounting for the interplay between SOI,external magnetic field, and effects of Coulomb blockade inquantum dot is presented in Sec. III. We discuss the solution of the one-dimensional (1D) quantum-mechanical scatteringproblem in the presence of strong SOI in Sec. IV. An effective model describing a low-energy physics of the problem and itsexact solution is presented in Sec. V. The transport coefficients computed with the help of the exact solutions are discussedin Sec. VI. Section VIIis devoted to discussion of the key results of the paper including estimation for the parametersand definition of the conditions necessary for experimentalobservations of the NFL physics. Summary and Conclusionsare given in Sec. VIII. Details of derivation of the effective model are presented in Appendix. 1098-0121/2015/92(4)/045125(10) 045125-1 ©2015 American Physical SocietyT. K. T. NGUYEN AND M. N. KISELEV PHYSICAL REVIEW B 92, 045125 (2015) Vg V× B T+ΔT IJT FIG. 1. (Color online) Typical setup for thermoelectric measure- ments: “cold” contact (light orange area) at reference temperature T, quantum dot–quantum point contact electrostatically defined by gates (blue boxes), separated by a tunnel barrier from a “hot” (deep orange) contact at temperature T+/Delta1T. The voltage Vis applied across the device for zero current measurements (see text for the details). II. PROPOSED EXPERIMENTAL SETUP We consider a two-terminal nanodevice (see Fig. 1) de- signed to be used for thermoelectric measurements [ 19,20]. The QD–QPC contains 2-d electron gas (2DEG) confined inthezdirection (light orange area on the Fig. 1). The open QPC connects it to the drain at the reference temperatureT. We assume that the Rashba SOI [ 21,22] (caused by the gradient of the confining potential in the zdirection) leads to appreciable effects which we will discuss in thispaper. The source is separated from the QD by a tunnelbarrier with low transparency |t|/lessmuch1. The temperature of the source (deep orange) is adjusted by the Joule heat controlledby the current I Jflowing along the lead (black arrow). The temperature difference /Delta1T across the tunnel barrier is assumed to be small compared to the reference temperatureTto guarantee the linear response operation regime for the device. The QD is electrostatically controlled by two plungergates (blue rectangles) to adjust the size of the electronisland. The device is operated in the steady state of zerosource-drain current I sd=0=G/Delta1V th+GT/Delta1T, controlled by applying a thermo-voltage /Delta1V thbetween the source and the drain. The QPC (denoted by the cross in the light orangearea) is tuned to the single mode regime characterized by acontrollable small reflectivity |r|/lessmuch1. Under this assumption and neglecting the resistance of the “metallic” QD we assumethat the voltage difference /Delta1V tharises across the tunnel barrier between the source and QD. The transport coefficients, electricconductance Gand thermoelectric coefficient G T(measured independently), define the thermoelectric power (TP) S: GT=∂Isd ∂/Delta1T,G=∂Isd ∂V,S=−/Delta1V th /Delta1T/vextendsingle/vextendsingle/vextendsingle/vextendsingle Isd=0=GT G. We assume that the magnetic field (blue arrow) is applied parallel to the plane of 2DEG to avoid orbital effects. III. THEORETICAL MODEL The theoretical description of setup (see Fig. 1)i sf o r m u - lated in terms of the Hamiltonian: H=Hs+Hd+Htun+Hz. (1) HereHsandHdare the Hamiltonians of the source (“hot” contact) and the drain (“cold” contact), respectively. Htundescribes tunneling between the source and the drain and Hz accounts for the Zeeman effect in both contacts. We assume that the source can be described by a standard FL approach, Hs=/summationdisplay k,σ/epsilon1kσc† kσckσ, (2) where c†andcare creation/annihilation operators of quasi- particles (we adopt a system of units /planckover2pi1=kB=1). The drainHd=Hc+HQPCincludes the Coulomb blockaded QD described by charging Hamiltonian Hcand QPC represented by HQPC=H0+HSOI+HBS. (3) We assume that the charge ˆQ=e(ˆns+ˆnd) in the QD is weakly quantized (mesoscopic CB regime [ 27]) and controlled by the gate voltage Vg: Hc=Ec[ˆns+ˆnd−N(Vg)]2, (4) where ˆnsandˆndare the operators of the number of electrons that entered the dot through the source and the drain,respectively, and E c∼e2/LQDis the charging energy of QD with geometric size LQD. Below we ignore effects associated with finite mean-level spacing in the dot. While charge isonly weakly quantized in the mesoscopic CB regime, thespin remains a good quantum number in the absence of SOI.However, when the SOI is present, two spin sub-bands are splithorizontally in k space and while spin is no more conserved, the sub-band index characterizes quantized states instead. Thesingle mode QPC being a short quantum wire can be viewed asa 1D electron system in the presence of Rashba SOI [ 23–26] H SOI=αR[/vectork×/vectornz]·/vectorσ: H0=−ivF/summationdisplay λσλ/integraldisplay∞ −∞dy/Psi1† λ,σ(y)∂y/Psi1λ,σ(y), (5) HSOI=αRkF/summationdisplay λσλ/integraldisplay∞ −∞dy[/Psi1† λ,↑/Psi1λ,↓+/Psi1† λ,↓/Psi1λ,↑].(6) We denote here by /Psi1λ,σthe left ( λ=−) and right ( λ=+) movers with spin σ=↑,↓. The constant αRcharacterizes Rashba SOI strength. The kFandvF=kF/m∗correspond to the Fermi momentum and Fermi velocity (here m∗is a fermion’s mass). The 1D electron transport through theQPC is along the yaxis (see Fig. 1). The Rashba SOI H SOI=αRkyσxis associated with the electric field gradient along the zaxis and can be characterized by the effective SOI field gμB/vectorBSOI/2=αRkF/vectorexperpendicular to the direction of electron transport ( gis the Lande factor; μBis the Bohr magneton). Notice that the SOI field alone does not lead to theTRS breaking. The backscattering (BS) Hamiltonian describes a scattering of electrons with momentum transfer 2 k Fon a nonmagnetic quantum impurity located at the origin and characterized by ashort-range potential V(y): H BS=/summationdisplay λ,σ/integraldisplay dy/Psi1† λ,σ(y)V(y)/Psi1¯λ,σ(y)e−2iλkFy. (7) 045125-2PROTECTION OF A NON-FERMI LIQUID BY SPIN- . . . PHYSICAL REVIEW B 92, 045125 (2015) The Hamiltonian Htunrepresents the weak tunneling |tk|= |t|/lessmuch1 of the electrons from the left contact to QD: Htun=/summationdisplay kλσ[tkc† kσ/Psi1λσ(−∞)+H.c.]. (8) The Zeeman Hamiltonian Hzdescribes the effects of the external magnetic field Hz=−gμB/vectorB(/vectorss+/vectorsd), where /vectorssand /vectorsdare the spin densities of electrons in the source and drain, respectively. We consider a situation when both sizesof the QD ( L QD) and QPC ( LQPC) are small compared to the SOI length scale LQD∼LQPC/lessmuchlSOI=1/(m∗αR). Since the effective energy scale determining the behavior of thetransport coefficients of the model ( 2)–(8) which will be referred to below as the Kondo temperature T Kis [14]∼Ec (see Appendix), the condition lSOI/greatermuchLQDis equivalent to gμBBSOI/lessmuchTK(see a discussion about interplay between the Kondo effect and SOI in Ref. [ 28]). We also assume that the SOI effects in the QD are already taken into account by usingthe approach developed in Ref. [ 29]. IV . SCATTERING IN THE PRESENCE OF SOI AND MAGNETIC FIELD We consider the 1D scattering problem in the presence of SOI [ 21] and Zeeman field applied parallel to the plane of 2DEG. The Hamiltonian is given by H=H0+V(y)=k2 2m∗+αRσxk−γ/vectorσ·/vectorB+V(y).(9) The short-range potential V(y) describes a nonmagnetic impurity located at the origin [ 24,25]. The electron’s transport is along the ydirection, k=ky. Angle ϕcharacterizes the orientation of magnetic field /vectorBwith respect to the yaxis (see Fig.2, left bottom panel insert). The kinetic energy term of ( 9) is given by H0=/parenleftBigg k2 2m∗ αRk+iγBeiϕ αRk−iγBe−iϕ k2 2m∗/parenrightBigg . (10) The Hamiltonian H0is promptly diagonalized in kspace. The eigenvalues (spectra) describe two sub-bands ( ν=+ and ν=−) split both horizontally due to the SOI and vertically due to the Zeeman effect (Fig. 2): E±(k)=k2 2m∗±/radicalBig (αRk)2+(γB)2−2αRγkB sinϕ.(11) (We use the short-hand notation γ=gμB/2.) In the presence of both fields there are two sub-bands while the spin polariza-tion changes continuously as one moves from one Fermi pointto the other along each sub-band. We assume that magneticfield is applied parallel to the plane of the 2DEG to decoupleit from the orbital degrees of freedom and concentrate on theZeeman effect only. The angle ϕcharacterizes the orientation of/vectorBwith respect to the axis of 1D transport ( y). The spectra for ϕ=0, perpendicular orientation of /vectorBand/vectorB SOI, describe the situation when magnetic field /vectorBis oriented along the direction of the transport. If Bis larger than BSOI, the most important effects on thermoelectric transport are due to the Zeemansplitting of two sub-bands (Fig. 2, upper left panel) [ 17]. In that limit the effects of Bare associated with breaking of theB/B SOI B/B cr↑ r↓ r±=r∓ r+ r−E(k)F r+r−r∓r±kk k B/B SOIB⊥BSOI B<B B<B−π/2<ϕ<π/ 2 BSOIB xyϕ FIG. 2. (Color online) (Top panels) Two sub-band spectra: (left) Zeeman splitting in the absence of SOI; (center) /vectorB⊥/vectorBSOI, angle between magnetic field and ydirection ϕ=0; (right) arbitrary angle −π/2<ϕ<π / 2. (Bottom panels) Magnetic field dependence of reflection amplitudes |rμν|for the spectra shown in top panels. For illustration we performed all calculations with model barrier V(y)= V0exp(−|y|/LQPC),k0FLQPC=3.6, and height of the barrier V0is tuned to get r2 0=0.1 (see details in the text). Insert shows relative orientation of /vectorBand/vectorBSOI. channel symmetry, |r↑|/negationslash=|r↓|(Fig. 2, lower left panel) which is crucial for the fate of NFL [ 17]. For the case B<B SOIwe distinguish two cases: (i) ϕ=0 (Fig. 2, central panel) and (ii) −π/2<ϕ<π / 2,ϕ/negationslash=0 (Fig. 2, right panel). Since the orbital effects are negligible for small magnetic fields if B<B SOI,t h e case (i) ϕ=0 (Fig. 2, central panel) can also be realized when magnetic field is perpendicular to the plane of 2DEG. Besides,the theory discussed in the paper is also applicable when bothRashba and Dresselhaus SOI [ 21,22] are present. The transport coefficients for the generic situation of the in-plane Bfield are fully determined by the angle /Phi1=ϕ 0−ϕbetween /vectorBand /vectorBSOI, where ϕ0(ϕ) are the angles between /vectorBSOI(/vectorB) and axis of 1D motion, respectively. The eigenfunctions of H1dare momentum-dependent spinors /Psi1ν(y)=eik·yχν(k), χ±(k)=1√ 2/parenleftbigg ±ie−iϑ(k) 1/parenrightbigg , (12) where ϑ(k)=arctan/parenleftbiggαRk−γBsinϕ γBcosϕ/parenrightbigg . (13) The four reflection amplitudes in the first order of the backscattering potential are determined by 2 kFmomentum transfer and given by the matrix elements of V(y) in spinor basis/Psi1ν. The diagonal matrix elements, |rμμ|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleV/parenleftbig k μ F+−kμ F−/parenrightbig v0Fcos/parenleftbiggϑ/parenleftbig kμ F+/parenrightbig −ϑ/parenleftbig kμ F−/parenrightbig 2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle(14) characterize the intraband scattering (we shall use the short- hand notations |r μμ|≡|rμ|below). Here kμ F+>0 andkμ F−< 0 stand for the right and left Fermi points of a sub-band μ, respectively, v0F∼(m∗a)−1originates from the high energy 045125-3T. K. T. NGUYEN AND M. N. KISELEV PHYSICAL REVIEW B 92, 045125 (2015) cutoff, and ais a lattice constant. The off-diagonal matrix elements ( |r+−|≡|r±|and|r−+|≡|r∓|): |rμν|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleV/parenleftbig k μ F+−kν F−/parenrightbig v0Fsin/parenleftbiggϑ/parenleftbig kμ F+/parenrightbig −ϑ/parenleftbig kν F−/parenrightbig 2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle(15) describe the interband scattering. The backscattering Hamiltonian ( 7) in the basis of eigen- functions ( 12) casts the following form: H BS=vF/summationdisplay λμν|rμν|[/Psi1† λ,μ(0)/Psi1¯λ,ν(0)+H.c.]. (16) Let us analyze various limits of the backscattering correspond- ing to different orientation of the in-plane magnetic field /vectorBin the regime of strong interplay with the effects of SOI. For the most generic case of interplay between SOI and Zeeman magnetic field there exist four independent scatteringprocesses resulting in four different reflection amplitudes(Fig. 2, right lower panel). The reflection amplitudes for the intraband scattering |r +|and|r−|(black and red dashed arrow on Fig. 2, right upper panel) in the first order of the backscattering potential are proportional to the amplitude ofB[25], |r +/−|=r0/bracketleftbiggk0F k0F∓δ/bracketrightbigg/parenleftbiggB BSOI/parenrightbigg cosϕ, (17) where k0Fis the Fermi momentum at zero splitting ( δ= 0),δ=m∗αR,r0∝|V(2k0F)m∗a|/lessmuch1 is a coefficient char- acterizing the transparency of the barrier. The intrabandscattering completely suppresses for ϕ=π/2 since the angle ϑ(k F±)=±π/2 and the eigenfunctions do not depend on B. The intraband reflection amplitudes |r+|=|r−|=0 while interband ( |r±|,|r∓|)/negationslash=0. Thus, for ϕ=π/2w eh a v eo n l y two nonzero reflection amplitudes |r±|and|r∓|and therefore the thermoelectric transport can be described by equations ofRef. [ 17] if replacing |r ↑|→|r±|and|r↓|→|r∓|. The interband scattering amplitudes (blue dashed arrows on central upper panel of Fig. 2) for the case −π/2<ϕ<π / 2 are given by |r±/∓|=r0/bracketleftbigg 1∓b/parenleftbiggBsinϕ BSOI/parenrightbigg −c(ϕ)/parenleftbiggB BSOI/parenrightbigg2/bracketrightbigg .(18) Here coefficients (b,c(ϕ))∼1 depend on the geometry of the QPC. One can see that for ϕ=0, the scattering term linear in B(linear Zeeman effect) disappears and additional symmetry |r±|=|r∓|emerges (Fig. 2, central panel). The reflection amplitudes depend on magnetic field quadratically (quadraticZeeman effect). Thus, the scattering Hamiltonian in that casecontains three independent scattering parameters. V . EFFECTIVE MODEL We recapitulate briefly the main steps of the derivation of transport coefficients (for details see Appendix): (i) webosonize the 1D Hamiltonian ( 5)–(7) using a standard ap- proach [ 31,32]. The effective bosonic Hamiltonian gives us a boundary sine-Gordon (BSG) model [ 32] with four different backscattering amplitudes. The high- Tresults are obtained by perturbative expansion (in reflection amplitudes)around the strong-coupling fixed point of the model. (ii) Thenonperturbative results in the low- Tregime are obtained by the re-fermionization procedure through mapping the BSG modelonto the effective Anderson model [ 12,15–17]. (iii) The effects of the Zeeman field at the QPC resulting in TRS breakingcaused by the asymmetry of reflection amplitudes [ 17]a r e accounted by a magnetic-field-dependent resonance width/Gamma1at the CB peaks [ 17]. The resonance width /Gamma1in the presence of the Zeeman field remains finite for a wholerange of the gate voltages, cuts the temperature-dependentlogarithm, and therefore restores FL properties. The width /Gamma1 (Refs. [ 12,14,32]) is attributed to a single local Majorana mode interacting with a single mode of chiral fermions [ 12,16]i nt h e theory containing only two (intraband) scattering processes.The scattering on quantum impurity in the presence of SOIand Zeeman fields involves four “2 k F” processes which can be accounted for by two local Majorana modes interacting with four modes of twospecies of the chiral fermions. As a result, two different resonance widths enter the transportcoefficients. The interplay between two widths associated withinter- and intraband processes leads to remarkable effects inthermoelectric transport. The effective Anderson model which describes a hybridiza- tion of two local Majorana fermions η 1andη2with two species ofchiral fermions (see Appendix) is a direct generalization of [12,14,16,17] for a case of interplay between Zeeman and SOI fields: Hτ(t)=/integraldisplay∞ −∞dk/bracketleftbigg/summationdisplay α=1,2(k·vF)c† α,kcα,k −√ 2(ωsτ(t)η1(c1,k−c† 1,k)−iωaτ(t)η1(c1,k+c† 1,k) +ωmsτ(t)η2(c2,k−c† 2,k)−iωmaτ(t)η2(c2,k+c† 2,k))/bracketrightbigg , (19) where following Ref. [ 16] we define ωs/a τ(t)=/Omega1s/afs/a τ(t), (20) ωms/ma τ (t)=/Omega1ms/mafs/a, τ (t), with /Omega1s/a=/radicalbigg vFEceC 2π3||r+|±|r−||, (21) /Omega1ms/ma =/radicalbigg vFEceC 2π3||r±|±|r∓||, and the time-dependent functions, fs/a τ=(−1)nτ(t)Re/Im[exp {i(δχτ(t)−πN)}]. (22) Function δχτ(t) describes the deviation of the phase of the charge mode mean value from πnτ(t)( R e f .[ 16]): δχτ(t)≈π2T 2Ec(cot[πT(t−τ)]−cot[πTt]), (23) where Nis a function of a gate voltage Vg:Nis integer in the Coulomb blockade valleys and half-integer in the Coulombblockade peaks, n τ(t)=θ(t)θ(τ−t)(θ(t) is a step function ), andC≈0.577 is the Euler’s constant. 045125-4PROTECTION OF A NON-FERMI LIQUID BY SPIN- . . . PHYSICAL REVIEW B 92, 045125 (2015) The original Matveev model [ 12] corresponds to the case/Omega1a=/Omega1ma=/Omega1ms=0 and describes a single Majo- rana fermion η1coupled to the odd combination of cre- ation/annihilation operators of chiral fermions. In contrast to the conventional Anderson model which preserves U(1) symmetry, the model [ 12,14] is characterized by Z2symmetry instead. As a result, the NFL properties associated with thetwo-channel Kondo physics emerge. The NFL behavior of thetwo-channel Kondo model are attributed to the overscreened regime realized when the number of orbital channels N exceeds twice the spin of a quantum impurity. As it was shownin Ref. [ 17], the Zeeman in-plane magnetic field restores theU(1) symmetry through appearance of nonzero /Omega1 aand therefore leads to the restoration of the FL behavior char-acteristic for the single-channel fully screened Kondo modelin both thermodynamic [ 15] and transport [ 17] coefficients. The Kondo temperature T Kis of the order of the charging energy Ec(see Appendix). The effective Hamiltonian ( 19) describing scattering in the presence of both Zeeman and SOIfields has a structure of two copies of the two-channel Kondomodel where coupling constants ω iτdepend on the magnetic field. Thus, when all reflection amplitudes are different, themodel ( 19) is characterized by generic FL properties. However, ifaccidental degeneracy fine-tuned by the orientation of the in-plane magnetic field appears, one of the nonidentical copiesof the two-channel Kondo model preserves the NFL properties. The effects of interplay between the backscattering at the QPC and Coulomb interaction in the QD can be ac-counted by the correlator K(τ)=/angbracketleftT τF(τ)F†(0)/angbracketright(see details in Refs. [ 16,17]). The operator F(τ) accounts for the weak charge quantization in the mesoscopically Coulomb blockaded QD. Following Ref. [ 16] we define the charge of QD ˆQ= e(ˆnτ+ˆnd), where ˆnτis an integer valued operator which commutes with the annihilation operator of the electron in thedot at the position of the source Ref. [ 14]. Since by definition [F(τ),ˆn τ]=F(τ), the role of operator F(τ) is to account for the effects of interaction in QD: /Psi1λ(τ)=F(τ)/Psi10λ(τ) where /Psi1λand/Psi10λcorrespond to interacting and nonin- teracting left/right fermions, respectively. Thus, the dressed Green’s function (GF) G(τ)=− /angbracketleftTτ/Psi1λ(τ)/Psi1† λ(0)/angbracketrightand free fermionic Green’s function G0(τ)=− /angbracketleftTτ/Psi10λ(τ)/Psi1† 0λ(0)/angbracketright= −πν0T/sin(πTτ ) are connected [ 16] by the simple relation G(τ)=K(τ)G0(τ) (here ν0is a density of states in QD). The transport coefficients of the model are determined by theGreen’s function G(see next section). In order to compute the Green’s function G(or, equivalently, compute the correlator K) we define the operator U τ= (−1)d†dwhere d=(η1+iη2)/√ 2 and d†=(η1−iη2)/√ 2. We keep notations of the Matveev and Andreev work Ref. [ 16] forUτ=2iη2η1. Notice that the “spin” and charge are completely disentangled in the correlator K(τ)=Kc(τ)Ks(τ). While Kc(τ)=π2Te−C/(2Ec|sin(πTτ )|) (see Refs. [ 14,16] for details of calculations), the Ks(τ)i n zeroth order in Hτ(t)−Hτ=0(t) is defined by the correlator, K(0) s(τ)=/angbracketleftTtU(τ)U(0)/angbracketright0. (24) Here/angbracketleft.../angbracketright0denotes an averaging with ( 19) taken at τ=0. (Notice obvious correspondence η1→σx/√ 2,η2→σy/√ 2,and 2iη2η1→σz, where σiare spin s=1/2 operators ( i= x,y,z ). Therefore, K0 s(τ)=/angbracketleftTtσz(τ)σz(0)/angbracketright.) The first nonvanishing order in Hτ(t)−Hτ=0(t) correction to the correlator Ks(τ)i sg i v e nb y K(1) s(τ)=−/integraldisplay1/T 0dt/angbracketleftTtH/prime τ(t)U(τ)U(0)/angbracketright0, (25) where the Hamiltonian H/prime τ(t) has the form, H/prime τ(t)=−2iδχτ(t)(/Omega1ssin(πN)η1ς1−/Omega1acos(πN)η1ζ1 +/Omega1mssin(πN)η2ς2−/Omega1macos(πN)η2ζ2).(26) Here we define four additional Majorana fermions ( α=1,2) through a k-Fourier transform of the even/odd combinations of creation/annihilation operators of two species of chiral fermions cα,ktaken at the position of the quantum impurity y=0: ζα=/integraldisplay∞ −∞dkζαk=1√ 2/integraldisplay∞ −∞dk(cα,k+c† α,k), (27) ςα=/integraldisplay∞ −∞dkςαk=1 i√ 2/integraldisplay∞ −∞dk(cα,k−c† α,k). In order to compute the correlators ( 24) and ( 25) we apply the Wick’s theorem to the product of an even number of fermionsand express the result in terms of the products of the single-particle GFs. The imaginary time (Matsubara) GFs form a6×6 matrix with 21 independent components (six diagonal and 15 off-diagonal): G ηη μν(τ)=− /angbracketleftTτημ(τ)ην(0)/angbracketright,Gζζ μν(τ)=− /angbracketleftTτζμ(τ)ζν(0)/angbracketright, Gςς μν(τ)=− /angbracketleftTτςμ(τ)ςν(0)/angbracketright,Gζς μν(τ)=− /angbracketleftTτζμ(τ)ςν(0)/angbracketright, Gζη μν(τ)=− /angbracketleftTτζμ(τ)ην(0)/angbracketright,Gςη μν(τ)=− /angbracketleftTτςμ(τ)ην(0)/angbracketright. The GF s of the quadratic Anderson-type Hamiltonian ( 19) can be found exactly (e.g., by solving equations of motion forthe Majorana fermions). For computing the correlators ( 24) and ( 25) we need only six GFs, namely, two diagonal local Majorana’s GF (here Rdenotes the retarded GFs): Gηη 11,R(/epsilon1)=1 /epsilon1+i/Gamma1B,Gηη 22,R(/epsilon1)=1 /epsilon1+i/Gamma1A, (28) and four off-diagonal hybridized GFs: Gζη 11,R(/epsilon1)=/Omega1asin(πN)2π/vF /epsilon1+i/Gamma1B, Gςη 11,R(/epsilon1)=/Omega1scos(πN)2π/vF /epsilon1+i/Gamma1B, (29) Gζη 22,R(/epsilon1)=/Omega1masin(πN)2π/vF /epsilon1+i/Gamma1A, Gςη 22,R(/epsilon1)=/Omega1mscos(πN)2π/vF /epsilon1+i/Gamma1A. Here we denote the resonance widths associated with the symmetric and antisymmetric combinations of reflectionamplitudes as /Gamma1 s/ms=/Omega12 s/mscos2(πN)4π/vF, (30) /Gamma1a/ma=/Omega12 a/ma sin2(πN)4π/vF. 045125-5T. K. T. NGUYEN AND M. N. KISELEV PHYSICAL REVIEW B 92, 045125 (2015) Two resonance Kondo widths entering the transport coeffi- cients are given by /Gamma1A=/Gamma1ms+/Gamma1ma,/Gamma1 B=/Gamma1s+/Gamma1a. (31) Notice that 10 GFs, namely Gζζ μν,Gςς μν, and Gζς μνdo not depend on the local Majorana fermions describing the quantumimpurity. These GF renormalize the correlations between theconduction electrons, but do not enter Eqs. ( 24) and ( 25). Another five GFs allowed by the symmetry of the Hamil-tonian ( 19) do not appear in Eqs. ( 24) and ( 25) due to specific form of fermionic correlations in the HamiltonianH /prime τ(t). VI. TRANSPORT COEFFICIENTS The thermoelectric coefficient GTand electric conductance G, GT=−iπ2GLT 2e/integraldisplay∞ −∞sinh(πTt) cosh3(πTt)K/parenleftbigg1 2T+it/parenrightbigg dt, (32)G=πGLT 2/integraldisplay∞ −∞1 cosh2(πTt)K/parenleftbigg1 2T+it/parenrightbigg dt, (33) are here calculated by accounting for interaction effects in the QD through the correlator K(τ) defined in the previous section. The conductance of the barrier between the source andQDG L=2πe2ν0νL|t|2is expressed through Fermi’s golden rule as a function of the density of states (DoS) of the sourceν L, the DoS of the QD ν0, and the weak tunneling amplitude |t|(Ref. [ 30]). The correlator K(0) s(1/(2T)+it)defined by ( 24)i sa ne v e n function of time. This correlator determines the behavior ofthe electric conductance G, but does not contribute to G T: G(0)=GL/Gamma1A/Gamma1Be−C 32πTE cFG/parenleftbigg/Gamma1A T,/Gamma1B T/parenrightbigg . (34) The equation for the thermoelectric coefficient GTis given by an odd function K(1) s(1/(2T)+it)defined by ( 25): G(1) T=−GL/Gamma1A/Gamma1Bsin(2πN) 6eπ2Ec/bracketleftbigg|r+r−| /Gamma1Bln/parenleftbiggEc T+/Gamma1B/parenrightbigg F/parenleftbigg/Gamma1B T,/Gamma1A T/parenrightbigg +|r±r∓| /Gamma1Aln/parenleftbiggEc T+/Gamma1A/parenrightbigg F/parenleftbigg/Gamma1A T,/Gamma1B T/parenrightbigg/bracketrightbigg . (35) The ratio of GTandGdefines the thermoelectric power: S=−16eCsin(2πN)T 3eπ/bracketleftbigg|r+r−| /Gamma1Bln/parenleftbiggEc T+/Gamma1B/parenrightbiggF/parenleftbig/Gamma1B T,/Gamma1A T/parenrightbig FG/parenleftbig/Gamma1A T,/Gamma1B T/parenrightbig+|r±r∓| /Gamma1Aln/parenleftbiggEc T+/Gamma1A/parenrightbiggF/parenleftbig/Gamma1A T,/Gamma1B T/parenrightbig FG/parenleftbig/Gamma1A T,/Gamma1B T/parenrightbig/bracketrightbigg . (36) The functions FGandFuniversally depend on the ratio of the resonance Kondo widths /Gamma1A,/Gamma1Band the temperature: FG(x,y)=/integraldisplay∞ −∞/integraldisplay∞ −∞dzdz/prime[(z+z/prime)2+π2] [(z/prime)2+x2][z2+y2]1 cosh/parenleftbigz 2/parenrightbig cosh/parenleftbigz/prime 2/parenrightbig cosh/parenleftbigz+z/prime 2/parenrightbig, (37) F(x,y)=/integraldisplay∞ −∞/integraldisplay∞ −∞dzdz/primez(z+z/prime)[(z+z/prime)2+π2] [z2+x2][(z/prime)2+y2]1 cosh/parenleftbigz 2/parenrightbig cosh/parenleftbigz/prime 2/parenrightbig cosh/parenleftbigz+z/prime 2/parenrightbig. (38) VII. RESULTS AND DISCUSSION A. Four main regimes of thermoelectric transport The nonperturbative equation for the resonance width /Gamma1 related to the interband scattering ( 30) and ( 31) demonstrates a weak dependence of /Gamma1on the magnetic field away from the CB peaks (see Fig. 3, left panel): /Gamma1A∝/Gamma10/bracketleftbig/parenleftbig 1−/Lambda12 A/parenrightbig cos2(πN)+/Delta12 A/bracketrightbig , (39) where /Gamma10=r2 0Ec,/Delta1A(B,ϕ)=b(B/B SOI)s i nϕ, and /Lambda12 A= /Delta12 A+2c(ϕ)(B/B SOI)2. In contrast, the resonance width /Gamma1associated with the intraband scattering ( 30) and ( 31) strongly depends on Bat all gate voltages: /Gamma1B∝/Gamma10/bracketleftbig/parenleftbig 1−/Delta12 B/parenrightbig cos2(πN)+/Delta12 B/bracketrightbig/parenleftbiggBcosϕ BSOI/parenrightbigg2 .(40) The/Gamma1min B≡/Gamma1B(N=1 2)∝/Gamma10(Bcosϕ/B c)2is a minimal reso- nance width, /Delta1B=δ/k 0F, andBc∼D, where D∼(m∗a2)−1 is the bandwidth. Thus, Bccorresponds to the field strength that is necessary to reach full spin polarization of the conductionchannel.Varying the temperature, gate voltage, amplitude, and direction of the magnetic field one can achieve four differentregimes of thermoelectric transport (Fig. 3, right panel): (A) (/Gamma1 A,/Gamma1B)/lessmuchT, fully perturbative NFL regime. While /Gamma1Bis gapped and the gap is /Gamma1min B∼B2,/Gamma1Acould be gapless B/B SOIN(Vg)Γ/Ec B/B SOIN(Vg) FIG. 3. (Color online) (Left panel) Gate voltage and magnetic field dependence of /Gamma1α/Ec:r e d , /Gamma1A;b l u e , /Gamma1B. (Right panel) A “phase diagram” as follows: Four main regimes of the thermoelectric transport are inside the green, red, blue, and magenta domains; A, perturbative NFL, B, weak partial NFL, C, strong partial NFL, D,nonperturbative FL (see details in Sec. VII). Domain boundaries are defined by the crossover condition /Gamma1 A(B,N )=Tand/Gamma1B(B,N )= T. For all plots αR=0.15vF,ϕ=π/4,r2 0=0.1,k0FLQPC=3.6, T=0.05Ec. 045125-6PROTECTION OF A NON-FERMI LIQUID BY SPIN- . . . PHYSICAL REVIEW B 92, 045125 (2015) if the gate voltage is fine-tuned to the positions of CB peaks N→1/2 andϕ→0 .T h eT P( 34)–(36) demonstrates finger- prints of weak NFL logarithmic behavior: S∝r2 0ln/parenleftbiggEc T/parenrightbigg sin(2πN). (41) (B)/Gamma1B/lessmuchT/lessmuch/Gamma1A, perturbative in /Gamma1B/Tand nonpertur- bative in /Gamma1A/T[see ( 34)–(36)]. This regime can be reached either by fine-tuning the gate voltage away from the CB peaksor by tuning the direction of the Zeeman field to be parallel toSOI in order to suppress the intraband scattering: S∝/bracketleftbigg |r +r−|ln/parenleftbiggEc T/parenrightbigg +|r±r∓|T /Gamma1Aln/parenleftbiggEc /Gamma1A/parenrightbigg/bracketrightbigg sin(2πN). (42) The weak NFL effects are manifested in the TP log behavior originated from the intraband scattering. The interband pro-cesses result in appreciable FL corrections to the TP ( 34)–(36). The NFL effects are weak since the amplitude of intraband scattering is small at B<B SOI. (C)/Gamma1A/lessmuchT/lessmuch/Gamma1B, perturbative in /Gamma1A/Tand nonper- turbative in /Gamma1B/T[see ( 34)–(36)]. This regime is achieved in the vicinity of CB peaks and characterized by strong NFL effects due to a weak magnetic field dependence of |r±|and|r∓|protected by SOI. Thus, by fine-tuning the orientation of magnetic field perpendicular to SOI ϕ=0 one can controllably protect the NFL behavior of TP in the regimeB<B SOI. The magnetic field controlled gap associated with the intraband scattering weakly depends on the gate voltageand results in small FL corrections to TP (compared to NFLeffects): S∝/bracketleftbigg |r +r−|T /Gamma1Bln/parenleftbiggEc /Gamma1B/parenrightbigg +|r±r∓|ln/parenleftbiggEc T/parenrightbigg/bracketrightbigg sin(2πN). (43) (D)T/lessmuch(/Gamma1A,/Gamma1B), FL nonperturbative regime. The NFL logs associated with the intra- and interband scattering pro-cesses are cut by the corresponding resonance widths ( 34)– (36): S∝T/bracketleftbigg|r +r−| /Gamma1Bln/parenleftbiggEC /Gamma1B/parenrightbigg +|r±r∓| /Gamma1Aln/parenleftbiggEC /Gamma1A/parenrightbigg/bracketrightbigg sin(2πN). (44) The TP is a linear function of the temperature. However, thecoefficient in front of Tstrongly depends on both gate voltage and magnetic field. B. Possible experimental realization and “smoking gun” predictions Choice of a material. We suggest using narrow-gap semi- conductors, e.g., InSb or InAs for the observation of theNFL fingerprints in the quantum transport. Both materialsare characterized by large bulk gfactors, e.g., |g|∼10 in InAs (see Ref. [ 34]) and |g|∼50 in InSb (see Ref. [ 35]). The domain of parameters favorable for the observation of theNFL regime is defined as δ 0<T <γB SOI<E c</epsilon1F, where δ0∼1/(ν0VQD) is a single-particle mean-level spacing in the QD of the size LQDand “volume” VQD:δ2D 0∼/planckover2pi12/(m∗L2 QD) andδ3D 0∼(kFLQD)−1/planckover2pi12/(m∗L2 QD),m∗is an effective mass of the carrier in a semiconductor, and other parameters are definedin the previous sections. The condition LQD<l SOIallows one to disregard the effects of the SOI in the QD (Ref. [ 29]), while the condition LQPC<lmfpdefines a ballistic regime of quantum transport through the QPC (here lmfpis an elastic mean-free path). According to Ref. [ 35], the parameters for InSb QD- QPC are as follows: /planckover2pi1αR∼0.1e V ˚A÷0.2e V ˚A,lSOI= /planckover2pi1/(m∗αR)∼200 nm ÷400 nm, |m∗|∼0.015me(meis elec- tron’s mass), ESOI=m∗α2 R/2∼50μeV, the typical charg- ing energy Ec∼1 meV, and typical Fermi velocities are /planckover2pi1vF∼0.5e V ˚A÷1e V ˚A, while the mean level spacing δ0<10μeV for LQD∼lSOI. The mean-free path of the QPC of a width dQPC∼10 nm is lmfp∼300 nm ÷1μm. Typical parameters for the InAs QD-QPC are not muchdifferent [ 34]:|m ∗|∼0.03me,/planckover2pi1αR∼0.05 eV ˚A÷0.3e V ˚A, lSOI∼200 nm ÷1μm, and lmfp∼300 nm ÷1μm. There- fore, if we assume that LQD≈LQPC∼300 nm ÷500 nm, our predictions could be verified at magnetic fields B< 500 mT and temperatures T∼100 mK ÷300 mK for typical densities of 2DEG n2DEG∼1011cm−2÷1012cm−2. This estimation for parameters is taken from available literature (to our bestknowledge), but may vary due to anisotropic character of thegfactor which in turn depends on external magnetic field [ 34] and may also be strongly reduced in confined geometries ofthe nano-structures. Testing a Mott-Cutler law. The first important test of the interplay between effects of the SOI and Zeeman field is toverify the Mott-Cutler (MC) law [ 33] at external in-plane magnetic field. The MC law is a standard benchmark for the FLproperties [ 16]. The MC law says that the TP is proportional to a log derivative of the electric conductance with respect toa position of the chemical potential (gate voltage V g): S∝T Ec∂lnG ∂N(Vg). (45) In the limit T/lessmuch(/Gamma1A,/Gamma1B)/lessmuchEccorresponding to the FL regime we get ∂lnG ∂N(Vg)∝Ec/bracketleftbigg|r+r−| /Gamma1B+|r±r∓| /Gamma1A/bracketrightbigg sin(2πN), (46) while the TP is given by ( 44). Thus, a strong deviation from the MC law in the FL regime at finite in-plane magnetic fieldsis a pre-cursor for the NFL behavior discussed in the paper.Notice that break down of the MC law indicates that thereexists no equivalent classic electric circuit consisting of theresistances connected in parallel or in series and therefore theeffects of both intra- and interband scattering play an importantrole in the quantum transport. The violation of MC law in thethermoelectric transport through a single-electron transistorhave been reported in Ref. [ 19]. We are not aware of existence of theoretical explanation of this effect in the framework ofthe FL theory. Thermopower in the presence of the external B field. The next step is to measure the thermopower of a prototypenanodevice (Fig. 1). The magnitude and orientation of the in-plane magnetic field can be controlled in a standard wayby four magnetic coils (not shown in the picture). The TPmaximum eS max(B) demonstrates a nonmonotonic magnetic field dependence (strong NFL) which is most pronounced 045125-7T. K. T. NGUYEN AND M. N. KISELEV PHYSICAL REVIEW B 92, 045125 (2015)eSmax eSmaxeSmax eSmaxB/B SOI ϕπ/2 π/4NN eS eS B/B SOI T/Γmin B FIG. 4. (Color online) (Top left) eSmaxas a function of B/B SOI for different angles ϕatT=0.001EcandαR=0.15vF, from top to bottom ϕ=π/12,π /6,π /3,5π/12. (Insert) eS(N)f o rϕ=5π/12 forB=0.5BSOI(black), B=BSOI(red),B=1.5BSOI(green). (Top right) eSmaxas a function of the angle ϕfor different amplitudes ofB/B SOIatαR=0.15vFandT=0.001Ecfrom top to bottom: B/B SOI=0.6,0.8,1.0,1.2.(Insert) eS(N)f o rϕ=5π/12,B= 0.5BSOI,T=0.001 (black), 0 .01 (red), 0 .1 (blue). (Bottom left) eSmaxas a function of B/B SOIfor different T/E catαR=0.15vF andϕ=π/6;T/E c=10−1, blue (regime A); 10−2, green (crossover A→C); 10−3,r e d(B→D→C); 10−4, black ( B→D). (Bottom right) eSmaxas a function of T//Gamma1min B[/Gamma1min B=/Gamma1B(1/2)] with αR= 0.15vF,B=0.5BSOI,f o rϕ=5π/12 (black), ϕ=π/3 (red), ϕ= π/6 (green), ϕ=π/12 (blue), ( r2 0=0.1a n dk0FLQPC=3.6). when /vectorBis orthogonal to /vectorBSOI(black curve on Fig. 4,t o p left and right panels.) This has to be contrasted to almostmonotonic TP maximum behavior (blue curves) characteristicfor weak NFL-FL regimes. The nonmonotonic behavior ofTP maximum as a function of magnetic field is a centralprediction of our paper. The nonmonotonicity indicates thatthe NFL regime of TP is protected by SOI contrast toFL-like behavior demonstrating rapid decrease of TP whenmagnetic field increases [ 17]. Another indication of the NFL behavior is attributed to the gate voltage dependence (Fig. 4 inserts). According to [ 16,17] it is characterized by strongly nonsinusoidal form (Fig. 4inserts). The TP maximum at zero field, NFL regime, scales according to [ 16]a seS max∼ r0√T/E cln(Ec/T). The TP maximum in the FL regime scales as eS∼T/T effwithTeff/Ec=B/B cln−1(Bc/(B|r0|) (Ref. [ 17]). The B-field dependence of the TP maximum measured at different temperatures (Fig. 4, left bottom panel) allows one to distinguish between four main regimes A–Ddiscussed in the previous subsection. This measurement canbe used for identification of crossovers between differentdomains. The TP maximum depends linearly on Tin the FL regime for T< /Gamma1 min B=/Gamma1B(N=1/2) (Ref. [ 17]). This regime holds for ϕ→π/2. In contrast to the FL regime, the temperature dependence of the TP maximum pronouncedlydeparts from the linear behavior (see Fig. 4, right bottom panel) when ϕis detuned from π/2. We suggest testing experimentally this effect as a benchmark for the NFL physics.VIII. SUMMARY AND CONCLUSIONS We have demonstrated that the theory describing scattering of electrons characterized by two orbital degrees of freedomon a spin s=1/2 quantum impurity (two channel Kondo model) is strongly modified in the presence of both appreciablespin-orbit interaction and Zeeman splitting. It is shown that,on the one hand, the lack of spin conservation due to SOIleads to the appearance of new (extra) scattering channelswhich potentially enhance the thermoelectric transport. On theother hand, the Zeeman splitting produces nonzero resonancewidth of Majorana modes describing the quantum impurityand thus suppresses the NFL effects. The interplay betweenthese two tendencies can be controlled by fine-tuning the anglebetween Zeeman and SOI fields. Our calculations predict astrong dependence of the thermoelectric power on the angle between /vectorBand/vectorB SOIand thus open the possibility of controling the scattering mechanism by changing between four, threeor two independent scattering processes. While the cases offour and two weak back-scattering do favor the FL behavior,the additional degeneracy in scattering amplitudes appearingfor three scattering models due to SOI effects protects theNFL behavior for the range of magnetic fields B<B SOI. We conclude therefore, that SOI can indeed protect the NFLagainst the destructive effects associated with breaking ofchannel symmetry. ACKNOWLEDGMENTS We are grateful to J. C. Egues, V . Fal’ko, L. Glazman, K. Kikoin, A. Komnik, S. Ludwig, C. Marcus, K. Matveev,L. W. Molenkamp, and O. Starykh for illuminating discus-sions. T.K.T.N. acknowledges support through the short-termvisiting program of ICTP. This research in Hanoi is fundedby Vietnam National Foundation for Science and TechnologyDevelopment (NAFOSTED) under the Grant No. 103.01-2014.24. APPENDIX: EFFECTIVE HAMILTONIAN 1. Backscattering: from fermions to bosons The backscattering Hamiltonian mixes the left- and right- moving fermions: HBS=vF/summationdisplay λμν|rμν|[/Psi1† λ,μ(0)/Psi1¯λ,ν(0)+H.c.]. (A1) The Hamiltonians Eqs. ( 3) and ( 4) of the main text and the Hamiltonian ( A1) can be bosonized [ 31,32] in terms of dual fields φν(y) andθν(y) satisfying commutation relations [φν(y),θμ(y/prime)]=−iπδνμsgn(y−y/prime)/2 (Refs. [ 31,32]): /Psi1λ,ν(y)=uλ,ν√ 2πaeiλkν Fyexp{i[−λφν(y)+θν(y)]},(A2) where uλ,νare Klein factors [ 31,32] introduced to ensure proper anticommutation relations between the right- andleft-moving fermions. Using a standard procedure [ 31,32] we introduce the symmetric (charge) and antisymmetric (“spin”) dual variables φ c,s(y)=[φ+(y)±φ−(y)]/√ 2 and θc,s(y)= [θ+(y)±θ−(y)]/√ 2 satisfying the commutation relations 045125-8PROTECTION OF A NON-FERMI LIQUID BY SPIN- . . . PHYSICAL REVIEW B 92, 045125 (2015) [φc/s(y),θc/s(y/prime)]=−iπsgn(y−y/prime)/2 (notice that we still refer to the antisymmetric in the band index bosonic field as “spin”). We rewrite the backscattering Hamiltonian ( A1)i n terms of the charge and spin bosonic fields as follows: HBS=−2D π(rscos[√ 2φc(0)] cos[√ 2φs(0)] +rasin[√ 2φc(0)] sin[√ 2φs(0)] +rmscos[√ 2φc(0)] cos[√ 2θs(0)] +rmasin[√ 2φc(0)] sin[√ 2θs(0)]), (A3) where rs=| |r+|+|r−||/2,ra=| |r+|−|r−||/2,rms= ||r±|+|r∓||/2,rma=| |r±|−|r∓||/2. 2. Backscattering: Majorana fermions As a first step we replace the charge mode by its mean value averaged over fast charge degrees of freedom using thefunctional integral technique developed in Ref. [ 16] and obtain the Hamiltonian: H τ(t)=vF 2π/integraldisplay∞ −∞{[∂yθs(y)]2+[∂yφs(y)]2}dy −/radicalBigg 4D vF(ωsτ(t) cos[√ 2φs(0)]+ωaτ(t)s i n [√ 2φs(0)] +ωmsτ(t) cos[√ 2θs(0)]+ωmaτ(t)s i n [√ 2θs(0)]), (A4) where we use the notations ( 20) and ( 21) of the Sec. V.As a next step we introduce the even and odd combina- tions of the “spin” (aka sub-band) bosonic fields φe/o(y)= [φs(y)±φs(−y)]/√ 2,θe/o(y)=[θs(y)±θs(−y)]/√ 2. As a result, we obtain new chiral fields /Phi11/2(y)=θo/e(y)−φe/o(y) satisfying the commutation relations: [ /Phi1α(y),/Phi1α/prime(y/prime)]= iπδαα/primesgn(y−y/prime) where α,α/prime=1,2. We define new fermionic fields /Psi1α(y)=(ηα/√ 2πa)e x p(−i/Phi1α(y) with a help of two local Majorana fermions η1=(d+ d†)/√ 2 and η2=(d−d†)/(i√ 2) representing the quantum impurity [ 16]. Finally, we integrate out the fluctuations of the spin degree of freedom with the frequencies exceeding Ec(Ref. [ 16]). This procedure is equivalent to the poor man’s scaling approach originally used for the Kondo problem [ 36] and leads to replacement of the bandwidth Dby the new bandwidth TK∼Ec. As a result, we derive the effective Anderson model which describes a hybridization of two local Majoranafermions η 1andη2with two species of conduction electrons. The effective Hamiltonian ( 19) has a structure of two copies of the two-channel Kondo model where coupling constants ωiτ depend on both Zeeman and SOI fields: Hτ(t)=/integraldisplay∞ −∞dk/bracketleftbigg/summationdisplay α=1,2(kvF)c† α,kcα,k −√ 2(ωsτ(t)η1(c1,k−c† 1,k)−iωaτ(t)η1(c1,k+c† 1,k) +ωmsτ(t)η2(c2,k−c† 2,k)−iωmaτ(t)η2(c2,k+c† 2,k))/bracketrightbigg . (A5) [1] L. D. Landau, Sov. Phys. JETP 3, 920 (1957); , 5, 101 (1957). [2] P. W. Anderson, The Theory of Superconductivity in the High-Tc Cuprate Superconductors (Princeton University Press, Princeton, 1997). [ 3 ] G .R .S t e w a r t , Rev. Mod. Phys. 73,797(2001 ). [4] D. C. Ralph and R. A. Buhrman, P h y s .R e v .L e t t . 69,2118 (1992 ); D. C. Ralph, A. W. W. Ludwig, J. von Delft, and R. A. Buhrman, ibid.72,1064 (1994 ). [5] R. M. Potok, I. G. Rau, H. Shtrikman, Y . Oreg, and D. Goldhaber- Gordon, Nature (London) 446,167(2007 ). [6] P. Nozieres and A. Blandin, J. Phys. (Paris) 41,193(1980 ). [7] A. Zawadowski, Phys. Rev. Lett. 45,211(1980 ). [8] A. W. W. Ludwig and I. Affleck, P h y s .R e v .L e t t . 67,3160 (1991 ). [9] D. Boese and R. Fazio, Europhys. Lett. 56,576(2001 ). [10] T. A. Costi and V . Zlatic, Phys. Rev. B 81,235127 (2010 ). [11] R. Zitko, J. Mravlje, A. Ramsak, and T. Rejec, New J. Phys. 15, 105023 (2013 ). [12] K. A. Matveev, P h y s .R e v .B 51,1743 (1995 ). [13] K. Flensberg, Phys. Rev. B 48,11156 (1993 ). [14] A. Furusaki and K. A. Matveev, P h y s .R e v .L e t t . 75,709(1995 ); ,Phys. Rev. B 52,16676 (1995 ). [15] K. Le Hur, Phys. Rev. B 64,161302 (R) ( 2001 ); K. Le Hur and G. Seelig, ibid.65,165338 (2002 ).[16] A. V . Andreev and K. A. Matveev, Phys. Rev. Lett. 86,280 (2001 ); K. A. Matveev and A. V . Andreev, P h y s .R e v .B 66, 045301 (2002 ). [17] T. K. T. Nguyen, M. N. Kiselev, and V . E. Kravtsov, Phys. Rev. B82,113306 (2010 ). [18] S. Amasha, I. G. Rau, M. Grobis, R. M. Potok, H. Shtrikman, and D. Goldhaber-Gordon, P h y s .R e v .L e t t . 107,216804 (2011 ). [19] R. Scheibner, H. Buhmann, D. Reuter, M. N. Kiselev, and L. W. Molenkamp, Phys. Rev. Lett. 95,176602 (2005 ). [20] R. Scheibner, E. G. Novik, T. Borzenko, M. Konig, D. Reuter, A. D. Wieck, H. Buhmann, and L. W. Molenkamp, Phys. Rev. B75,041301 (2007 ). [21] R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, Berlin, 2003). [22] I. Zituc, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76,323 (2004 ). [23] V . Gritsev, G. I. Japaridze, M. Pletyukhov, and D. Baeriswyl, Phys. Rev. Lett. 94,137207 (2005 ). [24] J. Sun, S. Gangadharaiah, and O. A. Starykh, Phys. Rev. Lett. 98,126408 (2007 ); S. Gangadharaiah, J. Sun, and O. A. Starykh, Phys. Rev. B 78,054436 (2008 ). [25] R. G. Pereira and E. Miranda, Phys. Rev. B 71,085318 (2005 ). [26] F. Cheng, K. S. Chan, and K. Chang, New J. Phys. 14,013016 (2012 ). 045125-9T. K. T. NGUYEN AND M. N. KISELEV PHYSICAL REVIEW B 92, 045125 (2015) [27] I. L. Aleiner and L. I. Glazman, Phys. Rev. B 57,9608 (1998 ). [28] K. Kikoin and Y . Avishai, Phys. Rev. B 86,155129 (2012 ). [29] I. L. Aleiner and V . I. Fal’ko, P h y s .R e v .L e t t . 87,256801 (2001 ). [30] We quantize the conduction electrons in the source using the SOI basis. The FL and tunneling Hamiltonians HsandHtunpreserve the form under a replacement σ→ν. [31] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, Oxford, 2004).[32] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosoniza- tion and Strongly Correlated Systems (Cambridge University Press, Cambridge, 2004). [33] M. Cutler and N. F. Mott, Phys. Rev. 181,1336 (1969 ). [34] M. D. Schroer, K. D. Petersson, M. 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PhysRevB.70.205115.pdf

Random-phase approximation of core-electron scattering in solids K. Nuroh* Department of Mathematical Sciences, Kent State University, Salem, Ohio 44460, USA (Received 13 November 2003; revised manuscript received 23 June 2004; published 12 November 2004 ) The interference between bound-bound transition and bound-free transition in an electron-impact excitation of a core-electron in systems with narrow bands is discussed within the context of diagrammatic perturbationtheory. The earlier treatment of the phenomenon in terms of autoionization and characteristic widths of thecore-hole via perturbation ladder diagrams is extended to include ring diagrams as well, in random-phaseapproximation (RPA). The associated characteristic line shape for the intensity of the inelastically scattered electrons is formulated in terms of Fano line shape parameters, and theory is compared with electron-energyloss measurements for metallic La and Ce. DOI: 10.1103/PhysRevB.70.205115 PACS number (s): 71.27. 1a, 71.28. 1d, 79.20.Fv, 79.20.Uv I. INTRODUCTION The response of electrons to some external perturbation in a metal in the Fermi-sea picture is usually adequately ana-lyzed via linear response formalism or the Kubo 1formula approach, which in turn requires the evaluation of a suscep-tibility tensor in general. If for example one is interested inthe transport properties of the solid, then the susceptibility isdefined in terms of the current-current correlation function.Embedded in the susceptibility are various interactions thatintroduce electron correlations of the system. One way ofevaluating the susceptibility is the use of diagrammatic per-turbation theory. An exact evaluation of the susceptibility isimpossible within this approach. Rather an approximation ofsome sort is resorted to depending on the physical propertyof interest. For an extended system, it is usually sufficient todescribe many of the physical properties with the knowledgeof the band structure of the solid. To evaluate the approxi-mate susceptibility in this Coulomb long-range interactionlimit is equivalent to summing a particular set of bubble orring diagrams describing the repeated electron-hole interac-tions in the random-phase approximation (RPA). 2However, if electron correlations are to be properly accounted for, par-ticularly if the interactions involve core electrons, then it isnecessary to include the atomic structure of the solid in somefashion. Such an approach was taken, for example, in theHubbard 3model in order to unveil the dominant short-range part of the Coulomb interactions responsible for the instabili-ties ind-electrons in transition metals. To evaluate the sus- ceptibility in the Hubbard interaction model is equivalent to considering a generalized Hartree-Fock approximation de-scribing the repeated electron-hole interactions resulting in aset of ladder diagrams that is summed in the RPA. 4 The foregoing preamble establishes the fact that whenever the RPA is invoked, the ladder and ring diagrams must inprinciple be considered on the same footing. Depending onthe observable quantity of interest however, one subset ofdiagrams may be more relevant than the other. For example,the RPA result for the electron correlation energy in the jel-lium model of a solid employs a series of ring diagrams. 5On the other hand, in the calculation of the ground state energyof an imperfect Fermi gas modeled as a dilute gas interactingwith strong short-range repulsive potentials, contributionsfrom two-particle scattering processes necessitate the sum-mation of ladder diagrams. 6The RPA has also found successful application in the study of the electronic structure of some highly polarizableatomic and solid-state systems. Beginning with the work ofAmusia, Cherepkov, and Chernysheva 7in the discussion of the photoionization of noble-gas atoms using the diagram-matic random-phase approximation with exchange (RPAE ), Wendin 8used RPAring diagrams to discuss the photoioniza- tion cross sections of xenon. He later extended the same RPAdiagrammatic approach to calculate photoionization crosssections of metallic lanthanum, thorium, and uranium usinglocal density functional-based approximated (LDA )orbitals for the wave functions. 9,10Calculations based on the RPA approach usually yield results that mimic those based ontime-dependent density-functional approximation (TDLDA ). TheTDLDAcalculations for barium and cerium by Zangwilland Soven 11on one hand, and for the barium ionized series on the other by Nuroh, Stott, and Zaremba12compared with the RPAcalculations of Ref. 8 exemplify the above assertion.Indeed, the uncanny similarity in the numerical photoabsorp-tion cross-section results for xenon, barium, and cerium us-ing RPA and TDLDA separately, led Nuroh 13to formally demonstrate the mathematical equivalence between the ringdiagram-based RPAand the TDLDAas far as calculations ofdynamic polarizability of atoms are concerned. The ladder-based RPA has been used to discuss the 4d-subshell electron energy-loss spectra (EELS )in lantha- num and cerium metals. 14There, the successive electron-hole pair interactions have been formulated in terms of an effec-tive interaction through the introduction of a self-energy in aDyson-type equation so as to make comparative spectralanalysis between 3 dand 4dEELS apparent. In this paper, we extend the above earlier discussion to include ring-based RPA as well. The formulation will besymmetric in the introduction of self-energies for both thering and ladder diagrams. Moreover, unlike the earlier dis-cussion, the self-energies will be quadratic in the electron-hole pair bare interactions, albeit the interactions appear dif-ferently in the respective self-energies. The obviousadvantage to such decomposition is that it allows for trans-parent comparison in the interactions involved in the ringand ladder diagrams, as well as the relative intensities ema-nating from the two respective excitation channels.PHYSICAL REVIEW B 70, 205115 (2004 ) 1098-0121/2004/70 (20)/205115 (8)/$22.50 ©2004 The American Physical Society 70205115-1II. FORMULATION We employ Raleigh-Schrödinger perturbation theory for the description of the electron impact excitation processes. Inthe one-electron picture, and incident electron of energy E scatters to a state u elabove the Fermi level of the metal while a core electron of state label uiIlis excited to a state ue8labove the Fermi level. (An underlined state label means that the state propagates as a hole. )The Coulomb interaction respon- sible for this scattering process is represented by the ampli-tude graph of Fig. 1 (a)in which the dashed line indicates the Coulomb interaction matrix element labeled V eE. Figures 1(b)and 1 (c)describe the general structure of resonant con-tributions from ring and ladder graphs to the basic excitation amplitude of Fig. 1 (a).They may be viewed as an initial state uElcoupling to a two-particle and one-hole final state uiIe8el consistent with Fig. 1 (a). We note in passing that Figs. 1 (b) and 1 (d)–1(h)that constitute the ring graphs are corrections to the ladder graphs of Figs. 1 (a),1(c), and 1 (i)–1(k)used in the earlier treatment.14 In Fig. 1 (b),VDEis a Coulomb interaction matrix element of state uEland a discrete state involving the core-hole uiIl i.e., the state uiIkel.The shaded ring graph plus the interaction vertex lines in Fig. 1 (b)is reconstituted in Fig. 1 (d). Figure 1(d)represents all possible ring graph insertions that begin with the Coulomb interaction matrix element or the interac-tion vertex line V DEand end up with the interaction vertex lineveD. In other words, we are looking at infinite order perturbation in interaction between the discrete state uiIkel and the continuum state uiInel, and such a basic interaction between the two states is represented by Fig. 1 (b).(A thick hole-line indicates the renormalized orbital that gives rise tocharacteristic decay width of the state uiIl.)The decomposi- tion of Fig. 1 (d)is made up of all the graphs of Figs. 1(e)–1(h). Further, a regrouping of the ring graphs is made to consist of those that begin and end with a thick hole-line orrenormalized state (i.e., those containing oddnumber of ring graphs—Fig. 1 (e), Fig. 1 (f), and higher orders ), and those that begin with a renormalized discrete state and end with acontinuum state (i.e., those containing evennumber of ring graphs—Fig. 1 (g), Fig. 1 (h), and higher orders. In the above decomposition S DsEdindicates a self-energy of the discrete state. The shaded rectangular graph in Fig. 1 (c)is reconstituted in Fig. 1 (i)that is made up of graphs of Figs. 1 (j)and 1 (k) and higher orders. Figure 1 (j)is the basic or lowest order ladder graph to Fig. 1 (i). In Fig. 1 (j),vDE8is a Coulomb interaction matrix element between the state uEland the dis- crete state uiIknl. Notice that the interaction matrix elements vDEandvDE8differ not only because the orbitals uelandunl for the respective discrete states uiIkelanduiIjnlin which they appear differ, but also in that they would have different an- gular factors. Here SD8sEdin Fig. 1 (k)is the self-energy of the discrete state for the ladder diagrams.We note in passing that Figs. 1 (a),1(j), and 1 (k)were the graphs used essentially in the discussion of the EEL spectra of lanthanum and ceriumin the previous work. 14The present work is therefore an extension of that work with the inclusion of contributionfrom ring graphs. We may therefore say that the analysisdescribed thus far is a simpler derivation of the more com-plicated multichannel Fano-type calculations that have beenused variously to discuss photoionization cross sections ofsome materials. 15–17 First, we consider the contribution to the excitation am- plitude that emanates from the ring graphs and the nonreso-nant graph (Fig. 1 (a))and denote this by asEd. The nonreso- nant graph, Fig. 1 (a), was part of the ladder graphs in Ref. 14. Since it is the basic interaction, we make it part of thering graphs in the present formulation. Subsequently, if thering graphs are to be neglected, we simply set vDE=0, or equivalently, g=0 in Eq. (19)to be consistent with the earlier results that makes Fig. 1 (a)part of the ladder graphs. Thus from Figs. 1 (a)and 1 (b)and Figs. 1 (d)–1(h),w eg e t FIG. 1. Amplitude diagrams made up of (a): the basic excitation amplitude; (b),(d)–(h)the resonant contributions from the ring dia- grams; (c),(i)–(k)the resonant contributions from ladder diagrams.K. NUROH PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-2asEd=FveE+veDvDE ED−E+vDE ED−ESSDsEd ED−EDveD+flG +FvDE ED−Eo nvenvnD En−E−id +vDE ED−ESSDsEd ED−EDo nvenvnD En−E−id+flG =veE+veDvDE ED−EH1+SSDsEd ED−ED+flJ +vDE ED−Eo nvenvnD En−E−idH1+SSDsEd ED−ED+flJ, s1d with the following definitions for EDandSDsEd: ED=e+Ek−SisEd, s2d SDsEd=o nvnD2 En−E−id=o n8vnD2 En−E+ipvED2. s3d The interacting self-energy SDsEdis represented by the an- gular momentum diagram in Fig. 2 (a)for the ring graphs,8 while Fig. 2 (b)is the corresponding diagram for the self- energySD8sEdfor the ladder graphs14to be considered later. In Eq. (3)and hereafter, a prime on a summation sign means that a principal part summation/integration is implied. In Eq.(2),S iis the self-energy of the core state defined bySisEd=o nvn2 En−Ek−Ek−E−id =o n8vn2 En−Ek−Ek−E+ipvE−2Ek2. s4d Hereafter, we will use the shortened notation vE2forvE−2Ek2in Eq.(4)above. The Coulomb interaction matrix element vn2is obtained from the self-energy graphs of Figs. 2 (c)and 2 (d). Figure 2(c)is a Feynman diagram of the core-hole self-energy oisEd,18while Fig. 2 (d)is its corresponding exchange dia- gram, constructed using angular momentum graphical techniques.19If we introduce an effective interaction matrix elementVeDsEdby the equation VeDsEd=veD+o nvenvnD En−E−id =veD+o n8venvnD En−E+ipveEvED, s5d then by summing to infinity the terms in the two curly brack- ets in Eq. (1)which form geometric series, we get the fol- lowing compact expression for the excitation amplitude: asEd=veE+VeDsEdvDE ED−SDsEd−E. s6d Further, if we introduce the Fano20line width parameters qsEdandhsEdby qsEd=ReVeDsEd ImVeDsEd=SveD+o n8venvnD En−ED/pveEvED, s7ad hsEd=−ResE−SDsEd−Ed ImsE−SDsEd−Ed, s7bd and define the autoionizing, characteristic, and total line width parameters, respectively, by Ga,Gc, and Gt,14through the equations Ga/2p=vED2,Gc/2p=vE2,Gt=Ga+Gc, s8d then with some algebra, the excitation amplitude reduces to the simpler form asEd=veEF1+Ga GtHq+i h−iJG. s9d The matrix element vE2that appears in Eq. (8)is defined in Eq.(4)and the statement after the equation. The contribution to the excitation amplitude arising from the ladder graphs is denoted by bsEd. Thus from Fig. 1 (c) and Figs. 1 (i)–1(k),w eg e t FIG. 2. Self-energy diagrams made up of (a)the self-energy insertion for ring diagrams, (b)the self-energy insertion for ladder diagrams, (c)Coulomb interaction for the characteristic decay of the discrete state, (d)exchange contribution to (c).RANDOM-PHASE APPROXIMATION OF CORE-ELECTRON PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-3bsEd=vDE8 ED8−Eo mvmD8vem Em8−E−id +vDE8 ED8−ESD8sEd ED8−Eo mvmD8vem Em8−E−id+fl =vDE8 ED8−Eo mvmD8vem Em8−E−idF1+SD8sEd ED8−E+flG, s10d where ED8=En+Ek−SisEd. s11d By introducing the self-energy SD8sEdand an effective inter- actionVDe8sEdthrough the equations SD8sEd=o mvED82 Em−E−id=o m8vED82 Em−E+ipvED82,s12d VD«8=o mvmD8vem Em−E−id=o m8vmD8vem Em−E+ipvED8veE,s13d and summing the infinite geometric series in the square brackets of Eq. (10)gives a simpler form for bsEdas bsEd=vDE8VD«8sEd ED8−SD8sEd−E. s14d Once more, we introduce new Fano line width parameters q8sEd=ReVDe8sEd ImVD«8=o m8vmD8vem Em−E/pvED8veE s15ad h8sEd=−ResED8−SD8sEd−Ed ImsED8−SD8sEd−Ed. s15bd Again if we introduce the autoionizing, characteristic, and total line widths, respectively, by Ga8,Gc8, and Gt8for the new channel by the equations Ga8/2p=vED82,Gc8/2p=vE2=Gc/2p,Gt8=Ga8+Gc8, s16d thenbsEdtakes on the form, bsEd=veEGa8 Gt8Hq8+i h8−iJ. s17d Again, the matrix element vE2that appears in Eq. (16)is defined in Eq. (4). Notice that the characteristic line width is the same for both the ring and ladder channels as indicated inEq.(16). Finally, by introducing the dimensionless param- eters gandg8as ratios of the autoionizing to the total line widths in the respective channels by g=Ga/Gt,g8=Ga8/Gt8. s18d Equations (9)and (17)can be combined to give the total excitation amplitude AsEdthat is the sum of asEdandbsEdasAsEd=veEF1+gHq+i h−iJ+g8Hq8+i h8−iJG. s19d Eachq-value ultimately gives rise to a partial excitation cross section as a function of the energy lost by the incidentelectron to the system. From Eqs. (7b)and(15b)we find that D h=hsEd−h8sEd=2ResED−SDsEd−Ed Gt −2ResED8−SD8sEd−Ed Gt8.s20d I will return to this general expression later in the article. It is useful to first examine some limiting cases. First setting h8 =hin Eq. (19)givesAsEdas AsEd=veEFsh+gq+g8q8d+sg+g8−1di h−i G.s21d The scattering intensity IsEdis proportional to the square of the absolute value of the excitation amplitude. Thus IsEd,uAsEdu2=veE2Fsh+gq+g8q8d2+sg+g8−1d2 h2+1 G. s22d We consider different physical scenarios that would arise from Eq. (22). First we set g8=0 to exclude contributions from ladder graphs, and g=1, appropriate for La when Gc =0. In such a situation the scattering intensity becomes (where the subscript Rdenotes ring ) IRsEd=veE2fsh+qd2/sh2+1dg. s23d This is essentially a Fano line profile excluding the non- resonant interaction matrix veE2. If we still exclude the con- tributions from the ladder graphs by demanding that gÞ1, then the scattering intensity becomes (appropriate for Ce ) IRsEd=veE2Fsh+gqd2+sg−1d2 h2+1G. s24d Excluding contributions from ring graphs, we have to set g =0. Then with g8=1sGc8=0d, the scattering intensity be- comes (where the subscript Ldenotes ladder ), and appropri- ate for La, ILsEd=veE2fsh+q8d2/sh2+1dg. s25d This is the result obtained in form in the earlier work, albeit q8is defined in terms of different matrix elements. Finally, by still excluding the ring graphs but now demanding that g8Þ1, the scattering intensity takes on the form (appropriate for Ce ), ILsEd=vzeE2Fsh+g8q8d2+sg8−1d2 h2+1G. s26d In a situation in which Dhis not approximately equal to zero, then we have to set h8=h−Dh. In such a case, the excitation amplitude of Eq. (19)has to be modified to take its complete form ACsEdgiven byK. NUROH PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-4ACsEd=veEF1+gHq+i h−iJ+Hq8+i h−Dh−iJG.s27d By taking the square of the absolute value of the above, we get the complete scattering intensity ICsEdas ICsEd,veE2HFgsh+qd h2+1+g8q8+g8sh−Dhd sh−Dhd2+1G2 +Fh2+hgq−g+1 h2+1+g8q8sh−Dhd−g8 sh−Dhd2+1G2J. s28d Note that if Dhis set equal to zero, Eq. (28)reduces to Eq. (22). The rest of the discussion will be devoted to using the above equations in model calculations. III. MODEL CALCULATIONS For the theoretical model and the formalism presented here, the target system on which the electron scatters couldbe an atom, a molecule or a solid. Here we perform a modelcalculation for metallic lanthanum and cerium that belong tothe lanthanide group.We envisage an electron scattering pro-cess in which a 4 d-subshell electron is excited. An atom from this group in a metallic environment is usually modeledas triply ionized, so the ground state configurations for thelanthanum and cerium ions would be 4 d 104f0and 4d104f1, respectively. This means that in Figs. 1 (a)–1(c), the follow- ing assignments are made for the states with labels e,i, and e8as,e!ef,i!4d, and e8!4f. The state uElrepresenting the incident electron is a plane wave that may be decom-posed into its partial waves. Only the g- andf-partial waves can couple to the 4 dI4fef-state. However, the higher partial wave provides larger numerical values to the matrix element veE. Thus an assignment of E!Egis made for that state label. As a consequence, assignments of k!4f,n!nf, and m!mgshould be made for these state labels in Figs. 1(e)–1(h),1(j), and 1 (k). From Fig. 1 (a), we get the matrix element veEin terms of Slater integrals as veE=2˛3R1sef4f;Eg4dd+2s˛66/11 dR3sef4f;Eg4dd +10s˛390/143 dR5sef4f;Eg4dd. s29d From the angular momentum graph representing the self- energySDsEdin Fig. 1 (f), we get the interaction matrix ele- ment vnD2from Fig. 2 (a)in terms of Slater integral as vnD2=6R1s4f4d;4dnfd2+22 63R3s4f4d;4dnfd +5000 11979R5s4f4d;4dnfd2. s30d Similarly, from the angular momentum graph representing the self-energy SDsEdin Fig. 1 (k), we get the interaction matrix element vED82from Fig. 2 (b)in terms of Slater integralsvED82=4R1s4fnf;4dmgd2+24 77R3s4fnf;4dmgd2 +3000 17303R5s4fnf;4dmgd2. s31d Finally, from the self-energy S4dfor the angular momentum graph represented by Figs. 2 (c)and 2 (d), we get the interac- tion matrix element vm2in terms of Slater integrals as vm2=44 55R1s4f4f;4dmgd2+12 77R3s4f4f;4dmgd2 +8100 121121R5s4f4f;4dmgd2 −8 35R1s4f4f;4dmgdR3s4f4f;4dmgd −8 77R1s4f4f;4dmgdR5s4f4f;4dmgd −48 847R3s4f4f;4dmgdR5s4f4f;4dmgd. s32d The expressions in Eqs. (29)–(32)above have been obtained using the rules for evaluating angular momentum graphs.21 With the above explicit expressions for the matrix elementswe proceed to get numerical values for the relevant param-eters that are needed to calculate the various spectral lines inSec. II. The continuum Estates have been calculated in the Hartree-Fock potential of the triply ionized ion of cerium. Inthe following discussion, the Slater integrals for cerium willbe used for lanthanum as well. (While there is no 4 f-state in the ground configuration of lanthanum atom/ion, there is anavailable localized 4 f-state below the Fermi level whenever there is a vacancy in a 3 d-o r4d-subshell. )Any of the E-dependent matrix elements that appear in the principal part summation/integration is slowly varying. We therefore repre-sent any such function as fsEdwith its appropriate quadratic fit: fsEd=a+bE+cE 2,E0łEłE0+D. s33d Then its principal part summation/integration represented by FsEdtakes the analytic form: FsEd=o n8fsEnd En−E!PEfsed e−Edx =fa+bE+cE2glnUD+E0−E E0−EU +Dfb+csD/2+Edg. s34d The contributions from the logarithmic component would be small. In the above Dreflects the 4 dI4fN-multiplet width for LasN=1dand Ce sN=2d. The energy E0is the 4d!4fex- citation energy. The parameters E0andDmay be taken from experiment or multiplet calculations,22as displayed in Table I. With these values, any E-dependent quantity will be evalu- ated at some mean energy E¯=E0+D/2, say. In such a situa-RANDOM-PHASE APPROXIMATION OF CORE-ELECTRON PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-5tion, the logarithmic contribution in Eq. (34)vanishes and we get for FsE¯dthe simple expression FsE¯d=Dfb+csE¯+D/2dg. s35d Thus from Eqs. (7a)and(15a), the computed values of qsE¯d andq8sE¯dare presented in Table I, together with the values ofgandg8that have been calculated using Eqs. (18)and (35). IV. RESULTS AND DISCUSSION In Fig. 3, the curve labeled (a)represents the theoretical line shape for Ce calculated using the complete expressiondefined by Eq. (28), with the appropriate parameters from Table I. The slowly varying nonresonant excitation matrix element veE2has been excluded. The corresponding theoreti- cal curve for La using Eq. (28)with g=g8=1 is similar in shape to the Ce curve with subtle differences in the reso-nance peaks and widths. Rather than comparing their relativeline shapes, we have instead calculated the ratio of the lineshape of Ce to La to reveal any elusive differences. Theresult of this ratio in which the La line shape was first nor-malized to the Ce peak is the curve labeled (b)in Fig. 3. Thecurve is troughlike with suppressed ridges. The flattened ridges approach the expected theoretical constant value ofunity far away from the zero energy-loss regions. The width of the trough is ,3 eV. The curve labeled (c)is the experi- mental ratio of electron energy-loss spectra of Ce and La. 23 In computing this ratio the experimental data abscissas weretranslated so that their decreasing portions, i.e., their peak totrough portions are coincident as was done in Ref. 14. Thewell-shaped distribution of the computed data points wasthen fitted to a biphasic function so that the experimentaltrough minimum was aligned with the theoretical one in en-ergy. We observe that the curve for the experimental ratio isalso troughlike, but unlike the theoretical one, the ridges arepronounced. The width of the trough is ,8 eV. While the general trend in the theoretical curve is found in the experimental one, there are two major discrepancies. Thefirst is the pronounced ridges in the experimental curve asmentioned above. The second is the width of the experimen-tal curve which is ,5 eV larger than the theoretical one. The former dissimilarity in the theoretical ratio arises mainly be-cause spin-orbit interactions are not included in the theoret-ical formulation, while electron-energy and absorption spec-tra are known to be strongly dependent on spin-orbiteffects. 22,24The cause for the latter dissimilarity is inherent in calculated atomic-based theoretical resonance scatteringspectra whose widths are typically narrower than those fromexperiment. 25Notwithstanding the above remarks, the over- all agreement of theory with experiment is fair to good. Figure 4 is presented to elucidate the importance of the interaction parameter Dh. In the figure we make comparison of only theoretical line shapes of Ce with La along the linesas was done in Fig. 3. Figure 4 (a)and 4 (b)are the same as Fig. 3 (a)and 3 (b), respectively. Figure 4 (c)is the ratio of the line shape of Ce to La using Eq. (28), but in the case of La, gis set equal to zero to indicate that only contributions from ladder diagrams are retained from the expression. The result-ing expression for La that is dependent on the parameter D h is numerically approximate to unity for all h-values. As a result this ratio is indistinguishable from the Ce line shape ofFig.3 (a)—hencethereasonthecurvebearsthetwolabels“a, c.” Figure 4 (d)is the calculated line shape of Ce to La, again using Eq. (28), but this time it is g8that is set equal to zero for La to indicate retention of only contributions from ringdiagrams. The resulting expression for La is independent ofthe parameter D h, and the curve for the ratio is highly reso- nant. (The maximum of this curve is 4.9 units occurring at the energy loss of −2 eV )This curve is contrasted with Fig. 4(e), which is the ratio of the Ce line shape defined by Eq. (28)to the La line shape defined by the Fano expression ofTABLE I. Computed q-values and ratios of autoionizing line widths to total line widths in the ring and ladder channels as described in the text. Elements (configuration )E0a (eV)Da (eV) q g q8 g8 Las4dI4fd 97.2 19.8 −0.548 −0.0565 Ces4dI4f2d 101.8 23.8 −0.521 0.950 −0.0628 0.783 aData extracted from Ref. 22. FIG. 3. Comparison of theoretical line shapes for metallic La and Ce with experiment: (a)The calculated full line shape for Ce; (b)ratio of the calculated full line shape for Ce to that of La; (c) electron energy-loss intensity ratio of Ce to La, computed fromexperimental data (Ref. 23 ).K. NUROH PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-6Eq.(23), i.e., due to contributions emanating from only ring diagrams assuming nonexisting ladder diagrams. We notethat Eq. (23)is independent of D hand Fig. 4 (e)also shows a resonance but not as pronounced as the one in Fig. 4 (d). The point being made through Fig. 4 is that while the reten-tion of only ring diagrams gives spurious picture for theratio, its inclusion in the formalism is necessary to providethe needed interference between the autoionizing and char-acteristic channels to produce the troughlike semblance for the ratio to better compare with experiment. We note alsothat although retaining only ladder diagrams produces asomewhat troughlike picture for the ratio in Fig. 4 (c)or equivalently in Fig. 3 (a), it does not compare better with the experimental features than what Fig. 4 (b)or Fig. 3 (b)does. Finally, the relative insignificance of ring to ladder dia- grams can be seen as follows.The leading contribution to theexcitation amplitude from ladder diagrams comes via Figs. 1(a)and 1 (c)that are first order in Coulomb interaction while the leading contribution from the ring diagrams via Fig. 1 (b) is second order in Coulomb interaction. Consequently, con-tributions from ring diagrams to core-electron impact excita-tion spectrum would be smaller than those from ladder dia-grams. In instances where the diagrammatic RPA has beenused with success, the exchange or ladder diagrams aretreated as corrections to ring diagrams, and sometimes ne-glected altogether. 7–10Also, as has been remarked in the RPA treatment for the charge fluctuation in metals using the Hub-bard model, ladder graphs contribute more to the energy ofthe electrons, but little to their dynamic properties such ascorrelation. 26Thus the minimal dynamic effects exhibited rather by ring diagrams in comparison to ladder diagrams isa novel observation manifested only in doing RPA with re-spect to core-electron impact excitation in metals. The final theoretical expressions of Sec. II may be used to approximately fit experimental electron energy-loss data withtrial and error choices of the parameters g,g8,q, andq8such that 0 łg,g8ł1. But as the foregoing discussion has indi- cated, for any meaningful fit, Dhwould have to be consid- ered as well. The Fano parameters qandq8can also take on positive values, as would be the case if the core transitionhad been 2 p!3dinstead, as analyzed in Ref. 14. The for- mulation presented here may be extended to the transitionmetals. However, two philosophical difficulties presentthemselves. One is some proper decomposition of the pertur-bation diagrams for the 3 d-open shell system or wide bands, and the other is the appropriate wave functions to be used forthe 3d-states in numerical computations. Work in this direc- tion is in progress. To summarize, we have extended thehybrid model that was based on ladder diagrams of diagram-matic perturbation theory to include ring diagrams as well,thereby providing a coherent random-phase approximationapproach to electron scattering in solids. ACKNOWLEDGMENT The author wishes to thank Kent State University’s Re- search Council for partial support. *Electronic address: nuroh@salem.kent.edu 1R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957 );Lectures in Theoret- ical Physics (Wiley-Interscience, New York, 1959 ), Vol. I, pp. 120–123. 2See, for example, S. Doniach and E. H. Sondheimer, Green’s Functions for Solid State Physicists (Benjamin, Reading, 1974 ), pp. 144–146. 3J. Hubbard, Proc. R. Soc. London, Ser. A 276, 238 (1963 );277, 237(1964 );281, 401 (1964 ). 4In Ref. 2, pp. 158–165. 5See, for example, A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971 ), pp. 154–156. 6In Ref. 5, pp. 131–139.7M. Ya Amusia, N. A. Cherepkov, and L. V. Chernysheva, Sov. Phys. JETP 33,9 0 (1971 ). 8G. Wendin, J. Phys. B 3, 466 (1970 ). 9G. Wendin, Phys. Rev. Lett. 53, 724 (1984 ). 10G. Wendin and N. Kerr Del Grande, Phys. Scr. 32, 286 (1985 ). 11A. Zangwill and P. Soven, Phys. Rev. A 21, 1561 (1980 ). 12K. Nuroh, M. J. Stott, and E. Zaremba, Phys. Rev. Lett. 49, 862 (1982 ). 13K. Nuroh, Phys. Scr. 56,5 6 (1997 ). 14K. Nuroh, Phys. Rev. B 66, 155126 (2002 ). 15J. L. Dehmer, A. F. Starace, U. Fano, J. Sugar, and J. W. Cooper, Phys. Rev. Lett. 26, 1521 (1971 ). 16A. F. Starace, Phys. Rev. B 5, 1773 (1972 ). 17L. C. Davis and L. A. Feldkamp, Phys. Rev. B 23, 6239 (1981 ). FIG. 4. Comparison of theoretical line shapes of Ce with La: (a) same as in Fig. 3 (a);(b)same as in Fig. 3 (b);(c)ratio of the calculated full line shape of Ce to La (including only ladder contri- butions );(d)ratio of the calculated full line shape of Ce to La (including only ring contributions );(e)ratio of the calculated full line width of Ce to La [Eq.(23)].RANDOM-PHASE APPROXIMATION OF CORE-ELECTRON PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-718L. S. Cederbaum, W. Domcke, J. Schirmer, and W. von Niessen, Phys. Scr. 21, 481 (1980 ). 19I. Lindgren and J. Morrison, Atomic Many-Body Theory (Springer-Verlag, New York, 1982 ), pp. 238–241. 20U. Fano, Phys. Rev. 124, 1866 (1961 ). 21In Ref. 19, pp. 80–84. 22J. Sugar, Phys. Rev. B 5, 1785 (1972 ).23G. Strasser, G. Rosina, J. A. D. Matthew, and F. P. Netzer, J. Phys. F: Met. Phys. 15, 739 (1985 ). 24K. Nuroh, Phys. Scr. 61, 589 (2000 ). 25See, for example, Ref. 24, p. 593. 26Gerald D. Mahan, Many-Particle Physics (Kluwer Academic/ Plenum Publishers, New York, 2000 ),p .4 1 1 .K. NUROH PHYSICAL REVIEW B 70, 205115 (2004 ) 205115-8

PhysRevB.76.113401.pdf

Magnetotransport in transparent single-wall carbon nanotube networks Manu Jaiswal,1,*Wei Wang,2K. A. Shiral Fernando,2Ya-Ping Sun,2and Reghu Menon1 1Department of Physics, Indian Institute of Science, Bangalore 560012, India 2Department of Chemistry and Laboratory for Emerging Materials and Technology, Clemson University, Clemson, South Carolina 29634-0973, USA /H20849Received 17 February 2007; revised manuscript received 19 May 2007; published 6 September 2007 /H20850 An experimental study of the low temperature magnetotransport in optically transparent single-wall carbon nanotube /H20849SWNT /H20850networks is reported. The SWNT network shows Coulomb gap variable-range hopping conduction at low temperatures. The magnetoresistance /H20849MR /H20850involves the interplay of two phenomena: a forward interference process leading to negative MR together with shrinkage of electronic wave functioncontributing to the positive MR. These two mechanisms fit the low-field data. The analysis of magnetotransportdata gives an estimate for intrinsic parameters including localization length and Coulomb gap. The temperaturedependence of the forward interference mechanism is shown to follow an inverse power-law dependence withan exponent close to 1, indicating the weak scattering process involved in the transport. DOI: 10.1103/PhysRevB.76.113401 PACS number /H20849s/H20850: 73.63.Fg, 73.43.Qt The electronic properties of single-wall carbon nanotube /H20849SWNT /H20850systems have attracted considerable interest due to their ideal one-dimensional structure and the possibilities fora wide range of disorder. A significant variation in the trans-port properties, from diffusive to ballistic, has been reported,and this has led to an ambiguous scenario. Since it is wellknown that one-dimensional systems are highly susceptibleto disorder-induced localization and Coulomb interactions,low temperature magnetotransport measurements can giveinsight into this subtle interplay. The magnetotransport hasbeen investigated in graphite nanotubules, 1macroscopic bundles of intercalated multiwall carbon nanotubes/H20849MWNTs /H20850, 2strongly disordered MWNTs,3entangled SWNT mats,4macroscopic doped SWNTs,5and disordered indi- vidual SWNTs.6Especially in the case of SWNT, minute variations in structural disorder can lead from weak to stronglocalization, and this yields complex and intriguing featuresin the analysis of magnetotransport data. Further, there islack of detailed understanding of the physics in the latterregime unlike in the case of the interference process in theweak localization regime near the metal-insulator /H20849M-I /H20850tran- sition. In this Brief Report, we report the magnetotransport mea- surements on a transparent network of SWNTs. As alreadyreviewed by Grüner and co-workers 7,8and Zhang et al. ,9 these networks have important applications as flexible trans- parent electrodes in organic optoelectronics. Our networksshow 85% transparency at 550 nm wavelength, which iscomparable to that of indium tin oxide transparent elec-trodes. The SWNTs were prepared by carbon-arc method re-sulting in a mixture of metallic and semiconducting nano-tubes of which the latter are two-thirds. The semiconductingnanotubes were separated from the metallic ones by meansof separating agents that preferentially associate with theformer. This yields about 90% enriched semiconductingSWNTs, and the process is described in detail elsewhere. 10,11 In the preparation of the transparent nanotube film, 5 mg semiconducting SWNTs was dispersed in 50 ml dimethylfor-mamide /H20849DMF /H20850via homogenization for 30 min, followed by sonication for 60 min. Then, 15 ml of the dispersion solutionwas sprayed onto a heated clean glass slide on a hot plate at150 °C. A typical scanning electron microscope /H20849SEM /H20850im- age of the network is shown in Fig. 1. SWNTs were ran- domly laid on the glass surface to form an interconnectednanotube network. The nanotubes have a typical bundle di-ameter of 10–20 nm and an average length of well beyond1 /H9262m. The magnetotransport experiments were performed with a standard four-probe dc method in a Janis variable-temperature cryogenic system equipped with an 11 T super-conducting magnet. The magnetic field was applied both per-pendicular and parallel to the network plane. The currentused in low temperature transport measurements are below0.5 /H9262A, and the heat dissipation is typically less than 30 nW. The temperature was stable to within 20 mK during the fieldsweep. The temperature dependence of resistance of the network is shown in Fig. 2/H20849a/H20850. The resistance ratio is R 4.2 K/R300 K /H1101157, and it increases significantly for T/H1102120 K, suggesting a “hopping-type” conduction mechanism. Recently, Skákalováet al. have reported current-voltage /H20849I-V/H20850characteristics and temperature dependence of conductance on transparent SWNT networks and found that the transport is consistentwith hopping conduction with important contribution fromthe disordered metallic tubes. 12To understand the exact na- ture of charge transport in such systems, our data are ana- FIG. 1. SEM image of the transparent SWNT network /H20849the scale bar is 1 /H9262m/H20850.PHYSICAL REVIEW B 76, 113401 /H208492007 /H20850 1098-0121/2007/76 /H2084911/H20850/113401 /H208494/H20850 ©2007 The American Physical Society 113401-1lyzed using the resistivity curve derivative analysis method.13The reduced activation energy /H20849W/H20850is estimated as W/H20849T/H20850=−/H11509lnR/H20849T/H20850//H11509lnT. The Wplot /H20851inset of Fig. 2/H20849a/H20850/H20852 yields a negative slope indicating that the network is in the insulating regime at low temperatures, with an exponentialtemperature dependence /H20849exponent p=0.45–0.5 /H20850,a sd e - scribed below. In Fig. 2/H20849b/H20850, the logarithmic resistance is plot- ted indicating the linear relationship with the appropriate power of temperature. The exponent p=1 2can arise from various transport regimes such as Mott variable-range hop-ping /H20849VRH /H20850in one dimension and Coulomb Gap /H20849CG/H20850 VRH. 13One-dimensional Mott VRH is an unlikely scenario in this case as the value obtained for the obtained localiza-tion length /H20849/H110110.2 /H9262m/H20850would be physically unrealistic for the disordered system, as earlier noted by Vavro et al.5The CG-type transport is also consistent with the same exponentbut involves the formation of an energy gap at the Fermilevel on account of Coulomb interactions. Near the Fermilevel, the density of states /H20849DOS /H20850follows a power-law de- pendence, N/H20849E/H20850=N 0/H20841E−EF/H20841/H9253with/H9253=1 for two dimensions and/H9253=2 for three dimensions.13The Coulomb correlations in this transparent network are much more significant due tothe unscreened nature of interactions with respect to the bulkSWNTs. 14Usually, this is a low temperature effect since at higher temperatures, the thermal energy is greater than thesize of the gap. The CG VRH has also been reported recentlyon an isolated single tube of SWNTs with some disorder 6 and previously on SWNT-polymer composites with low vol-ume packing fraction of tubes.15Alternatively, the same en- ergy gap can also arise from charging energy effect when thenetwork is considered as a granular metal. In this case, thetunnel junction across the tubes comprises a parallel platecapacitor and a resistor, with tunneling taking place acrossthe metallic grains. The value of the Coulomb gap is usefulto distinguish these mechanisms. The temperature depen-dence of resistance for CG VRH is universal for both two-dimensional and three-dimensional /H208493D/H20850disordered elec- tronic systems and is given by R/H20849T/H20850=R 0exp/H20851/H20849T0/T/H208501/2/H20852, /H208491/H20850 where T0=/H9252e2/kB/H9260/H9261,/H9260is the dielectric constant, /H9261is the localization length, and the numerical coefficient /H9252has the value 6.2 in two dimensions and 2.8 in three dimensions. Thevalues of R 0andT0are 2.275 k /H9024and 128 K, respectively. The value of characteristic “Coulomb gap” temperature TCG can be estimated as T0/TCG=/H9252/H208494/H9266/H208501/2;TCG=12.9 K. Since the linear regime extends to a higher temperature /H20849/H1101130 K /H20850as seen from the Wplot, the possibility of correlations between grains contributing to the transport should be considered, asdiscussed below. The magnetoresistance /H20849MR /H20850data in the CG-VRH regime are shown in Figs. 3/H20849a/H20850and3/H20849b/H20850as a function of transverse field at low temperatures. At low fields, the MR is negativeand linear with magnetic field /H20851Fig.3/H20849b/H20850/H20852forT/H333564.2 K. At high field, the MR decreases and even the sign changes topositive at lower temperatures, being entirely positive at1.3 K /H20851Fig.3/H20849a/H20850/H20852. Conventionally, this type of negative MR is attributed to the suppression of quantum localization cor-rections to the Drude conductivity in systems near the M-Itransition. 16However, this interpretation of the negative MR is not plausible in disordered SWNT since the temperaturedependence of resistance is in the CG-VRH regime. Hence,alternate explanations based on the strongly localized natureof the system should be considered. This negative MR instrong localization regime is attributed to forward interfer-ence effects of electrons undergoing tunneling hops. 17–19In the model by Nguyen et al. ,17the MR was shown to be linear with field, originating from multisite hops that interfere de-structively in the absence of the field. An identical result wasalso obtained by Raikh and Wessels by taking into accountthe forward interference process between sites with scatter-ing process involving a third, randomly distributed site. 18 Schirmacher also arrived at a similar result by considering an“interference hole” or a region with no destructive interfer-ence in the three-site hopping process. 19In this complex sce- nario, the choice of the exact model would depend on thenature of disorder /H20849both intra- and intertube /H20850and scattering process in the system. In addition to the above mentioned negative MR, the posi- tive MR observed at lower temperatures and high magneticfields can be attributed to the wave-function shrinkageprocess. 6The increased resistance results from a decrease in the probability of hopping on account of the contraction ofthe wave function’s exponential tails in the presence of thefield. For CG-VRH transport, the low-field /H20849 /H9257/H11271/H9261 /H20850positive MR is governed by the equation20FIG. 2. /H20849a/H20850Temperature dependence of the zero-field resistance. Inset: Reduced activation energy WvsT./H20849b/H20850lnRvsT−pplot show- ing the linear relationship /H20849arrows indicate limit of linear regime fit/H20850.BRIEF REPORTS PHYSICAL REVIEW B 76, 113401 /H208492007 /H20850 113401-2ln/H20851/H9267/H20849B/H20850//H9267/H208490/H20850/H20852= 0.0015 /H20849/H9261//H9257/H208504/H20849T0/T/H208503/2, /H208492/H20850 where /H9257=/H20881/H6036/eBis the magnetic length. The MR at 1.3 K is positive even at the lowest fields and a fit to Eq. /H208492/H20850/H20851see Fig. 3/H20849a/H20850/H20852gives the value of localization length /H9261=6 nm. This value is comparable to the results obtained on disorderedindividual SWNTs, where the reported localization lengthsare 4.5 and 15 nm for as-deposited and annealed tubes,respectively. 6Substituting /H9261in Eq. /H208491/H20850, the obtained dielec- tric constant /H9260for our network of randomly oriented tubes is 61, which is intermediate between the values for graphite/H20849 /H9260=12 /H20850and annealed SWNT with disorder measured along the tube axis /H20849/H9260=80 /H20850.6The Coulomb gap20can be estimated as/H9004CG=e3N/H20849EF/H208501/2//H92603/2=1.1 meV, where the dielectric con- stant is given by /H9260=/H208514/H9266/H92550+4/H9266e2N/H20849EF/H20850/H92612/H20852andN/H20849EF/H20850is the unperturbed DOS at the Fermi level.21The value of charac- teristic Coulomb gap temperature TCGobtained from the gap isTCG=/H9004CG/kB=12.8 K, in good agreement with the value obtained from resistance data. The charging energy Ec =e2/2Ccan be estimated from the geometrical capacitances of the network of bundles.15The estimate of /H9004CGis in excel- lent agreement with the value for Coulomb charging energy/H208491.3±0.3 meV /H20850, indicating that charging energy plays adominant role in the gap formation. This soft gap is occur- ring in metallic tubes /H20849/H1101110% of tubes /H20850, which play an im- portant role in the low temperature transport. 5,12It should be noted, however, that such a comparison is based on the as-sumption that the fluctuations in grain work function arelarger than E c. It then follows that at low temperature, the grains will be charged and correlation between the charges ofdifferent grains will give rise to a Coulomb gap. 20The value of DOS for the network obtained from this analysis is0.09 eV −1nm−3, which correlates with that for metallic tubes upon scaling with the typical estimates of volume packingfraction of tubes /H2084920%–25% /H20850for transparent networks. 15A correlation of the localization length with the size of SWNTbundles in the network has been suggested by Benoit et al. , 15 and this compares well for our system /H2084910–20 nm bundle diameters /H20850. Incidentally, this length scale is also of the same order of magnitude as the disorder along the tube length ofan individual SWNT; 22however, that system shows dis- tinctly different transport properties in comparison to ournetwork. The values of /H9261and/H9004 CGobtained from this analy- sis are quite significant to get an insight into the extent ofdisorder in SWNTs. However, a precise definition for local-ization length is a subject of debate for nanotube systems andthese parameters are rather indicative values. 5At high fields, at 1.3 K, the positive MR saturates /H20851Fig.3/H20849a/H20850/H20852, as expected from the VRH theory.5,20 For the experimental data at 4.2 K and higher tempera- tures, the MR involves contributions from the forward inter-ference mechanism, 17–19as well as from wave-function shrinkage. As previously shown,23–25these contributions are additive and the total MR is given by ln/H20851/H20849/H9267/H20849B/H20850//H9267/H208490/H20850/H20850/H20852=−a1B+a2B2+a3, /H208493/H20850 where a1is the coefficient in forward interference mecha- nism, a2is the coefficient of the wave-function shrinkage term, and a3accounts for the complex nature of hops at very low fields.18A fit to the experimental data for low and inter- mediate fields /H20849B/H110214T/H20850is carried out with the above equa- tion, as shown in Fig. 3/H20849b/H20850. It is necessary to include the term a3in order to obtain proper fits to low-field data but the fits are not extended to 0 T. Moreover, Raikh and Wessels haveshown that the energy dependence of electron scattering andabsorption or emission of phonon at the scatterer are tworelevant processes at very low fields. 18These effects can pro- duce an initial quadratic field dependence that changes overto the linear one. Our data are suggestive of this process forB/H110210.6 T, as observed from the fits by taking into account the third term in Eq. /H208493/H20850. The temperature dependence of the coefficients of linear negative MR can be useful to identifythe nature of the scattering process. The coefficient of thelinear term /H20849a 1/H20850, obtained from the fitting, is plotted as a function of temperature in a log-log scale /H20851Fig. 3/H20849c/H20850/H20852. For “strong scattering limit,” a1is expected to be constant, while it increases with temperature in the “weak scattering limit.”25 The temperature dependence of a1points to the latter process in our network. The theoretical model19predicts T−sbehavior fora1and the exponent sis sensitive to the nature of disorder and dimensionality involved in the transport. The estimatedvalues are s=7/8 for dimension d=3 and s=1 for d=2. 19FIG. 3. /H20849a/H20850Transverse field dependence of MR at various tem- peratures. The solid line is the wave-function shrinkage fit to low-field /H20849B/H110214T/H20850positive MR at 1.3 K. /H20849b/H20850Low-field MR at various temperatures. The solid lines are fits to Eq. /H208493/H20850described in text. /H20849c/H20850 Coefficient of term linear in field /H20849a 1/H20850vsT.BRIEF REPORTS PHYSICAL REVIEW B 76, 113401 /H208492007 /H20850 113401-3The observed exponent for our network is s=1.04±0.15, a value that is close to 1. Since the exponents for both dimen-sions are close, an interpretation for dimensionality must relyon the anisotropy of MR. Furthermore, this model indicatesthat the numerical value of a 1has some relation to the den- sity of scattering sites, which in this context comprises theextent of defects, tube-tube intersections, volume packingfraction, etc. The physical relevance of a 2, in terms of local- ization length, is already described in Eq. /H208492/H20850. A comparison of the MR for transverse and longitudinal fields is shown in Fig. 4. The forward interference mecha- nism, which is relevant at low fields, shows a weak aniso-tropy, as seen from the MR at 4.2 K /H20851Fig.4/H20849a/H20850/H20852;a t1 0K ,t h e MR is nearly independent of the field direction. According tothe forward interference models, 17–19this is expected when the effective dimensionality is between 2 and 3, closer to thelatter. These models rely on the concept of averaging thehops in a “cigar-shaped” area through which the magneticflux penetrates. While moving from three to two dimensions,the hopping paths get increasingly confined to the plane ofthe sample, thereby hardly any area is present to the longi- tudinal magnetic field and this causes the anisotropy in nega-tive MR. However, the anisotropy of wave-function shrink-age is much more pronounced, as seen from the MR at 1.3 K/H20851Fig.4/H20849b/H20850/H20852. The difference in the extent of anisotropy further confirms that two different mechanisms are responsible forthe observed MR, each being dominant at different tempera-ture intervals. In summary, we have presented low temperature magne- totransport measurements on transparent SWNT films. Inter-play of two effects—a forward interference mechanism andwave-function shrinkage—are observed in a Coulomb gapvariable-range hopping regime. The temperature dependenceof low-field negative magnetoresistance, with exponent s /H110111, points to the network being in the weak scattering limit. The data analysis gives an estimate for both localizationlength and Coulomb gap. The behavior of magnetotransportindicates the extent of disorder and tube intersections in thenanotube network, both can serve as scattering centers forthe electrons undergoing hops. A study of the scattering pro-cess from MR data in transparent SWNT networks is quiteessential toward the understanding of transport in this formof SWNT. This particular data analysis consistently takesinto account the various processes involved in the low tem-perature transport in these systems. These measurements were performed at the DST National Facility for Low Temperature and High Magnetic Field, Ban-galore /H20849the authors thank V. Prasad /H20850. M.J. would like to thank CSIR, New Delhi, for financial assistance, and Y.-P.S. ac-knowledges financial support from the U.S. National ScienceFoundation. *manu@physics.iisc.ernet.in 1S. N. Song et al. , Phys. Rev. Lett. 72, 697 /H208491994 /H20850. 2M. Baxendale et al. , Phys. Rev. B 56, 2161 /H208491997 /H20850. 3R. Tarkiainen et al. , Phys. Rev. B 69, 033402 /H208492004 /H20850. 4G. T. Kim et al. , Phys. Rev. B 58, 16064 /H208491998 /H20850. 5J. Vavro et al. , Phys. Rev. B 71, 155410 /H208492005 /H20850. 6D. P. Wang et al. , arXiv:cond-mat/0610747 /H20849unpublished /H20850. 7G. Grüner, J. Mater. Chem. 16, 3533 /H208492006 /H20850. 8D. Hecht et al. , Appl. Phys. Lett. 89, 133112 /H208492006 /H20850. 9D. Zhang et al. , Nano Lett. 6, 1880 /H208492006 /H20850. 10H. Li et al. , J. Am. Chem. Soc. 126, 1014 /H208492004 /H20850. 11Y.-P. Sun, U.S. Patent Application No. 2006/0054555 A1 /H20849pend- ing/H20850. 12V. Skákalová et al. , Phys. Rev. B 74, 085403 /H208492006 /H20850. 13A. G. Zabrodskii, Philos. Mag. B 81, 1131 /H208492001 /H20850. 14A possible comparison of our unscreened network can be made with the nonoriented undoped pulsed laser vaporized SWNT/H20849Vavro et al. , Ref. 5/H20850; the latter is a “bulk” sample showing 3D Mott VRH. 15J. M. Benoit et al. , Phys. Rev. B 65, 241405 /H20849R/H20850/H208492002 /H20850. 16M. Reghu et al. , Phys. Rev. B 49, 16162 /H208491994 /H20850. 17V. L. Nguyen et al. , Sov. Phys. JETP 62, 1021 /H208491985 /H20850. 18M. E. Raikh and G. F. Wessels, Phys. Rev. B 47, 15609 /H208491993 /H20850. 19W. Schirmacher, Phys. Rev. B 41, 2461 /H208491990 /H20850. 20B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors /H20849Springer-Verlag, Berlin, 1984 /H20850. 21This estimate for the gap assumes a 3D electronic system; this is justified by the nearly isotropic negative MR for parallel andperpendicular fields. 22B. Gao et al. , Phys. Rev. B 74, 085410 /H208492006 /H20850. 23R. Rosenbaum et al. , Phys. Rev. B 63, 094426 /H208492001 /H20850. 24M. Benzaquen et al. , Phys. Rev. B 38, 10933 /H208491988 /H20850. 25N. V. Agrinskaya and V. I. Kozub, Phys. Status Solidi B 205,1 1 /H208491998 /H20850.FIG. 4. /H20849a/H20850Field dependence of MR for transverse /H20849open sym- bols /H20850and longitudinal fields /H20849filled symbols /H20850at 4.2 K /H20849triangles /H20850and 10 K /H20849squares /H20850./H20849b/H20850Field dependence of MR for transverse /H20849open symbols /H20850and longitudinal fields /H20849filled symbols /H20850at 1.3 K. /H20849Solid curves are a guide to the eye. /H20850BRIEF REPORTS PHYSICAL REVIEW B 76, 113401 /H208492007 /H20850 113401-4

PhysRevB.90.245409.pdf

PHYSICAL REVIEW B 90, 245409 (2014) Strain-induced modifications of transport in gated graphene nanoribbons Diana A. Cosma,1,*Marcin Mucha-Kruczy ´nski,2Henning Schomerus,1and Vladimir I. Fal’ko1 1Department of Physics, Lancaster University, Lancseter LA1 4YB, United Kingdom 2Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom (Received 24 September 2014; revised manuscript received 21 November 2014; published 3 December 2014) We investigate the effects of homogeneous and inhomogeneous deformations and edge disorder on the conductance of gated graphene nanoribbons. Under increasing homogeneous strain the conductance of suchdevices initially decreases before it acquires a resonance structure and, finally, becomes completely suppressedat higher strain. Edge disorder induces mode mixing in the contact regions, which can restore the conductanceto its ballistic value. The valley-antisymmetric pseudomagnetic field induced by inhomogeneous deformationsleads to the formation of additional resonance states, which originate either from the coupling into Fabry-P ´erot states that extend through the system or from the formation of states that are localized near the contacts, wherethe pseudomagnetic field is largest. In particular, the n=0 pseudo-Landau level manifests itself via two groups of conductance resonances close to the charge neutrality point. DOI: 10.1103/PhysRevB.90.245409 PACS number(s): 73 .22.Pr,62.20.−x,71.70.Di I. INTRODUCTION Monolayer graphene [ 1] is a unique material capable of sustaining reversible deformations in excess of several percent[2–6]. The effects of strain in this one-atom-thick crystalline membrane [ 7,8] attract attention due to the peculiar way in which they affect the already unusual electronic propertiesof this material [ 1,9,10]. Pristine graphene displays a conical dispersion (Dirac points; DPs) at the gapless edge between thevalence and the conduction bands. The DPs are replicated atthe inequivalent KandK /primecorners of the hexagonal Brillouin zone (BZ), and the effect of lattice deformations on electronsis equivalent to that of an effective gauge field with the signinverted in the opposite valleys [ 2,3,11–14]. Consequently, homogeneous deformations result in a small shift of theDirac cones from the corners of the BZ [ 2,20], whereas inhomogeneous strain influences electron motion similarly to avalley-dependent effective pseudomagnetic field [ 9,12,15–19]. Recent scanning-tunneling experiments on graphene nanobub-bles [ 21] revealed that even small inhomogeneous defor- mations can induce pseudomagnetic fields that reach valuesequivalent to hundreds of teslas. Such strong fields result in thelocalization of the electronic states and lead to the formationof a discrete Landau level (LL) spectrum with the peculiarn=0 LL state positioned at zero Fermi energy ( E F=0) [12,15–17,21–23]. In this paper, we perform a systematic analysis of the conductance of gated armchair graphene nanoribbons (GNRs),which are subjected to both homogeneous and inhomogeneouslongitudinal deformations, as well as to various types of edgedisorder. Our calculations are carried out within a tight-bindingmodel that incorporates the strained-induced modificationsof the couplings [ 24]. The conductance is then obtained in the Landauer-B ¨uttiker approach [ 25], where the transmission probabilities are obtained using the recursive Green’s functiontechnique [ 26,27]. Under homogeneous deformations of increasing strength, the conductance of such ribbons first decreases, then acquires *d.cosma@lancaster.ac.uka resonant structure, and, finally, becomes completely sup-pressed in a large range of energies. These effects arise froma combination of a strain-induced mismatch of the Fermisurfaces in the leads and the strained regions [ 19,20] and the finite-size quantization of the transverse momentum. We foundthat these transport features are washed out by single-atomedge defects, while double-atom defects (consisting of theremoval of a dimer at the edge) do not alter the resonantstructure significantly and can even restore the ballistictransport properties of the ribbon in the regime where theconductance is completely suppressed by the deformations inthe absence of disorder. For completeness, we compare the above-mentioned results with those obtained for suspended GNRs [ 6,28–32], which display inhomogeneous strain distributions [ 15–17]. This includes an extended discussion of additional resonancesformed by LL quantization in the pseudomagnetic field, firstreported in shorter form in Ref. [ 33]. The nature of these resonances is revealed through the local density of states(LDOS) profiles, which we calculate at the resonance energies.We found that these features can be attributed either toFabry-P ´erot-like standing waves or to resonant transmission via pseudomagnetic LL states that form in the contact regionsof the GNR. The zeroth LL is identified by its sublatticepolarization [ 12,23] and is found to result in resonance states close to the charge neutrality point. The above-mentioned results are described in detail in Sec. III. The preceding Sec. IIintroduces the tight-binding model for strained graphene ribbons and identifies the un-derlying physics of homogeneously and inhomogeneouslystrained armchair GNRs, while Sec. IVsummarizes their consequences. II. MODELING OF STRAINED GRAPHENE NANORIBBONS A. Hamiltonian We consider a narrow and long-strained GNR, clamped to unstrained graphitic leads and suspended over metalliccontacts. The ribbon is assumed to have free-standing armchair 1098-0121/2014/90(24)/245409(10) 245409-1 ©2014 American Physical SocietyCOSMA, MUCHA-KRUCZY ´NSKI, SCHOMERUS, AND FAL’KO PHYSICAL REVIEW B 90, 245409 (2014) FIG. 1. (Color online) (a) Sketch of a graphene nanoribbon (GNR) of aspect ratio L/W=4, where Lis the length and Wis the width of the system. The color code indicates the pseudomagnetic fieldB(in teslas) for electrons in the Kvalley; at w=0.05 inhomogeneous tensile strain in the middle of a system of widthW/similarequal40 nm. We also sketch the honeycomb lattice corresponding to the tight-binding model in Eq. ( 1). The leads are heavily doped by imposing an on-site potential V=−200 meV . This potential step can be controlled via electrostatic gates. In the central region the strain modulates the hopping matrix elements γ ijand the on-site energy Vi. (b) Shift of the Dirac cones from the corners KandK/primein the Brillouin zone of a homogeneously strained armchair GNR. (c) Fermi surfaces atEF=100 meV in the vicinity of a Kpoint in the BZ of the GNR shown in (a). We contrast the situation without strain ( w=0; red circles) to externally imposed homogeneous strain ( w=0.015 and 0.024; blue circles). Green lines represent the quantized transverse momentum values of the unstrained GNR. edges along the transport direction yand contacts with bulk electrodes along the xaxis, as sketched in Fig. 1(a). Such ribbons can be obtained by oriented growth on patterned SiCsubstrates [ 34], etching of graphene samples with catalytic nanoparticles [ 35], or use of chemical derivation [ 36]. Within the tight-binding model, the ribbon can be described by theHamiltonian [ 1] H=/summationdisplay iVic† ici+/summationdisplay /angbracketrightij/angbracketrightγijc† icj, (1)where ciis a fermionic annihilation operator acting on site i, while /angbracketleftij/angbracketrightdenotes pairs of nearest neighbors. In pristine, unstrained graphene with carbon-carbon bond lengths r= 1.42˚A, the hopping matrix elements take the constant value γij=γ0≈−3 eV . The system can be doped via electrostatic gates, which induce a potential step of size Vat the contacts. We account for this effect in Eq. ( 1) by setting the on-site potential in the leads to Vi=V, while Vi=0 in the central region of an unstrained system. For strained monolayermembranes, both the on-site potential V iand the hopping matrix elements γijare modified by the deformation of the lattice. (Note that in our tight-binding approach the lattice itselfremains unchanged and all deformations are absorbed intothe modified hoppings, as recalculated from the microscopictheory. This procedure automatically isolates the physicallyrelevant effects of the strain.) The on-site potential thenacquires an additional contribution, V i=1 2r∂/epsilon1c ∂rdivu(ri), (2) where u=(ux,uy) is the displacement field of the membrane and/epsilon1ca characteristic energy function. This contribution vanishes for homogeneous strain and, furthermore, is typicallywell screened by the electrons in the flake and in theelectrostatic environment [ 24]. We therefore focus on the hopping matrix elements, which must now be renormalizedto [2] γ ij=γ0eη0(lij/r−1),lij/similarequalr(1+nij·ˆwnij). (3) Herelijis the strain-modified distance between lattice sites, η0=∂γ0 ∂rr γ0≈−3 relates the change of the nearest-neighbor coupling to the change of the bond length [ 37],ˆwis the 2 ×2 strain tensor wαβ=1 2(∂αuβ+∂βuα) with α,β=xory, and nij=(0,1), (√ 3 2,−1 2), and ( −√ 3 2,−1 2) are the unit vectors along the carbon-carbon bonds in the unstrained honeycomblattice. The strain-induced asymmetry in the hoppings between neighboring carbon sites is equivalent to the effect of a valley-dependent gauge vector potential [ 12], eA=ξ/planckover2pi1η 0 2r/parenleftbigg wxx−wyy −2wxy/parenrightbigg , (4) written for the states near one of the corners of the BZ, where ξ=1(ξ=−1) for valleys K(K/prime). B. Homogeneous strain For an externally imposed homogeneous deformation, where the GNR is elongated along the yaxis, the elements of the strain tensor are wxx=−σw,wyy=w, andwxy=0, where σ=0.165 is the Poisson ratio for graphite [ 38] andw parameterizes tensile strain. In this case, both the scalar and thevector potentials V iandAare constant. The scalar potential merely introduces a shift of the energy scale, which cannot bedistinguished from the effect of electrostatic gating. The vectorpotential shifts the nonequivalent Dirac cones from the Kand K /primecorners of the BZ into opposite directions [ 12,20], as shown in Fig. 1(b). (Note that higher order effects induce a distortion of the Dirac cone [ 39,40] in addition to the shift, which is neglected in this work.) Infinitely wide samples are robust 245409-2STRAIN-INDUCED MODIFICATIONS OF TRANSPORT IN . . . PHYSICAL REVIEW B 90, 245409 (2014) against such deformations and their spectrum remains gapless for strains below 20% [ 2]. In contrast, GNRs behave markedly differently due to quantum confinement effects, which allowfor an opening of the gap even for small strains ( w/lessmuch20%) [41–43]. Figure 1(c)shows a comparison between the Fermi surfaces around a Kpoint in the BZ, for an unstrained ribbon ( w=0; red circles) and homogeneously strained ribbons ( w=0.015 and 0.024; blue circles) of width W/similarequal40 nm, at energy E F=100 meV from the DP. When the strain is smoothly increased from w=0t ow=0.024, the DP (filled black circle) crosses several quantized momentum lines (green lines)and the system undergoes multiple semiconducting-metallic-semiconducting phase transitions. Therefore, the size of thegap in the spectrum of armchair GNRs is controllable by theamount of deformation [ 41,42], within a range determined by the width of the ribbon. C. Inhomogeneous strain To model a more realistic deformation, we assume that a suspended ribbon is clamped at the leads and stretched alongtheyaxis. Because of the clamping, the resulting deformation is inhomogeneous [ 44]. We neglect spontaneous wrinkling of the ribbon [ 45,46] and consider this simplified problem within two-dimensional linear elasticity theory [ 47]. With the origin of the coordinate system chosen in the center of the ribbon,the displacement is then prescribed by two equations [ 24], 2∂ xxux+(1−σ)∂yyux+(1+σ)∂xyuy=0, 2∂yyuy+(1−σ)∂xxuy+(1+σ)∂xyux=0,(5) accompanied by clamped boundary conditions for the left and right edge as well as free boundary conditions for the top andbottom edge: clamped/braceleftBigg u x(x,±L/2)=0, uy(x,±L/2)=±1 2wL, free/braceleftBigg [∂xux+σ∂yuy]x=±W 2=0, [∂xuy+∂yux]x=±W 2=0.(6) Despite its simplicity, the problem of finding the displace- ment field satisfying Eqs. ( 5) and ( 6) does not have an analytic solution, so that we apply the finite-element method [ 48] with a nine-point element to determine u(x,y). Having obtained the displacement [ 45], we calculate numerically the vector potential A(x,y) as predicted by the continuum model, Eq. ( 4). The corresponding pseudomagnetic field B(x,y)=rotA(x,y) in theKvalley of a GNR with width W/similarequal40 nm, aspect ratio L/W=4, and inhomogeneous tensile strain w=0.05 in the central part is illustrated in Fig. 1(a). The pseudomagnetic field is the largest positive (blue area) or negative (red area)near the contacts at the right and left ends and is small in themiddle part of the ribbon, where the strain is approximatelyhomogeneous. This is in contrast to the system considered inRef. [ 17], where the flake is overlayed on top of a ridge and high pseudomagnetic fields develop in the central regions. Such strong pseudomagnetic fields can lead to the quan- tization of electronic states into LLs and the appearance ofgaps in the electronic spectrum [ 12,21,24]. Furthermore, thesefields should be capable of deflecting the electrons into states that are inaccessible at homogeneous strain. In the followingsection, we explore these effects via the transport propertiesof the GNR. III. CONDUCTANCE Having established the effects of both homogeneous and inhomogeneous strains on the electronic structure of theGNRs, we now turn to the main goal of this paper and discussthe conductance of the two-terminal device sketched in Fig. 1. In our numerical procedure, we first map the displacementdirectly onto the crystalline lattice of the ribbon and calculatethe positions of the carbon atoms after the deformation. Wethen recalculate the nearest-neighbor couplings according toEq. ( 3) and use this information as input for the tight-binding Hamiltonian, ( 1). As mentioned above, we ignore the on-site scalar potential V i, as it is screened by the electrons in the flake and the electrostatic environment [ 24]. The phase-coherent transport properties of such two- terminal devices are encoded in the scattering matrix [ 49–51], S=/parenleftbigg rt/prime tr/prime/parenrightbigg , (7) which we evaluate using the recursive Green’s function technique [ 26,27] applied to the tight-binding model. Here, t,t/prime(r,r/prime) are the transmission (reflection) amplitudes of charge carriers incident from the source or the drain leads,respectively. Using the Landauer-B ¨uttiker formalism [ 25], we calculate the conductance at zero temperature, G(E F,T=0)=2e2 hTr(t†t), (8) as a function of the Fermi level EF. We also consider the effects of finite temperatures, where G(μ,T)=2e2 h/integraldisplay dEF/parenleftbigg −∂f(EF−μ) ∂EF/parenrightbigg Tr(t†t). (9) Here μis the chemical potential, which enters together with the temperature into the Fermi distribution f(ε)=(1+ exp(ε/k BT))−1. Throughout the following, we set the height of the gate- controlled potential-energy step between the doped grapheneleads and the suspended part to V=−200 meV . The resulting device is a p-p /prime-pjunction ( EF<−200 meV), an n-p-n junction ( −200<E F<0 meV), or an n-n/prime-njunction ( EF> 0 meV). In such systems, most of the conductance featuresare determined by scattering from the strain-modified p-p /prime, n-p,o rn-n/primeinterfaces, a behavior which can be investigated by analyzing the spatial distribution of the electronic states.Within the used formalism, this can be revealed via the LDOS[52], LDOS =i 4πTr/parenleftbigg S†∂S ∂Vi−∂S† ∂ViS/parenrightbigg , (10) which corresponds to the response of the scattering amplitudes to a small local perturbation δViadded to the Hamiltonian in Eq. ( 1). 245409-3COSMA, MUCHA-KRUCZY ´NSKI, SCHOMERUS, AND FAL’KO PHYSICAL REVIEW B 90, 245409 (2014) FIG. 2. (Color online) (a)–(i) Linear-response conductance as a function of chemical potential μat several fixed temperatures T,f o ra homogeneously strained GNR of width W/similarequal40 nm and aspect ratio L/W=3. The leads are doped via an on-site potential V=−200 meV . Each panel corresponds to a different value of externally imposed homogeneous strain. (a), (e), (h) Spatial structure of electron wave amplitudesat energy E F=−129.2 meV , evaluated using Eq. ( 10). A. Transport across homogeneously strained armchair GNRs Figure 2shows the numerically evaluated conductance, Eq. ( 9), for a GNR of width W/similarequal40 nm and aspect ratio L/W=3, as a function of chemical potential, for various values of homogeneous strain and temperature. The unstrainedGNR [Fig. 2(a)] is semiconducting with a gap of /similarequal30 meV , as determined by the quantization of the transverse momentum discussed above. The conductance exhibits two minima, atμ=−200 and 0 meV , and a local maximum at −100 meV . The conductance oscillations away from the two DPs aredue to the Fabry-P ´erot-like standing-wave resonances in the electron transmission across the potential barrier [ 19,53,54]. This can be seen in the LDOS profile shown in the inset, which we calculated using Eq. ( 10) at the resonance energy E F=−129.2m e V . For the homogeneously strained GNRs in Figs. 2(b)–2(i), the results show that the conductance continues to exhibit theminima at μ=−200 and 0 meV . The effect of the strain ismost noticeable in the energy range −200 meV <μ< 0m e V , where the system constitutes an n-p-njunction. For deformations w< 0.015 [Figs. 2(b) and 2(c)], the conductance in this range is broadly suppressed. This canbe attributed to the strain-induced shift of the DP, whichresults in a misalignment between the Fermi surfaces in theunstrained leads and the strained central region, as illustratedby the example in Fig. 1(c). Only the quantized momenta that cross the overlapping area of the two Fermi surfacescorrespond to propagating modes in the leads that coupleto propagating modes in the suspended region and thereforecontribute to transport. With increasing strain, the area ofthe overlap decreases, and the conductance is reduced as anincreasing number of conducting channels become blocked. For strain 0 .018/lessorequalslantw< 0.024 [Figs. 2(d)–2(h)] the con- ductance exhibits a series of well-defined resonances. Inthis range of strain, the area of the overlap between theFermi surfaces in the leads and in the central region isnarrower than the separation between neighboring quantized 245409-4STRAIN-INDUCED MODIFICATIONS OF TRANSPORT IN . . . PHYSICAL REVIEW B 90, 245409 (2014) momenta lines. For a fixed strain w, the width of the overlap remains constant with varying energy, but the overlap itself isshifted in the momentum plane along the k xaxis. Therefore, zero-conductance plateaus appear periodically in the range ofenergies when there is no quantized-momentum line crossingthe area of the overlap. In this case, the propagating modesin the central device only couple to evanescent modes inthe leads, leading to the formation of transport gaps in thesystem. The finite-conductance resonances are entirely due toFabry-P ´erot-like standing wave patterns, as illustrated by the LDOS profiles in Figs. 2(e)and2(h). For strains w/greaterorequalslant0.024 [Fig. 2(i)], the conductance in the range −200 meV <μ< 0 meV is completely suppressed, which results from the complete misalignment between theFermi surfaces in the two regions at such strong deformations[19]. This threshold for the insulating behavior is controlled by the parameters used in Fig. 2and can be lowered (raised) by reducing (increasing) the height of the potential step V between the central part of the ribbon and the contacts. The finite-conductance resonances are characteristic for junctions between regions of different polarity ( n-p-njunc- tions) and are absent in junctions between regions of the samepolarity ( n-n /prime-nandp-p/prime-pjunctions). This is because for μ<−200 meV and μ> 0 meV the region of overlap of the Fermi surfaces increases with increasing energy and containsan increasing number of quantized momentum lines. With larger strains ( w> 0.03) the two Fermi surfaces will only start overlapping at energies farther away from the DPs ( E F< −200 meV or EF>0 meV), which results in a widening of the transport gap in Fig. 2(i). For example, at w=0.05 we find that the conductance Gvanishes in the entire energy range |EF|/lessorequalslant100 meV around the DP of the suspended region. B. Influence of edge disorder in GNRs Ideal ribbons with perfectly cut edges are not realistic, as most fabricated structures present a certain degree of rough-ness at the edges [ 36,55–58]. Therefore, in this subsection we establish the robustness of the strain-induced conductanceresonances against edge defects. We introduce edge disorderby randomly removing a fraction fof individual atoms within a strip of width 2 rfrom the edges in the strained region (single-atom vacancies) [ 26,59–61] and compare this to the removal of carbon-carbon dimers in the outermost rows of theedges (double-atom vacancies) [ 59,62]. The missing atoms are modeled by setting all the nearest-neighbor hopping elementsγ ijto 0. Figure 3shows the effect on the conductance at a fixed temperature T=20 K for the homogeneously strained GNR of width W/similarequal40 nm and aspect ratio L/W=3, for several strains wand percentages fof single-atom and double-atom vacancies as indicated in each panel. Figure 3(a) shows that single-atom defects induce smearing and suppression of thefinite-conductance resonances. Previous studies have shownthat in the absence of strain, such edge disorder gives riseto drastic changes in the transport properties of armchairGNRs, by inducing large fluctuations in the conductance evenfor small percentages of defects. By breaking the sublatticesymmetry [ 59] and acting as short-range scatterers [ 60,61], such edge defects induce backscattering, Anderson-type FIG. 3. (Color online) Linear-response conductance Gas a func- tion of chemical potential μat a fixed temperature T=20 K, for homogeneously strained GNRs, of width W=40 nm and aspect ratioL/W=3, subjected to various types of disorder. (a) Effect of f=1% and 5% single-atom vacancies for strain w=0.02, 0.021, 0.022, and 0 .023; (b) corresponding effect of double-atom vacancies. (c) Conductance for various fractions of double-atom vacancies for a fixed strain w=0.024. localization, and even the formation of conduction gaps. Sim- ilarly, our calculations show that the conductance rapidly de-grades with increasing edge disorder, as an increasing numberof conductive paths become blocked. Compared to the resultsfor a defect-free system [Figs. 2(d)–2(h)], the conductance is already greatly reduced in the presence of f=1% edge disor- der, and the resonances become barely visible when f=5%. Double-atom edge defects, on the other hand, preserve the sublattice symmetry and therefore are expected to induce only small changes in the conductance [ 59]. This is confirmed by the results in Fig. 3(b). Compared to the defect-free ribbon [Figs. 2(d)–2(h)], the conductance for f=1% and f=5% disorder shows remarkably little changes. Even at higher degrees ofdisorder, the resonances are still visible. The most significanteffect is obtained for w=0.024 strain, depicted in Fig. 3(c), where we show the conductance calculated for various per- centages of edge disorder. In this case, the transport properties of the device observed in the ballistic regime are restored bylarge percentages of double-atom edge defects. This behaviorcan be understood by comparing the two theoretical disorderextremes: f=0% and f=100%. At f=0% (no edge disorder), the central device and leads are perfectly matched,both having width Wand the same transverse momentum 245409-5COSMA, MUCHA-KRUCZY ´NSKI, SCHOMERUS, AND FAL’KO PHYSICAL REVIEW B 90, 245409 (2014) FIG. 4. (Color online) (a) Suspended GNRs of width W/similarequal40 nm and aspect ratios L/W=2, 3, and 4, which are clamped at the highly doped contacts. The color code shows the strength of the pseudomagnetic fields B(in teslas) for electrons in the Kvalley, for inhomogeneous strainw=0.05. (b) Spatial structure of resonance states for selected resonances identified in Fig. 5. quantization. The requirement for conservation of transverse momenta leads to the complete suppression of the conductancesince the Fermi surfaces in the leads and the strained suspendedregion do not overlap. At f=100% edge disorder the outer- most rows of dimer lines at the top and bottom edges of the sus-pended region are completely removed. Therefore this regionhas a smaller width and correspondingly different quantized transverse momenta than the leads. This mismatch induces a mode mixing mechanism at the interfaces with the contacts,leading to the appearance of finite-conductance resonanceseven if the Fermi surfaces do not overlap. Other degrees ofedge disorder will induce a random mixture of local boundaryconditions at the edges [ 62] and therefore yield intermediate conductance results. Similar results were obtained by Ref.[20], where, using the continuum model, the authors showed that residual disorder restores a small finite conductivity. C. Transport across inhomogeneously strained armchair GNRs We now study the transport in suspended GNRs, which display inhomogeneous strain distributions. In contrast toclean and disordered homogeneously strained GNRs, wherethe conductance vanishes around the neutrality point of thesuspended part, we now find that the conductance featuresseveral additional resonances, including resonances close to the neutrality point [ 33]. Since previous works predict the formation of pseudomagnetic LLs in such systems [ 12,21,24], we aim to determine whether any of the observed new featuresin the conductance reflect this quantization of the electronicstates, without the addition of an external magnetic field asused in Ref. [ 17]. We focus our study on the energy range |E F|<100 meV around the DP in the suspended region, where, if present, the first few LLs are well resolved. Outsideof this energy range, the states are likely to be broadened andsmeared [ 12]. We consider three inhomogeneously strained ribbons, of width W/similarequal40 nm and aspect ratios L/W=2, 3, and 4. The pseudomagnetic field distributions for inhomogeneous tensilestrain w=0.05 are shown in Fig. 4(a).U s i n gE q .( 8), we calculate the zero-temperature conductance, which is shownin the left panels in Fig. 5. In contrast to the results obtained in the previous subsections, where the conductance wascompletely suppressed for homogeneous strains w/greaterorequalslant0.024, here we find four groups of sharp and clearly defined resonanceconductance peaks for each considered aspect ratio. The twogroups positioned farthest from the DP, at E F/similarequal−70 and /similarequal40 meV , contain several resonances, with their number being 245409-6STRAIN-INDUCED MODIFICATIONS OF TRANSPORT IN . . . PHYSICAL REVIEW B 90, 245409 (2014) FIG. 5. (Color online) Left: Zero-temperature conductance Gas a function of Fermi energy for the GNRs shown in Fig. 4. Top,L/W=2; middle, L/W=3; bottom, L/W=4. Right: Highly resolved results for the groups of peaks identified in the panels at the left. proportional to the aspect ratio of the respective ribbons. For the other two groups, positioned in the energy range−25 meV <E F<0 meV just below the DP, the highly resolved conductance results in the right panels in Fig. 5 reveal that these resonances always occur in pairs of two.Furthermore, the splitting of the two peaks in each groupdecreases with increasing aspect ratio. To uncover the origin of each group of peaks, we analyze the spatial distribution of the corresponding electronic states usingEq. ( 10) and arrive at the LDOS profiles shown in Fig. 4(b). As illustrated in the top two rows, the states away from theDP correspond to Fabry-P ´erot-like standing waves that form due to multiple electron reflections from the left and rightinterfaces. Similarly to the LDOS profiles in Fig. 2, such states are confined to the central part of the structure, where the straindistribution is approximately homogeneous. The inhomogene-ity near the contacts is still important, as it mixes states with dif-ferent transverse momenta and thus allows the charge carriersto overcome the misalignment of the Fermi surfaces describedin Sec. II. For the two groups in the energy range −25 meV < E F<0 meV , where the resonances occur in almost-degenerate pairs, the LDOS profiles shown in the bottom four rows in Fig.4(b) do, however, point towards a very different behavior. Un- like any of the resonances we have found up to now, the spatialstructure of these states clearly resembles the pseudomagneticfield distributions, which is an indicator of the formationof LLs. As demonstrated next, this quadruplet of resonances (two groups, each containing two conductance peaks) can beattributed to the n=0 pseudomagnetic LL induced by the inhomogeneity at the interfaces. We exploit a unique feature ofthis LL in armchair GNRs, namely, that the electron amplituderesides on either the Aor theBsublattice [ 12,15–17,22,23,63]. This sublattice polarization can be seen from the low-energyHamiltonian [ 1], H=v F/parenleftbigg 0 ˆπ† ˆπ 0/parenrightbigg ,ˆπ=ˆp+e cA, ˆp=px+ipy.(11) 245409-7COSMA, MUCHA-KRUCZY ´NSKI, SCHOMERUS, AND FAL’KO PHYSICAL REVIEW B 90, 245409 (2014) FIG. 6. (Color online) Sublattice-resolved electron amplitude for the resonances in Fig. 5(L/W=2, 3, and 4, at energies EF=−5.84, −6.57, and −23.83 meV respectively), obtained from Eq. ( 10) by placing the probing perturbation δViontoAsites (top) or onto Bsites (bottom). HerevFis the Fermi velocity, ˆpparameterizes the in-plane momentum relative to the KorK/primepoint, and Ais the vector potential in Eq. ( 4). The operator ˆπfulfills [ ˆπ,ˆπ†]=const and acts as an annihilation operator if the pseudomagnetic field ispositive. Acting by this Hamiltonian on the state /parenleftbigg |0/angbracketright 0/parenrightbigg , (12) where ˆπ|0/angbracketright=0, we obtain the eigenvalue E=0. This eigen- state has a finite amplitude on the Asublattice but a vanishing amplitude on the Bsublattice. For a negative value of the pseudomagnetic field the sublattice polarization moves onto theBsublattice. However, in all cases the selected sublattice is independent of the valley [ 12]. In contrast, higher order LLs and Fabry-P ´erot-like resonances occupy both sublattices equally [ 63]. By placing the probing perturbation δV iin Eq. ( 10) on either theAor theBsites, we find that the low-energy resonances are localized on the Asites near the left interface (where B<0) and on the Bsites near the right interface (where B>0). This is illustrated in Fig. 6. Further evidence that supports our interpretation of the origin of these states is the fact that we find four suchlow-energy resonances, with the separation between each pairinversely proportional to the aspect ratio of the ribbon. Thex→−xreflection symmetry of the system maps the Kand K /primevalleys onto each other, which results in the formation of a symmetric and an antisymmetric superposition of thetwo valley manifestations of the n=0 LL. This leads to a splitting of the n=0 LL into two branches, corresponding to each of the two groups of resonances. The branch located atE F≈−24 meV is valley symmetric and displays a maximum on the symmetry axis. The branch located at EF≈−7m e Vi s valley antisymmetric and displays a nodal line on the symmetryaxis. Two states appear in each of the branches due to thehybridization of states localized at the two contacts. The tunnelcoupling of these states is provided by the evanescent tails of the electronic wave functions in the central part of the systemwhere Bis small. Naturally, since the overlap of the evanescent tails decreases with increasing aspect ratio, the splitting in eachof these pairs is smaller in a longer ribbon, thus explaining thetrend we highlighted in our discussion of the highly resolvedconductance result in the right panels in Fig. 5. IV . CONCLUSION In conclusion, we performed a systematic study of the transport characteristics of homogeneously and inhomoge-neously strained suspended armchair GNRs. The combinationof strain-induced shifts of the DP in the momentum plane andsize-confinement effects leads to significant modifications inthe transport of homogeneously strained systems. In particular,an uncommon resonance structure appears when both of theseeffects compete. Large percentages of single-atom vacanciesdestroy the observed resonant structure. In contrast, “double-site” vacancies do not suppress the conductance and can evenrestore the ballistic transport properties. For inhomogeneousdeformations, we have found that the inhomogeneity devel-oped near the contacts aids the resonant transmission of chargecarriers, either through a mode mixing mechanism or throughtunneling via the sublattice-polarized n=0 pseudomagnetic LL. The mode mixing leads to the coupling to Fabry-P ´erot-like standing waves in the central part of the ribbon, which resultsin the formation of additional conductance peaks far from theDP. The states associated with the n=0 pseudomagnetic LL form near the contact regions and give rise to two pairs ofconductance peaks near the DP. ACKNOWLEDGMENTS We thank F. Guinea and H. Ochoa for useful discussions. 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PhysRevB.102.245402.pdf

PHYSICAL REVIEW B 102, 245402 (2020) Fano-Kondo resonance versus Kondo plateau in an Aharonov-Bohm ring with an embedded quantum dot Mikio Eto Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan Rui Sakano Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Received 10 September 2020; revised 4 November 2020; accepted 5 November 2020; published 1 December 2020) We theoretically examine the transport through an Aharonov-Bohm ring with an embedded quantum dot (QD), the so-called QD interferometer, to address two controversial issues regarding the shape of the Coulomb peaksand measurement of the transmission phase shift through a QD. We extend a previous model [B. R. Bulka and P.Stefa ´nski, P h y s .R e v .L e t t . 86, 5128 (2001) ; W. Hofstetter, J. König, and H. Schoeller, ibid.87, 156803 (2001) ]t o consider multiple conduction channels in two external leads, LandR. We introduce a parameter p α(|pα|/lessorequalslant1) to characterize a connection between the two arms of the ring through lead α(=L,R), which is the overlap integral between the conduction modes coupled to the two arms. First, we study the shape of a conductance peak as afunction of energy level in the QD, in the absence of electron-electron interaction U. We show an asymmetric Fano resonance for |p L,R|=1 in the case of single conduction channel in the leads and an almost symmetric Breit-Wigner resonance for |pL,R|<0.5 in the case of multiple channels. Second, the Kondo effect is taken into account by the Bethe ansatz exact solution in the presence of U. We precisely evaluate the conductance at temperature T=0 and show a crossover from an asymmetric Fano-Kondo resonance to the Kondo plateau with changing pL,R. Our model is also applicable to the multiterminal geometry of the QD interferometer. We discuss the measurement of the transmission phase shift through the QD in a three-terminal geometry by a “double-slitexperiment.” We derive an analytical expression for the relation between the measured value and the intrinsicvalue of the phase shift. DOI: 10.1103/PhysRevB.102.245402 I. INTRODUCTION In the mesoscopic physics, an Aharonov-Bohm (AB) ring with an embedded quantum dot (QD), the so-called QD in-terferometer, has been intensively studied to elucidate the coherent transport through a QD with discrete energy levels and strong Coulomb interaction [ 1–4]. Controversial issues still remain regarding the transport through the interferometerdespite long-term experimental and theoretical studies. Wetheoretically revisit these issues by generalizing a previousmodel to consider multiple conduction channels in externalleads and a multiterminal geometry. We first discuss the shape of Coulomb peaks, i.e., con- ductance Gas a function of gate voltage attached to the QD to control the energy levels electrostatically. Kobayashiet al. observed an asymmetric shape of the Coulomb peaks, which has a peak and dip in accordance with the Fanoresonance, using a QD interferometer [ 5]. The Fano res- onance stems from the interference between a discrete energy level in the QD and continuum energy states in the ring [ 6,7]. Remarkably the resonant shape of the Coulomb peaks changes with a magnetic flux penetrating the ring.However, the other groups observed symmetric Coulombpeaks, which can be fitted to the Lorentzian function ofBreit-Wigner resonance [ 8]. No criteria has been elucidatedregarding the Fano or Breit-Wigner resonance in the QD interferometer. The second issue concerns the measurement of the trans- mission phase shift through a QD using the QD interferometeras a double-slit experiment. It is well known that the phaseshift cannot be observed by the interferometer in the two-terminal geometry [ 1]. This is due to the restriction by the Onsager’s reciprocity theorem: Conductance Gsatisfies G(B)=G(−B) for magnetic field B,o rG(φ)=G(−φ)f o r the AB phase φ=2π/Phi1/ (h/e) with magnetic flux /Phi1penetrat- ing the ring [ 3,4]. The phase measurement was first reported using the interferometer in a four-terminal geometry [ 2]. In the Kondo regime, the phase shift through the QD shouldbe locked at π/2[9–11]. This phase locking was also inves- tigated experimentally using four- or three-terminal devices[8,12–17]. It is nontrivial, however, how precisely the phase shift is measured using the multiterminal interferometer. Theoretically, Bulka and Stefa ´nski studied Fano and Kondo resonances using a model for the two-terminal QD interfer-ometer, in which a QD is coupled to leads LandR and the leads are directly coupled to each other [ 18]. Hofstetter et al. found an asymmetric Fano-Kondo resonance by applying thenumerical renormalization group calculation to an equivalentmodel [ 19]. Their works were followed by many theoretical studies, e.g., to elucidate various aspects of the Kondo effect 2469-9950/2020/102(24)/245402(16) 245402-1 ©2020 American Physical SocietyMIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) FIG. 1. (a) Model for an AB ring with an embedded quantum dot (QD), the so-called QD interferometer, in the two-terminal geometry.The lower arm of the ring involves a QD with single energy level ε d, whereas the upper arm directly connects leads LandR. The tunnel couplings between the QD and leads, VL,k,VR,k/primeand that through the upper arm Wk/prime,k, depend on states kin lead Land state k/primein lead R. A magnetic flux /Phi1penetrating the ring is taken into account by the AB phase φ=2π/Phi1/ (h/e). The electron-electron interaction U works in the QD. (b) Model for the QD interferometer in a four- terminal geometry, with leads L(1),L(2),R(1), and R(2). State k[k/prime] belongs to lead L(1) or L(2) [R(1) or R(2)]. The chemical potentials in the leads are denoted by μ(1) L,μ(2) L,μ(1) R,a n dμ(2) R, respectively, in the formulation in Appendix A.W efi x μ(1) L=μ(2) L≡μLandμ(1) R= μ(2) R≡μRwithμL−μR=eVin our calculations. [20–29], fluctuation theorem [ 30], and dynamics of electronic states [ 31]. Recently, the Fano resonance was proposed to detect the Majorana bound states [ 32,33]. Although the model in Refs. [ 18,19] was widely used, it is insufficient to describe experimental situations with mul-tiple conduction channels in the leads. In the present paper,we propose an extended model for the QD interferometer toresolve the above-mentioned problems. As shown in Fig. 1(a), our model is the same as the previous model except the tunnelcouplings, V L,VR, and W, depend on the states in leads Land R. We show that the state dependence can be disregarded only in the case of single conduction channel in the leads. Our model yields a parameter pα(|pα|/lessorequalslant1) to char- acterize a connection between the two arms of the ringthrough lead α(=L,R), which is the overlap integral be- tween the conduction modes coupled to the upper and lowerarms of the ring. First, we examine the shape of a conduc-tance peak in the two-terminal geometry, in the absence ofelectron-electron interaction Uin the QD. We show an asym- metric Fano resonance for |p L,R|=1 in the case of single conduction channel in the leads and an almost symmetricBreit-Wigner resonance at |p L,R|<0.5 in the case of multiple channels. Hence our model could explain the experimentalresults of both the asymmetric Fano resonance [ 5] and almostsymmetric Breit-Wigner resonance [ 8], with fitting parame- tersp L,Rto their data. Second, the transport in the Kondo regime is examined by exploiting the Bethe ansatz exact solution. This methodprecisely gives us the conductance at temperature T=0i n the presence of U. We show a crossover from an asymmetric Fano-Kondo resonance [ 19] to the Kondo plateau with chang- ingp L,R. Our model is also applicable to the multiterminal geometry, where state k[k/prime] belongs to lead L(1) or L(2) [R(1) or R(2)], as depicted in Fig. 1(b). We discuss the measurement of the transmission phase shift through the QD by a “double-slitexperiment” using a three-terminal interferometer. We derivean analytical relation between the observed phase shift andintrinsic phase shift in the absence of U. Using a simple model to represent the experiment by Takada et al. [8,16,17], we evaluate the measured phase shift in both the absence andpresence of U.F o r U/negationslash=0, we show that the phase locking at π/2 is observable in the Kondo regime although it is slightly different from the behavior of the intrinsic phase shift thatsatisfies the Friedel sum rule. The organization of the present paper is as follows. In Sec. II, we present our model and calculation method. The parameters p LandpRare introduced, which are relevant to the shape of a conductance peak. We explain the calculationmethod of the conductance at T=0, taking into account the Kondo effect exactly. In Sec. III, the calculated results are given for the shape of the conductance peak in a two-terminalgeometry. We discuss the asymmetric Fano resonance versussymmetric Breit-Wigner resonance in the absence of U,b y changing p L,R. We also study the conductance in the Kondo regime in the presence of Uand show a crossover from an asymmetric Fano-Kondo resonance to the Kondo plateau.In Sec. IV, we examine the phase measurement in a three- terminal geometry by a double-slit interference experiment.We derive an analytical relation between the measured valueand intrinsic value for the transmission phase shift throughthe QD in the absence of U. Two specific models are studied to see a crossover from two- to three-terminal measurementand to simulate the experimental situation using two quantumwires to form the QD interferometer [ 8,16,17]. Section V is devoted to the discussion regarding the justification andgenerality of our model. The conclusions are given in Sec. VI. II. MODEL AND CALCULATION METHOD A. Model Let us consider a model for the QD interferometer in a two- terminal geometry, depicted in Fig. 1(a). The Hamiltonian is given by H=Hdot+Hleads+HT, (1) where Hdot=εd/summationdisplay σnσ+Un↑n↓, (2) Hleads=/summationdisplay α=L,R/summationdisplay kσεka† α,kσaα,kσ (3) 245402-2FANO-KONDO RESONANCE VERSUS KONDO PLATEAU IN … PHYSICAL REVIEW B 102, 245402 (2020) HT=/summationdisplay α=L,R/summationdisplay kσ(Vα,ka† α,kσdσ+H.c.) +/summationdisplay k,k/prime,σ(Wk/prime,keiφa† R,k/primeσaL,kσ+H.c.). (4) Here nσ=d† σdσis the number operator for electrons with spin σin the QD with energy level εd, where d† σanddσare creation and annihilation operators, respectively. a† α,kσandaα,kσare those for conduction electrons in lead α(=L,R) with state k and spin σwhose energy is denoted by εk.Uis the electron- electron interaction in the QD. The tunnel Hamiltonian HT connects the QD and state kin lead αbyVα,kthrough the lower arm of the ring, whereas it connects state kin lead Land state k/primein lead RbyWk/prime,kthrough the upper arm of the ring. The AB phase is defined by φ=2π/Phi1/ (h/e) for a magnetic flux/Phi1penetrating the ring. To make the calculation simple, we decompose Wk/prime,kinto the contributions from state kin lead Land state k/primein lead Ras Wk/prime,k=√wR,k/primewL,k. (5) This separable form is justified for the tight-binding models, as discussed in Sec. V. For lead α, we introduce the following three parameters to describe the contribution to the transport: /Gamma1α(ε)=π/summationdisplay k(Vα,k)2δ(ε−εk), (6) xα(ε)=π/summationdisplay kwα,kδ(ε−εk), (7) /radicalbig /Gamma1α(ε)xα(ε)pα(ε)=π/summationdisplay kVα,k√wα,kδ(ε−εk).(8) We assume that the ε-dependence of these parameters is weak around the Fermi level and simply express /Gamma1α,xα, and pαfor ε≈EF./Gamma1α(xα) characterizes the strength of tunnel coupling to the QD (coupling through the upper arm of the ring). Usingx=x LxR, the transmission probability through the upper arm of the ring is given by Tupper=4x (1+x)2. (9) Concerning xLandxR, the physical quantities are always writ- ten in terms of x=xLxRin our model [ 34]. The parameter pα(|pα|/lessorequalslant1) defined by Eq. ( 8) character- izes a connection between the two arms of the ring throughleadα(=L,R). Namely, p α(ε) is an overlap integral between the conduction mode coupled to the QD and that coupledto the upper arm of the ring in lead αat a given energy ε. The tunnel Hamiltonian H Tin Eq. ( 4) indicates that these modes are given by |ψQD α(ε)/angbracketright∝/summationtext kVα,k|α,k/angbracketrightδ(ε−εk) and |ψupper α(ε)/angbracketright∝/summationtext k√wα,k|α,k/angbracketrightδ(ε−εk), respectively, where |α,k/angbracketrightis the state kin lead α.F o r|ψ(ε)/angbracketright=/summationtext kCk|α,k/angbracketrightδ(ε− εk) and |ϕ(ε)/angbracketright=/summationtext kDk|α,k/angbracketrightδ(ε−εk), we denote the in- ner product by /angbracketleftψ(ε)|ϕ(ε/prime)/angbracketright=/angbracketleftψ|ϕ/angbracketrightεδ(ε−ε/prime), or/angbracketleftψ|ϕ/angbracketrightε=/summationtext kC∗ kDkδ(ε−εk). Then pα(ε)=/angbracketleftbig ψQD α/vextendsingle/vextendsingleψupper α/angbracketrightbig ε/radicalBig/angbracketleftbig ψQD α/vextendsingle/vextendsingleψQD α/angbracketrightbig ε/angbracketleftbig ψupper α/vextendsingle/vextendsingleψupper α/angbracketrightbig ε. (10)The interference by the AB effect is maximal when |pL|= |pR|=1, whereas it completely disappears when pL=0o r pR=0. In the previous model [ 18,19],|ψQD α/angbracketright=|ψupper α/angbracketrightand thus pα=1 since Vα,kand√wα,kare constant, irrespective of state k. As seen in the following sections, pLand pR play a crucial role in determining the shape of conductance peaks. Although pLandpRshould be given by the details of experimental systems, we treat them as parameters as well as/Gamma1 L,/Gamma1R, and x. As an example, let us consider quasi-one-dimensional leads, or leads of a quantum wire. The state in lead αis speci- fied by k=qin the case of single conduction channel and by k=(q,i) in the presence of multiple channels, where qis the momentum along the wire and iis the index of the subbands. In the former, Vα,k=Vα(εk) and wα,k=wα(εk), which yield /Gamma1α(ε)=πρα(ε)[Vα(ε)]2with density of states ραin the lead, xα(ε)=πρα(ε)wα(ε), and |pα|=1f r o mE q s .( 6)t o( 8). In the case of multiple channels, |pα|<1, as shown in Sec. V. Note that a similar parameter to pαwas introduced in the study on a double quantum dot in parallel and was evaluated forthree- or two-dimensional leads with a flat surface [ 35]. For the multiterminal geometry, lead αis divided into leads α(1) and α(2), as depicted in Fig. 1(b). The Hamiltonian in Eq. ( 1) is applicable even to this case, in which the summation over kis taken in both lead α(1) [denoted by/summationtext (1) k] and leadα(2) [by/summationtext(2) k]. We define /Gamma1(j) αusing Eq. ( 6) with the summation over kin lead α(j) only /Gamma1(j) α(ε)=π(j)/summationdisplay k|Vα,k|2δ(ε−εk) (11) forα=L,Randj=1, 2. Similarly, we define x(1) α,x(2) α,p(1) α, andp(2) α. They satisfy the following relations: /Gamma1α=/Gamma1(1) α+/Gamma1(2) α, (12) xα=x(1) α+x(2) α, (13) /radicalbig /Gamma1αxαpα=/radicalBig /Gamma1(1) αx(1) αp(1) α+/radicalBig /Gamma1(2) αx(2) αp(2) α. (14) B. Formulation of electric current We formulate the electric current using the Keldysh Green’s functions along the lines of Ref. [ 19] (see Appendix A). For example, the current from lead L(1) in Fig. 1(b) is given by I(1) L=−e/angbracketleftbig˙N(1) L/angbracketrightbig =−e i¯h/angbracketleftbig/bracketleftbig N(1) L,H/bracketrightbig/angbracketrightbig , (15) where N(1) L=(1)/summationdisplay kσa† L,kσaL,kσ (16) is the number operator for electrons in the lead. In the sta- tionary state, I(1) Lis expressed in terms of the retarded Green’s function Gr d,d(ε) and lesser Green’s function G< d,d(ε)o ft h e QD, in Eq. ( A22) in Appendix A. Next, we eliminate G< d,d(ε) from the expression and write the current using Gr d,d(ε) only. We restrict ourselves to the 245402-3MIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) case of μ(1) L=μ(2) L≡μL,μ(1) R=μ(2) R≡μR, (17)withμL−μR=eV, to simplify the current expression. Then the current conservation is written as follows in the stationarystate: 0=I(1) L+I(2) L+I(1) R+I(2) R =4e h/integraldisplay dε/braceleftBigg −˜/Gamma1/bracketleftbig [fL(ε)+fR(ε)]ImGr d,d(ε)+ImG< d,d(ε)/bracketrightbig +[fL(ε)−fR(ε)]/bracketleftbigg −(/Gamma1L−/Gamma1R)−4√/Gamma1L/Gamma1RxpLpR (1+x)2sinφ+x(x+3) (1+x)2/parenleftbig /Gamma1Lp2 L−/Gamma1Rp2 R/parenrightbig/bracketrightbigg ImGr d,d(ε)/bracerightBigg , (18) where fα(ε)=[(ε−μα)/(kBT)+1]−1is the Fermi distribution function in lead α(1) or α(2) [ ˜/Gamma1will be given in Eq. ( 24)]. Using Eq. ( 18), we eliminate G< d,d(ε) from the current expression, e.g., Eq. ( A22)f o r I(1) L. C. Electric current in two-terminal systems Now we present the expression for the electric current in the two-terminal systems. The current from the left lead IL[= I(1) L+I(2) Lusing the results in Appendix A] is expressed as IL=2e h/integraldisplay dε[fL(ε)−fR(ε)]T(ε), (19) with the transmission probability T(ε)=4x (1+x)2+81−x (1+x)3/radicalbig /Gamma1L/Gamma1RxpLpRcosφReGr d,d(ε)+4C1 (1+x)3˜/Gamma1ImGr d,d(ε). (20) Here, the coefficient C1is given by C1=x3 1+x/bracketleftbig/parenleftbig /Gamma1Lp2 L/parenrightbig2+/parenleftbig /Gamma1Rp2 R/parenrightbig2/bracketrightbig +x(1−x)[(/Gamma1LpL)2+(/Gamma1RpR)2] −/Gamma1L/Gamma1R/bracketleftBigg (1+x)3+4x 1+x(pLpR)2sin2φ+x2(x2+4x+9) 1+x(pLpR)2−x(x2+3x+4)/parenleftbig p2 L+p2 R/parenrightbig/bracketrightBigg . (21) Note that (i) for pL=pR=1, where a single conduction channel is effective in each lead, Eq. ( 20) coincides with the current expression derived in Ref. [ 19]. (ii) For pL=pR=0, the transmission probability is given by T(ε)=4x (1+x)2−4/Gamma1L/Gamma1R /Gamma1L+/Gamma1RImGr d,d(ε). (22) This is the summation of the transmission probability through the upper arm, Tupper in Eq. ( 9), and that through the QD, indicating no interference effect between the two paths in theQD interferometer. For multiterminal systems, the current is expressed in terms of the retarded Green’s function G r d,d(ε) in a similar way. The expression is given in Eqs. ( A29) and ( A30)i n Appendix A. D. Exact calculation for Kondo effect In the absence of Coulomb interaction, U=0, the re- tarded Green’s function of the QD is given by Gr d,d(ε)=1/ (ε−εd−/Sigma1d), where the self-energy by the tunnel couplings is /Sigma1d=−2√/Gamma1L/Gamma1Rx 1+xpLpRcosφ−i˜/Gamma1, (23)with an effective linewidth ˜/Gamma1=/Gamma1L/parenleftbigg 1−x 1+xp2 L/parenrightbigg +/Gamma1R/parenleftbigg 1−x 1+xp2 R/parenrightbigg . (24) This expression is common to two- and multiterminal sys- tems. In the presence of U,Gr d,d(ε) is evaluated exactly in the following way. The Green’s function at U=0 indicates that our models are equivalent to the situation in which a QD withan energy level ˜ε d(φ)=εd−2√/Gamma1L/Gamma1Rx 1+xpLpRcosφ (25) is connected to a lead with linewidth ˜/Gamma1,a ss h o w ni n Appendix B. In the Fermi liquid theory, the Green’s function is written as Gr d,d(0)=z −˜ε∗ d+iz˜/Gamma1=˜/Gamma1∗ ˜/Gamma11 −˜ε∗ d+i˜/Gamma1∗, (26) atε=EF=0, where ˜ ε∗ dis the renormalized value of ˜ εd(φ) in Eq. ( 25),˜/Gamma1∗=z˜/Gamma1is that of ˜/Gamma1in Eq. ( 24), and zis a fac- tor of wave-function renormalization by the electron-electroninteraction U[36–38]. Since the phase shift θ QDat the QD is given by tan θQD=˜/Gamma1∗/˜ε∗ d, the Green’s function is determined 245402-4FANO-KONDO RESONANCE VERSUS KONDO PLATEAU IN … PHYSICAL REVIEW B 102, 245402 (2020) byθQDas Gr d,d(0)=−1 ˜/Gamma1eiθQDsinθQD. (27) θQDis related to the electron number per spin in the QD through the Friedel sum rule, θQD=π/angbracketleftnσ/angbracketright./angbracketleftnσ/angbracketrightis evaluated at temperature T=0 using the Bethe ansatz exact solution [39,40]. Hence we can precisely calculate Gr d,d(0) and thus the conductance G=dIL/dV(V→0) at T=0 in the presence ofU. It is worth mentioning that the effective energy level ˜ εd(φ) in the QD gives rise to the φ-dependent Kondo temperature [20]. It is written as kBTK(φ)≈D/bracketleftbigg˜/Gamma1U |˜εd(φ)|[˜εd(φ)+U]/bracketrightbigg1/2 e−π|˜εd(φ)|[˜εd(φ)+U]/(2˜/Gamma1U), (28) with Dbeing the bandwidth [ 36,41] although TK(φ) is irrele- vant to our study on the transport properties at T=0. III. CALCULATED RESULTS IN TWO-TERMINAL GEOMETRY In this section, we present the calculated results for the two-terminal system, paying attention to the shape of a con-ductance peak as a function of energy level ε din the QD. We find that parameters pLandpRare relevant in both the cases ofU=0 and U/negationslash=0. A. Fano versus Breit-Wigner resonance We begin with the case of no electron-electron interaction in the QD, U=0. Figure 2shows the conductance GatT=0 as a function of energy level εdin the QD for (a) pL=pR=1, (b) 0.75, and (c) 0.5. The AB phase is φ=0 (solid line), ±π/2 (broken line), and π(dotted line). G(φ)=G(−φ) holds by the Onsager’s reciprocal theorem. In Fig. 2(a) with pL=pR=1, the conductance Gshows an asymmetric resonant shape with dip and peak in the ab-sence of magnetic field ( φ=0). This is known as the Fano resonance, which is ascribable to the interference between thetunneling through a discrete level and that through continuousstates [ 6,7]. A magnetic field changes the resonant shape to be symmetric at φ=±π/2 and asymmetric with peak and dip atφ=π. This Fano resonance is characterized by a complex Fano factor [ 5]. Indeed the conductance can be analytically expressed [ 42] in the form of G=2e 2 h4x (1+x)2|e+q|2 e2+1(29) with e=[εd−˜εd(φ)]/˜/Gamma1, where ˜ εd(φ)=εd−2√/Gamma1L/Gamma1Rxcosφ/ (1+x) and ˜/Gamma1=(/Gamma1L+/Gamma1R)/(1+x)[ E q s .( 25) and ( 24)f o r pL=pR=1]. The complex Fano factor is given by q=√/Gamma1L/Gamma1R ˜/Gamma1√x/parenleftbigg1−x 1+xcosφ−isinφ/parenrightbigg . (30) With a decrease in pLand pR, the conductance peak becomes more symmetric and its φ-dependence is less prominent, as shown in Figs. 2(b) and 2(c). The shape ofFIG. 2. Calculated results for the conductance Gin the two- terminal system in the absence of U.Gat temperature T=0i s plotted as a function of energy level εdin the quantum dot. /Gamma1L= /Gamma1R=/Gamma1/2,x=0.09 (xL=xR=0.3), and (a) pL=pR=1, (b) 0.75, and (c) 0.5. The AB phase for the magnetic flux penetrating the ring isφ=0 (solid line), φ=±π/2 (broken line), and φ=π(dotted line). conductance peak is closer to that of the Lorentzian function of Breit-Wigner resonance as pLandpRgo to zero. Note that the conductance Gcan exceed unity in units of 2e2/hwhen pL,pR<1, reflecting the multiple conduction channels in the leads. See Eq. ( 22) in the limit of pL=pR= 0: The upper limit of G/(2e2/h) is the sum of the transmis- sion probability through the QD (unity if /Gamma1L=/Gamma1R) and that through the upper arm, Tupper in Eq. ( 9). B. Fano-Kondo resonance versus Kondo plateau In the presence of U, the Kondo effect is exactly taken into account in the evaluation of the conductance at T=0, as described in the previous section. In Fig. 3, the conductance Gis shown as a function of energy level εdin the QD, for U//Gamma1=8 and/Gamma1L=/Gamma1R=/Gamma1/2; (a) pL=pR=1, (b) 0.75, and 245402-5MIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) FIG. 3. Calculated results for the conductance Gin the two- terminal system in the presence of U.Gat temperature T=0i s plotted as a function of energy level εdin the quantum dot. /Gamma1L= /Gamma1R=/Gamma1/2,x=0.09 (xL=xR=0.3), and (a) pL=pR=1, (b) 0.75, and (c) 0.5. U=8/Gamma1. The AB phase for the magnetic flux penetrating the ring is φ=0 (solid line), φ=±π/2 (broken line), and φ=π (dotted line). (c) 0.5. The AB phase is φ=0 (solid line), ±π/2 (broken line), and π(dotted line). ForpL=pR=1,Gbehaves as a “Fano-Kondo resonance” proposed by Hofstetter et al. [19], which stems from an in- terplay between the Kondo resonance ( G∼2e2/hat−U< εd<0) and the Fano resonance. When φ=0(π),Gshows a dip and peak (peak and dip) with a gradual slope aroundthe center of the Kondo valley, i.e., Coulomb blockade regimew i t has p i n1 /2 in the QD. When φ=π/2,Gis almost constant at 2 e 2/hin the Kondo valley and symmetric with respect to the valley center. With decreasing pLandpR, the asymmetric shape of the Fano-Kondo resonance changes to a conductance plateau, theso-called Kondo plateau, in the Kondo valley: G→2e 2/h+ Tupper aspL,R→0 when /Gamma1L=/Gamma1R. Besides, Gis less depen- dent on φ.IV . CALCULATED RESULTS IN THREE-TERMINAL GEOMETRY In this section, we examine a three-terminal system to discuss the measurement of transmission phase shift throughthe QD by a “double-slit interference experiment.” We assumetwo leads R(1) and R(2) on the right side and a single lead L on the left side in Fig. 1(b). We evaluate the conductance from leadLtoR(1) or to R(2), G (1)=−dI(1) R dV,G(2)=−dI(2) R dV, (31) foreV=μL−μR→0(μ(1) R=μ(2) R=μR)a t T=0, as a function of AB phase φ. We define the measured phase shift by the AB phase φmaxat which the conductance G(1)(φ)i s maximal. As an intrinsic transmission phase shift through the QD, we introduce θ(0) QDandθQDby tanθ(0) QD=/Gamma1L+/Gamma1R εd−EF, (32) tanθQD=˜/Gamma1 ˜εd(φ)−EF, (33) respectively, in the absence of U.θ(0) QDis the phase shift through the QD without the upper arm of the ring, whereasθ QDsatisfies the Friedel sum rule θQD=π/angbracketleftnσ/angbracketrightfor the QD embedded in the ring. This last depends on the AB phase φfor the magnetic flux penetrating the ring. In the next subsection,we derive an analytical relation between the measured phaseφ maxandθ(0) QDin Eq. ( 32) in the absence of U. In Secs. IV .B and C, we examine two specific models depicted in Fig. 4.I nF i g . 4(a), leads LandR(1) are connected FIG. 4. Two specific models for the three-terminal system. (a) Leads LandR(1) are connected to both the quantum dot and upper arm of the ring, whereas lead R(2) is connected to the quantum dot only. (b) Leads R(1) and R(2) are quantum wires which are tunnel-coupled to each other at the hatched region. Lead Lis con- nected to both the quantum dot and upper arm of the ring, whereas lead R(1) [R(2)] is connected to the quantum dot [upper arm of the ring] only at the end of the leads. 245402-6FANO-KONDO RESONANCE VERSUS KONDO PLATEAU IN … PHYSICAL REVIEW B 102, 245402 (2020) to both the QD and upper arm of the ring, whereas lead R(2) is connected to the QD only. We vary the strength of tunnel cou-p l i n gt ol e a d R(2),/Gamma1 (2) R, to investigate a crossover from two- to three-terminal phase measurement. In Fig. 4(b), we model the experimental situation by Takada et al. , in which leads R(1) andR(2) are partly coupled quantum wires [ 8,16,17]. A. Measured phase shift for U=0 For the three-terminal model in Fig. 1(b) with leads L, R(1), and R(2), we introduce the following dimensionless parameters: γ(j) R=/Gamma1(j) R /Gamma1R,y(j) R=x(j) R xR,q(j) R=/radicalBig /Gamma1(j) Rx(j) Rp(j) R√/Gamma1RxRpR(34) forj=1 and 2. They are the ratios of contribution from lead R(j)t o/Gamma1R,xR, and√/Gamma1RxRpR, respectively, and satisfy the relations of γ(1) R+γ(2) R=y(1) R+y(2) R=q(1) R+q(2) R=1. In the absence of U,E q s .( A29) and ( A30) yield the con- ductance in the form of G(1)=2e2 h1 [EF−˜εd(φ)]2+˜/Gamma12 ×/bracketleftbigg8√/Gamma1L/Gamma1RxpLpR (1+x)2F(φ)+(φ-indep. terms)/bracketrightbigg ,(35) where F(φ)=q(1) R(ε−εd) cosφ+/bracketleftbig x/parenleftbig q(1) R−y(1) R/parenrightbig /Gamma1L/parenleftbig 1−p2 L/parenrightbig +/parenleftbig γ(1) R−q(1) R/parenrightbig /Gamma1R/bracketrightbig sinφ. (36) If we neglect the φ-dependence in ˜ εd(φ) in the denominator in Eq. ( 35), the measured phase φmaxis given by tanφmax=x/parenleftbig y(1) R−q(1) R/parenrightbig /Gamma1L(1−p2 L)+/parenleftbig q(1) R−γ(1) R/parenrightbig /Gamma1R q(1) R(/Gamma1L+/Gamma1R) ×tanθ(0) QD, (37) where θ(0) QDis defined in Eq. ( 32). This is an approximate for- mula for the relation between the measured value and intrinsicvalue of the transmission phase shift through the QD. In the two-terminal geometry, lead R(2) is absent and thus γ (1) R=y(1) R=q(1) R=1. Then Eq. ( 37) yields tan φmax=0, i.e., φmax=0o rπin accordance with the Onsager’s reciprocal theorem. B. Model in Fig. 4(a)withU=0 To elucidate a crossover from two- to three-terminal measurement of the transmission phase shift through the QD,we examine the model depicted in Fig. 4(a) with U=0. In this model, leads LandR(1) are connected to both the QD and upper arm of the ring, whereas lead R(2) is connected to the QD only. From x (2) R=0 and/Gamma1R=/Gamma1(1) R+/Gamma1(2) R, dimensionless parameters in the previous subsection become γ(1) R=/Gamma1(1) R//Gamma1R andy(1) R=q(1) R=1. In the QD, the effective energy level and linewidth are ˜ εd(φ)=εd−2/radicalBig /Gamma1L/Gamma1(1) RxpLp(1) Rcosφ/(1+x) and ˜/Gamma1=/Gamma1L[1−xp2 L/(1+x)]+/Gamma1(1) R[1−xp(1)2 R/(1+x)]+/Gamma1(2) R,FIG. 5. Calculated results for the three-terminal model depicted in Fig. 4(a) in the absence of U. In the left panels, the conductance G(1)to lead R(1) at temperature T=0 is plotted as a function of energy level εdin the quantum dot. /Gamma1L=/Gamma1R=/Gamma1/2,x=0.09 (xL=xR=0.3), and pL=pR=0.5. The tunnel coupling to lead R(2) is increased from (a) to (c): (a) /Gamma1(2) R//Gamma1R=0.2, (b) 0.5, and (c) 0.8 with /Gamma1(1) R+/Gamma1(2) R=/Gamma1R. The AB phase for the magnetic flux penetrating the ring is φ=0 (solid line), φ=π/2 (broken line), φ=π(dotted line), and φ=−π/2 (thin solid line). In the right panels, the measured phase shift φmaxis plotted as a function of εd (solid line), which is numerically evaluated as the AB phase when G(1)(φ) is maximal. φmaxgiven by the formula in Eq. ( 38) is plotted by broken line, whereas the transmission phase shift θ(0) QDthrough the quantum dot without the upper arm of the ring is plotted by dotted line. respectively. Equation ( 37) yields an approximate relation of tanφmax=/Gamma1(2) R /Gamma1L+/Gamma1(1) R+/Gamma1(2) Rtanθ(0) QD, (38) which indicates that the measured phase shift φmaxapproaches the intrinsic phase shift θ(0) QDwith an increase in /Gamma1(2) R. Figure 5presents the calculated results for the model in Fig. 4(a). In the left panels, the conductance G(1)to lead R(1) at T=0 is plotted as a function of energy level εd in the QD. /Gamma1L=/Gamma1R=/Gamma1/2 and (a) /Gamma1(2) R//Gamma1R=0.2, (b) 0.5, and (c) 0.8. For small /Gamma1(2) R//Gamma1R[Fig. 5(a)],G(1)is almost t h es a m ea t φ=±π/2 corresponding to the Onsager’s re- ciprocal theorem in the two-terminal system. With increasing/Gamma1 (2) R//Gamma1R[Figs. 5(b) and5(c)], the deviation from the theorem becomes more prominent. The peak height of G(1)is reduced by stronger tunnel coupling to lead R(2). The right panels in Fig. 5showφmaxthat is numerically evaluated from G(1)(φ), as a function of εd(solid lines). The intrinsic phase shift θ(0) QDin Eq. ( 32) is plotted by dotted lines. Broken lines show φmaxin Eq. ( 38), indicating that the formula is a good approximation to estimate φmaxfrom 245402-7MIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) FIG. 6. Measured phase shift φmaxas a function of energy level εdin the quantum dot, in the three-terminal model depicted in Fig.4(a)in the absence of U. The data for φmaxare the same as in the right panels in Fig. 5./Gamma1(2) R//Gamma1R=0.2 (solid line), 0.5 (dotted line), and 0.8 (broken line). A thin solid line indicates the transmission phase shift θ(0) QDthrough the quantum dot without the upper arm of the ring. θ(0) QD.I nF i g . 5(a) with/Gamma1(2) R//Gamma1R=0.2,φmaxchanges almost abruptly from zero to πaround εd=EF=0, which is close to the behavior in the two-terminal system. For larger /Gamma1(2) R//Gamma1R, φmaxchanges more gradually with εdand closer to the in- trinsic phase shift θ(0) QDalthough φmaxdoes not go to θ(0) QDas /Gamma1(2) R//Gamma1R→1 under the condition of /Gamma1L=/Gamma1R. To illustrate the crossover from the two- to three-terminal phase measurement, we replot φmaxfor three values of /Gamma1(2) R//Gamma1Rin a graph in Fig. 6. C. Model in Fig. 4(b) Now we study the model shown in Fig. 4(b) to examine the experimental situation using partly coupled quantum wires toform a mesoscopic ring [ 8,16,17]. We assume that leads R(1) andR(2) consist of two equivalent wires aandbof single conduction channel. They are tunnel-coupled to each other inthe vicinity of their edges, which mixes states |a,k /prime/angbracketrightin lead a and|b,k/prime/angbracketrightin lead b. As a result, the edge states in leads R(1) andR(2) are given by /vextendsingle/vextendsingleψ(1) Rk/prime/angbracketrightbig =αR|a,k/prime/angbracketright+βR|b,k/prime/angbracketright, (39) /vextendsingle/vextendsingleψ(2) Rk/prime/angbracketrightbig =βR|a,k/prime/angbracketright−αR|b,k/prime/angbracketright, (40) respectively, with real coefficients αRandβR(α2 R+β2 R=1). Far from the edges, |ψ(1) Rk/prime/angbracketright→| a,k/prime/angbracketrightin lead R(1) and |ψ(2) Rk/prime/angbracketright→| b,k/prime/angbracketrightin lead R(2) in an asymptotic way. As shown in Fig. 4(b),|ψ(1) Rk/prime/angbracketrightin Eq. ( 39) is coupled to the QD while |ψ(2) Rk/prime/angbracketrightin Eq. ( 40) is connected to the upper arm of the ring. In the tunnel Hamiltonian HTin Eq. ( 4), VR,k/prime=VRαRand√wR,k/prime=√wRβRwhen state k/primebelongs to lead R(1) while VR,k/prime=VRβRand√wR,k/prime=−√wRαRwhen state k/primebelongs to lead R(2). Thus /Gamma1(1) R=α2 R/Gamma1R,/Gamma1(2) R=β2 R/Gamma1R, x(1) R=β2 RxR, and x(2) R=α2 RxR. In this model, pR=0(p(1) R=1,p(2) R=−1) as explained in Appendix Cand in consequence ˜ εd(φ)=εdin Eq. ( 25). ForU=0, Eq. ( 37) exactly holds, which yields tanφmax=−x/Gamma1L/parenleftbig 1−p2 L/parenrightbig +/Gamma1R /Gamma1L+/Gamma1Rtanθ(0) QD. (41) In addition, the phase shift θQDcan be defined independently ofφ, which satisfies the Friedel sum rule in the QD embedded in the ring [see Eq. ( 33) in the case of U=0]. For both U=0 andU/negationslash=0, we obtain an exact relation of tanφmax=−x/Gamma1L/parenleftbig 1−p2 L/parenrightbig +/Gamma1R ˜/Gamma1tanθQD, (42) with ˜/Gamma1=/Gamma1L[1−xp2 L/(1+x)]+/Gamma1R.t a nθQD=˜/Gamma1/ε din the absence of Uand tan θQD=˜/Gamma1∗/˜ε∗ din the presence of Uwhen EF=0. Neither Eq. ( 41) nor Eq. ( 42) depend on αRandβR. We show the calculated results for U=0i nF i g . 7.I n Figs. 7(a) and7(b), the conductance G(1)is shown as a func- tion of energy level εdin the QD, for (a) β2 R=0.1 and (b) 0.5. The height of G(1)depends on φmore largely in Fig. 7(b) than in Fig. 7(a)though φmaxdoes not depend on βR. Figure 7(c) plotsφmaxthat is numerically evaluated from G(1)(φ). It changes smoothly from zero to πviaπ/2a tεd= 0.φmaxquantitatively deviates from θ(0) QDandθQD(dotted and thin solid lines). Their relations are exactly given by Eqs. ( 41) and ( 42). It should be mentioned that the sum of the currents to leads R(1) and R(2),I(1) R+I(2) R, does not depend on the AB phase φ, reflecting pR=0 in this model (see Appendix C). Therefore, the AB oscillation of G(1)(φ) is out-of-phase to that ofG(2)(φ), as indicated in the insets in Fig. 7.φmaxevaluated from G(1)behaves similarly to θ(0) QD, while that from G(2)sim- ilarly to −θ(0) QD, irrespective of the absence or presence of U. This agrees with the experimental observation by Takada et al. [8,16]. Finally, the measured phase is discussed in the Kondo regime with U/negationslash=0. In Fig. 8,w ep l o t φmaxthat is numerically evaluated from G(1), as a function of energy level εdin the QD; (a)U//Gamma1=8 and (b) 16 with /Gamma1L=/Gamma1R=/Gamma1/2. In the Kondo valley ( −U<ε d<0), the phase locking at π/2 is observable by a “double-slit experiment” using the QD interferometer.We calculate the intrinsic phase shift θ QDusing the Friedel sum rule θQD=π/angbracketleftnσ/angbracketright, where /angbracketleftnσ/angbracketrightis given by the Bethe ansatz exact solution (dotted line). φmaxandθQDare related to each other by Eq. ( 42). The phase locking seems smeared in the curve of the measured phase shift φmax, in comparison with the intrinsic phase shift θQD. V . DISCUSSION In our models shown in Figs. 1(a) and1(b), we assume a separable form for the tunnel coupling between the leadsin Eq. ( 5). Here, we discuss the justification of this form using a tight-binding model. We also show that |p α|<1i n the presence of multiple conduction channels in lead α. As a simple example, let us consider the model depicted in Fig. 9(a). The leads consist of two sites in width and N sites in length ( N/greatermuch1). The eigenvalues of the Hamiltonian for leads LandRform two subbands ε±(q), where qis the 245402-8FANO-KONDO RESONANCE VERSUS KONDO PLATEAU IN … PHYSICAL REVIEW B 102, 245402 (2020) FIG. 7. Calculated results for the conductance and measured phase shift in the three-terminal model depicted in Fig. 4(b) in the absence of U. In the upper two panels, the conductance G(1)to lead R(1) at temperature T=0 is plotted as a function of energy level εd in the quantum dot. /Gamma1L=/Gamma1R=/Gamma1/2,x=0.09 (xL=xR=0.3), and pL=0.5. (a) β2 R=0.1, (b) 0.5 ( α2 R+β2 R=1;/Gamma1(1) R=α2 R/Gamma1R,/Gamma1(2) R= β2 R/Gamma1R,x(1) R=β2 RxR,a n d x(2) R=α2 RxR, see text). The AB phase for the magnetic flux penetrating the ring is φ=0 (solid line), φ=π/2 (broken line), φ=π(dotted line), and φ=−π/2 (thin solid line). In panel (c), the measured phase shift φmaxis plotted as a function ofεd(solid line), which is defined by the AB phase when G(1)(φ)i s maximal. φmaxdoes not depend on βR. The phase shift θ(0) QDthrough the QD without the upper arm of the ring is plotted by dotted line, which is almost overlapped by θQD(thin solid line) that satisfies the Friedel sum rule in the QD embedded in the ring. Insets in panels(a) and (b): G (1)andG(2)[conductance to lead R(2)] as a function of the AB phase φ,a tεd=0. wave number in the xdirection (0 <q<π / a) with abeing the lattice constant [Fig. 9(b)]. The corresponding states are |L;q,±/angbracketright =−1√N+1−1/summationdisplay j=−N(|j,1/angbracketright±| j,2/angbracketright)sinqja,(43) |R;q,±/angbracketright =1√N+1N/summationdisplay j=1(|j,1/angbracketright±| j,2/angbracketright)sinqja,(44) where |j,/lscript/angbracketrightis the Wannier function at site ( j,/lscript). The tun- nel coupling between |L;q,γ/angbracketrightand|R;q/prime,γ/prime/angbracketright(γ,γ/prime=±)FIG. 8. Calculated results for the measured phase shift in the three-terminal model depicted in Fig. 4(b) in the presence of U.T h e measured phase shift φmaxis plotted by solid line as a function of en- ergy level εdin the quantum dot. φmaxis numerically evaluated as the AB phase at which the conductance G(1)(φ) to lead R(1) is maximal at temperature T=0./Gamma1L=/Gamma1R=/Gamma1/2,x=0.09 (xL=xR=0.3), and pL=0.5. (a) U//Gamma1=8 and (b) 16. θQDcalculated from the Friedel sum rule, θQD=π/angbracketleftnσ/angbracketright, is plotted by dotted line. φmaxand θQDare related to each other by Eq. ( 42). is expressed as Wq/prime,γ/prime;q,γ=ψR;q/prime,γ/prime(1,2)WψL;q,γ(−1,2) using the wave functions at the edge of the leads, ψL;q,±(−1,2)= /angbracketleft−1,2|L;q,±/angbracketrightandψR;q/prime,±(1,2)=/angbracketleft1,2|R;q/prime,±/angbracketright. In conse- quence Wq/prime,γ/prime;q,γhas a separable form in Eq. ( 5) with √wL;q,γ=√ WψL;q,γ(−1,2), (45) √wR;q/prime,γ/prime=√ WψR;q/prime,γ/prime(1,2). (46) When the Fermi level intersects both the subbands, there are two conduction channels, labeled by k=(q,±), as indi- cated in Fig. 9(b). Then pL=pR=sinq+a−sinq−a sinq+a+sinq−a, (47) where q±are the intersections between the subband ±and Fermi level, as derived in Appendix D. Thus |pL,R|<1. On the other hand, pL,R=±1, in the case of single conduction channel when EFcrosses one of the subbands. Although we considered a specific model in Fig. 9(a),t h e separable form of Wk/prime,kin Eq. ( 5) should be justified when the system is described by a tight-binding model in general.Then√ wL,k(√wR,k/prime) is proportional to the wave function 245402-9MIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) FIG. 9. (a) A tight-binding model for the QD interferometer. A QD is connected to sites ( −1,1) and (1,1) by transfer integrals VL andVR, respectively, whereas the upper arm of the ring couples sites (−1,2) and (1,2) by We±iφin±xdirection, where φis the AB phase for the magnetic flux penetrating the ring (central rectangular region including the QD). Leads LandRconsist of two sites in width ( y direction) and Nsites in length ( xdirection; N/greatermuch1), in which the transfer integral is −t(−t1)i nt h e x(y) direction and the lattice con- stant is a. (b) Two subbands in the leads, ε±(q)=∓t1−2tcosqa,a s a function of wave number qin the xdirection (0 <q<π / a). There are two conduction channels when the Fermi level EFintersects both the subbands at q=q±. ψL,k(ψR,k/prime) at the edge of the lead, as in Eqs. ( 45) and ( 46). We could also claim that pL,R<1 for the leads of multiple conduction channels and pL,R=1 for the leads of single channel in the usual cases. Precisely speaking, the presenceof multiple channels is a necessary condition for p L,R<1:pα is determined by the detailed shape of the system around a junction between the ring and lead αthrough Eq. ( 10). We comment on the generality of our models. In this sec- tion, we examined a model in which the subbands ( ±)a r e well defined in the leads. Then the state in the leads is labeledbyk=(q,±) in the presence of two conduction channels. This is not the case in experimental systems of various shape.We believe that /Gamma1 α,xα, and pαcan be defined in Eqs. ( 6) to (8) using state-dependent tunnel couplings without loss of generality. In our models in Figs. 1(a) and1(b), we assume a single conduction channel in the upper arm of the ring. Themultiple channels in the arm should be beyond the scope ofour study. VI. CONCLUSION We theoretically examined the transport through an Aharonov-Bohm ring with an embedded quantum dot (QD),the so-called QD interferometer, to address two controversialissues, one concerns the shape of the conductance peak as a function of energy level ε din the QD and the other is about the phase measurement in the multiterminal geometry as adouble-slit experiment. For this purpose, we generalized a pre-vious model in Refs. [ 18,19] to consider multiple conduction channels in leads LandR. In our model, the tunnel couplings between the QD and leads and that between the leads dependon the states in the leads, as shown in Figs. 1(a) and1(b). This gives rise to a parameter p α(|pα|/lessorequalslant1) to characterize a connection between the two arms of the ring through lead α (=L,R), which is equal to the overlap integral between the conduction modes coupled to the upper and lower arms of thering. First, we examined the shape of the conductance peak in the two-terminal geometry, in the absence of electron-electroninteraction Uin the QD. We showed an asymmetric Fano res- onance at |p L,R|≈1 and an almost symmetric Breit-Wigner resonance at |pL,R|<0.5. Hence our model could explain the experimental results of both an asymmetric Fano resonance[5] and almost symmetric Breit-Wigner resonance [ 8], with fitting parameters p L,Rto their data. Second, we took into account the Kondo effect in the pres- ence of U, using the Bethe ansatz exact solution, and precisely evaluated the conductance at temperature T=0. We showed a crossover from an asymmetric Fano-Kondo resonance [ 19] to the Kondo plateau with changing pL,R. Our model is also applicable to the multiterminal geom- etry to address the second issue on the measurement of thetransmission phase shift through the QD by a double-slit ex-periment. We studied the measured phase φ max, the AB phase at which the conductance G(1)(φ) to lead R(1) is maximal in Fig. 1(b). In the absence of U,E q .( 37) indicates the relation of φmaxto an intrinsic phase shift θ(0) QDthat is the phase shift through the QD without the upper arm of thering. We examined two specific models in the three-terminalgeometry, depicted in Fig. 4. We discussed a crossover from two- to three-terminal phase measurement in the former andsimulated the experimental system consisting of two quantumwires [ 8,16,17] in the latter. Using the latter model, we showed how precisely the phase locking at π/2 is measured in the Kondo regime. ACKNOWLEDGMENTS We appreciate fruitful discussions with Dr. Akira Oguri. This work was partially supported by JSPS KAKENHI GrantsNo. JP26220711 and No. JP15H05870 and JST-CREST GrantNo. JPMJCR1876. APPENDIX A: CURRENT FORMULATION USING KELDYSH GREEN’S FUNCTIONS The current is formulated for the multiterminal model depicted in Fig. 1(b), using the Keldysh Green’s functions [43–45]. The chemical potential in lead L(j)[R(j)] is de- noted by μ(j) L[μ(j) R]. The spin index σis omitted in this Appendix. 245402-10FANO-KONDO RESONANCE VERSUS KONDO PLATEAU IN … PHYSICAL REVIEW B 102, 245402 (2020) 1. Keldysh Green’s functions The retarded, advanced, and lesser Green’s functions are defined by Gr d,Lk(t,t/prime)=1 i¯h/angbracketleft{d(t),a† Lk(t/prime)}/angbracketrightθ(t−t/prime), (A1) Ga d,Lk(t,t/prime)=−1 i¯h/angbracketleft{d(t),a† Lk(t/prime)}/angbracketrightθ(t/prime−t), (A2) G< d,Lk(t,t/prime)=−1 i¯h/angbracketlefta† Lk(t/prime)d(t)/angbracketright, (A3) respectively, where {A,B}=AB+BAandθ(t)i st h eH e a v - iside step function. The other Green’s functions, Gλ d,d, Gλ Lk,Rk/prime, and so on ( λ=r, a, <), are defined in a similar manner. The average is taken for the station-ary state and hence all the Green’s functions depend on t−t /primeonly. Note that G< d,Lk(t−t/prime)=−[G< Lk,d(t/prime−t)]∗and G< d,d(t−t/prime)=−[G< d,d(t/prime−t)]∗. The Fourier transformation (t−t/prime→ω) yields G< d,Lk(ω)=−[G< Lk,d(ω)]∗andG< d,d(ω)= −[G< d,d(ω)]∗. We also introduce the Green’s functions in isolate leads L andR, in the absence of tunnel coupling, HTin Eq. ( 4). For example, gr Lk(t,t/prime)=1 i¯h/angbracketleftbig/braceleftbig aLk(t),a† Lk(t/prime)/bracerightbig/angbracketrightbig θ(t−t/prime) =1 i¯he−iεk(t−t/prime)/¯hθ(t−t/prime), (A4)g< Lk(t,t/prime)=−1 i¯h/angbracketleftbig a† Lk(t/prime)aLk(t)/angbracketrightbig =−1 i¯hf(j) L(εk)e−iεk(t−t/prime)/¯h, (A5) where f(j) L(ε)=[(ε−μ(j) L)/(kBT)+1]−1is the Fermi distri- bution function in lead L(j) that state kbelongs to ( j=1o r 2). The Fourier transformation leads to gr Lk(ω)=1 ¯hω−εk+iδ =P1 ¯hω−εk−iπδ(¯hω−εk), (A6) g< Lk(ω)=2πif(j) L(¯hω)δ(¯hω−εk). (A7) In the following calculations, the real part (principal value) of gr αk(ω) and ga αk(ω)=[gr αk(ω)]∗is disregarded in the summa- tion over k, assuming a wide band limit. In the next subsection, G< d,Lkis replaced by Gr d,dandG< d,d. For this purpose, their relation is derived in the following.In the Baym-Kadanoff-Keldysh nonequilibrium techniques, acomplex-time contour is considered from t=− ∞ tot=t 0 just above the real axis and from t=t0tot=− ∞ just below the real axis. For the contour-ordered Green’s function, GC d,Lk(τ,τ/prime)=1 i¯h/angbracketleftTCd(τ)a† Lk(τ/prime)/angbracketright, (A8) the equation-of-motion method yields [ 44,45] GC d,Lk(τ,τ/prime)=/integraldisplay dτ1/bracketleftBigg GC d,d(τ,τ 1)VLk+(1),(2)/summationdisplay k/primeGC d,Rk/prime(τ,τ 1)Wk/prime,keiφ/bracketrightBigg gC Lk(τ1,τ/prime). (A9) According to the Langreth’s theorem [ 45,46], this results in Gr d,Lk(t,t/prime)=/integraldisplay dt1/bracketleftBigg Gr d,d(t,t1)VLk+(1),(2)/summationdisplay k/primeGr d,Rk/prime(t,t1)Wk/prime,keiφ/bracketrightBigg gr Lk(t1,t/prime), (A10) and G< d,Lk(t,t/prime)=/integraldisplay dt1/braceleftBigg/bracketleftBigg Gr d,d(t,t1)VLk+(1),(2)/summationdisplay k/primeGr d,Rk/prime(t,t1)Wk/prime,keiφ/bracketrightBigg g< Lk(t1,t/prime) +/bracketleftBigg G< d,d(t,t1)VLk+(1),(2)/summationdisplay k/primeG< d,Rk/prime(t,t1)Wk/prime,keiφ/bracketrightBigg ga Lk(t1,t/prime)/bracerightBigg . (A11) Similar relations are obtained for Gr d,Rk/prime, and so on. 2. Current formula using Gr d,dandG< d,d We express the current from lead L(1) in terms of Gr d,dandG< d,d. The substitution of the Hamiltonian in Eq. ( 1) into Eq. ( 15) results in I(1) L=−2e i¯h(1)/summationdisplay k/bracketleftBigg VLk/angbracketleftbig a† Lkd−d†aLk/angbracketrightbig +(1),(2)/summationdisplay k/primeWk/prime,k/angbracketleftbig e−iφa† LkaRk/prime−eiφa† Rk/primeaLk/angbracketrightbig/bracketrightBigg . (A12) 245402-11MIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) We added a factor of 2 by the summation over spin index σ. This equation is rewritten as I(1) L=4eRe(1)/summationdisplay k/bracketleftBigg VLkG< d,Lk(t,t)+(1),(2)/summationdisplay k/primeWk/prime,ke−iφG< Rk/prime,Lk(t,t)/bracketrightBigg =4e 2πRe/integraldisplay dω(1)/summationdisplay k/bracketleftBigg VLkG< d,Lk(ω)+(1),(2)/summationdisplay k/primeWk/prime,ke−iφG< Rk/prime,Lk(ω)/bracketrightBigg . (A13) Hence we need to calculate two terms in the integral, X0=(1)/summationdisplay kVLkG< d,Lk(ω), (A14) Y0=(1)/summationdisplay k(1),(2)/summationdisplay k/primeWk/prime,ke−iφG< Rk/prime,Lk(ω). (A15) Let us consider X0. Using the Fourier transformation of Eq. ( A11), we obtain X0=i/Gamma1(1) L/bracketleftbig 2f(1) L(¯hω)Gr d,d(ω)+G< d,d(ω)/bracketrightbig +i˜p(1) Leiφ(1),(2)/summationdisplay k/prime√wRk/prime/bracketleftbig 2f(1) L(¯hω)Gr d,Rk/prime(ω)+G< d,Rk/prime(ω)/bracketrightbig , (A16) where ˜ p(j) α=/radicalBig /Gamma1(j) αx(j) αp(j) α. Then we need X1=(1),(2)/summationdisplay k/prime√wRk/primeGr d,Rk/prime(ω), (A17) X2=(1),(2)/summationdisplay k/prime√wRk/primeG< d,Rk/prime(ω). (A18)For X1, we use an equation for Gr d,Rk/primecorresponding to Eq. ( A10)f o r Gr d,Lk, which leads to X1=−i/bracketleftbig ˜p(1) R+˜p(2) R/bracketrightbig Gr d,d(ω)−ixRe−iφY1 (A19) with Y1=(1),(2)/summationdisplay k√wLkGr d,Lk(ω). (A20) Using the Fourier transformation of Eq. ( A10), we obtain Y1=−i/bracketleftbig ˜p(1) L+˜p(2) L/bracketrightbig Gr d,d(ω)−ixLeiφX1. (A21) From Eqs. ( A19) and ( A21), we express X1in terms of Gr d,d(ω). In the same way, X2can be written using Gr d,d(ω) and G< d,d(ω). A similar procedure is adopted for Y0. The final result is so lengthy that we show the current expression in the case ofEq. ( 17), i.e., μ (1) L=μ(2) L≡μLandμ(1) R=μ(2) R≡μR.A f t e r the variable conversion of ¯ hω→ε, I(1) L=4e h/integraldisplay dε/braceleftbigg −/Gamma1(1) L/bracketleftbig 2fL(ε)ImGr d,d(ε)+ImG< d,d(ε)/bracketrightbig +x(1) LxR2 (1+x)2[fL(ε)−fR(ε)] +˜p(1) L/bracketleftbig A1ReGr d,d(ε)+A2ImGr d,d(ε)+A3ImG< d,d(ε)/bracketrightbig +x(1) L/bracketleftbig B1ReGr d,d(ε)+B2ImGr d,d(ε)+B3ImG< d,d(ε)/bracketrightbig/bracerightbigg ,(A22) where A1=4 (1+x)2˜pRcosφ[fL(ε)−fR(ε)], (A23) A2=4 (1+x)2{fL(ε)[x˜pRsinφ+(2+x)xR˜pL]+fR(ε)[˜pRsinφ−xR˜pL]}, (A24) A3=2 1+x(˜pRsinφ+xR˜pL), (A25) B1=−8 (1+x)3xR˜pL˜pRcosφ[fL(ε)−fR(ε)], (A26) B2=2 (1+x)3/braceleftbig fL(ε)/bracketleftbig −2(1+x)xR˜pL˜pRsinφ+(1−x)˜p2 R−(3+x)x2 R˜p2 L/bracketrightbig −2fR(ε)/parenleftbig ˜p2 R−x2 R˜p2 L/parenrightbig/bracerightbig , (A27) B3=−1 (1+x)2/parenleftbig 2xR˜pL˜pRsinφ+˜p2 R+x2 R˜p2 L/parenrightbig , (A28) with ˜ pα=˜p(1) α+˜p(2) α=√/Gamma1αxαpα. The current I(2) Lfrom lead L(2) is given by replacing (1) →(2) in Eq. ( A22). The current I(j) Rfrom lead R(j) is obtained from I(j) Lby replacing L↔Randφ→−φ. These equations yield Eq. ( 18) for the current conservation. 245402-12FANO-KONDO RESONANCE VERSUS KONDO PLATEAU IN … PHYSICAL REVIEW B 102, 245402 (2020) 3. Current formula in terms of Gr d,d For the two-terminal model in Fig. 1(a), the current from lead LisIL=I(1) L+I(2) L. The elimination of G< d,dusing Eq. ( 18) results in its expression in Eq. ( 20). As a three-terminal model, we examine the model in Fig. 1(b) consisting of leads L,R(1), and R(2). We introduce parameters, γ(j) R,y(j) R, and q(j) Rin Eq. ( 34). The current into lead R(1) is given by −I(1) R. Eliminating G< d,dusing Eq. ( 18), we obtain −I(1) R=2e h/integraldisplay dε[fL(ε)−fR(ε)]T(1)(ε)dε, (A29) T(1)(ε)=4x (1+x)2y(1) R+8(1+x)q(1) R−2xy(1) R (1+x)3/radicalbig /Gamma1L/Gamma1RxpLpRcosφReGr d,d(ε)+4C2 (1+x)3˜/Gamma1ImGr d,d(ε), (A30) where C2=−2(1+x)/radicalbig /Gamma1L/Gamma1RxpLpRsinφ/bracketleftbig x/parenleftbig q(1) R−y(1) R/parenrightbig /Gamma1L/parenleftbig 1−p2 L/parenrightbig +/parenleftbig γ(1) R−q(1) R/parenrightbig /Gamma1R/bracketrightbig +x3 1+xy(1) R/bracketleftbig/parenleftbig /Gamma1Lp2 L/parenrightbig2+/parenleftbig /Gamma1Rp2 R/parenrightbig2/bracketrightbig +x(1−x)y(1) R(/Gamma1LpL)2+x/bracketleftbig (1+x)/parenleftbig −γ(1) R+2q(1) R/parenrightbig −2xy(1) R/bracketrightbig (/Gamma1RpR)2−/Gamma1L/Gamma1RD2, (A31) with D2=(1+x)3γ(1) R+4x(1+x)q(1) R−xy(1) R 1+x(pLpR)2sin2φ+x2/bracketleftbig 2(x+3)(x+1)q(1) R−(x2+4x−3)y(1) R/bracketrightbig 1+x(pLpR)2 −x/bracketleftbig (x+1)(x+2)γ(1) R+2y(1) R/bracketrightbig p2 L−x/bracketleftbig 2(x+1)(x+2)q(1) R−x(x+3)y(1) R/bracketrightbig p2 R. (A32) Regarding the φ-dependence of the conductance at T=0, Eqs. ( A29) and ( A30) yield Eqs. ( 35) and ( 36) in the absence ofU. In the presence of U, however, we cannot obtain such a simple form in general. APPENDIX B: GREEN’S FUNCTION IN THE PRESENCE OF U For our models shown in Figs. 1(a) and1(b), the Green’s function of the QD is solvable in the case of U=0. As discussed in Sec. II.D, the retarded Green’s function is given by Gr d,d(ε)=1 ε−˜εd(φ)+i˜/Gamma1(B1) with the effective energy level ˜ εd(φ)i nE q .( 25) and effective linewidth ˜/Gamma1in Eq. ( 24). The renormalization due to the direct tunneling between the leads and the Aharonov-Bohm effectby the magnetic flux is included in these effective parameters. In the presence of U, we formulate the perturbation with respect to the electron-electron interaction in the QD,H U=Un↑n↓. The Hamiltonian in Eq. ( 1) is divided into the noninteracting part H0and HU;H=H0+HU.T h e contour-ordered Green’s function of the QD, GC d,d(τ,τ/prime)= /angbracketleftTCdσ(τ)d† σ(τ/prime)/angbracketright/(i¯h), is written as GC d,d(τ,τ/prime) =1 i¯htr/braceleftbigg ρ0TCdI,σ(τ)d† I,σ(τ/prime)e x p/bracketleftbigg/integraldisplay Cdτ/prime/primeHI,U(τ/prime/prime)/bracketrightbigg/bracerightbigg , (B2) where ρ0is the density matrix for U=0 and index I indicates the operator in the interaction picture, OI(τ)= eiH0τ/¯hOe−iH0τ/¯h. In the perturbative expansion, the unper- turbed Green’s function is given by Eq. ( B1). This problem is equivalent to that of the conventional Anderson impu-rity model, in which an impurity with energy level ˜ εd(φ) and Coulomb interaction Uis connected to an energy- band of conduction electrons via the effective hybridization ˜/Gamma1: HAnderson =˜εd(φ)/summationdisplay σnσ+Un↑n↓+/summationdisplay kσεka† k,σak,σ +/summationdisplay kσ(va† k,σdσ+H.c.), (B3) where ˜/Gamma1=πρ|v|2, with the density of states ρfor the con- duction electrons. In the equilibrium with eV=0, the physical quantities of electrons in our model can be evaluated by exploiting theestablished methods for the Anderson impurity model [ 19]. The retarded Green’s function is given by G r d,d(ε)=1 ε−˜εd(φ)+i˜/Gamma1−/Sigma1U(ε)(B4) with use of the self-energy /Sigma1U(ε) due to the electron-electron interaction in the QD. Note that z=[1−d/Sigma1U dε(0)]−1and ˜ε∗ d= z[˜εd(φ)+/Sigma1U(0)] in Eq. ( 26).Gr d,d(0) is expressed in Eq. ( 27) using the phase shift θQD. The Friedel sum rule connects the phase shift to the electron occupation per spin in the QD,θ QD=π/angbracketleftnσ/angbracketright, where /angbracketleftnσ/angbracketright=1 2−1 πtan−1/parenleftbigg˜εd(φ)+/Sigma1U(0) ˜/Gamma1/parenrightbigg . (B5) We use the Bethe ansatz exact solution to evaluate /angbracketleftnσ/angbracketright [39,40]. APPENDIX C: CURRENT IN THREE-TERMINAL MODEL IN FIG. 4(b) We apply the current formula in Eqs. ( A29) and ( A30)t o the model in Fig. 4(b). 245402-13MIKIO ETO AND RUI SAKANO PHYSICAL REVIEW B 102, 245402 (2020) As mentioned in Sec. IV C ,VR,k/prime=VRαRand√wR,k/prime=√wRβRwhen state k/primebelongs to lead R(1) while VR,k/prime=VRβR and√wR,k/prime=−√wRαRwhen state k/primebelongs to lead R(2) in the tunnel Hamiltonian HT. This results in /Gamma1(1) R=α2 R/Gamma1R, /Gamma1(2) R=β2 R/Gamma1R,x(1) R=β2 RxR, and x(2) R=α2 RxR. We also find that p(1) R=1,p(2) R=−1, and hence pR=p(1) R+p(2) R=0. From pR=0, ˜εd(φ)=εdin Eq. ( 25), which is indepen- dent of the AB phase φfor the magnetic flux. The Green’s function in the absence of Ubecomes Gr d,d(ε)=1 ε−εd+i˜/Gamma1(C1) with ˜/Gamma1=/Gamma1L/parenleftbigg 1−x 1+xp2 L/parenrightbigg +/Gamma1R. (C2) The substitution of γ(1) R=α2 R,y(1) R=β2 R, and q(1) RpR=αRβR (q(1) R=∞ ) into Eq. ( A30) results in T(1)(ε)=4x (1+x)2β2 R+8αRβR (1+x)2/radicalbig /Gamma1L/Gamma1RxpLcosφReGr d,d(ε) +4C/prime 2 (1+x)3˜/Gamma1ImGr d,d(ε), (C3) where C/prime 2=−2(1+x)/radicalbig /Gamma1L/Gamma1RxpLsinφ/bracketleftbig x/Gamma1L/parenleftbig 1−p2 L/parenrightbig −/Gamma1R/bracketrightbig αRβR +x3 1+xβ2 R/parenleftbig /Gamma1Lp2 L/parenrightbig2+x(1−x)β2 R(/Gamma1LpL)2 −/Gamma1L/Gamma1R/braceleftbig (1+x)3α2 R−x/bracketleftbig (x+1)(x+2)α2 R+2β2 R/bracketrightbig p2 L/bracerightbig . (C4) Since ˜ εd(φ)=εdin this model, Eq. ( 37) exactly holds in the absence of U, which leads to Eq. ( 41). In addition, even in the presence of U, a relation between φmaxandθQDis derived in the following. The substitution of Eq. ( 26) into Eq. ( C3) yields T(1)(0)=8αRβR (1+x)2/radicalbig /Gamma1L/Gamma1RxpL˜/Gamma1∗ ˜/Gamma11 (˜ε∗ d)2+(˜/Gamma1∗)2F1(φ) +(φ-indep. terms) (C5) atε=EF=0, where F1(φ)=− ˜ε∗ dcosφ+/bracketleftbig x/Gamma1L/parenleftbig 1−p2 L/parenrightbig −/Gamma1R/bracketrightbig˜/Gamma1∗ ˜/Gamma1sinφ.(C6) Forφ=φmaxat which F1(φ) is maximal, tanφmax=−x/Gamma1L(1−p2 L)+/Gamma1R ˜/Gamma1tanθQD, (C7) where tan θQD=˜/Gamma1∗/˜ε∗ d.θQDsatisfies the Friedel sum rule in the presence of U. The current to lead R(2),−I(2) R, is given by replacing (1)→(2) in Eq. ( A29).T(2) Ris obtained from T(1) Rin Eq. ( C3), replacing αR→βRandβR→−αR.I nT(1) RandT(2) R, coef- ficients of cos φand sin φare the same in magnitude and opposite in sign. As a result, the total current to leads R(1)andR(2) does not depend on the AB phase φfor the magnetic flux: −I(1) R−I(2) R=2e h/integraldisplay [fL(ε)−fR(ε)]T(ε)dε, (C8) where T(ε)=T(1) R+T(2) R=4x (1+x)2+4C1 (1+x)3˜/Gamma1ImGr d,d(ε), (C9) with C1=x3 1+x/parenleftbig /Gamma1Lp2 L/parenrightbig2+x(1−x)(/Gamma1LpL)2 −/Gamma1L/Gamma1R/bracketleftbig (1+x)3−x(x2+3x+4)p2 L/bracketrightbig .(C10) This coincides with Eq. ( 20) for the current in the two- terminal system with pR=0. APPENDIX D: TIGHT-BINDING MODEL IN FIG. 9 In the tight-binding model in Fig. 9(a), leads Land Rconsist of two sites in width and Nsites in length (N/greatermuch1). There are two subbands in the leads, as depicted in Fig. 9(b), ε±(q)=∓t1−2tcosqa, (D1) where t(t1) is the transfer integral in x(y) direction and ais the lattice constant. qis the wave number in the xdirection, q=πn/[(N+1)a] with n=1,2,..., N. The corresponding states are given by Eqs. ( 43) and ( 44). Let us consider the case of two conduction channels in the leads when the Fermi level intersects both the two subbands.They are labeled by k=(q,±). In the tunnel Hamil- tonian H Tin Eq. ( 4),VL;q,±=VLψL;q,±(−1,1),VR;q,±= VRψR;q,±(1,1), and Wq/prime,γ/prime;q,γ=√wR;q/prime,γ/primewL;q,γ, where wL;q,γ andwR;q/prime,γ/primeare given by Eqs. ( 45) and ( 46), respec- tively, for γ,γ/prime=±. Here, ψα;q,γ(j,/lscript)=/angbracketleftj,/lscript|α;q,γ/angbracketright is the wave function of the conduction mode ( q,γ) in lead α:ψL;q,±(−1,1)=ψR;q,±(1,1)=sinqa/√N+1 andψL;q,±(−1,2)=ψR;q,±(1,2)=± sinqa/√N+1 from Eqs. ( 43) and ( 44). We calculate /Gamma1α,xα, and pαin Eqs. ( 6)t o( 8)a tε=EF. 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PhysRevB.79.060502.pdf

Interband superconductivity: Contrasts between Bardeen-Cooper-Schrieffer and Eliashberg theories Oleg V. Dolgov Max-Planck-Institut für Festkörperforschung, D-70569 Stuttgart, Germany Igor I. Mazin and David Parker Naval Research Laboratory, 4555 Overlook Avenue SW, Washington, DC 20375, USA Alexander A. Golubov Faculty of Science and Technology, University of Twente, 7500 AE Enschede, The Netherlands /H20849Received 7 October 2008; revised manuscript received 25 November 2008; published 4 February 2009 /H20850 The recently discovered iron pnictide superconductors apparently present an unusual case of interband- channel pairing superconductivity. Here we show that in the limit where the pairing occurs within the interbandchannel, several surprising effects occur quite naturally and generally: different density of states on the twobands leads to several unusual properties, including a gap ratio which behaves inversely to the ratio of densityof states; the weak-coupling limits of the Eliashberg and the BCS theories, commonly taken as equivalent, infact predict qualitatively different dependence of the /H9004 1//H90042and/H9004/Tcratios on coupling constants. We show analytically that these effects follow directly from the interband character of superconductivity. Our resultsshow that in the interband-only pairing model the maximal gap ratio is/H20881N2/N1as strong-coupling effects act only to reduce this ratio. Our results show that pnictide BCS calculations must use renormalized couplingconstants to get accurate results. Our results also suggest that if the large experimentally reported gap ratios /H20849up to a factor 2 /H20850are correct, the pairing mechanism must include more intraband interaction than what is usually assumed. DOI: 10.1103/PhysRevB.79.060502 PACS number /H20849s/H20850: 74.20.Rp, 76.60. /H11002k, 74.25.Nf, 71.55. /H11002i Although first proposed 50 years ago, multiband super- conductivity where the order parameter is different in differ-ent bands had not attracted much interest until 2001 whenMgB 2was found to be a two-band superconductor. MgB 2 represents a particular case where one “leading” band enjoysthe strongest pairing interactions, while the interband-pairinginteraction, as well as the intraband pairing in the other band,is weak. There is growing evidence that the recently discov-ered superconducting ferropnictides represent another limit-ing case: the pairing interaction is predominantly interband,while the intraband pairing in both bands is weak. This leadsto a number of interesting and unexpected effects, includingthe fact that a repulsive interband interaction may be nearlyas effective in creating superconductivity as an attractiveone. In this Rapid Communication we will show another un- usual feature of the two-band “interband” superconductivity/H20849meaning superconductivity induced predominantly by inter- band interactions /H20850: entirely counterintuitively, the BCS theory for such superconductors is not the weak-couplinglimit of the Eliashberg theory , and the difference is not only quantitative but qualitative. This fact holds for either repul- sive /H20849as, presumably, in pnictides /H20850or attractive interactions. Specifically, we will concentrate on the dependence of the superconducting gaps in the two bands on the ratio of thedensities of states and the magnitude of the superconductingcoupling. We will show that the gap ratio is always smallerin the Eliashberg theory than in the BCS theory, the devia-tion grows with coupling strength and with temperature, andit is largest just below T c.Let us start with the BCS equations1and their multiband generalization as proposed by Suhl et al.2For a two-band interband-only case, with gap parameters given on the twobands as /H9004 1and/H90042, the BCS gap equations take the form /H90041=/H20858 kV/H90042tanh /H20849E2,k/2kBT/H20850 2E2,k, /H90042=/H20858 kV/H90041tanh /H20849E1,k/2kBT/H20850 2E1,k, /H208491/H20850 where Ei,kis the usual quasiparticle energy in band igiven by/H20881/H20849/H9255i,k−/H9262/H208502+/H9004i2, the normal-state electron energy is /H9255i,k,/H9262is the chemical potential, and Vis the interband interaction causing the superconductivity. Vcan be either attractive /H20849/H110220 in this convention /H20850or repulsive /H20849as presumably in the pnictides3,4/H20850. In the case of an attractive interband interaction the gap will have the same sign on both Fermi surfaces,while for a repulsive interaction the signs will be opposite.Otherwise all equations are exactly the same. For simplicity,in the rest of this Rapid Communication, whenever V,/H9261,/H9004 1, or/H90042is used, these should be understood as absolute values: /H20841V/H20841,/H20841/H9004/H20841, and so on. The BCS theory assumes Vto be constant up to the cutoff energy /H9275c. Following the BCS prescription, we can convert the momentum sums to energy integrals up to a cutoff energy /H9275cand assume Fermi-level density of states /H20849DOS /H20850N1and N2. Near Tcthese equations can be linearized givingPHYSICAL REVIEW B 79, 060502 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 1098-0121/2009/79 /H208496/H20850/060502 /H208494/H20850 ©2009 The American Physical Society 060502-1/H90041=/H90042/H926112ln/H208491.136/H9275c/Tc/H20850, /H90042=/H90041/H926121ln/H208491.136/H9275c/Tc/H20850, /H208492/H20850 where /H926112=N2V—the dimensionless coupling constant with a similar expression for /H926121. These equations readily yield /H9261eff=/H20881/H926112/H926121and/H90042//H90041=/H20881N1/N2/H11013/H92510. This result has been obtained before.5,6Similarly, at T=0 in the weak-coupling limit, /H90041=/H90042/H926112sinh−1/H20849/H9275c//H90042/H20850, /H90042=/H90041/H926121sinh−1/H20849/H9275c//H90041/H20850. /H208493/H20850 Obviously, for /H9261eff→0 we have Tc→0 and the relation /H90042//H90041=/H20881N1/N2should hold. The same is not true for /H9261eff/H110220. For clarity in all of the following we assume that N1/H11022N2so that /H90042/H11022/H9004 1. We denote N1/N2as/H9252so that /H92510=/H20881/H9252. Our first-principles calculations4suggest for the pnictides /H9252/H11013N1/N2/H113511.4, corresponding to the gap ratio /H9251/H113511.2. Ex- perimental estimates for the gaps differ wildly, yielding gapratios /H9251ranging from 1.3 to 3.4. Since the goal of this Rapid Communication is to address the effect of the density-of-states difference on the gap ratio, we will use an intermediatenumber 7/H9252=2.6 /H20849/H92510=1.6 /H20850. The fact that the band with the larger DOS ends up with asmaller gap is somewhat counterintuitive. This is a direct result of the interband-only pairing—the pairing amplitudeon one band is generated by the DOS on the other . The numerical solution of Eq. /H208492/H20850atT=0/H20849Fig.1/H20850gives, as ex- pected, /H9251=/H20881/H9252=1.6 at /H9261eff→0. As a function of /H9261effit in-creases linearly, reaching /H110152.3 at/H9261eff/H110151.6/H20849note that as dis- cussed below, it will ultimately saturate at /H9252=2.6 in the superstrong limit /H20850. This increase can be easily explained. Let us define xsuch that /H9251=/H92521/2/H208491+x/H20850=/H92510/H208491+x/H20850, so that x/H112701a t/H9261eff/H112701, and substitute sinh−1/H20849/H9275c//H9004/H20850→ln/H208492/H9275c//H9004/H20850.A few lines of algebra then lead to x=ln/H9252 2/H208491+2 //H20881/H926112/H926121/H20850/H11229/H9261effln/H9252 4. /H208494/H20850 This result was also obtained by Bang and Choi.6The qua- dratic in /H9261term can also be worked out and reads /H90042 /H90041=/H20881/H9252/H208751+/H9261effln/H9252 4+/H9261eff2/H208494l n/H9252+l n2/H9252/H20850 32/H20876. /H208495/H20850 As Fig. 1shows, this expression describes the numerical so- lution at small /H9261very well. Although not apparent from the plots, the /H90042//H90041ratio will saturate at large /H9261, as shown by Bang and Choi,6and can also be seen from Eq. /H208493/H20850since /H90041=/H90042/H926112sinh−1/H20849/H9275c//H90042/H20850→/H926112/H9275cfor/H90042/H11271/H9275c. Similarly, in this limit /H90042=/H926121/H9275cso that /H90042//H90041=/H926121//H926112 =N1/N2. All these BCS results, however, are inconsistent with a known analytical result9that in the superstrong /H20849Eliashberg /H20850limit/H9261eff/H112711, the gap ratio /H9251→1 is independent of/H9252. Let us now move to Eliashberg10theory. In this theory, the BCS gap function /H90040is replaced by a complex energy-dependent quantity /H90040/H20849/H9275/H20850, which must be determined along with a mass-renormalization parameterZ/H20849 /H9275/H20850. One commonly formulates the equations in terms of /H9278/H20849/H9275/H20850=Z/H20849/H9275/H20850/H9004/H20849/H9275/H20850, and these equations can be solved either on the real frequency axis or on the imaginary axis /H20849using Matsubara frequencies /H20850. These equations are formulated in a two-band interband-pairing case on the imaginary axis asfollows /H20849some of the notations are repeated from Ref. 8/H20850: /H9004 1/H20849i/H9275n/H20850Z1/H20849i/H9275n/H20850=/H9266T/H20858 mK12/H20849i/H9275m−i/H9275n/H20850/H90042/H20849i/H9275m/H20850 /H20881/H9275m2+/H900422/H20849i/H9275m/H20850, /H208496/H20850 Z1/H20849i/H9275n/H20850=1+/H9266T /H9275n/H20858 mK12/H20849i/H9275m−i/H9275n/H20850/H9275m /H20881/H9275m2+/H900422/H20849i/H9275m/H20850./H208497/H20850 Here the kernel K12is given by K12/H20849i/H9275m−i/H9275n/H20850=2/H20885 0/H11009/H9024B12/H20849/H9024/H20850d/H9024 /H90242+/H20849/H9275m−/H9275n/H208502. This B12represents the electron-boson coupling function which supplants the pairing potential used in the BCS theory,and there is an exactly analogous equation for band 2. HereB 12/H20849/H9024/H20850/B21/H20849/H9024/H20850=N2/N1=1 //H9252. First we assume a simple Einstein-type electron-boson coupling function. The numerical solution of the Eliashbergequation /H208497/H20850finds that the ratio of the gaps decreases with/H9261, which is opposite to the BCS prediction that the ratio of thegaps increases with increasing coupling. This can be under- stood analytically as well. First of all, we observe that neglecting the mass renormal-0 0.3 0.6 0.9 1.2 1.5 λeff11.522.5∆1/∆2 0 0.2 0.4 λeff1.251.51.752∆1/∆20.8 1 1.2 1.4 1.6 λeff1.31.4 ∆1/∆2 FIG. 1. /H20849Color online /H20850The ratio of the gap functions in an interband-pairing case, as a function of /H9261eff, for the BCS /H20849dashed line/H20850and Eliashberg-Einstein /H20849line/H20850spectrum and spin-fluctuation /H20849triangle /H20850spectrum cases. The dotted line represents numerical Eliashberg-Einstein spectrum results in which the mass-renormalization parameter has been artificially taken as 1, showingthat the difference between BCS and Eliashberg is mainly a mass-renormalization effect. The dashed-dotted line /H20849lower part, main panel, and upper inset /H20850is the renormalized BCS /H20849RBCS /H20850/H20849Ref. 8/H20850 approximation to the Eliashberg numerical results. Inset: analyticapproximations to numerical results; diamonds are BCS, Eq. /H208495/H20850, and circles are Eliashberg, Eq. /H2084915/H20850.DOLGOV et al. PHYSICAL REVIEW B 79, 060502 /H20849R/H20850/H208492009 /H20850 RAPID COMMUNICATIONS 060502-2ization by setting Z=1 in Eq. /H208496/H20850yields results very close to the BCS solution /H20851in fact, the deviation from the lowest- order approximation of Eq. /H208494/H20850is mainly due to the increas- ing difference between sinh−1/H20849/H9275c//H9004/H20850and ln /H208492/H9275c//H9004/H20850/H20852. Let us now work out the effect of the mass renormalization. Assuming an Einstein spectrum with the frequency /H9024,a t T=0 Eqs. /H208496/H20850and /H208497/H20850reduce to /H90041/H20849/H9275/H20850Z1/H20849/H9275/H20850=/H926112/H90242 2/H20885 −/H11009/H11009d/H9275/H11032/H90042/H20849/H9275/H11032/H20850 /H20851/H90242+/H20849/H9275−/H9275/H11032/H208502/H20852/H20851/H20881/H9275/H110322+/H900422/H20849/H9275/H20850/H20852 and Z1/H20849/H9275/H20850=1+1 2/H9275/H926112/H90242/H20885 −/H11009/H11009 d/H9275/H11032 /H11003/H9275/H11032 /H20851/H90242+/H20849/H9275−/H9275/H11032/H208502/H20852/H20851/H20881/H9275/H110322+/H900422/H20849/H9275/H20850/H20852 with a similar equation for /H90042andZ2. In the popular “square- well” approximation11,12the equations become /H90041/H20849/H9275/H20850Z1/H20849/H9275/H20850=/H926112/H9258/H20849/H9024−/H20841/H9275/H20841/H20850 2/H20885 −/H11009/H11009 d/H9275/H11032/H9258/H20849/H9024−/H20841/H20849/H9275/H11032/H20850/H20841/H20850 /H11003/H90042/H20849/H9275/H11032/H20850 /H20851/H90242+/H20849/H9275−/H9275/H11032/H208502/H20852/H20849/H20881/H9275/H110322+/H900422/H20850, /H208498/H20850 Z1/H20849/H9275/H20850=1+1 2/H9275/H926112/H20885 −/H11009/H11009 d/H9275/H11032/H9258/H20849/H9024−/H20841/H9275−/H9275/H11032/H20841/H20850 /H11003/H90042/H20849/H9275/H11032/H20850 /H20851/H90242+/H20849/H9275−/H9275/H11032/H208502/H20852/H20849/H20881/H9275/H110322+/H900422/H20850, /H208499/H20850 which may be readily integrated to yield the following renor- malization behavior for Z/H20849/H9275/H20850: Z1/H20849/H9275/H20850=1+/H926112for/H9275/H11021/H9024 /H2084910/H20850 =1 +/H926112/H9024//H9275for/H9024/H11021/H9275/H110212/H9024 /H2084911/H20850 =1 +/H926112/2 for /H9275/H110222/H9024. /H2084912/H20850 This mass-renormalization behavior can then be incorporated in the previous BCS equations yielding a natural result, /H90041/H208491+/H926112/H20850=/H90042/H926112sinh−1/H20849/H9275c//H90042/H20850, /H2084913/H20850 /H90042/H208491+/H926121/H20850=/H90041/H926121sinh−1/H20849/H9275c//H90041/H20850, /H2084914/H20850 reducing to Eq. /H208493/H20850if/H926112in Eq. /H208493/H20850would be substituted by /H926112//H208491+/H926112/H20850, and similarly for /H926121. In Ref. 8this approxima- tion /H20849termed “RBCS” /H20850has been used, and we will see that this approximation captures the full Eliashberg results ex-tremely well. In the linear order in /H9261, we have /H9004 2 /H90041=/H20881/H9252/H208731+/H9261effln/H9252 4+/H926112−/H926121 2/H20874. /H2084915/H20850 The last term is negative and always larger than the previous one /H20849independent of /H9252/H20850. Thus, the net effect is always oppo-site to what the unrenormalized BCS theory predicts. We have plotted up the above analytic approximation in Fig. 1 /H20849solid line in the inset /H20850and find good agreement for /H9261eff/H110210.4, showing that the mass renormalization is respon- sible for the lessening of the gap ratios with increasing cou-pling in Eliashberg theory. Also contained in Fig. 1, main panel and upper inset, is a comparison of the results of theRBCS approximation with the Eliashberg numerical results,which agree nearly perfectly. The lessening of the gap ratios with increasing coupling might in hindsight have been expected given that the Fermisurface with the larger gap at weak coupling can be expectedto have larger self-energy interactions in Eliashberg theory,reducing the gap anisotropy. This trend is also consistentwith the superstrong-coupling limit of equal gaps, as men-tioned previously. Of course, the effect of mass renormaliza-tion on the superconductivity in the sense of a coupling-constant renormalization by a factor of 1 //H208491+/H9261/H20850is well known /H20849see, e.g., Ref. 13/H20850, as well as the fact that this renor- malization can be different for different bands in a multibandsuperconductor /H20849see, e.g., Ref. 14/H20850. It is interesting, however, that this fact leads to unexpected effects in the case of inter-band superconductivity. The above suggests that BCS-type analyses of the pnic- tides in an interband-pairing limit must use the Eliashberg/ renormalized BCS approximation /H20849i.e., letting /H9261 i→/H9261i 1+/H9261i/H20850in order to get even qualitatively correct answers. In particular, omitting this renormalization has led to a belief that Eliash-berg theory predicts an enhancement of gap ratios relative toBCS when in fact the opposite is true. As we mentioned in the beginning, Eq. /H2084915/H20850remains ex- actly the same whether the interband interaction is repulsiveor attractive as long as all /H9004’s and /H9261’s are understood as absolute values. The reason is, of course, that the mass renor-malization is always positive, independent of the sign of theinteraction. 15One may ask another question, whether the mass-renormalization constants are the same as the pairingcoupling constants. Generally speaking, for unconventionalsuperconductivity where /H9004/H20849k/H20850/HS11005const, this is not true /H20849cf. Ref. 16/H20850. In the case of purely interband interaction, though, these constants only differ in sign. Interestingly, this strong-coupling renormalization effect remains operative at all temperatures up to T c, while the previous term in Eq. /H2084915/H20850vanishes at Tc. Therefore /H20849cf. Fig. 2/H20850the actual gap ratio is even closer to 1 near Tcthan at T=0. Finally, we note that the above Eliashberg results were obtained using an Einstein spectral function for simplicity,but as indicated on the plot the use of a typical spin-fluctuation spectrum /H20851/H11011 /H9275/H9024//H20849/H92752+/H90242/H20850/H20852does not alter the re- sults. Another interesting observation to be made concerns the /H9004/H208490/H20850/Tcratios predicted by BCS and Eliashberg theories. In the conventional weak-coupling one-band BCS theory thisratio does not depend on /H9261. This is no longer the case in the two-band BCS with the interband coupling only. In the low-est order the reduced gaps are simply /H9004 2/H208490/H20850/Tc=1.76/H92521/4and /H90041/H208490/H20850/Tc=1.76/H9252−1 /4. The next order can be worked out using Eq. /H208495/H20850,INTERBAND SUPERCONDUCTIVITY: CONTRASTS … PHYSICAL REVIEW B 79, 060502 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 060502-3/H90042/H208490/H20850 Tc= 1.76/H92521/4/H208731+/H9261eff4l n/H9252−l n2/H9252 32/H20874, /H2084916/H20850 /H90041/H208490/H20850 Tc= 1.76/H9252−1 /4/H208731−/H9261eff4l n/H9252+l n2/H9252 32/H20874. /H2084917/H20850 This is confirmed by numerical calculations /H20849Fig. 2/H20850: the smaller gap ratio decreases with /H9261eff, while the other gap increases. Since the Eliashberg equation makes the gapscloser with increased coupling, this odd behavior does notshow up; both reduced gaps grow with /H9261. For completeness, we also show in Fig. 3the behavior /H20849in Eliashberg theory /H20850of the reduced gaps as a function of the DOS ratio N 1/N2=/H926121//H926112. As might be expected, as the DOS ratio becomes very small the gap ratios move apartappreciably. Interestingly, T c/H20849shown in the right panel /H20850is notconstant as it would be in a weak-coupling regime but varies significantly for coupling-constant ratios far from 1. This is aresult of the use of comparatively large coupling constantson one band when the other coupling constant is small, sothatT csuppression due to thermal excitation of real phonons /H20849an effect not present in the BCS formalism /H20850is stronger. To conclude, in this work we have shown for the interband-only pairing that the two-band superconductivity isqualitatively incorrectly described by the BCS formalismeven for the weak-coupling limit. BCS and Eliashberg theo-ries predict qualitatively different behavior /H20849as a function of coupling constant /H20850for such basic characteristics as the gap ratio /H9251=/H90042//H90041as well as for the reduced gaps /H9004/Tc.I n particular, the sign ofd/H9251/d/H9261changes from BCS to Eliash- berg theory. We have found this result analytically and nu-merically by solving Eliashberg equations for the modelspectra. This finding is relevant to the superconducting pnic-tides where the interband-pairing regime is believed to berealized. 1J. R. Schrieffer, Theory of Superconductivity /H20849Perseus, Reading, MA, 1999 /H20850. 2H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev. Lett. 3, 552 /H208491959 /H20850. 3K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki, Phys. Rev. Lett. 101, 087004 /H208492008 /H20850. 4I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Phys. Rev. Lett. 101, 057003 /H208492008 /H20850. 5D. Parker, O. V. Dolgov, M. M. Korshunov, A. A. Golubov, and I. I. Mazin, Phys. Rev. B 78, 134524 /H208492008 /H20850. 6Y. Bang and H.-Y. Choi, Phys. Rev. B 78, 134523 /H208492008 /H20850. 7Ł. Malone, J. D. Fletcher, A. Serafin, A. Carrington, N. D. Zhi- gadlo, Z. Bukowksi, S. Katrych, and J. Karpinski,arXiv:0806.3908 /H20849unpublished /H20850. 8E. J. Nicol and J. P. Carbotte, Phys. Rev. B 71, 054501 /H208492005 /H20850. 9O. V. Dolgov and A. A. Golubov, Phys. Rev. B 77, 214526 /H208492008 /H20850.10G. M. Eliashberg, Sov. Phys. JETP 11, 696 /H208491960 /H20850. 11P. B. Allen and B. Mitrovic, Solid State Phys. 37,1/H208491982 /H20850. 12The square-well model is not a consistent approximation as dif- ferent functional forms are assumed for /H9004in the first and in the second Eliashberg equations. Sometimes this may lead to quali-tative errors /H20851e.g., O. V. Dolgov, I. I. Mazin, A. A. Golubov, S. Y. Savrasov, and E. G. Maksimov, Phys. Rev. Lett. 95, 257003 /H208492005 /H20850/H20852. In this particular case, however, it can be shown that using more accurate and consistent functional forms, /H9004 1/H20849/H9275/H20850 =/H90041/H208490/H20850/H90242//H20849/H92752+/H90242/H20850andZ1/H20849/H9275/H20850=1+ /H20849/H926112/H9024//H9275/H20850tan−1/H20849/H9275//H9024/H20850, leads to essentially the same result. 13J. P. Carbotte, Rev. Mod. Phys. 62, 1027 /H208491990 /H20850. 14I. I. Mazin and V. P. Antropov, Physica C 385,4 9 /H208492003 /H20850. 15P. Monthoux and G. G. Lonzarich, Phys. Rev. B 63, 054529 /H208492001 /H20850. 16D. J. Scalapino, E. Loh, Jr., and J. E. Hirsch, Phys. Rev. B 34, 8190 /H208491986 /H20850.0 0.3 0.6 0.9 1.2 1.5 λeff11.522.53∆(T=0)/Tc BCS, ∆1/Tc BCS, ∆2/Tc Eliash., ∆1/Tc Eliash., ∆2/Tc FIG. 2. /H20849Color online /H20850/H9004/H20849T=0/H20850/Tcratios are shown as a function of the overall coupling constant.0 0.5 1 λ12/λ211.522.5∆(T=0)/Tc ∆1 ∆2 0 0.5 1 λ12/λ210.81 Tc(arb. units) FIG. 3. /H20849Color online /H20850./H20849Left/H20850The behavior of the Eliashberg /H9004/H208490/H20850/Tcratios as a function of the ratio of coupling constants. /H20849Right /H20850The behavior of Tcin this case. For both cases /H9261effis fixed at 1.DOLGOV et al. PHYSICAL REVIEW B 79, 060502 /H20849R/H20850/H208492009 /H20850RAPID COMMUNICATIONS 060502-4

PhysRevB.88.085413.pdf

PHYSICAL REVIEW B 88, 085413 (2013) Entanglement in quantum impurity problems is nonperturbative H. Saleur Institut de Physique Th ´eorique, CEA, IPhT, and CNRS, URA2306, Gif Sur Yvette F-91191, France and Department of Physics, University of Southern California, Los Angeles, California 90089-0484, USA P. Schmitteckert Institute for Nanotechnology, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany R. Vasseur Institut de Physique Th ´eorique, CEA Saclay, 91191 Gif Sur Yvette, France and LPTENS, 24 rue Lhomond, 75231 Paris, France (Received 13 May 2013; published 12 August 2013) We study the entanglement entropy of a region of length 2 Lwith the remainder of an infinite one-dimensional gapless quantum system in the case where the region is centered on a quantum impurity. The coupling to thisimpurity is not scale invariant, and the physics involves a crossover between weak- and strong-coupling regimes.While the impurity contribution to the entanglement has been computed numerically in the past, little is knownanalytically about it, since in particular the methods of conformal invariance cannot be applied because of thepresence of a crossover length. We show in this paper that the small coupling expansion of the entanglemententropy in this problem is quite generally plagued by strong infrared divergences, implying a nonperturbativedependence on the coupling. The large coupling expansion turns out to be better behaved, thanks to powerfulresults from the boundary CFT formulation and, in some cases, the underlying integrability of the problem.However, it is clear that this expansion does not capture well the crossover physics. In the integrable case—whichincludes problems such as an XXZ chain with a modified link, the interacting resonant level model or the anisotropic Kondo model—a nonperturbative approach is in principle possible using form factors. We adapt inthis paper the ideas of Cardy et al. [J. Stat. Phys. 130, 129 (2008) ] and Castro-Alvaredo and Doyon [ J. Stat. Phys. 134, 105 (2009) ] to the gapless case and show that, in the rather simple case of the resonant level model, and after some additional renormalizations, the form-factors approach yields remarkably accurate results for theentanglement all the way from short to large distances. This is confirmed by detailed comparison with numericalsimulations. Both our form factor and numerical results are compatible with a nonperturbative form at shortdistance. DOI: 10.1103/PhysRevB.88.085413 PACS number(s): 05 .70.Ln, 72 .15.Qm, 85 .35.Be I. INTRODUCTION Quantum entanglement has given rise to much work in the condensed-matter community as a new way to explore interesting aspects of physical systems. The Kondo problem,for instance, has been revisited along these lines, with studiesaddressing the interplay between the impurity screening andthe information shared between the impurity and the bath. 1,2It is certainly reasonable to expect that entanglement—together with other quantities inspired by quantum information theory,such as the Loschmidt echo or the work distribution—mightshed new light on, and offer new experimental/numericalprobes of, the key physical features of the Kondo and other problems. 1,3A particularly interesting question in this direction is whether the Kondo screening cloud—which hashad so elusive an appearance in standard thermodynamicquantities 4—might play a bigger role in quantum information aspects. Other aspects of interest in the context of two-level systems interacting with gapless excitations—generalizingthe Kondo problem—apply to the decoherence of qubitsinteracting with the environment. 5–7 A large part of the work combining entanglement and quantum impurities has been numerical so far. Indeed, apartfrom the scale invariant situations, where conformal invariancetechniques have led to spectacular progress, 8,9the generalsituations involving crossover are very difficult to tackle. This is mostly because the entanglement is a different kind ofquantity, not amenable to simple Bethe-ansatz calculations, forinstance. There is, however, another reason for the relative lackof analytical results in this area: entanglement, being a zero-temperature quantity, is naturally plagued by IR divergences,which make it nonperturbative in the impurity strength. In thatrespect, it does behave somehow like some properties of theKondo screening cloud studied in Refs. 4and10. In order to clarify the main features of entanglement in the presence of impurities—in particular its scaling properties, andflow from small to strong coupling—we focus in this paper on acouple of representative situations, which we handle by a mixof analytical and numerical techniques. The lessons learnedwill be put to use in forthcoming papers, with applications ofmore direct physical interest. The paper is organized as follows. In Sec. IIwe discuss the basic models we want to study, and define preciselythe entanglement entropy. In Sec. IIIwe put together the perturbative calculation of the entanglement at small coupling,and show that it is plagued by strong IR divergences. InSec. IVwe discuss this difficulty in a more general context. In Sec. Vwe show how the nonperturbative nature of the entanglement entropy can be obtained using general conformalfield-theoretic arguments. In Sec. VIwe recall the principles 085413-1 1098-0121/2013/88(8)/085413(14) ©2013 American Physical SocietyH. SALEUR, P. SCHMITTECKERT, AND R. V ASSEUR PHYSICAL REVIEW B 88, 085413 (2013) of the large coupling expansion proposed in Ref. 2and carried out to high order in Ref. 11. When the dimension of the perturbation is h=1 2, we develop in Sec. VII the form-factor approach using the results of Refs. 12and13, and obtain nonperturbative approximations for the entanglementextrapolating all the way from the UV to the IR limit. Finally, inSec. VIII we compare our results with those of exact numerical calculations on large spin chains. The conclusion containssome last comments and prospect for future work. Finally, theAppendix contains a discussion of the equivalence betweenour impurity models when the dimension of the perturbation ish= 1 2to the boundary Ising model with a boundary magnetic field at special values of the coupling. II. MODELS AND QUESTIONS The main problem we study in this paper—though it has various, mathematically equivalent formulations (seebelow)—is the calculation of the entanglement of a regionof length 2 Lcentered on an “impurity” in an otherwise one dimensional, gapless quantum system. We characterize thisentanglement by the von Neumann entropy S=− Trρlnρ, where ρis the reduced density matrix that has been formed by tracing over the degrees of freedom outside of the segment oflength 2 L. An example of this setup is obtained by taking two semi- infinite XXZ chains coupled by a weak link: H= −1/summationdisplay −∞J/bracketleftbig Sx iSx i+1+Sy iSy i+1+/Delta1Sz iSz i+1/bracketrightbig +∞/summationdisplay 1J/bracketleftbig Sx iSx i+1+Sy iSy i+1+/Delta1Sz iSz i+1/bracketrightbig +J/prime/bracketleftbig Sx 0Sx 1+Sy 0Sy 1+/Delta1Sz 0Sz 1/bracketrightbig . (1) The bulk chains are in a gapless Luttinger liquid phase for −1</Delta1/lessorequalslant1. We shall consider the case of anisotropy /Delta1< 0, where the tunneling between the two half infinite chainsis a relevant perturbation, and one observes healing at largescales. The case /Delta1=0 is exactly marginal. We parametrize /Delta1=− cos μπ 2,μ∈[0,1]. We focus on the physics at energies much smaller than the bandwidth, where field-theoretic resultscan be applied. Consider then the entanglement of a region of length 2 L centered on the modified link. We can easily surmise what thisentanglement will look like in the high- and low-energy limitsfrom the existing literature. Indeed, at high energy, the systemis effectively cut in half. Using the well-known formula forthe entropy of a region on the edge of a conformal invariant system we have S UV=2×/bracketleftbigg1 6ln2L /epsilon1+s1 2+lng/bracketrightbigg =1 3ln2L /epsilon1+s1+lngUV, (2) where we used that the central charge is unity, /epsilon1is a UV cutoff of the order of the lattice spacing a, and ln gUV=2l ngwhere lngis the boundary entropy14associated with the conformal boundary condition corresponding to an open XXZ spin chain.The remaining constant term s1is nonuniversal as it obviously depends on the definition of the cutoff /epsilon1. On the other hand, at low energy, healing has taken place, the system behaves just as one ordinary quantum spin chain,and the entropy obeys the general form for a region of length2Lin the bulk of a conformal invariant system: S IR=1 3ln2L /epsilon1+s1+lngIR. (3) Here, we have allowed for a term ln gIR, which can be thought of as a residual contribution of the weak link at low energy.In general, comparing entanglements for bulk and boundarytheories is indeed difficult, since the dependency of the cutoff/epsilon1on the physical cutoff (the lattice spacing in the spin chain a, which is the same in both geometries) is not universal, and not necessarily the same in the bulk and boundary cases.This important aspect is discussed in detail in Ref. 13;s e e in particular Sec. 6.2.1 in that reference. The point for usis that the quantity ln g UV−lngIRis well defined, and its value ln gUV−lngIR=−1 2lnμcan be easily obtained from the folded version of the system [see Eq. (6)below]. More generally, since the bulk behavior of the entanglement entropy is not modified, it is natural to expect the existence ofa scaling relation S(L)−S IR=Simp(LTB), (4) where the crossover scale TBis expected to be related to the coupling J/prime.Simpshould be a monotonic function extrapolating between −1 2ln(μ) at small values of the argument and 0 at large values. To proceed, and conveniently describe the field theory limit,15we first observe that the problem, at low energy, can be turned into a purely chiral one. Indeed, in the low-energylimit, each half chain is equivalent to a combination of Land Rmoving excitations, and we formally map via a canonical transformation the Lmoving sector into a Rmoving one so as to have two chiral “wires” representing the two half chains.The additional tunneling between the two chains becomes,in this language, a hopping term between two chiral wires.Bosonizing, forming odd and even combinations of the bosonsfor each wire, one finds that the odd combination decouples,while the even one obtains the simple Hamiltonian H=/integraldisplay ∞ −∞dx(∂xφR)2+λcosβφR(0), (5) whereβ2 8π=μ≡his the conformal weight of the perturbation, and we have set the Fermi velocity vF=1. The dimension of the perturbation being [length]−μ, we see that TB∝λ1/(1−μ)∝ (J/prime)1/(1−μ). One can also fold back this problem into the boundary sine-Gordon model (BSG) with Hamiltonian HBSG=/integraldisplay0 −∞dx1 2[(∂x/Phi1)2+/Pi12]+λcosβ 2/Phi1(0). (6) This shows equivalence to a large variety of other problems, including the one of tunneling between edge states in thefractional quantum Hall effect (FQHE). 16,17In this case, μ=ν is the filling fraction. The RG flows from Neumann ( λ=0) to Dirichlet ( λ=∞ ) boundary conditions (BCs), and the boundary entropy associated with these conformally invariantBCs satisfy ln g UV−lngIR=−1 2lnμ, as claimed earlier. 085413-2ENTANGLEMENT IN QUANTUM IMPURITY PROBLEMS IS ... PHYSICAL REVIEW B 88, 085413 (2013) An interesting variant involves modifying two successive links on the chain: H=−2/summationdisplay −∞J/bracketleftbig Sx iSx i+1+Sy iSy i+1+/Delta1Sz iSz i+1/bracketrightbig +∞/summationdisplay 1J/bracketleftbig Sx iSx i+1+Sy iSy i+1+/Delta1Sz iSz i+1/bracketrightbig +J/prime/bracketleftbig Sx −1Sx 0+Sy −1Sy 0+/Delta1Sz −1Sz 0+Sx 0Sx 1 +Sy 0Sy 1+/Delta1Sz 0Sz 1/bracketrightbig . (7) This is equivalent to tunneling through a resonant level at the origin. This time, the dimension of the tunneling operator ishalf what it is in the previous situation, J /primeis always relevant for −1</Delta1/lessorequalslant1, and the system is always healed at low energy. The same series of manipulations—“unfolding the two halfchains,” forming odd and even combinations, decoupling theodd one and bosonizing—leads to the Hamiltonian formula-tion H=/integraldisplay ∞ −∞dx(∂xφR)2+λ[ei(β/√ 2)φR(0)S−+e−i(β/√2)φR(0)S+], (8) where we recall thatβ2 8π=μ. In this case, the dimension of the perturbation is h=μ 2(note the factor1 2compared with the first case). The problem can also be folded back into theanisotropic Kondo problem H AK=/integraldisplay0 −∞dx1 2[(∂x/Phi1)2+/Pi12]+λ[ei(β/2√ 2)/Phi1(0)S− +e−i(β/2√ 2)/Phi1(0)S+]. (9) Of particular interest is the case /Delta1=0,μ=1 which corresponds to free fermions. While the chain with one weaklink is marginal, the chain with two weak links describes aninteresting flow, and is in fact equivalent to a widely studiedproblem—that of the resonant level model (RLM). Indeed,fermionization in this case leads to H=−J/parenleftBigg −2/summationdisplay −∞c† m+1cm+H.c./parenrightBigg −J/parenleftBigg∞/summationdisplay 1c† m+1cm+H.c./parenrightBigg −J/prime(c† −1c0+c† 0c−1+c† 0c1+c† 1c0), (10) where we have redefined the couplings J→− 2JandJ/prime→ −2J/prime. When going to the continuum limit, the i=0 site behaves like a two-level impurity, and the Hamiltonian reads H=/integraldisplay0 −∞i[ψ† 1L∂xψ1L−ψ† 1R∂xψ1R]dx +/integraldisplay∞ 0i[ψ† 2L∂xψ2L−ψ† 2R∂xψ2R]dx +λ[(ψ† 1(0)+ψ† 2(0))d+H.c.], (11) withψ1L(0)=ψ1R(0)≡ψ1(0), same for the second species, λ∝J/prime.18In contrast with the case of the XX chain with a single defect, this noninteracting problem is not scale invariant . The coupling λflows, and the system again exhibits healing : at low energy, the impurity level is completely hybridized with the two half chains.Let us go back to the general case /Delta1=− cosπμ 2.P r o - ceeding like before, we can write the limiting behaviors ofthe entanglement entropy. At low energy, the impurity ishybridized, the system behaves just as onenonchiral wire and a hybridized impurity, and the entropy obeys the general formfor a region of length 2 Lin the bulk of a conformal invariant system decoupled from the two baths, so S IR=1 3ln2L /epsilon1+s1+lngIR, (12) where once again, we included a term gIRthat accounts for the remaining boundary condition at x=0 of the hybridized im- purity. Meanwhile, at high energy, the impurity is completelydecoupled from the wires, and one gets S UV=2×/bracketleftbigg1 6ln2L /epsilon1+s1 2+lng/bracketrightbigg =1 3ln2L /epsilon1+s1+lngUV. (13) Using the folded (boundary) version of the system (9), one can easily argue that ln gUV−lngIR=ln 2, as a decoupled impurity has two degrees of freedom. One thus expects abehavior entirely similar to (4), where the crossover scale T Bis expected to be proportional to a power of the coupling square,T B∝λ2/(2−μ), and Simpshould be a monotonic function extrapolating between ln 2 at small values of the argumentand 0 at large values. Finally, we note that in the boundary versions (6)and(9), the entanglement impurity we have discussed is now theentanglement of a region of length Lon the edge of the system with the rest. If one were to start from an (anisotropic) Kondoversion, this would be the most natural point of view. 1 There are of course other variants of the problem, for instance involving a slightly modified link in theantiferromagnetic XXZ chain with /Delta1> 0, interactions in the RLM model, etc. In all these cases, we should stress thatthe geometry we are considering is probably not the mostinteresting: considering the entanglement of the two halvesconnected by a weak link or a quantum dot is probably morephysical. This latter problem is however significantly moredifficult technically. We will discuss it in our next paper,relying on the present work as a stepping stone. III. UV PERTURBATION The most natural way to explore the behavior of Simp(LTB) between the fixed points is to use perturbation theory. Therequired calculation is a modification of the one proposed inRefs. 8,9, and 19. Using the well-known replica trick, one first observes that the entanglement entropy Scan be obtained from the Renyi entropies R n=Trρnby considering S= −limn→1∂ ∂nRn.The Renyi entropies in turn can be obtained asRn=Zn (Z1)n, where Znis the partition function on a n- sheeted Riemann surface Rn,1with the sheets joined at a cut corresponding to the segment of length 2 L. The difference between the problem at hand and the conformal case isthat now there is a perturbation inserted at the origin in theHamiltonian formulation, which corresponds to the insertionof a perturbation along an imaginary time line for each of 085413-3H. SALEUR, P. SCHMITTECKERT, AND R. V ASSEUR PHYSICAL REVIEW B 88, 085413 (2013) thensheets. The modified partition functions Zncan in principle be expanded in powers of the coupling constant,and perturbative corrections to the Renyi entropies and theentanglement entropies finally obtained.To see what happens in more detail, we consider first Hamiltonian (5). We start with n=2 sheets, and write formally the Renyi entropy as a functional integral for a pair of chiralbosons as R2=Z2 (Z1)2=/integraltext twist[Dφ1][Dφ2]e x p/braceleftbig −A[φ1]−A[φ2]−λ/integraltext∞ −∞[V1(x=0,y)+V2(x=0,y)]dy/bracerightbig /parenleftbig/integraltext [Dφ]e x p/braceleftbig −A[φ]−λ/integraltext∞ −∞V(x=0,y)dy/bracerightbig/parenrightbig2, (14) where Vi=cosβφiis the perturbation, and A[φ]=/integraltext d2xL[φ] is the free action. Note that we are working in the chiral version, but have suppressed the “ R” label in the fields for simplicity of notation. Finally, the label twist meansthe functional integral is evaluated with conditions around thecut, φ 1(−L/lessorequalslantx/lessorequalslantL,y=0+)=φ2(−L/lessorequalslantx/lessorequalslantL,y=0−), φ2(−L/lessorequalslantx/lessorequalslantL,y=0+)=φ1(−L/lessorequalslantx/lessorequalslantL,y=0−). (15) Ifλ=0, the ratio (14) is nothing but the correlation function of an (order 2) twist operator corresponding to (15), which we will write then as8,19 R2(λ=0)=z2 z2 1=/angbracketleftτ2(L,0) ˜τ2(−L,0)/angbracketrightL(2),C∝L−1/8,(16) where /angbracketleft ···/angbracketright L(2),Cmeans that the correlator is to be evaluated in the plane C=R2(worldsheet) with the Lagrangian L(2)= L[φ1]+L[φ2]. We also recall that in general, the scaling dimension of the twist operator τnreadshn=c 24(n−1/n). We now consider the perturbation expansion in powers of λ. For the denominator, we have immediately D=d2;d=z1[1+λ2 2/integraldisplay dydy/prime 1 2|y−y/prime|2μ+··· ], (17) where the factor 1 /2 in the integral comes from the 1 /2’s in the cosines, and we recall that μis the conformal weight μ=β2 8πof the perturbation. For the numerator, things are a little more complicated since we have two types of fields on the plane, with 1 −1,1−2, and 2−2 contractions, in the presence of the gluing conditions along the cut. To proceed, we uniformize. We start with thecomplex coordinates w=x+iy, and introduce z=/parenleftbiggw−u w−v/parenrightbigg1/2 , (18) where u=Landv=−Lare the complex coordinates of the cut’s extremities. This maps the whole two-sheeted Riemannsurface R 2,1to the z-complex plane C. We then write (14) as R2 R2(λ=0)=1+λ2 2/integraltext dwdw/prime/angbracketleftcosβφ(w) cosβφ(w/prime)/angbracketrightR2,1/parenleftbig 1+λ2 2/integraltext dydy/prime 1 2|y−y/prime|2μ/parenrightbig2 + ···, (19)where the spatial integrals in the numerator are now over R2,1 (worldsheet), and we have a unique boson φinstead of φ1and φ2. Here the integrals in the numerator correspond to insertions along twolines, corresponding to the two copies of the theory, so overall there are four possible terms (contractions). Theperturbation V=cosβφis a primary operator, so we can calculate the correlations on R 2,1by using the conformal mapping (18).W eh a v e 2×22μ/angbracketleftcosβφ(w) cosβφ(w/prime)/angbracketrightR2,1 =(u−v)2μ N2μ 12(w−u)μ/2(w−v)μ/2(w/prime−u)μ/2(w/prime−v)μ/2. (20) Here, N12=(w−u)1/2(w/prime−v)1/2−(w/prime−u)1/2(w−v)1/2.(21) In(19) we have to integrate w,w/primeboth over the imaginary axisw=iy, but also over the second sheet, which is obtained by sending ( w−u)→e2iπ(w−u) and the same for w/prime.T h i s means we end up with two integrals where w,w/primeare on the same sheet, and two where they are on different sheets. Replacing everything by the particular choice of coordi- nates, and expanding the denominator in (19) we get R2 R2(λ=0)=1+λ2 4×2/integraldisplay∞ −∞dydy/prime[Gsame(y,y/prime) +Gdiff(y,y/prime)]+···, (22) where the factor 2 comes since there are two sheets, and insertions can be on the same or different sheets, so Gsame(y,y/prime) =1 22μ|y−y/prime|2μ ×[(iy−L)1/2(iy/prime+L)1/2+(iy/prime−L)1/2(iy+L)1/2]2μ (y2+L2)μ/2[(y/prime)2+L2]μ/2 −1 |y−y/prime|2μ, (23) while Gdiffwill be the same expression with a minus in the numerator’s bracket, and no subtraction (the two point functionof the fields in different copies in the denominator of coursevanish identically). 085413-4ENTANGLEMENT IN QUANTUM IMPURITY PROBLEMS IS ... PHYSICAL REVIEW B 88, 085413 (2013) It is convenient to introduce new variables via tan θ=y L, so the integral becomes L2−2μ/integraldisplayπ/2 −π/2dθdθ/prime(cos2θ)μ−1(cos2θ/prime)μ−1/parenleftbig cos2θ−θ/prime 2/parenrightbigμ−1 [sin2(θ−θ/prime)]μ. (24) The second integral reads similarly L2−2μ/integraldisplayπ/2 −π/2dθdθ/prime(cos2θ)μ−1(cos2θ/prime)μ−1 1/parenleftbig cos2θ−θ/prime 2/parenrightbigμ.(25) Both integrals are UV convergent for a relevant perturbation μ< 1. They are however both IR divergent (here the IR region being θ≈±π 2). This means that, although formally the perturbation at small coupling looks like it should be anexpansion in powers of λ 2L2−2μ, this might actually not be the case and, as we shall see later, is not. In fact, we will see thatthe entanglement is simply nonperturbative in λ 2, and cannot be obtained via this perturbation theory. This result could appear as a surprise. On the one hand, the entanglement is a T=0 quantity, and such quantities are often plagued by IR divergences. On the other hand, weare looking for an L-dependent quantity, and it would be natural to expect that Lwould act as an effective IR cutoff, rendering the perturbation expansion finite. This is howeverdefinitely not what happens. The situation is reminiscent ofsimilar divergences encountered in the Kondo screening cloudproblem. 4 We stress finally that the argument applies almost without modification to the Hamiltonian (8). All that changes is that the perturbation is of the form eiβφS−+H.c.instead of eiβφ+ H.c., so exponentials of opposite signs have to alternate in the imaginary time insertions, modifying some of the numericalcoefficients, but not the integrals or their divergences. IV . IR DIVERGENCES IN QUANTUM IMPURITY PROBLEMS To gain a better understanding of the situation, it is useful to start by discussing another observable20than the entanglement entropy for (5). We turn briefly to the boundary formulation (6), and consider the one point function /angbracketleftcosβ/Phi1 2(x)/angbracketrightwhich appears, for instance, in the determination of Friedel oscillations forimpurities in Luttinger liquids. Simple scaling argumentssuggest the general form /angbracketleftbigg : cosβ/Phi1 2:/angbracketrightbigg =/parenleftbigg2 |x|/parenrightbiggμ/2 F(λ|x|1−μ), (26) where Fis a universal function obeying F(∞)=1, so the field sees Dirichlet boundary conditions, and the bulk normalizationhas been chosen appropriately. Determining the function Fis also a difficult problem. The most natural is once again to attempt perturbation theory in λ. This would share many features of the calculation of one andtwo point functions in the bulk sine-Gordon theory. 21There, it is well known that (in fact, the result essentially goes back toColeman), provided h/lessorequalslant1 (that is, the perturbing operator is not irrelevant), there are no UV divergences in the calculation. All divergences coming from bringing together two insertionsof the perturbing term cos β/Phi1 2are exactly canceled by similardivergences coming from the expansion of the denominator (the partition function and associated bubble diagrams). Ingeneral, the divergences are indeed controlled by the operatorproduct expansion (OPE), e iβ/Phi1(y)/2e−iβ/Phi1(y/prime)/2 =|y−y/prime|−2μ(1+···− πμ(y−y/prime)2[∂/Phi1(y)]2+··· ), with all fields at x=0,ythe coordinate along the boundary. The leading order comes from the contribution of the identityoperator and leads to a disconnected piece subtracted offby a similar term in the denominator. The ···stand for higher orders, or lower orders that vanish after integration.The overall singularity at order O(λ 2n−1) thus behaves as ∼/integraltext/producttextn−1 i=0d(y2i−y2i+1)×(y2i−y2i+1)2−2μ, it comes with dimension dUV=n(3−2μ),and for μ/lessorequalslant1 the integrals are UV finite. Other singularities (when several points are broughttogether at once, etc.) 21behave similarly. However, there will always appear IR divergences at a certain order, depending on the exact value of the conformalweight h. Of course, we expect in the end the scaling form to hold, and thus to depend only on xT B,TB∝λ1/1−μ. What will happen in general is that the divergences in the perturbativeexpansion have to be resummed before the proper scalingform can be obtained. The latter, in general, will thus behavenonperturbatively in the coupling λ. This is nicely illustrated in the case μ= 1 2, where the exact form of the one point function is known, thanks to a mappingto the boundary Ising model (see below), together with avery clever argument by Chatterjee and Zamolodchikov. 22One finds23 /angbracketleftbigg : cosβ/Phi1 2:/angbracketrightbigg (x)=4λ√π/parenleftbiggx 2/parenrightbigg1/4 /Psi1(1/2,1; 8πλ2x),(27) where /Psi1is the degenerate hypergeometric function. The asymptotics follows from /Psi1(1/2,1; 2x)=ex√πK0(x), where K0is the usual modified Bessel function, so that we find /angbracketleftbigg : cosβ/Phi1 2:/angbracketrightbigg (x)∼/parenleftbigg2 x/parenrightbigg1/4 ,x/greatermuch1, /angbracketleftbigg : cosβ/Phi1 2:/angbracketrightbigg (x)∼x−1/427/4λx1/2×− ln(λ2x),x/lessmuch1. (28) We thus see that this function exhibits a nonperturbative dependence at small coupling λ. The nonanalyticity in λarises from the IR divergence of the first perturbative integral. There is a general way to understand the nonanalyticity of course. Whenever a bulk operator (of conformal weights h,h) is sent to the boundary where it becomes a boundary field ofweight h B, one has O(x)≈xhB−2hOB+···. (29) In our case, the cosine of the bulk field simply goes over to the cosine of the boundary field. We have thus h=μ 4 andhB=μ, while [ OB]=L−hB∝ThB B∝λhB/1−μ. We thus expect that /angbracketleftOB/angbracketright=/angbracketleft : cosβ/Phi1(0) 2:/angbracketright∝λμ/1−μ. The dependence of the one point function of the boundary field on λis nonanalytic in λ, and nonperturbative—of course, because again of IR divergences. This problem is the cousin of a similar 085413-5H. SALEUR, P. SCHMITTECKERT, AND R. V ASSEUR PHYSICAL REVIEW B 88, 085413 (2013) problem in bulk massive theories, and has been studied in Refs. 24and25. We deduce from this that, to leading order, /angbracketleftO(x)/angbracketright∝xhB−2hλhB/1−μ(30) More generally, we can write O(x)≈/summationdisplay ixhi B−2hOi B (31) so /angbracketleftO(x)/angbracketright≈/summationdisplay ixhi B−2hci BThi B B=x−2h/summationdisplay ixhi Bci BThi B B ≡x−2hG(xTB) (32) withG(xTB)≡F(λx1−μ), F(y)∝yhB∝λhB/1−μxhB. (33) Going back to the case of Friedel oscillations, we have therefore F(y)∝yμ∝xμλμ/1−μ. This leading dependence inλreplaces the expected perturbative one, which would be linear in λ. The foregoing argument applies in the generic case. Whenever there are “resonances” and the parameter μtakes special rational values μ=1−1 2n, extra logarithmic terms appear in the one point functions of the operators right onthe boundary, which translates in logarithms in the one pointfunctions of operators at x/negationslash=0 as well. This is the case precisely when μ= 1 2. It is important to stress also that the IR divergences naturally disappear at finite temperature, 1 /Tproviding a natural cutoff. Once again this is illustrated in the μ=1 2case, where one finds,23,26for Friedel oscillations at finite temperature, /angbracketleftbigg : cosβ/Phi1 2:/angbracketrightbigg (x) =f(2λ2/T)/parenleftbigg4πT sinh(2 πTx )/parenrightbigg1/4 ×F/parenleftbigg1 2,1 2;1+2λ2 T,1−coth 2πxT 2/parenrightbigg .(34) HereFis the usual hypergeometric function, fis a function whose existence and value were determined in Ref. 26.T h e right-hand side admits a perturbative expansion in powers of λ, whose leading term, at fixed x, goes as λ/√ TwhenT→0. The coefficient of λthus diverges in the zero-temperature limit, in agreement with the fact that the true expansion is then inλlnλ. The general IR behavior can easily be investigated. One finds that at order O(λ 2n+1), there is no IR divergence provided μ>n+1/2 n+1. Only when μ=1—that is, the boundary perturbation is exactly marginal, and the bulk is a Fermiliquid—are all orders finite. In this case, the Friedel oscillationsadmit a perturbative expansion in powers of λ. 27 While the nature of the divergences is quite generic, the quantities for which they occur depend on the problem athand. For instance, for the screening cloud in the (anisotropic)Kondo model, divergences occur even when the boundaryperturbation has dimension 1—in that case, it is marginallyrelevant. 4V . SMALL COUPLING BEHA VIOR OF THE ENTANGLEMENT ENTROPY We now go back to the calculation of the entanglement entropy for Hamiltonian (5). We see that, to obtain the nonperturbative UV behavior, we must discuss twist fields andtheir OPEs. We follow the paper 28but focus more directly on the question at hand. Imagine that we have a single interval forwhich we want to calculate the entanglement with the rest ofthe system, and introduce accordingly the n-sheeted Riemann surface ( nreplicas) R n,1. In the limit where the interval of length Lshrinks, we expect the presence of the two sewing points to decompose like an operator product expansion of theform I=/summationdisplay {kj}C{kj}n/productdisplay j=1/Psi1kj(zj), (35) where we allowed for fields inserted at points zj, the point zon the jth sheet, and the set {/Psi1k}denotes a complete set of local fields for one copy of the CFT. Recall that the cutin the Riemann surface R n,1corresponds to the insertion of twist fields in the complex plane, so that I∼τn(L)˜τn(−L) and(35) should be considered as the OPE of these twist fields. What (35) means more precisely is that, if we have other operators inserted elsewhere, we can expect to have Zn(L) Zn 1/angbracketleftBiggn/productdisplay j=1Oj/angbracketrightBigg Rn,1 =/angbracketleftBigg In/productdisplay j=1Oj/angbracketrightBigg Cn=/summationdisplay {kj}C{kj}n/productdisplay j=1/angbracketleftbig /Psi1kj(zj)Oj/angbracketrightbig Cj,(36) where Ojdesignates operators inserted on the jth sheet, and Cjis thejth copy of the complex plane. Note indeed that the expectation on the right is taken in a fully factorized theory. Restricting now to the /Psi1kthat make an orthonormal basis (so in particular they are all quasiprimary), and choosing Oj= /Psi1kjshows that the structure constant C{kj}will not vanish only if the average of/producttext jOjon the Riemann surface Rn,1does not vanish. It is useful to make things concrete now, so for instancewe see that there is no term with a single primary operator onthe right-hand side of (35) since the corresponding one point function on R n,1vanishes. There is, however, at least one term with a single operator, the stress energy tensor, since we knowthat/angbracketleftT/angbracketright Rn,1/negationslash=0. Apart from this, the most important terms will be those involving the same primary operator on twodifferent sheets/producttext n j=1/Psi1kj=/Psi11/Psi12, whose average on Rn,1 will be nonzero in general. If the field /Psi1has conformal weights h,¯h, we will thus have that C∝L−4hnL2×(h+¯h), (37) where hn=c 24(n−1 n) is the conformal weight of the twist field. The crucial point is that Cinvolves twice the scaling dimension of primary fields, in contrast with ordinary OPEswhere only the scaling dimension would appear. The discussion carries over to the boundary case. One can, for instance, think of it after unfolding the system so as tokeep only chiral fields as in (5). Everything then formally goes through after setting ¯h=0. The question is then, what kind 085413-6ENTANGLEMENT IN QUANTUM IMPURITY PROBLEMS IS ... PHYSICAL REVIEW B 88, 085413 (2013) of fields ψ(the chiral part of /Psi1) can appear in the OPE of two twist fields? The one copy bulk theory is a compact bosonwhich allows for the fields exp( ±i β 2/Phi1) on the boundary. This means that the radius is R=2 β, and thus the bulk conformal weights are given by /Delta1wk=2π/parenleftbiggβk 8π−w β/parenrightbigg2 ,¯/Delta1wk=2π/parenleftbiggβk 8π+w β/parenrightbigg2 .(38) Restricting to scalar operators we get /Delta1=k2β2 32πor/Delta1=2π β2w2. For instance, the first values of /Delta1correspond to fields exp(±ikβ 2/Phi1), or, for the chiral part, exp( ±ikβ 2φR). We now go back to the entropy calculation in the folded, nonchiral theory (6). Upon folding, the chiral vertex operators e±ik(β/2)φR(0)become e±ik(β/4)/Phi1(0),a s/Phi1(0)=φR(0)+φL(0)= 2φR(0). Recall also that the nonchiral twist field in the folded version can be thought of as the chiral part of Iin the unfolded theory. Hence, going through the discussion of short-distanceexpansions we find, for the nonchiral twist field, τ n(L)≈L−2hn ×⎛ ⎜⎜⎜⎝1+/summationdisplay kL2/Delta1kcnn/summationdisplay i,j=1 i/negationslash=jeikβ 4/Phi1i(0)eikβ 4/Phi1j(0)+···⎞ ⎟⎟⎟⎠ (39) where we used the fact 28that the two fields ψin the twist OPEs must belong to different copies. We are only interestedin terms whose one point function acquires a nonzero value inthe presence of the perturbation. This means the first term withk=1 cannot contribute, and thus we need k=2,/Delta1 2=μ. Taking derivative with respect to ngives then the leading term for the entanglement correction, which should go as Simp−ln 2∝(LTB)2μ∝L2μλ2μ/(1−μ). (40) Forμ=1 2in particular, this can be corrected by a resonance, and it is tempting to speculate then that one has Simp−ln 2∝LTB[cst+cst ln(LTB)]. (41) Finally, we note once again that the RLM or the various (anisotropic) Kondo versions will behave identically, thepresence of the operators S +,S−not modifying in any essential way the OPE argument—but one will have to be careful withthe dimensions of the operators involved, and their relationshipwithμ. In the end, we find that for the RLM (41) is expected to hold as well. VI. LARGE COUPLING EXPANSION While the small coupling expansion is plagued with IR divergences, a large coupling expansion is possible. It isnow finite in the IR, and exhibits UV divergences whichare easily taken care of using integrability and analyticity.Let us recall how the calculation goes at leading order inthe anisotropic Kondo case 29(see also, e.g., Ref. 30). The leading IR perturbation is nothing but the stress energytensor H=H IR+1 πTBT(0)+···.The correction to the Renyientropy can therefore be expressed as −δZn=n πTB/integraldisplay+∞ −∞dτ/angbracketleftT(w=iτ)/angbracketrightRn,1 =n−n−1 24πTB/integraldisplay+∞ −∞(2L)2 (iτ−L)2(iτ+L)2dτ =1 12LTB/parenleftbigg n−1 n/parenrightbigg . (42) The first correction to the entanglement entropy thus reads Simp=1 6LTB+···. (43) It is quite remarkable that this result does not depend on the anisotropy parameterμ 2(recall that /Delta1=− cosπμ 2in the XXZ language). It turns out that this IR expansion can be generalized to higher orders.11The results for the Kondo case are as follows: Simp=1 6ln/parenleftbigg 1+1 LTB/parenrightbigg −18 35(πg4)2 (2LTB)6(4α4−8α2+9) +O[(LTB)−7], (44) where the coefficient g4has the following dependence on the dimension h=μ 2of the tunneling operator: g4=μ 12π2/parenleftbigg/Gamma1[μ/2(2−μ)] /Gamma1[1/(2−μ)]/parenrightbigg3/Gamma1[3/(2−μ)] /Gamma1[3μ/2(2−μ)], α=(2−μ)√2μ. (45) Note that in (44), the first term in the right-hand side has to be truncated at order 6. While in principle higher orders in the IR expansion could be determined, the complexity of the calculations increasesconsiderably. Moreover, the convergence properties of thisexpansion are not clear. Finally, we observe that, in this point ofview, the pure BSG case turns out to be quite different, becausedifferent operators appear in the IR effective description. Thecorresponding result has not even been worked out yet. Making analytical progress therefore requires developing nonperturbative approaches. The problems we are interestedin are indeed integrable, at least in their boundary versions.While it is natural to expect that this can be used in some way,integrability has been mostly used to calculate local propertiessuch as magnetization, energy, or impurity entropies. vonNeumann entanglement is nonlocal, and therefore much harderto obtain in general. VII. FORM FACTOR APPROACH TO THE ENTANGLEMENT ENTROPY We will in what follows restrict to the case where the dimension of the perturbation is h=1 2: this corresponds to /Delta1=−√ 2 2(μ=1 2) for the problem of tunneling between XXZ chains, and to /Delta1=0—the RLM ( μ=1)—for the tunneling through an impurity. These cases are closely relatedto the boundary Ising model with a boundary magneticfield (see Appendix). While the problem of calculating theentanglement nonperturbatively remains extremely difficult—entanglement still involving nonlocal observables in the 085413-7H. SALEUR, P. SCHMITTECKERT, AND R. V ASSEUR PHYSICAL REVIEW B 88, 085413 (2013) fermionic language—it can be tackled using the idea of form factors. It has been known for many years that correlation functions of local observables in massive integrable theories can be cal-culated using the form-factors approach, where the integrablequasiparticles provide a basis of the Hilbert space, and the formfactors (FFs)—that is, the matrix elements of the operatorsin that basis—can be obtained using an axiomatic approachbased on the knowledge of the Smatrix and the bootstrap. It is a natural idea to extend this approach to the case ofentanglement entropy. Indeed, the von Neumann entanglementis obtained form the Renyi entropy by taking an nderivative atn=1, and the Renyi entropies can be considered formally as correlation functions of twist operators that live in ncopies of the theory of interest. The integrability of a single theorycarries over to integrability of the ncopies, and a calculation similar to the one of ordinary correlators can be set up, aftersome additional work to determine the form factors of the twistoperators τ,˜τ. 12,13 We are interested here in a variant where the bulk is massless. The form-factors technique in this case is moredelicate to use, since particles can have arbitrarily low energies,and the convergence of the approach is not guaranteed. Variousregularization tricks have to be used in the calculation of localquantities (e.g., the charge density for Friedel oscillations), 31,32 and we will see below that the situation for the entanglement is not better. Nevertheless, Simp(LTB) can be calculated for h=1 2, by using the Ising model formulation, and relying heavily on the work.12,13 To fix ideas, and explore the feasibility of form-factors calculations in our problem, we first discuss briefly the bulkcase and the massless limit. One can find in 12,13the first order contribution to the two point function of the bulk Ising modeltwist field in the bulk, /angbracketleftτ(r)˜τ(0)/angbracketright=/angbracketleftτ/angbracketright 2+1 2n/summationdisplay i,j=1/integraldisplaydθ1 2πdθ2 2π/vextendsingle/vextendsingleFτ|ij 2(θ12,n)/vextendsingle/vextendsingle2 ×e−mr(coshθ1+coshθ2)+···, (46) where nis the number of copies, θiis the rapidity of the ith particle with energy e=mcoshθiand momentum p= msinhθi, andFτ|ij 2(θ12,n) is the two-particle form factor of the twist field τ, Fτ|ij 2(θ12,n)=/angbracketleft0|τ(0)Z† i(θ1)Z† j(θ2)|0/angbracketright. (47) In this last expression, we have used the notation Z† jfor the usual Faddeev-Zamolodchikov creation operators (here,the fermions) living in the jth copy. Since the theory is integrable, the form factors Fτ|ij 2(θ12,n) can be computed exactly and are conveniently expressed using the function K(θ)=Fτ|11 2 /angbracketleftτ/angbracketright=−icosπ 2nsinhθ 2n nsinhiπ+θ 2nsinhiπ−θ 2n, (48) which vanishes when n=1. The other form factors Fτ|ij 2(θ12,n) can then be obtained from Fτ|11 2(θ12,n) by shifting appropriately θ12by a factor of 2 πi. Going to variables θ1±θ2one can perform one integration, and be left with /angbracketleftτ(r)˜τ(0)/angbracketright =/angbracketleftτ/angbracketright2/parenleftbigg 1+n 4π/integraldisplay∞ −∞dθf(θ,n)K0[2mrcosh(θ/2)]/parenrightbigg + ···, (49) where /angbracketleftτ/angbracketright2f(θ)≡/vextendsingle/vextendsingleFτ|11 2(θ)/vextendsingle/vextendsingle2+n−1/summationdisplay j=1/vextendsingle/vextendsingleFτ|11 2(θ+2iπj)/vextendsingle/vextendsingle2.(50) Doyon et al. then argue the crucial result that d dnnf(θ,n)/vextendsingle/vextendsingle/vextendsingle/vextendsingle n=1=π2 2δ(θ). (51) Taking the derivative of the two point twist correlation function meanwhile should give, at short distances, the entanglemententropy of the CFT. Since (46) is only a first-order approxima- tion where contributions with a larger number of particleshave not been included, we get an approximation to theentanglement entropy of a segment of length rin the bulk with the rest of the system, 12 SA=··· −K0(2mr) 8+···≈···+lnr 8+···,(52) and thus the expected factorc 3=1 6is approximated by1 8at this order. Since in this paper we are interested in bulk CFTs, we need to take an m→0 limit. This corresponds formally to describing the CFT using massless particles and masslessscattering. We thus set m 2=Me−θ0and send θ0→∞ .O n l y two types of excitations remain at finite energies: those forwhich θ=±θ 0±βwithβfinite. In the first case, one obtains right moving particles with e=p=Meβand in the second case left moving particles with e=−p=Meβ. Conformal fields factorizing into left and right components are notexpected to mix the LandRsectors. Indeed, lim θ→∞K(θ)=0, (53) so only the LLandRRsectors will contribute in the massless limit of (46). Therefore, setting (say for the Rsector) θ1,2=θ0+β1,2, (54) and introducing β±≡β1±β2, we obtain /angbracketleftτ(r)˜τ(0)/angbracketright=/angbracketleftτ/angbracketright2+1 2n/summationdisplay i,j=1/integraldisplaydβ+ 2πdβ− 2π/vextendsingle/vextendsingleFτ|ij 2(β−,n)/vextendsingle/vextendsingle2 ×e−2Mreβ+/2cosh(β−/2)+···, (55) where the 1 /2 coming from the Jacobian was canceled by the fact that there are two integrals, the Land the Rone. Using (51) we get the correction to the entanglement entropy as SA=··· −1 16/integraldisplay∞ −∞dβ+e−2Mreβ+/2 =···−1 8/integraldisplay∞ 0dx xe−2Mrx. (56) This integral is divergent at small energy, a feature which is quite general in the use of massless form factors. We regularize 085413-8ENTANGLEMENT IN QUANTUM IMPURITY PROBLEMS IS ... PHYSICAL REVIEW B 88, 085413 (2013) by considering the integral /integraldisplay∞ 0dxxα−1e−2Mrx =1 (2Mr)α/Gamma1(α)=/parenleftbigg1 α+···/parenrightbigg [1−αln(2Mr)+··· ] (57) so the finite part of the integral is −ln(2Mr) and thus we recover SA=···+1 8lnr+···. (58) Let us now consider the Ising model with a boundary magnetic field as in (A5) , and to start assume that the bulk is massive. The form-factors approach can be applied to thiscase as well. The first nontrivial contribution reads then 13 /angbracketleft0|τ(r)|B/angbracketright=/angbracketleftτ/angbracketright+1 2n/summationdisplay i=1/integraldisplaydθ 2πR/parenleftbiggiπ 2−θ/parenrightbigg e−2mrcoshθ ×Fτ|11 2(−θ,θ,n )+··· (59) coming from the boundary state |B/angbracketright=exp⎡ ⎣1 4πn/summationdisplay j=1/integraldisplay dθR/parenleftbiggiπ 2−θ/parenrightbigg Z† j(−θ)Z† j(θ)⎤ ⎦|0/angbracketright, (60) where we recall that Z† jare the usual Faddeev-Zamolodchikov creation operators living in the jth copy, and R(θ)is the reflection matrix33of the Ising field theory with a boundary magnetic field hb[proportional to λin(A5) ]. Ultimately, we want once again to take the massless limit m→0. Notice that(59) involves F2instead of |F2|2. We write /angbracketleft0|τ(r)|B/angbracketright=/angbracketleftτ/angbracketright+n 4π/integraldisplay dθR/parenleftbiggiπ 2−θ/parenrightbigg Fτ|11 2(−θ,θ) ×e−2mrcoshθ+··· (61) and observe that the analytical continuation in nis trivial because the particle and its reflection must belong to the samecopy. To every order, contributions are linear in F. But there is a lot of similarity—e.g., between the term with four particleshere, and the term with two particles in the bulk entropy. In themassless limit case, since the boundary produces as many Las Rparticles, and since we need both these numbers to be even, only the terms with 2 lRmovers and 2 lLmovers contribute. Indeed, the first correction to the entanglement, after taking derivative with respect to natn=1, reads explicitly s 1=−1 4/integraldisplay∞ 0dθ/parenleftbiggκ+coshθ κ−coshθ/parenrightbigg/parenleftbiggcoshθ−1 cosh2θ/parenrightbigg e−2mLcoshθ, (62) withκ=1−h2 b/(2m). To obtain a scaling expression in the massless limit, we boost rapidities like in the bulk case, andwe obtain s 1≈1 4/integraldisplay∞ −∞dβeβ−h2 b 2M eβ+h2 b 2M×2e−θ0e−βe−2LMeβ→0,(63)a vanishing result—natural, since in this limit, the two-particle form factor factorizes onto one-particle form factors (one forthe left, one for the right), which both vanish. We thus need togo to the next order (corresponding to four particles), wherewe use formula (3.25) in Ref. 13: s 2=1 16/integraldisplay∞ 0dθ/parenleftbiggκ+coshθ κ−coshθ/parenrightbigg2/parenleftbigg1−coshθ 1+coshθ/parenrightbigg e−4mLcoshθ. (64) We obtain then s2≈−1 16/integraldisplay∞ −∞dβ/parenleftBigg eβ−h2 b 2M eβ+h2 b 2M/parenrightBigg2 e−4LMeβ =−1 16/integraldisplay∞ −∞dβ/parenleftbiggeβ−TB eβ+TB/parenrightbigg2 e−4Leβ, (65) where we have set TB≡h2 b 2(66) and we have shifted the βintegral. Now the expression (65) is divergent at low energies, just like (56). To regularize it, we consider the difference: s2(LTB)−s2(∞)=1 4/integraldisplay∞ −∞dβeβ (1+eβ)2e−4LTBeβ =1 4/integraldisplay∞ 0du (1+u)2e−4LTBu. (67) The UV value is1 4=0.25, to be compared with the exact value 1 2ln 2=0.346 574 .... This indicates that we are on the right track. To proceed, we now take Eq. (3.54) in Ref. 13and perform the appropriate limits and rescalings to get s4≈1 28π2/integraldisplay/productdisplay idβiδ/parenleftBig/summationdisplay βi/parenrightBig/bracketleftBigg e−2L/summationtexteβi/productdisplay ieβi−TB eβi+TB ×/productdisplay i1 coshβi−βi+1 2−β1,3→β1,3±iπ 4 β2,4→β2,4∓iπ 4/bracketrightBigg , (68) where products and sums run over i=1,..., 4 and we have setβ4+1≡β1. The second term is obtained by shifting the contours of integration in the imaginary direction as indicated.We observe the same divergence at low energy, and the sameregularization [subtracting the formal expression for s 4(∞)] also works like for s2. We find s4(LTB)−s4(∞) =/integraldisplay∞ 0du1du2du3 16π2/bracketleftBigg e−2LTB(u1+u2)(u2+u3)/u2 ×u2 (u1+u2)2(u2+u3)2/parenleftBigg/productdisplay i1−ui 1+ui−1/parenrightBigg −.../bracketrightBigg ,(69) where the dots correspond to the two other terms obtained by shifting the contours of integration as in (68).T h eU Vv a l u ei s s4(0)−s4(∞)=1 24, so at second order we have the UV value 1 4+1 24=0.291 667 ..., to be compared once again with the value1 2ln 2=0.346 574 .... 085413-9H. SALEUR, P. SCHMITTECKERT, AND R. V ASSEUR PHYSICAL REVIEW B 88, 085413 (2013) 10−310−210−110010110−210−1100 L TBSimp(L TB) IR expansion FF Order 1 FF Order 2ln 2 FIG. 1. (Color online) The form factor approximations together with the IR expansion. We find that higher orders can be dealt with in the same way, and that the UV values can be resummed exactly to yieldthe exact result, S UV−SIR=∞/summationdisplay l=1[s2l(0)−s2l(∞)]=∞/summationdisplay l=11 4l(2l−1)=1 2ln 2, (70) as expected. We now return to the RLM, whose results are obtained simply by multiplying those for Ising by a factor of 2.In the following, we will allow for an extra multiplicativerenormalization to obtain the UV result exactly, that isconsider, at lowest order, the ratio S (2) imp(LTB)≡ln 2s2(LTB)−s2(∞) s2(0)−s2(∞) =ln 2/integraldisplay∞ 0du (1+u)2e−4LTBu. (71) It is then interesting to consider the IR expansion of this quantity. Using s2(LTB)−s2(∞)=1 2[αeαEi(−α)+1],α=4LTB,(72) where Ei is the usual exponential integral function. One finds S(2) imp(LTB)=2l n2n/summationdisplay k=1(−1)k−1k! 2(4LTB)k+O/parenleftbigg1 (LTB)n+1/parenrightbigg , (73) where the expansion is only asymptotic. We see thus that our “renormalized” first-order approximation interpolates betweenln 2 and to ln 2 4LTB=0.173 287 /(LTB), while the exact result goes from ln 2 to1 6LTB=0.166 66 /(LTB), which is quite good. The next order approximation can be handled similarly, andwe will simply provide the corresponding results on the curvesbelow. We plot our results for the FF approach in Fig. 1where the dashed line is the IR expansion (see above and Ref. 11), and the full colored lines are form factors approximations. Clearly,on this scale, the (renormalized) FF expansion has convergedvery quickly. We shall soon see how close it is to the real datafrom numerical simulations on the XX chain. We now discuss briefly the UV behavior. For s 2, the standard tables give S(2) imp(LTB)=ln 2+4l n2×(LTB)[ln 4+γ+ln(LTB)] + ···, (74) 0 5 10 15 20 25 30 35 40 45 -4 -3 -2 -1 0 1 2((ln2- S) / ( T B L / v F ) log10( TB L / v F)RLM, a=-30.43, b=-7.718 All FF, order 1 FF, order 2 a + b*x*log10 FIG. 2. (Color online) Singularity in the UV , with /Delta1S≡Simp(LTB). 085413-10ENTANGLEMENT IN QUANTUM IMPURITY PROBLEMS IS ... PHYSICAL REVIEW B 88, 085413 (2013) 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 10S / ln(2) TB L/vF J'=0.010,M=100000 J'=0.015,M=50000 J'=0.02, M=30000 J'=0.03, M=30000 J'=0.04, M=30000J'=0.05, M=30000 J'=0.06, M=30000 J'=0.10, M=30000 J'=0.18, M=30000 IR, n=1IR, n=2 IR, n=3 FF FIG. 3. (Color online) Comparison of numerical results and various approximations (recall that /Delta1S=Simp(LTB)). where γ/similarequal0.5772... is the Euler constant. This pro- vides a leading-order correction in the UV which reads4l n2 (LT B)l n (LTB). Note that this is compatible with what was expected from the general discussion about the nonper-turbative behavior in the UV: we get a term linear in LT B, decorated by logarithmic corrections. The next order is moredifficult to handle analytically, but very accurate numericsshows that it does behave similarly, only leading to a correctionof the slope which goes from 4 ln 2 =2.772 59 to 3 .53. We plot in Fig. 2the leading correction divided by LT Bas a function of −ln(LTB), together with numerical data that shall be discussed in the next section. We see that the FF expansion convergesvery well except in the UV , where it seems to still converge,but more slowly, giving us only a rough approximation of theexact leading singularity. Finally, the results for the entanglement for the problem of weakly coupled XXZ chains at /Delta1=− √ 2 2would be identical but for an overall normalization by a factor 1 /2. VIII. NUMERICS We now turn to a numerical determination of the entangle- ment entropy in the RLM, going back to the formulation (10) where we will now also have to be careful with the overallfinite size of the system. We write H=−J−2/summationdisplay m=−M/prime(c† mcm+1+H.c.)−JM/prime−1/summationdisplay m=1(c† mcm+1+H.c.) −˜J(c† −1c0+c† 0c−1+c† 0c1+c† 1c0). (75) So the left and right leads have M/primesites, and the impurity sits at site 0. We can now switch to a representation ofsymmetric and antisymmetric combination of the lead sites,C(˜C)=(c j±c−j)/√ 2. Since the antisymmetric combination decouples from the rest, its contribution to the impurityentanglement will drop out. It is therefore sufficient to study asystem of M≡M /prime+1 sites: H=−JM/prime−1/summationdisplay m=1(C† mCm+1+H.c.) −˜J√ 2(C† 1C0+C† 0C1). (76) Here a single resonant level couples to a single chain of Msites. In order to compare with field theory we use the exactly halffilled system to exploit the linear regime of the cosine band. Inreturn we have to use an even number of M=M /prime+1 sites. The scale of the resonant level with a coupling of J/prime=√ 2˜J is TB/vF=J/prime2 2√ 1−J/prime2, (77) withvF=2 as we have chosen the normalization J=1. Following the recipe of Ref. 34we now calculate the reduced single-particle matrix ρI,L+1for the last L+1 sites, where the first site (labeled 0) corresponds to the impurity. In order toobtain the bulk result we determine the reduced density matrixρ B,Lfor the first Lsites of the chain. The reason for taking the bulk result from the opposite end of the chain is that wecannot just study a chain of M /primesites, as we would then have a degenerate ground state as M/primeis odd. The diagonalization is performed within double precision, while the trace for theentropy is performed using quadruple (128 bit) precision. The entanglement entropy corresponding to the single- particle reduced density matrix is now given by S=− Trρlnρ−Tr(1−ρ)ln(1−ρ), (78) 085413-11H. SALEUR, P. SCHMITTECKERT, AND R. V ASSEUR PHYSICAL REVIEW B 88, 085413 (2013) 0 0.1 0.2 0.3 0.4 0.5 1 10S / ln(2) TB L/vF J'=0.010,M=100000 J'=0.015,M=50000 J'=0.02, M=30000 J'=0.03, M=30000 J'=0.04, M=30000J'=0.05, M=30000 J'=0.06, M=30000 J'=0.10, M=30000 J'=0.18, M=30000 IR, n=1IR, n=2 IR, n=3 FF FIG. 4. (Color online) Comparison of numerical results and various approximations: focus on the IR. which finally leads to Simp,L=SI,L+1−SB,L. (79) In Fig. 3we plot the numerical results together with the first three orders of the IR expansion and the first order of the FFexpansion. We would like to remark that on the lattice we getsmall 2 k Foscillations on top of the continuum result. Figure 4 is a similar plot emphasizing the IR behavior. Finally, in Fig. 2—as commented on already—we focus on the singularity in the UV , comparing slopes obtained fromthe FF expansion. Our numerics on system sizes M=3× 10 4···105is consistent with a singularity Simp=ln 2+αLT Bln(LTB)+···, (80) with a slope −7.5/lessorsimilarα/lessorsimilar−8. By applying damped boundary conditions (DBCs),35one can access very small energy scales on system sizes which are accessible by numerics. By lookingat systems of M=4000 sites where we scale down the bulk hopping elements by a factor of /Lambda1=0.98 on each bond from site 2000 to 3000 and using a bulk hopping elementofJ/Lambda1 1000on the last 1000 sites we find an indication that the singularity is even slightly stronger. While we can excludeanL 2behavior, we cannot rule out the possibility of an LTBln2(LTB) contribution. Note that the DBCs change the form of the density of states at the Fermi surface; for detailssee Ref. 36. It is therefore possible that this additional increase is due to this modification of the level spacing at the Fermisurface. Due to the slow increase of the logarithm such aclarification is asking for multiprecision arithmetic. IX. CONCLUSION This study shows that the entanglement entropy of quantum impurities involved in an RG flow is a quantity which isdifficult to access. It is nonperturbative in the UV , and the IR perturbation, while well defined, does not capture the crossoverregime very well. The form-factors approach, on the otherhand, is remarkably successful. It is, however, difficult todevelop except in the simple case of the Ising model, and morework will have to be done in that direction. Nevertheless, webelieve that the essential features of S impare under control, although it would be useful to check the UV singularities forother values of the coupling (anisotropy). In conclusion, we emphasize that the geometry considered in this paper where the interval for the entropy is centeredon the impurity is probably not the most natural physically.To characterize the Kondo physics, one would rather beinterested in the entanglement of the two wires tunnelingthrough an impurity. This could be characterized physicallyby the entropy of an interval with the impurity at its boundaryor, for example, by the negativity of two intervals in thedifferent wires (see, e.g., Refs. 37–39for examples related to the Kondo problem). This situation is unfortunately muchmore complicated technically, mostly because the foldingprocedures described in this paper no longer apply. However,we still expect the conclusions of this paper to hold in that caseas well, namely, we expect the entanglement entropy (or otherentanglement estimators) to depend nonpertubatively on thecoupling to the impurity when this is weak. We believe thatimproper regularizations of the IR divergences encountered inperturbation theory led to some confusion in the literature. 40 We will report on this—together with a correct calculation—ina subsequent paper. ACKNOWLEDGMENTS We thank I. Affleck, B. Doyon, J. Cardy, and P. Calabrese for discussions. H.S. was supported by the French AgenceNationale pour la Recherche (ANR Project No. 2010 Blanc 085413-12ENTANGLEMENT IN QUANTUM IMPURITY PROBLEMS IS ... PHYSICAL REVIEW B 88, 085413 (2013) SIMI 4 : DIME) and the U.S. Department of Energy (Grant No. DE-FG03-01ER45908). Computations were performed onthe computer cluster of the YIG 8-18 group of P. Orth at theKarlsruhe Institute of Technology. APPENDIX: RELATIONSHIP BETWEEN THE CASE h=1 2 AND THE BOUNDARY ISING MODEL Let us begin by discussing the relationship between the RLM and the Ising model with a boundary magnetic field. Thiscan be seen in various ways. Start from the Hamiltonian (11), unfold the wires to get only right movers, and form thecombinations /Psi1 R≡1√ 2(ψ1R+ψ2R),˜/Psi1R≡1√ 2(ψ1R−ψ2R).(A1) The fermion ˜/Psi1Rdecouples from the impurity entirely, and we will mostly discard it from now on. The remaining dynamicsis then encoded in the Hamiltonian H=−i/integraldisplay ∞ −∞/Psi1† R∂x/Psi1Rdx+λ√ 2[/Psi1† R(0)d+H.c.].(A2) We then refold this Hamiltonian to map back to a boundary problem, introducing a /Psi1Lcomponent: H=−i/integraldisplay0 −∞[/Psi1† R∂x/Psi1R−/Psi1† L∂x/Psi1L]dx +λ√ 2[/Psi1†(0)d+H.c.], (A3) where /Psi1(0)≡/Psi1L(0)=/Psi1R(0). The next—and almost final step—is to go to a Majorana version of this problem. Wedecompose the fermions into real and imaginary parts as /Psi1 R=1√ 2(ξR+iηR),/Psi1 L=1√ 2(ξL+iηL),(A4)where ξ,ηare real and obey {ξR(x),ξR(x/prime)}=δ(x−x/prime), etc. We set similarly d=a+ib√ 2with{a,a}={b,b}=1. The prob- lem then decouples into two independent Majorana problems H=H1+H2, with H1=−i 2/integraldisplay0 −∞[ξR∂xξR−ξL∂xξL]dx+i√ 2λξ(0)b, (A5) H2=−i 2/integraldisplay0 −∞[ηR∂xηR−ηL∂xηL]dx−i√ 2λη(0)a, andξ(0)≡ξR(0)+ξL(0), the same for η. The problems correspond of course to two Ising models with a boundarymagnetic field proportional to ±λ(up to normalizations), the boundary spin operator being σ B(0)=i(ξR+ξL)(0)b. Note that this result is compatible with boundary entropy counting. The flow from UV to IR in the original problemleads to g UV/gIR=2 since a dot with two states is screened. In each of the Ising models meanwhile, we have a flow from free to fixed, with gfree/gfixed=√ 2, so the product of the two ratios—one for each Ising copy—is 2 indeed. Turning now to entanglement entropy, we see that the RLM entanglement for a region of size 2 Lcentered around the impurity is exactly twice the entanglement for a region oflength Lon the edge of the system in the boundary Ising model. This has been studied numerically, e.g., in Refs. 2 and41, and Sec. VIIpresents an analytical calculation of this quantity. 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